Integrand size = 34, antiderivative size = 1795 \[ \int \frac {(e+f x)^3 \text {csch}^3(c+d x) \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx =\text {Too large to display} \]
-3/4*f^3*polylog(4,exp(2*d*x+2*c))/a/d^4+2*(f*x+e)^3*arctanh(exp(2*d*x+2*c ))/a/d+3/4*f^3*polylog(4,-exp(2*d*x+2*c))/a/d^4-6*b*f^3*polylog(3,-exp(d*x +c))/a^2/d^4+6*b*f^3*polylog(3,exp(d*x+c))/a^2/d^4+3/4*b^2*f^3*polylog(4,e xp(2*d*x+2*c))/a^3/d^4-6*I*b^3*f^2*(f*x+e)*polylog(3,-I*exp(d*x+c))/a^2/(a ^2+b^2)/d^3-3*I*b^3*f*(f*x+e)^2*polylog(2,I*exp(d*x+c))/a^2/(a^2+b^2)/d^2+ b^4*(f*x+e)^3*ln(1+exp(2*d*x+2*c))/a^3/(a^2+b^2)/d+6*b*f*(f*x+e)^2*arctanh (exp(d*x+c))/a^2/d^2+6*b*f^2*(f*x+e)*polylog(2,-exp(d*x+c))/a^2/d^3-3/2*f* (f*x+e)^2*polylog(2,exp(2*d*x+2*c))/a/d^2+3/2*f^2*(f*x+e)*polylog(3,exp(2* d*x+2*c))/a/d^3+3/2*f*(f*x+e)^2*polylog(2,-exp(2*d*x+2*c))/a/d^2-6*b*f^2*( f*x+e)*polylog(2,exp(d*x+c))/a^2/d^3+3/2*b^2*f*(f*x+e)^2*polylog(2,exp(2*d *x+2*c))/a^3/d^2-2*b^3*(f*x+e)^3*arctan(exp(d*x+c))/a^2/(a^2+b^2)/d-3/2*b^ 2*f*(f*x+e)^2*polylog(2,-exp(2*d*x+2*c))/a^3/d^2+3/2*b^2*f^2*(f*x+e)*polyl og(3,-exp(2*d*x+2*c))/a^3/d^3+3/4*b^4*f^3*polylog(4,-exp(2*d*x+2*c))/a^3/( a^2+b^2)/d^4-6*b^4*f^3*polylog(4,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/a^3/(a ^2+b^2)/d^4-6*b^4*f^3*polylog(4,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/a^3/(a^ 2+b^2)/d^4-6*I*b*f^3*polylog(4,-I*exp(d*x+c))/a^2/d^4+6*I*b*f^2*(f*x+e)*po lylog(3,-I*exp(d*x+c))/a^2/d^3+6*I*b^3*f^3*polylog(4,-I*exp(d*x+c))/a^2/(a ^2+b^2)/d^4+3*I*b*f*(f*x+e)^2*polylog(2,I*exp(d*x+c))/a^2/d^2+3*I*b^3*f*(f *x+e)^2*polylog(2,-I*exp(d*x+c))/a^2/(a^2+b^2)/d^2+6*I*b^3*f^2*(f*x+e)*pol ylog(3,I*exp(d*x+c))/a^2/(a^2+b^2)/d^3+1/2*(f*x+e)^3/a/d-3/2*b^2*f^2*(f...
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(5813\) vs. \(2(1795)=3590\).
Time = 12.84 (sec) , antiderivative size = 5813, normalized size of antiderivative = 3.24 \[ \int \frac {(e+f x)^3 \text {csch}^3(c+d x) \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Result too large to show} \]
Time = 8.16 (sec) , antiderivative size = 1596, normalized size of antiderivative = 0.89, number of steps used = 24, number of rules used = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.676, Rules used = {6123, 5985, 27, 6123, 5985, 25, 6123, 5984, 3042, 26, 4670, 3011, 6107, 6095, 2620, 3011, 7163, 2720, 7143, 7292, 27, 7293, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {(e+f x)^3 \text {csch}^3(c+d x) \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx\) |
| \(\Big \downarrow \) 6123 |
| \(\displaystyle \frac {\int (e+f x)^3 \text {csch}^3(c+d x) \text {sech}(c+d x)dx}{a}-\frac {b \int \frac {(e+f x)^3 \text {csch}^2(c+d x) \text {sech}(c+d x)}{a+b \sinh (c+d x)}dx}{a}\) |
| \(\Big \downarrow \) 5985 |
| \(\displaystyle \frac {-3 f \int -\frac {1}{2} (e+f x)^2 \left (\frac {\coth ^2(c+d x)}{d}+\frac {2 \log (\tanh (c+d x))}{d}\right )dx-\frac {(e+f x)^3 \coth ^2(c+d x)}{2 d}-\frac {(e+f x)^3 \log (\tanh (c+d x))}{d}}{a}-\frac {b \int \frac {(e+f x)^3 \text {csch}^2(c+d x) \text {sech}(c+d x)}{a+b \sinh (c+d x)}dx}{a}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {3}{2} f \int (e+f x)^2 \left (\frac {\coth ^2(c+d x)}{d}+\frac {2 \log (\tanh (c+d x))}{d}\right )dx-\frac {(e+f x)^3 \coth ^2(c+d x)}{2 d}-\frac {(e+f x)^3 \log (\tanh (c+d x))}{d}}{a}-\frac {b \int \frac {(e+f x)^3 \text {csch}^2(c+d x) \text {sech}(c+d x)}{a+b \sinh (c+d x)}dx}{a}\) |
| \(\Big \downarrow \) 6123 |
| \(\displaystyle \frac {\frac {3}{2} f \int (e+f x)^2 \left (\frac {\coth ^2(c+d x)}{d}+\frac {2 \log (\tanh (c+d x))}{d}\right )dx-\frac {(e+f x)^3 \coth ^2(c+d x)}{2 d}-\frac {(e+f x)^3 \log (\tanh (c+d x))}{d}}{a}-\frac {b \left (\frac {\int (e+f x)^3 \text {csch}^2(c+d x) \text {sech}(c+d x)dx}{a}-\frac {b \int \frac {(e+f x)^3 \text {csch}(c+d x) \text {sech}(c+d x)}{a+b \sinh (c+d x)}dx}{a}\right )}{a}\) |
| \(\Big \downarrow \) 5985 |
| \(\displaystyle \frac {\frac {3}{2} f \int (e+f x)^2 \left (\frac {\coth ^2(c+d x)}{d}+\frac {2 \log (\tanh (c+d x))}{d}\right )dx-\frac {(e+f x)^3 \coth ^2(c+d x)}{2 d}-\frac {(e+f x)^3 \log (\tanh (c+d x))}{d}}{a}-\frac {b \left (\frac {-3 f \int -(e+f x)^2 \left (\frac {\arctan (\sinh (c+d x))}{d}+\frac {\text {csch}(c+d x)}{d}\right )dx-\frac {(e+f x)^3 \arctan (\sinh (c+d x))}{d}-\frac {(e+f x)^3 \text {csch}(c+d x)}{d}}{a}-\frac {b \int \frac {(e+f x)^3 \text {csch}(c+d x) \text {sech}(c+d x)}{a+b \sinh (c+d x)}dx}{a}\right )}{a}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {3}{2} f \int (e+f x)^2 \left (\frac {\coth ^2(c+d x)}{d}+\frac {2 \log (\tanh (c+d x))}{d}\right )dx-\frac {(e+f x)^3 \coth ^2(c+d x)}{2 d}-\frac {(e+f x)^3 \log (\tanh (c+d x))}{d}}{a}-\frac {b \left (\frac {3 f \int (e+f x)^2 \left (\frac {\arctan (\sinh (c+d x))}{d}+\frac {\text {csch}(c+d x)}{d}\right )dx-\frac {(e+f x)^3 \arctan (\sinh (c+d x))}{d}-\frac {(e+f x)^3 \text {csch}(c+d x)}{d}}{a}-\frac {b \int \frac {(e+f x)^3 \text {csch}(c+d x) \text {sech}(c+d x)}{a+b \sinh (c+d x)}dx}{a}\right )}{a}\) |
| \(\Big \downarrow \) 6123 |
| \(\displaystyle \frac {\frac {3}{2} f \int (e+f x)^2 \left (\frac {\coth ^2(c+d x)}{d}+\frac {2 \log (\tanh (c+d x))}{d}\right )dx-\frac {(e+f x)^3 \coth ^2(c+d x)}{2 d}-\frac {(e+f x)^3 \log (\tanh (c+d x))}{d}}{a}-\frac {b \left (\frac {3 f \int (e+f x)^2 \left (\frac {\arctan (\sinh (c+d x))}{d}+\frac {\text {csch}(c+d x)}{d}\right )dx-\frac {(e+f x)^3 \arctan (\sinh (c+d x))}{d}-\frac {(e+f x)^3 \text {csch}(c+d x)}{d}}{a}-\frac {b \left (\frac {\int (e+f x)^3 \text {csch}(c+d x) \text {sech}(c+d x)dx}{a}-\frac {b \int \frac {(e+f x)^3 \text {sech}(c+d x)}{a+b \sinh (c+d x)}dx}{a}\right )}{a}\right )}{a}\) |
| \(\Big \downarrow \) 5984 |
| \(\displaystyle \frac {\frac {3}{2} f \int (e+f x)^2 \left (\frac {\coth ^2(c+d x)}{d}+\frac {2 \log (\tanh (c+d x))}{d}\right )dx-\frac {(e+f x)^3 \coth ^2(c+d x)}{2 d}-\frac {(e+f x)^3 \log (\tanh (c+d x))}{d}}{a}-\frac {b \left (\frac {3 f \int (e+f x)^2 \left (\frac {\arctan (\sinh (c+d x))}{d}+\frac {\text {csch}(c+d x)}{d}\right )dx-\frac {(e+f x)^3 \arctan (\sinh (c+d x))}{d}-\frac {(e+f x)^3 \text {csch}(c+d x)}{d}}{a}-\frac {b \left (\frac {2 \int (e+f x)^3 \text {csch}(2 c+2 d x)dx}{a}-\frac {b \int \frac {(e+f x)^3 \text {sech}(c+d x)}{a+b \sinh (c+d x)}dx}{a}\right )}{a}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {3}{2} f \int (e+f x)^2 \left (\frac {\coth ^2(c+d x)}{d}+\frac {2 \log (\tanh (c+d x))}{d}\right )dx-\frac {(e+f x)^3 \coth ^2(c+d x)}{2 d}-\frac {(e+f x)^3 \log (\tanh (c+d x))}{d}}{a}-\frac {b \left (\frac {3 f \int (e+f x)^2 \left (\frac {\arctan (\sinh (c+d x))}{d}+\frac {\text {csch}(c+d x)}{d}\right )dx-\frac {(e+f x)^3 \arctan (\sinh (c+d x))}{d}-\frac {(e+f x)^3 \text {csch}(c+d x)}{d}}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x)^3 \text {sech}(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {2 \int i (e+f x)^3 \csc (2 i c+2 i d x)dx}{a}\right )}{a}\right )}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {\frac {3}{2} f \int (e+f x)^2 \left (\frac {\coth ^2(c+d x)}{d}+\frac {2 \log (\tanh (c+d x))}{d}\right )dx-\frac {(e+f x)^3 \coth ^2(c+d x)}{2 d}-\frac {(e+f x)^3 \log (\tanh (c+d x))}{d}}{a}-\frac {b \left (\frac {3 f \int (e+f x)^2 \left (\frac {\arctan (\sinh (c+d x))}{d}+\frac {\text {csch}(c+d x)}{d}\right )dx-\frac {(e+f x)^3 \arctan (\sinh (c+d x))}{d}-\frac {(e+f x)^3 \text {csch}(c+d x)}{d}}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x)^3 \text {sech}(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {2 i \int (e+f x)^3 \csc (2 i c+2 i d x)dx}{a}\right )}{a}\right )}{a}\) |
| \(\Big \downarrow \) 4670 |
| \(\displaystyle \frac {\frac {3}{2} f \int (e+f x)^2 \left (\frac {\coth ^2(c+d x)}{d}+\frac {2 \log (\tanh (c+d x))}{d}\right )dx-\frac {(e+f x)^3 \coth ^2(c+d x)}{2 d}-\frac {(e+f x)^3 \log (\tanh (c+d x))}{d}}{a}-\frac {b \left (\frac {3 f \int (e+f x)^2 \left (\frac {\arctan (\sinh (c+d x))}{d}+\frac {\text {csch}(c+d x)}{d}\right )dx-\frac {(e+f x)^3 \arctan (\sinh (c+d x))}{d}-\frac {(e+f x)^3 \text {csch}(c+d x)}{d}}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x)^3 \text {sech}(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {2 i \left (\frac {3 i f \int (e+f x)^2 \log \left (1-e^{2 c+2 d x}\right )dx}{2 d}-\frac {3 i f \int (e+f x)^2 \log \left (1+e^{2 c+2 d x}\right )dx}{2 d}+\frac {i (e+f x)^3 \text {arctanh}\left (e^{2 c+2 d x}\right )}{d}\right )}{a}\right )}{a}\right )}{a}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle \frac {\frac {3}{2} f \int (e+f x)^2 \left (\frac {\coth ^2(c+d x)}{d}+\frac {2 \log (\tanh (c+d x))}{d}\right )dx-\frac {(e+f x)^3 \coth ^2(c+d x)}{2 d}-\frac {(e+f x)^3 \log (\tanh (c+d x))}{d}}{a}-\frac {b \left (\frac {3 f \int (e+f x)^2 \left (\frac {\arctan (\sinh (c+d x))}{d}+\frac {\text {csch}(c+d x)}{d}\right )dx-\frac {(e+f x)^3 \arctan (\sinh (c+d x))}{d}-\frac {(e+f x)^3 \text {csch}(c+d x)}{d}}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x)^3 \text {sech}(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {2 i \left (-\frac {3 i f \left (\frac {f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )}{2 d}\right )}{2 d}+\frac {3 i f \left (\frac {f \int (e+f x) \operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )}{2 d}\right )}{2 d}+\frac {i (e+f x)^3 \text {arctanh}\left (e^{2 c+2 d x}\right )}{d}\right )}{a}\right )}{a}\right )}{a}\) |
| \(\Big \downarrow \) 6107 |
| \(\displaystyle \frac {\frac {3}{2} f \int (e+f x)^2 \left (\frac {\coth ^2(c+d x)}{d}+\frac {2 \log (\tanh (c+d x))}{d}\right )dx-\frac {(e+f x)^3 \coth ^2(c+d x)}{2 d}-\frac {(e+f x)^3 \log (\tanh (c+d x))}{d}}{a}-\frac {b \left (\frac {3 f \int (e+f x)^2 \left (\frac {\arctan (\sinh (c+d x))}{d}+\frac {\text {csch}(c+d x)}{d}\right )dx-\frac {(e+f x)^3 \arctan (\sinh (c+d x))}{d}-\frac {(e+f x)^3 \text {csch}(c+d x)}{d}}{a}-\frac {b \left (-\frac {b \left (\frac {b^2 \int \frac {(e+f x)^3 \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{a^2+b^2}+\frac {\int (e+f x)^3 \text {sech}(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}\right )}{a}+\frac {2 i \left (-\frac {3 i f \left (\frac {f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )}{2 d}\right )}{2 d}+\frac {3 i f \left (\frac {f \int (e+f x) \operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )}{2 d}\right )}{2 d}+\frac {i (e+f x)^3 \text {arctanh}\left (e^{2 c+2 d x}\right )}{d}\right )}{a}\right )}{a}\right )}{a}\) |
| \(\Big \downarrow \) 6095 |
| \(\displaystyle \frac {\frac {3}{2} f \int (e+f x)^2 \left (\frac {\coth ^2(c+d x)}{d}+\frac {2 \log (\tanh (c+d x))}{d}\right )dx-\frac {(e+f x)^3 \coth ^2(c+d x)}{2 d}-\frac {(e+f x)^3 \log (\tanh (c+d x))}{d}}{a}-\frac {b \left (\frac {3 f \int (e+f x)^2 \left (\frac {\arctan (\sinh (c+d x))}{d}+\frac {\text {csch}(c+d x)}{d}\right )dx-\frac {(e+f x)^3 \arctan (\sinh (c+d x))}{d}-\frac {(e+f x)^3 \text {csch}(c+d x)}{d}}{a}-\frac {b \left (-\frac {b \left (\frac {b^2 \left (\int \frac {e^{c+d x} (e+f x)^3}{a+b e^{c+d x}-\sqrt {a^2+b^2}}dx+\int \frac {e^{c+d x} (e+f x)^3}{a+b e^{c+d x}+\sqrt {a^2+b^2}}dx-\frac {(e+f x)^4}{4 b f}\right )}{a^2+b^2}+\frac {\int (e+f x)^3 \text {sech}(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}\right )}{a}+\frac {2 i \left (-\frac {3 i f \left (\frac {f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )}{2 d}\right )}{2 d}+\frac {3 i f \left (\frac {f \int (e+f x) \operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )}{2 d}\right )}{2 d}+\frac {i (e+f x)^3 \text {arctanh}\left (e^{2 c+2 d x}\right )}{d}\right )}{a}\right )}{a}\right )}{a}\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle \frac {\frac {3}{2} f \int (e+f x)^2 \left (\frac {\coth ^2(c+d x)}{d}+\frac {2 \log (\tanh (c+d x))}{d}\right )dx-\frac {(e+f x)^3 \coth ^2(c+d x)}{2 d}-\frac {(e+f x)^3 \log (\tanh (c+d x))}{d}}{a}-\frac {b \left (\frac {3 f \int (e+f x)^2 \left (\frac {\arctan (\sinh (c+d x))}{d}+\frac {\text {csch}(c+d x)}{d}\right )dx-\frac {(e+f x)^3 \arctan (\sinh (c+d x))}{d}-\frac {(e+f x)^3 \text {csch}(c+d x)}{d}}{a}-\frac {b \left (-\frac {b \left (\frac {b^2 \left (-\frac {3 f \int (e+f x)^2 \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right )dx}{b d}-\frac {3 f \int (e+f x)^2 \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right )dx}{b d}+\frac {(e+f x)^3 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}+\frac {(e+f x)^3 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}-\frac {(e+f x)^4}{4 b f}\right )}{a^2+b^2}+\frac {\int (e+f x)^3 \text {sech}(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}\right )}{a}+\frac {2 i \left (-\frac {3 i f \left (\frac {f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )}{2 d}\right )}{2 d}+\frac {3 i f \left (\frac {f \int (e+f x) \operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )}{2 d}\right )}{2 d}+\frac {i (e+f x)^3 \text {arctanh}\left (e^{2 c+2 d x}\right )}{d}\right )}{a}\right )}{a}\right )}{a}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle \frac {\frac {3}{2} f \int (e+f x)^2 \left (\frac {\coth ^2(c+d x)}{d}+\frac {2 \log (\tanh (c+d x))}{d}\right )dx-\frac {(e+f x)^3 \coth ^2(c+d x)}{2 d}-\frac {(e+f x)^3 \log (\tanh (c+d x))}{d}}{a}-\frac {b \left (\frac {3 f \int (e+f x)^2 \left (\frac {\arctan (\sinh (c+d x))}{d}+\frac {\text {csch}(c+d x)}{d}\right )dx-\frac {(e+f x)^3 \arctan (\sinh (c+d x))}{d}-\frac {(e+f x)^3 \text {csch}(c+d x)}{d}}{a}-\frac {b \left (-\frac {b \left (\frac {b^2 \left (-\frac {3 f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d}\right )}{b d}-\frac {3 f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d}\right )}{b d}+\frac {(e+f x)^3 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}+\frac {(e+f x)^3 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}-\frac {(e+f x)^4}{4 b f}\right )}{a^2+b^2}+\frac {\int (e+f x)^3 \text {sech}(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}\right )}{a}+\frac {2 i \left (-\frac {3 i f \left (\frac {f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )}{2 d}\right )}{2 d}+\frac {3 i f \left (\frac {f \int (e+f x) \operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )}{2 d}\right )}{2 d}+\frac {i (e+f x)^3 \text {arctanh}\left (e^{2 c+2 d x}\right )}{d}\right )}{a}\right )}{a}\right )}{a}\) |
| \(\Big \downarrow \) 7163 |
| \(\displaystyle \frac {-\frac {\coth ^2(c+d x) (e+f x)^3}{2 d}-\frac {\log (\tanh (c+d x)) (e+f x)^3}{d}+\frac {3}{2} f \int (e+f x)^2 \left (\frac {\coth ^2(c+d x)}{d}+\frac {2 \log (\tanh (c+d x))}{d}\right )dx}{a}-\frac {b \left (\frac {-\frac {\arctan (\sinh (c+d x)) (e+f x)^3}{d}-\frac {\text {csch}(c+d x) (e+f x)^3}{d}+3 f \int (e+f x)^2 \left (\frac {\arctan (\sinh (c+d x))}{d}+\frac {\text {csch}(c+d x)}{d}\right )dx}{a}-\frac {b \left (\frac {2 i \left (\frac {i \text {arctanh}\left (e^{2 c+2 d x}\right ) (e+f x)^3}{d}-\frac {3 i f \left (\frac {f \left (\frac {(e+f x) \operatorname {PolyLog}\left (3,-e^{2 c+2 d x}\right )}{2 d}-\frac {f \int \operatorname {PolyLog}\left (3,-e^{2 c+2 d x}\right )dx}{2 d}\right )}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )}{2 d}\right )}{2 d}+\frac {3 i f \left (\frac {f \left (\frac {(e+f x) \operatorname {PolyLog}\left (3,e^{2 c+2 d x}\right )}{2 d}-\frac {f \int \operatorname {PolyLog}\left (3,e^{2 c+2 d x}\right )dx}{2 d}\right )}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )}{2 d}\right )}{2 d}\right )}{a}-\frac {b \left (\frac {\left (-\frac {(e+f x)^4}{4 b f}+\frac {\log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right ) (e+f x)^3}{b d}+\frac {\log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right ) (e+f x)^3}{b d}-\frac {3 f \left (\frac {2 f \left (\frac {(e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d}-\frac {f \int \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )dx}{d}\right )}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d}\right )}{b d}-\frac {3 f \left (\frac {2 f \left (\frac {(e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d}-\frac {f \int \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )dx}{d}\right )}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d}\right )}{b d}\right ) b^2}{a^2+b^2}+\frac {\int (e+f x)^3 \text {sech}(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}\right )}{a}\right )}{a}\right )}{a}\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle \frac {-\frac {\coth ^2(c+d x) (e+f x)^3}{2 d}-\frac {\log (\tanh (c+d x)) (e+f x)^3}{d}+\frac {3}{2} f \int (e+f x)^2 \left (\frac {\coth ^2(c+d x)}{d}+\frac {2 \log (\tanh (c+d x))}{d}\right )dx}{a}-\frac {b \left (\frac {-\frac {\arctan (\sinh (c+d x)) (e+f x)^3}{d}-\frac {\text {csch}(c+d x) (e+f x)^3}{d}+3 f \int (e+f x)^2 \left (\frac {\arctan (\sinh (c+d x))}{d}+\frac {\text {csch}(c+d x)}{d}\right )dx}{a}-\frac {b \left (\frac {2 i \left (\frac {i \text {arctanh}\left (e^{2 c+2 d x}\right ) (e+f x)^3}{d}-\frac {3 i f \left (\frac {f \left (\frac {(e+f x) \operatorname {PolyLog}\left (3,-e^{2 c+2 d x}\right )}{2 d}-\frac {f \int e^{-2 c-2 d x} \operatorname {PolyLog}\left (3,-e^{2 c+2 d x}\right )de^{2 c+2 d x}}{4 d^2}\right )}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )}{2 d}\right )}{2 d}+\frac {3 i f \left (\frac {f \left (\frac {(e+f x) \operatorname {PolyLog}\left (3,e^{2 c+2 d x}\right )}{2 d}-\frac {f \int e^{-2 c-2 d x} \operatorname {PolyLog}\left (3,e^{2 c+2 d x}\right )de^{2 c+2 d x}}{4 d^2}\right )}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )}{2 d}\right )}{2 d}\right )}{a}-\frac {b \left (\frac {\left (-\frac {(e+f x)^4}{4 b f}+\frac {\log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right ) (e+f x)^3}{b d}+\frac {\log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right ) (e+f x)^3}{b d}-\frac {3 f \left (\frac {2 f \left (\frac {(e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d}-\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )de^{c+d x}}{d^2}\right )}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d}\right )}{b d}-\frac {3 f \left (\frac {2 f \left (\frac {(e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d}-\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )de^{c+d x}}{d^2}\right )}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d}\right )}{b d}\right ) b^2}{a^2+b^2}+\frac {\int (e+f x)^3 \text {sech}(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}\right )}{a}\right )}{a}\right )}{a}\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle \frac {-\frac {\coth ^2(c+d x) (e+f x)^3}{2 d}-\frac {\log (\tanh (c+d x)) (e+f x)^3}{d}+\frac {3}{2} f \int (e+f x)^2 \left (\frac {\coth ^2(c+d x)}{d}+\frac {2 \log (\tanh (c+d x))}{d}\right )dx}{a}-\frac {b \left (\frac {-\frac {\arctan (\sinh (c+d x)) (e+f x)^3}{d}-\frac {\text {csch}(c+d x) (e+f x)^3}{d}+3 f \int (e+f x)^2 \left (\frac {\arctan (\sinh (c+d x))}{d}+\frac {\text {csch}(c+d x)}{d}\right )dx}{a}-\frac {b \left (\frac {2 i \left (\frac {i \text {arctanh}\left (e^{2 c+2 d x}\right ) (e+f x)^3}{d}-\frac {3 i f \left (\frac {f \left (\frac {(e+f x) \operatorname {PolyLog}\left (3,-e^{2 c+2 d x}\right )}{2 d}-\frac {f \operatorname {PolyLog}\left (4,-e^{2 c+2 d x}\right )}{4 d^2}\right )}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )}{2 d}\right )}{2 d}+\frac {3 i f \left (\frac {f \left (\frac {(e+f x) \operatorname {PolyLog}\left (3,e^{2 c+2 d x}\right )}{2 d}-\frac {f \operatorname {PolyLog}\left (4,e^{2 c+2 d x}\right )}{4 d^2}\right )}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )}{2 d}\right )}{2 d}\right )}{a}-\frac {b \left (\frac {\left (-\frac {(e+f x)^4}{4 b f}+\frac {\log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right ) (e+f x)^3}{b d}+\frac {\log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right ) (e+f x)^3}{b d}-\frac {3 f \left (\frac {2 f \left (\frac {(e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d}-\frac {f \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d^2}\right )}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d}\right )}{b d}-\frac {3 f \left (\frac {2 f \left (\frac {(e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d}-\frac {f \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d^2}\right )}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d}\right )}{b d}\right ) b^2}{a^2+b^2}+\frac {\int (e+f x)^3 \text {sech}(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}\right )}{a}\right )}{a}\right )}{a}\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \frac {-\frac {\coth ^2(c+d x) (e+f x)^3}{2 d}-\frac {\log (\tanh (c+d x)) (e+f x)^3}{d}+\frac {3}{2} f \int \frac {(e+f x)^2 \left (\coth ^2(c+d x)+2 \log (\tanh (c+d x))\right )}{d}dx}{a}-\frac {b \left (\frac {-\frac {\arctan (\sinh (c+d x)) (e+f x)^3}{d}-\frac {\text {csch}(c+d x) (e+f x)^3}{d}+3 f \int \frac {(e+f x)^2 (\arctan (\sinh (c+d x))+\text {csch}(c+d x))}{d}dx}{a}-\frac {b \left (\frac {2 i \left (\frac {i \text {arctanh}\left (e^{2 c+2 d x}\right ) (e+f x)^3}{d}-\frac {3 i f \left (\frac {f \left (\frac {(e+f x) \operatorname {PolyLog}\left (3,-e^{2 c+2 d x}\right )}{2 d}-\frac {f \operatorname {PolyLog}\left (4,-e^{2 c+2 d x}\right )}{4 d^2}\right )}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )}{2 d}\right )}{2 d}+\frac {3 i f \left (\frac {f \left (\frac {(e+f x) \operatorname {PolyLog}\left (3,e^{2 c+2 d x}\right )}{2 d}-\frac {f \operatorname {PolyLog}\left (4,e^{2 c+2 d x}\right )}{4 d^2}\right )}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )}{2 d}\right )}{2 d}\right )}{a}-\frac {b \left (\frac {\left (-\frac {(e+f x)^4}{4 b f}+\frac {\log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right ) (e+f x)^3}{b d}+\frac {\log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right ) (e+f x)^3}{b d}-\frac {3 f \left (\frac {2 f \left (\frac {(e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d}-\frac {f \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d^2}\right )}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d}\right )}{b d}-\frac {3 f \left (\frac {2 f \left (\frac {(e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d}-\frac {f \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d^2}\right )}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d}\right )}{b d}\right ) b^2}{a^2+b^2}+\frac {\int (e+f x)^3 \text {sech}(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}\right )}{a}\right )}{a}\right )}{a}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {\coth ^2(c+d x) (e+f x)^3}{2 d}-\frac {\log (\tanh (c+d x)) (e+f x)^3}{d}+\frac {3 f \int (e+f x)^2 \left (\coth ^2(c+d x)+2 \log (\tanh (c+d x))\right )dx}{2 d}}{a}-\frac {b \left (\frac {-\frac {\arctan (\sinh (c+d x)) (e+f x)^3}{d}-\frac {\text {csch}(c+d x) (e+f x)^3}{d}+\frac {3 f \int (e+f x)^2 (\arctan (\sinh (c+d x))+\text {csch}(c+d x))dx}{d}}{a}-\frac {b \left (\frac {2 i \left (\frac {i \text {arctanh}\left (e^{2 c+2 d x}\right ) (e+f x)^3}{d}-\frac {3 i f \left (\frac {f \left (\frac {(e+f x) \operatorname {PolyLog}\left (3,-e^{2 c+2 d x}\right )}{2 d}-\frac {f \operatorname {PolyLog}\left (4,-e^{2 c+2 d x}\right )}{4 d^2}\right )}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )}{2 d}\right )}{2 d}+\frac {3 i f \left (\frac {f \left (\frac {(e+f x) \operatorname {PolyLog}\left (3,e^{2 c+2 d x}\right )}{2 d}-\frac {f \operatorname {PolyLog}\left (4,e^{2 c+2 d x}\right )}{4 d^2}\right )}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )}{2 d}\right )}{2 d}\right )}{a}-\frac {b \left (\frac {\left (-\frac {(e+f x)^4}{4 b f}+\frac {\log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right ) (e+f x)^3}{b d}+\frac {\log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right ) (e+f x)^3}{b d}-\frac {3 f \left (\frac {2 f \left (\frac {(e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d}-\frac {f \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d^2}\right )}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d}\right )}{b d}-\frac {3 f \left (\frac {2 f \left (\frac {(e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d}-\frac {f \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d^2}\right )}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d}\right )}{b d}\right ) b^2}{a^2+b^2}+\frac {\int (e+f x)^3 \text {sech}(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}\right )}{a}\right )}{a}\right )}{a}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {-\frac {\coth ^2(c+d x) (e+f x)^3}{2 d}-\frac {\log (\tanh (c+d x)) (e+f x)^3}{d}+\frac {3 f \int \left (\coth ^2(c+d x) (e+f x)^2+2 \log (\tanh (c+d x)) (e+f x)^2\right )dx}{2 d}}{a}-\frac {b \left (\frac {-\frac {\arctan (\sinh (c+d x)) (e+f x)^3}{d}-\frac {\text {csch}(c+d x) (e+f x)^3}{d}+\frac {3 f \int \left (\arctan (\sinh (c+d x)) (e+f x)^2+\text {csch}(c+d x) (e+f x)^2\right )dx}{d}}{a}-\frac {b \left (\frac {2 i \left (\frac {i \text {arctanh}\left (e^{2 c+2 d x}\right ) (e+f x)^3}{d}-\frac {3 i f \left (\frac {f \left (\frac {(e+f x) \operatorname {PolyLog}\left (3,-e^{2 c+2 d x}\right )}{2 d}-\frac {f \operatorname {PolyLog}\left (4,-e^{2 c+2 d x}\right )}{4 d^2}\right )}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )}{2 d}\right )}{2 d}+\frac {3 i f \left (\frac {f \left (\frac {(e+f x) \operatorname {PolyLog}\left (3,e^{2 c+2 d x}\right )}{2 d}-\frac {f \operatorname {PolyLog}\left (4,e^{2 c+2 d x}\right )}{4 d^2}\right )}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )}{2 d}\right )}{2 d}\right )}{a}-\frac {b \left (\frac {\left (-\frac {(e+f x)^4}{4 b f}+\frac {\log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right ) (e+f x)^3}{b d}+\frac {\log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right ) (e+f x)^3}{b d}-\frac {3 f \left (\frac {2 f \left (\frac {(e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d}-\frac {f \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d^2}\right )}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d}\right )}{b d}-\frac {3 f \left (\frac {2 f \left (\frac {(e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d}-\frac {f \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d^2}\right )}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d}\right )}{b d}\right ) b^2}{a^2+b^2}+\frac {\int \left (a (e+f x)^3 \text {sech}(c+d x)-b (e+f x)^3 \tanh (c+d x)\right )dx}{a^2+b^2}\right )}{a}\right )}{a}\right )}{a}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-\frac {\coth ^2(c+d x) (e+f x)^3}{2 d}-\frac {\log (\tanh (c+d x)) (e+f x)^3}{d}+\frac {3 f \left (\frac {4 \text {arctanh}\left (e^{2 c+2 d x}\right ) (e+f x)^3}{3 f}+\frac {2 \log (\tanh (c+d x)) (e+f x)^3}{3 f}+\frac {(e+f x)^3}{3 f}-\frac {\coth (c+d x) (e+f x)^2}{d}+\frac {\operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right ) (e+f x)^2}{d}-\frac {\operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right ) (e+f x)^2}{d}-\frac {(e+f x)^2}{d}+\frac {2 f \log \left (1-e^{2 (c+d x)}\right ) (e+f x)}{d^2}-\frac {f \operatorname {PolyLog}\left (3,-e^{2 c+2 d x}\right ) (e+f x)}{d^2}+\frac {f \operatorname {PolyLog}\left (3,e^{2 c+2 d x}\right ) (e+f x)}{d^2}+\frac {f^2 \operatorname {PolyLog}\left (2,e^{2 (c+d x)}\right )}{d^3}+\frac {f^2 \operatorname {PolyLog}\left (4,-e^{2 c+2 d x}\right )}{2 d^3}-\frac {f^2 \operatorname {PolyLog}\left (4,e^{2 c+2 d x}\right )}{2 d^3}\right )}{2 d}}{a}-\frac {b \left (\frac {-\frac {\arctan (\sinh (c+d x)) (e+f x)^3}{d}-\frac {\text {csch}(c+d x) (e+f x)^3}{d}+\frac {3 f \left (-\frac {2 \arctan \left (e^{c+d x}\right ) (e+f x)^3}{3 f}+\frac {\arctan (\sinh (c+d x)) (e+f x)^3}{3 f}-\frac {2 \text {arctanh}\left (e^{c+d x}\right ) (e+f x)^2}{d}+\frac {i \operatorname {PolyLog}\left (2,-i e^{c+d x}\right ) (e+f x)^2}{d}-\frac {i \operatorname {PolyLog}\left (2,i e^{c+d x}\right ) (e+f x)^2}{d}-\frac {2 f \operatorname {PolyLog}\left (2,-e^{c+d x}\right ) (e+f x)}{d^2}+\frac {2 f \operatorname {PolyLog}\left (2,e^{c+d x}\right ) (e+f x)}{d^2}-\frac {2 i f \operatorname {PolyLog}\left (3,-i e^{c+d x}\right ) (e+f x)}{d^2}+\frac {2 i f \operatorname {PolyLog}\left (3,i e^{c+d x}\right ) (e+f x)}{d^2}+\frac {2 f^2 \operatorname {PolyLog}\left (3,-e^{c+d x}\right )}{d^3}-\frac {2 f^2 \operatorname {PolyLog}\left (3,e^{c+d x}\right )}{d^3}+\frac {2 i f^2 \operatorname {PolyLog}\left (4,-i e^{c+d x}\right )}{d^3}-\frac {2 i f^2 \operatorname {PolyLog}\left (4,i e^{c+d x}\right )}{d^3}\right )}{d}}{a}-\frac {b \left (\frac {2 i \left (\frac {i \text {arctanh}\left (e^{2 c+2 d x}\right ) (e+f x)^3}{d}-\frac {3 i f \left (\frac {f \left (\frac {(e+f x) \operatorname {PolyLog}\left (3,-e^{2 c+2 d x}\right )}{2 d}-\frac {f \operatorname {PolyLog}\left (4,-e^{2 c+2 d x}\right )}{4 d^2}\right )}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )}{2 d}\right )}{2 d}+\frac {3 i f \left (\frac {f \left (\frac {(e+f x) \operatorname {PolyLog}\left (3,e^{2 c+2 d x}\right )}{2 d}-\frac {f \operatorname {PolyLog}\left (4,e^{2 c+2 d x}\right )}{4 d^2}\right )}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )}{2 d}\right )}{2 d}\right )}{a}-\frac {b \left (\frac {\left (-\frac {(e+f x)^4}{4 b f}+\frac {\log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right ) (e+f x)^3}{b d}+\frac {\log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right ) (e+f x)^3}{b d}-\frac {3 f \left (\frac {2 f \left (\frac {(e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d}-\frac {f \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d^2}\right )}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d}\right )}{b d}-\frac {3 f \left (\frac {2 f \left (\frac {(e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d}-\frac {f \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d^2}\right )}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d}\right )}{b d}\right ) b^2}{a^2+b^2}+\frac {\frac {b (e+f x)^4}{4 f}+\frac {2 a \arctan \left (e^{c+d x}\right ) (e+f x)^3}{d}-\frac {b \log \left (1+e^{2 (c+d x)}\right ) (e+f x)^3}{d}-\frac {3 i a f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right ) (e+f x)^2}{d^2}+\frac {3 i a f \operatorname {PolyLog}\left (2,i e^{c+d x}\right ) (e+f x)^2}{d^2}-\frac {3 b f \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right ) (e+f x)^2}{2 d^2}+\frac {6 i a f^2 \operatorname {PolyLog}\left (3,-i e^{c+d x}\right ) (e+f x)}{d^3}-\frac {6 i a f^2 \operatorname {PolyLog}\left (3,i e^{c+d x}\right ) (e+f x)}{d^3}+\frac {3 b f^2 \operatorname {PolyLog}\left (3,-e^{2 (c+d x)}\right ) (e+f x)}{2 d^3}-\frac {6 i a f^3 \operatorname {PolyLog}\left (4,-i e^{c+d x}\right )}{d^4}+\frac {6 i a f^3 \operatorname {PolyLog}\left (4,i e^{c+d x}\right )}{d^4}-\frac {3 b f^3 \operatorname {PolyLog}\left (4,-e^{2 (c+d x)}\right )}{4 d^4}}{a^2+b^2}\right )}{a}\right )}{a}\right )}{a}\) |
(-1/2*((e + f*x)^3*Coth[c + d*x]^2)/d - ((e + f*x)^3*Log[Tanh[c + d*x]])/d + (3*f*(-((e + f*x)^2/d) + (e + f*x)^3/(3*f) + (4*(e + f*x)^3*ArcTanh[E^( 2*c + 2*d*x)])/(3*f) - ((e + f*x)^2*Coth[c + d*x])/d + (2*f*(e + f*x)*Log[ 1 - E^(2*(c + d*x))])/d^2 + (2*(e + f*x)^3*Log[Tanh[c + d*x]])/(3*f) + (f^ 2*PolyLog[2, E^(2*(c + d*x))])/d^3 + ((e + f*x)^2*PolyLog[2, -E^(2*c + 2*d *x)])/d - ((e + f*x)^2*PolyLog[2, E^(2*c + 2*d*x)])/d - (f*(e + f*x)*PolyL og[3, -E^(2*c + 2*d*x)])/d^2 + (f*(e + f*x)*PolyLog[3, E^(2*c + 2*d*x)])/d ^2 + (f^2*PolyLog[4, -E^(2*c + 2*d*x)])/(2*d^3) - (f^2*PolyLog[4, E^(2*c + 2*d*x)])/(2*d^3)))/(2*d))/a - (b*((-(((e + f*x)^3*ArcTan[Sinh[c + d*x]])/ d) - ((e + f*x)^3*Csch[c + d*x])/d + (3*f*((-2*(e + f*x)^3*ArcTan[E^(c + d *x)])/(3*f) + ((e + f*x)^3*ArcTan[Sinh[c + d*x]])/(3*f) - (2*(e + f*x)^2*A rcTanh[E^(c + d*x)])/d - (2*f*(e + f*x)*PolyLog[2, -E^(c + d*x)])/d^2 + (I *(e + f*x)^2*PolyLog[2, (-I)*E^(c + d*x)])/d - (I*(e + f*x)^2*PolyLog[2, I *E^(c + d*x)])/d + (2*f*(e + f*x)*PolyLog[2, E^(c + d*x)])/d^2 + (2*f^2*Po lyLog[3, -E^(c + d*x)])/d^3 - ((2*I)*f*(e + f*x)*PolyLog[3, (-I)*E^(c + d* x)])/d^2 + ((2*I)*f*(e + f*x)*PolyLog[3, I*E^(c + d*x)])/d^2 - (2*f^2*Poly Log[3, E^(c + d*x)])/d^3 + ((2*I)*f^2*PolyLog[4, (-I)*E^(c + d*x)])/d^3 - ((2*I)*f^2*PolyLog[4, I*E^(c + d*x)])/d^3))/d)/a - (b*(-((b*((b^2*(-1/4*(e + f*x)^4/(b*f) + ((e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2 ])])/(b*d) + ((e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])...
3.5.91.3.1 Defintions of rubi rules used
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/ ((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp [((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si mp[d*(m/(b*f*g*n*Log[F])) Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x )))^n/a)], x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct ionOfExponentialQ[u, x] && !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ [{a, m, n}, x] && IntegerQ[m*n]] && !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) *(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.) *(x_))^(m_.), x_Symbol] :> Simp[(-(f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Simp[g*(m/(b*c*n*Log[F])) Int[(f + g*x)^( m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e , f, g, n}, x] && GtQ[m, 0]
Int[csc[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x _Symbol] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^((-I)*e + f*fz*x)]/(f*fz*I)), x] + (-Simp[d*(m/(f*fz*I)) Int[(c + d*x)^(m - 1)*Log[1 - E^((-I)*e + f*fz*x )], x], x] + Simp[d*(m/(f*fz*I)) Int[(c + d*x)^(m - 1)*Log[1 + E^((-I)*e + f*fz*x)], x], x]) /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[m, 0]
Int[Csch[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.)*Sech[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Simp[2^n Int[(c + d*x)^m*Csch[2*a + 2*b*x ]^n, x], x] /; FreeQ[{a, b, c, d}, x] && RationalQ[m] && IntegerQ[n]
Int[Csch[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.)*Sech[(a_.) + (b_.)*(x_)]^(p_.), x_Symbol] :> With[{u = IntHide[Csch[a + b*x]^n*Sech[a + b*x]^p, x]}, Simp[(c + d*x)^m u, x] - Simp[d*m Int[(c + d*x)^(m - 1)*u, x], x]] /; FreeQ[{a, b, c, d}, x] && IntegersQ[n, p] && GtQ[m, 0] && NeQ[n , p]
Int[(Cosh[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sin h[(c_.) + (d_.)*(x_)]), x_Symbol] :> Simp[-(e + f*x)^(m + 1)/(b*f*(m + 1)), x] + (Int[(e + f*x)^m*(E^(c + d*x)/(a - Rt[a^2 + b^2, 2] + b*E^(c + d*x))) , x] + Int[(e + f*x)^m*(E^(c + d*x)/(a + Rt[a^2 + b^2, 2] + b*E^(c + d*x))) , x]) /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && NeQ[a^2 + b^2, 0]
Int[(((e_.) + (f_.)*(x_))^(m_.)*Sech[(c_.) + (d_.)*(x_)]^(n_.))/((a_) + (b_ .)*Sinh[(c_.) + (d_.)*(x_)]), x_Symbol] :> Simp[b^2/(a^2 + b^2) Int[(e + f*x)^m*(Sech[c + d*x]^(n - 2)/(a + b*Sinh[c + d*x])), x], x] + Simp[1/(a^2 + b^2) Int[(e + f*x)^m*Sech[c + d*x]^n*(a - b*Sinh[c + d*x]), x], x] /; F reeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && NeQ[a^2 + b^2, 0] && IGtQ[n, 0 ]
Int[(Csch[(c_.) + (d_.)*(x_)]^(n_.)*((e_.) + (f_.)*(x_))^(m_.)*Sech[(c_.) + (d_.)*(x_)]^(p_.))/((a_) + (b_.)*Sinh[(c_.) + (d_.)*(x_)]), x_Symbol] :> S imp[1/a Int[(e + f*x)^m*Sech[c + d*x]^p*Csch[c + d*x]^n, x], x] - Simp[b/ a Int[(e + f*x)^m*Sech[c + d*x]^p*(Csch[c + d*x]^(n - 1)/(a + b*Sinh[c + d*x])), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0] && IGtQ[p, 0]
Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d , e, n, p}, x] && EqQ[b*d, a*e]
Int[((e_.) + (f_.)*(x_))^(m_.)*PolyLog[n_, (d_.)*((F_)^((c_.)*((a_.) + (b_. )*(x_))))^(p_.)], x_Symbol] :> Simp[(e + f*x)^m*(PolyLog[n + 1, d*(F^(c*(a + b*x)))^p]/(b*c*p*Log[F])), x] - Simp[f*(m/(b*c*p*Log[F])) Int[(e + f*x) ^(m - 1)*PolyLog[n + 1, d*(F^(c*(a + b*x)))^p], x], x] /; FreeQ[{F, a, b, c , d, e, f, n, p}, x] && GtQ[m, 0]
\[\int \frac {\left (f x +e \right )^{3} \operatorname {csch}\left (d x +c \right )^{3} \operatorname {sech}\left (d x +c \right )}{a +b \sinh \left (d x +c \right )}d x\]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 23903 vs. \(2 (1636) = 3272\).
Time = 0.77 (sec) , antiderivative size = 23903, normalized size of antiderivative = 13.32 \[ \int \frac {(e+f x)^3 \text {csch}^3(c+d x) \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \]
Timed out. \[ \int \frac {(e+f x)^3 \text {csch}^3(c+d x) \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Timed out} \]
\[ \int \frac {(e+f x)^3 \text {csch}^3(c+d x) \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{3} \operatorname {csch}\left (d x + c\right )^{3} \operatorname {sech}\left (d x + c\right )}{b \sinh \left (d x + c\right ) + a} \,d x } \]
-(b^4*log(-2*a*e^(-d*x - c) + b*e^(-2*d*x - 2*c) - b)/((a^5 + a^3*b^2)*d) + 2*b*arctan(e^(-d*x - c))/((a^2 + b^2)*d) - a*log(e^(-2*d*x - 2*c) + 1)/( (a^2 + b^2)*d) + 2*(b*e^(-d*x - c) - a*e^(-2*d*x - 2*c) - b*e^(-3*d*x - 3* c))/((2*a^2*e^(-2*d*x - 2*c) - a^2*e^(-4*d*x - 4*c) - a^2)*d) + (a^2 - b^2 )*log(e^(-d*x - c) + 1)/(a^3*d) + (a^2 - b^2)*log(e^(-d*x - c) - 1)/(a^3*d ))*e^3 + (3*a*f^3*x^2 + 6*a*e*f^2*x + 3*a*e^2*f + 2*(b*d*f^3*x^3*e^(3*c) + 3*b*d*e*f^2*x^2*e^(3*c) + 3*b*d*e^2*f*x*e^(3*c))*e^(3*d*x) - (2*a*d*f^3*x ^3*e^(2*c) + 3*a*e^2*f*e^(2*c) + 3*(2*d*e*f^2 + f^3)*a*x^2*e^(2*c) + 6*(d* e^2*f + e*f^2)*a*x*e^(2*c))*e^(2*d*x) - 2*(b*d*f^3*x^3*e^c + 3*b*d*e*f^2*x ^2*e^c + 3*b*d*e^2*f*x*e^c)*e^(d*x))/(a^2*d^2*e^(4*d*x + 4*c) - 2*a^2*d^2* e^(2*d*x + 2*c) + a^2*d^2) - 3*(b*d*e^2*f + a*e*f^2)*x/(a^2*d^2) + 3*(b*d* e^2*f - a*e*f^2)*x/(a^2*d^2) + 3*(b*d*e^2*f + a*e*f^2)*log(e^(d*x + c) + 1 )/(a^2*d^3) - 3*(b*d*e^2*f - a*e*f^2)*log(e^(d*x + c) - 1)/(a^2*d^3) - (d^ 3*x^3*log(e^(d*x + c) + 1) + 3*d^2*x^2*dilog(-e^(d*x + c)) - 6*d*x*polylog (3, -e^(d*x + c)) + 6*polylog(4, -e^(d*x + c)))*(a^2*f^3 - b^2*f^3)/(a^3*d ^4) - (d^3*x^3*log(-e^(d*x + c) + 1) + 3*d^2*x^2*dilog(e^(d*x + c)) - 6*d* x*polylog(3, e^(d*x + c)) + 6*polylog(4, e^(d*x + c)))*(a^2*f^3 - b^2*f^3) /(a^3*d^4) - 3*(a^2*d*e*f^2 - b^2*d*e*f^2 - a*b*f^3)*(d^2*x^2*log(e^(d*x + c) + 1) + 2*d*x*dilog(-e^(d*x + c)) - 2*polylog(3, -e^(d*x + c)))/(a^3*d^ 4) - 3*(a^2*d*e*f^2 - b^2*d*e*f^2 + a*b*f^3)*(d^2*x^2*log(-e^(d*x + c) ...
Timed out. \[ \int \frac {(e+f x)^3 \text {csch}^3(c+d x) \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {(e+f x)^3 \text {csch}^3(c+d x) \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {{\left (e+f\,x\right )}^3}{\mathrm {cosh}\left (c+d\,x\right )\,{\mathrm {sinh}\left (c+d\,x\right )}^3\,\left (a+b\,\mathrm {sinh}\left (c+d\,x\right )\right )} \,d x \]