Integrand size = 32, antiderivative size = 454 \[ \int \frac {(e+f x) \sinh (c+d x) \tanh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {e x}{b}+\frac {f x^2}{2 b}-\frac {a f \arctan (\sinh (c+d x))}{b^2 d^2}+\frac {a^3 f \arctan (\sinh (c+d x))}{b^2 \left (a^2+b^2\right ) d^2}-\frac {a^3 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right )^{3/2} d}+\frac {a^3 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right )^{3/2} d}-\frac {a^2 f \log (\cosh (c+d x))}{b^3 d^2}+\frac {f \log (\cosh (c+d x))}{b d^2}+\frac {a^4 f \log (\cosh (c+d x))}{b^3 \left (a^2+b^2\right ) d^2}-\frac {a^3 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right )^{3/2} d^2}+\frac {a^3 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right )^{3/2} d^2}+\frac {a (e+f x) \text {sech}(c+d x)}{b^2 d}-\frac {a^3 (e+f x) \text {sech}(c+d x)}{b^2 \left (a^2+b^2\right ) d}+\frac {a^2 (e+f x) \tanh (c+d x)}{b^3 d}-\frac {(e+f x) \tanh (c+d x)}{b d}-\frac {a^4 (e+f x) \tanh (c+d x)}{b^3 \left (a^2+b^2\right ) d} \]
e*x/b+1/2*f*x^2/b-a*f*arctan(sinh(d*x+c))/b^2/d^2+a^3*f*arctan(sinh(d*x+c) )/b^2/(a^2+b^2)/d^2-a^2*f*ln(cosh(d*x+c))/b^3/d^2+f*ln(cosh(d*x+c))/b/d^2+ a^4*f*ln(cosh(d*x+c))/b^3/(a^2+b^2)/d^2-a^3*(f*x+e)*ln(1+b*exp(d*x+c)/(a-( a^2+b^2)^(1/2)))/b/(a^2+b^2)^(3/2)/d+a^3*(f*x+e)*ln(1+b*exp(d*x+c)/(a+(a^2 +b^2)^(1/2)))/b/(a^2+b^2)^(3/2)/d-a^3*f*polylog(2,-b*exp(d*x+c)/(a-(a^2+b^ 2)^(1/2)))/b/(a^2+b^2)^(3/2)/d^2+a^3*f*polylog(2,-b*exp(d*x+c)/(a+(a^2+b^2 )^(1/2)))/b/(a^2+b^2)^(3/2)/d^2+a*(f*x+e)*sech(d*x+c)/b^2/d-a^3*(f*x+e)*se ch(d*x+c)/b^2/(a^2+b^2)/d+a^2*(f*x+e)*tanh(d*x+c)/b^3/d-(f*x+e)*tanh(d*x+c )/b/d-a^4*(f*x+e)*tanh(d*x+c)/b^3/(a^2+b^2)/d
Result contains complex when optimal does not.
Time = 5.67 (sec) , antiderivative size = 358, normalized size of antiderivative = 0.79 \[ \int \frac {(e+f x) \sinh (c+d x) \tanh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {\frac {(c+d x) (c f-d (2 e+f x))}{b}+\frac {2 f \arctan \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )}{a-i b}+\frac {2 f \arctan \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )}{a+i b}+\frac {f \log (\cosh (c+d x))}{i a-b}-\frac {f \log (\cosh (c+d x))}{i a+b}+\frac {2 a^3 \left (-2 d e \text {arctanh}\left (\frac {a+b e^{c+d x}}{\sqrt {a^2+b^2}}\right )+2 c f \text {arctanh}\left (\frac {a+b e^{c+d x}}{\sqrt {a^2+b^2}}\right )+f (c+d x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )-f (c+d x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )+f \operatorname {PolyLog}\left (2,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )-f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )\right )}{b \left (a^2+b^2\right )^{3/2}}+\frac {2 d (e+f x) \text {sech}(c+d x) (-a+b \sinh (c+d x))}{a^2+b^2}}{2 d^2} \]
-1/2*(((c + d*x)*(c*f - d*(2*e + f*x)))/b + (2*f*ArcTan[Tanh[(c + d*x)/2]] )/(a - I*b) + (2*f*ArcTan[Tanh[(c + d*x)/2]])/(a + I*b) + (f*Log[Cosh[c + d*x]])/(I*a - b) - (f*Log[Cosh[c + d*x]])/(I*a + b) + (2*a^3*(-2*d*e*ArcTa nh[(a + b*E^(c + d*x))/Sqrt[a^2 + b^2]] + 2*c*f*ArcTanh[(a + b*E^(c + d*x) )/Sqrt[a^2 + b^2]] + f*(c + d*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b ^2])] - f*(c + d*x)*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])] + f*Pol yLog[2, (b*E^(c + d*x))/(-a + Sqrt[a^2 + b^2])] - f*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))]))/(b*(a^2 + b^2)^(3/2)) + (2*d*(e + f*x)*Se ch[c + d*x]*(-a + b*Sinh[c + d*x]))/(a^2 + b^2))/d^2
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.
| \(\displaystyle \int \frac {(e+f x) \sinh (c+d x) \tanh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx\) |
| \(\Big \downarrow \) 6115 |
| \(\displaystyle \frac {\int (e+f x) \tanh ^2(c+d x)dx}{b}-\frac {a \int \frac {(e+f x) \tanh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {a \int \frac {(e+f x) \tanh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\int -\left ((e+f x) \tan (i c+i d x)^2\right )dx}{b}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {a \int \frac {(e+f x) \tanh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}-\frac {\int (e+f x) \tan (i c+i d x)^2dx}{b}\) |
| \(\Big \downarrow \) 4203 |
| \(\displaystyle -\frac {a \int \frac {(e+f x) \tanh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}-\frac {\frac {i f \int i \tanh (c+d x)dx}{d}-\int (e+f x)dx+\frac {(e+f x) \tanh (c+d x)}{d}}{b}\) |
| \(\Big \downarrow \) 17 |
| \(\displaystyle -\frac {a \int \frac {(e+f x) \tanh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}-\frac {\frac {i f \int i \tanh (c+d x)dx}{d}+\frac {(e+f x) \tanh (c+d x)}{d}-\frac {(e+f x)^2}{2 f}}{b}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {a \int \frac {(e+f x) \tanh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}-\frac {-\frac {f \int \tanh (c+d x)dx}{d}+\frac {(e+f x) \tanh (c+d x)}{d}-\frac {(e+f x)^2}{2 f}}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {a \int \frac {(e+f x) \tanh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}-\frac {-\frac {f \int -i \tan (i c+i d x)dx}{d}+\frac {(e+f x) \tanh (c+d x)}{d}-\frac {(e+f x)^2}{2 f}}{b}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {a \int \frac {(e+f x) \tanh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}-\frac {\frac {i f \int \tan (i c+i d x)dx}{d}+\frac {(e+f x) \tanh (c+d x)}{d}-\frac {(e+f x)^2}{2 f}}{b}\) |
| \(\Big \downarrow \) 3956 |
| \(\displaystyle -\frac {a \int \frac {(e+f x) \tanh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}-\frac {-\frac {f \log (\cosh (c+d x))}{d^2}+\frac {(e+f x) \tanh (c+d x)}{d}-\frac {(e+f x)^2}{2 f}}{b}\) |
| \(\Big \downarrow \) 6101 |
| \(\displaystyle -\frac {a \left (\frac {\int (e+f x) \text {sech}(c+d x) \tanh (c+d x)dx}{b}-\frac {a \int \frac {(e+f x) \text {sech}(c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}-\frac {-\frac {f \log (\cosh (c+d x))}{d^2}+\frac {(e+f x) \tanh (c+d x)}{d}-\frac {(e+f x)^2}{2 f}}{b}\) |
| \(\Big \downarrow \) 5974 |
| \(\displaystyle -\frac {a \left (\frac {\frac {f \int \text {sech}(c+d x)dx}{d}-\frac {(e+f x) \text {sech}(c+d x)}{d}}{b}-\frac {a \int \frac {(e+f x) \text {sech}(c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}-\frac {-\frac {f \log (\cosh (c+d x))}{d^2}+\frac {(e+f x) \tanh (c+d x)}{d}-\frac {(e+f x)^2}{2 f}}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {-\frac {f \log (\cosh (c+d x))}{d^2}+\frac {(e+f x) \tanh (c+d x)}{d}-\frac {(e+f x)^2}{2 f}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x) \text {sech}(c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {-\frac {(e+f x) \text {sech}(c+d x)}{d}+\frac {f \int \csc \left (i c+i d x+\frac {\pi }{2}\right )dx}{d}}{b}\right )}{b}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle -\frac {a \left (\frac {\frac {f \arctan (\sinh (c+d x))}{d^2}-\frac {(e+f x) \text {sech}(c+d x)}{d}}{b}-\frac {a \int \frac {(e+f x) \text {sech}(c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}-\frac {-\frac {f \log (\cosh (c+d x))}{d^2}+\frac {(e+f x) \tanh (c+d x)}{d}-\frac {(e+f x)^2}{2 f}}{b}\) |
| \(\Big \downarrow \) 6117 |
| \(\displaystyle -\frac {a \left (\frac {\frac {f \arctan (\sinh (c+d x))}{d^2}-\frac {(e+f x) \text {sech}(c+d x)}{d}}{b}-\frac {a \left (\frac {\int (e+f x) \text {sech}^2(c+d x)dx}{b}-\frac {a \int \frac {(e+f x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}\right )}{b}-\frac {-\frac {f \log (\cosh (c+d x))}{d^2}+\frac {(e+f x) \tanh (c+d x)}{d}-\frac {(e+f x)^2}{2 f}}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {-\frac {f \log (\cosh (c+d x))}{d^2}+\frac {(e+f x) \tanh (c+d x)}{d}-\frac {(e+f x)^2}{2 f}}{b}-\frac {a \left (\frac {\frac {f \arctan (\sinh (c+d x))}{d^2}-\frac {(e+f x) \text {sech}(c+d x)}{d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\int (e+f x) \csc \left (i c+i d x+\frac {\pi }{2}\right )^2dx}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 4672 |
| \(\displaystyle -\frac {-\frac {f \log (\cosh (c+d x))}{d^2}+\frac {(e+f x) \tanh (c+d x)}{d}-\frac {(e+f x)^2}{2 f}}{b}-\frac {a \left (\frac {\frac {f \arctan (\sinh (c+d x))}{d^2}-\frac {(e+f x) \text {sech}(c+d x)}{d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x) \tanh (c+d x)}{d}-\frac {i f \int -i \tanh (c+d x)dx}{d}}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {a \left (\frac {\frac {f \arctan (\sinh (c+d x))}{d^2}-\frac {(e+f x) \text {sech}(c+d x)}{d}}{b}-\frac {a \left (\frac {\frac {(e+f x) \tanh (c+d x)}{d}-\frac {f \int \tanh (c+d x)dx}{d}}{b}-\frac {a \int \frac {(e+f x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}\right )}{b}-\frac {-\frac {f \log (\cosh (c+d x))}{d^2}+\frac {(e+f x) \tanh (c+d x)}{d}-\frac {(e+f x)^2}{2 f}}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {-\frac {f \log (\cosh (c+d x))}{d^2}+\frac {(e+f x) \tanh (c+d x)}{d}-\frac {(e+f x)^2}{2 f}}{b}-\frac {a \left (\frac {\frac {f \arctan (\sinh (c+d x))}{d^2}-\frac {(e+f x) \text {sech}(c+d x)}{d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x) \tanh (c+d x)}{d}-\frac {f \int -i \tan (i c+i d x)dx}{d}}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {-\frac {f \log (\cosh (c+d x))}{d^2}+\frac {(e+f x) \tanh (c+d x)}{d}-\frac {(e+f x)^2}{2 f}}{b}-\frac {a \left (\frac {\frac {f \arctan (\sinh (c+d x))}{d^2}-\frac {(e+f x) \text {sech}(c+d x)}{d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x) \tanh (c+d x)}{d}+\frac {i f \int \tan (i c+i d x)dx}{d}}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3956 |
| \(\displaystyle -\frac {a \left (\frac {\frac {f \arctan (\sinh (c+d x))}{d^2}-\frac {(e+f x) \text {sech}(c+d x)}{d}}{b}-\frac {a \left (\frac {\frac {(e+f x) \tanh (c+d x)}{d}-\frac {f \log (\cosh (c+d x))}{d^2}}{b}-\frac {a \int \frac {(e+f x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}\right )}{b}-\frac {-\frac {f \log (\cosh (c+d x))}{d^2}+\frac {(e+f x) \tanh (c+d x)}{d}-\frac {(e+f x)^2}{2 f}}{b}\) |
| \(\Big \downarrow \) 6107 |
| \(\displaystyle -\frac {a \left (\frac {\frac {f \arctan (\sinh (c+d x))}{d^2}-\frac {(e+f x) \text {sech}(c+d x)}{d}}{b}-\frac {a \left (\frac {\frac {(e+f x) \tanh (c+d x)}{d}-\frac {f \log (\cosh (c+d x))}{d^2}}{b}-\frac {a \left (\frac {b^2 \int \frac {e+f x}{a+b \sinh (c+d x)}dx}{a^2+b^2}+\frac {\int (e+f x) \text {sech}^2(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}\right )}{b}\right )}{b}\right )}{b}-\frac {-\frac {f \log (\cosh (c+d x))}{d^2}+\frac {(e+f x) \tanh (c+d x)}{d}-\frac {(e+f x)^2}{2 f}}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {-\frac {f \log (\cosh (c+d x))}{d^2}+\frac {(e+f x) \tanh (c+d x)}{d}-\frac {(e+f x)^2}{2 f}}{b}-\frac {a \left (\frac {\frac {f \arctan (\sinh (c+d x))}{d^2}-\frac {(e+f x) \text {sech}(c+d x)}{d}}{b}-\frac {a \left (\frac {\frac {(e+f x) \tanh (c+d x)}{d}-\frac {f \log (\cosh (c+d x))}{d^2}}{b}-\frac {a \left (\frac {\int (e+f x) \text {sech}^2(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}+\frac {b^2 \int \frac {e+f x}{a-i b \sin (i c+i d x)}dx}{a^2+b^2}\right )}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3803 |
| \(\displaystyle -\frac {a \left (\frac {\frac {f \arctan (\sinh (c+d x))}{d^2}-\frac {(e+f x) \text {sech}(c+d x)}{d}}{b}-\frac {a \left (\frac {\frac {(e+f x) \tanh (c+d x)}{d}-\frac {f \log (\cosh (c+d x))}{d^2}}{b}-\frac {a \left (\frac {2 b^2 \int -\frac {e^{c+d x} (e+f x)}{-2 e^{c+d x} a-b e^{2 (c+d x)}+b}dx}{a^2+b^2}+\frac {\int (e+f x) \text {sech}^2(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}\right )}{b}\right )}{b}\right )}{b}-\frac {-\frac {f \log (\cosh (c+d x))}{d^2}+\frac {(e+f x) \tanh (c+d x)}{d}-\frac {(e+f x)^2}{2 f}}{b}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {a \left (\frac {\frac {f \arctan (\sinh (c+d x))}{d^2}-\frac {(e+f x) \text {sech}(c+d x)}{d}}{b}-\frac {a \left (\frac {\frac {(e+f x) \tanh (c+d x)}{d}-\frac {f \log (\cosh (c+d x))}{d^2}}{b}-\frac {a \left (\frac {\int (e+f x) \text {sech}^2(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}-\frac {2 b^2 \int \frac {e^{c+d x} (e+f x)}{-2 e^{c+d x} a-b e^{2 (c+d x)}+b}dx}{a^2+b^2}\right )}{b}\right )}{b}\right )}{b}-\frac {-\frac {f \log (\cosh (c+d x))}{d^2}+\frac {(e+f x) \tanh (c+d x)}{d}-\frac {(e+f x)^2}{2 f}}{b}\) |
| \(\Big \downarrow \) 2694 |
| \(\displaystyle -\frac {a \left (\frac {\frac {f \arctan (\sinh (c+d x))}{d^2}-\frac {(e+f x) \text {sech}(c+d x)}{d}}{b}-\frac {a \left (\frac {\frac {(e+f x) \tanh (c+d x)}{d}-\frac {f \log (\cosh (c+d x))}{d^2}}{b}-\frac {a \left (\frac {\int (e+f x) \text {sech}^2(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}-\frac {2 b^2 \left (\frac {b \int -\frac {e^{c+d x} (e+f x)}{2 \left (a+b e^{c+d x}-\sqrt {a^2+b^2}\right )}dx}{\sqrt {a^2+b^2}}-\frac {b \int -\frac {e^{c+d x} (e+f x)}{2 \left (a+b e^{c+d x}+\sqrt {a^2+b^2}\right )}dx}{\sqrt {a^2+b^2}}\right )}{a^2+b^2}\right )}{b}\right )}{b}\right )}{b}-\frac {-\frac {f \log (\cosh (c+d x))}{d^2}+\frac {(e+f x) \tanh (c+d x)}{d}-\frac {(e+f x)^2}{2 f}}{b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {a \left (\frac {\frac {f \arctan (\sinh (c+d x))}{d^2}-\frac {(e+f x) \text {sech}(c+d x)}{d}}{b}-\frac {a \left (\frac {\frac {(e+f x) \tanh (c+d x)}{d}-\frac {f \log (\cosh (c+d x))}{d^2}}{b}-\frac {a \left (\frac {\int (e+f x) \text {sech}^2(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}-\frac {2 b^2 \left (\frac {b \int \frac {e^{c+d x} (e+f x)}{a+b e^{c+d x}+\sqrt {a^2+b^2}}dx}{2 \sqrt {a^2+b^2}}-\frac {b \int \frac {e^{c+d x} (e+f x)}{a+b e^{c+d x}-\sqrt {a^2+b^2}}dx}{2 \sqrt {a^2+b^2}}\right )}{a^2+b^2}\right )}{b}\right )}{b}\right )}{b}-\frac {-\frac {f \log (\cosh (c+d x))}{d^2}+\frac {(e+f x) \tanh (c+d x)}{d}-\frac {(e+f x)^2}{2 f}}{b}\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle -\frac {a \left (\frac {\frac {f \arctan (\sinh (c+d x))}{d^2}-\frac {(e+f x) \text {sech}(c+d x)}{d}}{b}-\frac {a \left (\frac {\frac {(e+f x) \tanh (c+d x)}{d}-\frac {f \log (\cosh (c+d x))}{d^2}}{b}-\frac {a \left (\frac {\int (e+f x) \text {sech}^2(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}-\frac {2 b^2 \left (\frac {b \left (\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}-\frac {f \int \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right )dx}{b d}\right )}{2 \sqrt {a^2+b^2}}-\frac {b \left (\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}-\frac {f \int \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right )dx}{b d}\right )}{2 \sqrt {a^2+b^2}}\right )}{a^2+b^2}\right )}{b}\right )}{b}\right )}{b}-\frac {-\frac {f \log (\cosh (c+d x))}{d^2}+\frac {(e+f x) \tanh (c+d x)}{d}-\frac {(e+f x)^2}{2 f}}{b}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle -\frac {a \left (\frac {\frac {f \arctan (\sinh (c+d x))}{d^2}-\frac {(e+f x) \text {sech}(c+d x)}{d}}{b}-\frac {a \left (\frac {\frac {(e+f x) \tanh (c+d x)}{d}-\frac {f \log (\cosh (c+d x))}{d^2}}{b}-\frac {a \left (\frac {\int (e+f x) \text {sech}^2(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}-\frac {2 b^2 \left (\frac {b \left (\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}-\frac {f \int e^{-c-d x} \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right )de^{c+d x}}{b d^2}\right )}{2 \sqrt {a^2+b^2}}-\frac {b \left (\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}-\frac {f \int e^{-c-d x} \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right )de^{c+d x}}{b d^2}\right )}{2 \sqrt {a^2+b^2}}\right )}{a^2+b^2}\right )}{b}\right )}{b}\right )}{b}-\frac {-\frac {f \log (\cosh (c+d x))}{d^2}+\frac {(e+f x) \tanh (c+d x)}{d}-\frac {(e+f x)^2}{2 f}}{b}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle -\frac {a \left (\frac {\frac {f \arctan (\sinh (c+d x))}{d^2}-\frac {(e+f x) \text {sech}(c+d x)}{d}}{b}-\frac {a \left (\frac {\frac {(e+f x) \tanh (c+d x)}{d}-\frac {f \log (\cosh (c+d x))}{d^2}}{b}-\frac {a \left (\frac {\int (e+f x) \text {sech}^2(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}-\frac {2 b^2 \left (\frac {b \left (\frac {f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b d^2}+\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}\right )}{2 \sqrt {a^2+b^2}}-\frac {b \left (\frac {f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b d^2}+\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}\right )}{2 \sqrt {a^2+b^2}}\right )}{a^2+b^2}\right )}{b}\right )}{b}\right )}{b}-\frac {-\frac {f \log (\cosh (c+d x))}{d^2}+\frac {(e+f x) \tanh (c+d x)}{d}-\frac {(e+f x)^2}{2 f}}{b}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {a \left (\frac {\frac {f \arctan (\sinh (c+d x))}{d^2}-\frac {(e+f x) \text {sech}(c+d x)}{d}}{b}-\frac {a \left (\frac {\frac {(e+f x) \tanh (c+d x)}{d}-\frac {f \log (\cosh (c+d x))}{d^2}}{b}-\frac {a \left (\frac {\int \left (a (e+f x) \text {sech}^2(c+d x)-b (e+f x) \text {sech}(c+d x) \tanh (c+d x)\right )dx}{a^2+b^2}-\frac {2 b^2 \left (\frac {b \left (\frac {f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b d^2}+\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}\right )}{2 \sqrt {a^2+b^2}}-\frac {b \left (\frac {f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b d^2}+\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}\right )}{2 \sqrt {a^2+b^2}}\right )}{a^2+b^2}\right )}{b}\right )}{b}\right )}{b}-\frac {-\frac {f \log (\cosh (c+d x))}{d^2}+\frac {(e+f x) \tanh (c+d x)}{d}-\frac {(e+f x)^2}{2 f}}{b}\) |
3.5.13.3.1 Defintions of rubi rules used
Int[(c_.)*((a_.) + (b_.)*(x_))^(m_.), x_Symbol] :> Simp[c*((a + b*x)^(m + 1 )/(b*(m + 1))), x] /; FreeQ[{a, b, c, m}, x] && NeQ[m, -1]
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[((F_)^(u_)*((f_.) + (g_.)*(x_))^(m_.))/((a_.) + (b_.)*(F_)^(u_) + (c_.) *(F_)^(v_)), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[2*(c/q) Int [(f + g*x)^m*(F^u/(b - q + 2*c*F^u)), x], x] - Simp[2*(c/q) Int[(f + g*x) ^m*(F^u/(b + q + 2*c*F^u)), x], x]] /; FreeQ[{F, a, b, c, f, g}, x] && EqQ[ v, 2*u] && LinearQ[u, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[m, 0]
Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Simp[1/(d*e*n*Log[F]) Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) ))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 , (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + (Complex[0, fz_])* (f_.)*(x_)]), x_Symbol] :> Simp[2 Int[(c + d*x)^m*(E^((-I)*e + f*fz*x)/(( -I)*b + 2*a*E^((-I)*e + f*fz*x) + I*b*E^(2*((-I)*e + f*fz*x)))), x], x] /; FreeQ[{a, b, c, d, e, f, fz}, x] && NeQ[a^2 - b^2, 0] && IGtQ[m, 0]
Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d *x], x]]/d, x] /; FreeQ[{c, d}, x]
Int[((c_.) + (d_.)*(x_))^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symb ol] :> Simp[b*(c + d*x)^m*((b*Tan[e + f*x])^(n - 1)/(f*(n - 1))), x] + (-Si mp[b*d*(m/(f*(n - 1))) Int[(c + d*x)^(m - 1)*(b*Tan[e + f*x])^(n - 1), x] , x] - Simp[b^2 Int[(c + d*x)^m*(b*Tan[e + f*x])^(n - 2), x], x]) /; Free Q[{b, c, d, e, f}, x] && GtQ[n, 1] && GtQ[m, 0]
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Int[csc[(e_.) + (f_.)*(x_)]^2*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp [(-(c + d*x)^m)*(Cot[e + f*x]/f), x] + Simp[d*(m/f) Int[(c + d*x)^(m - 1) *Cot[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]
Int[((c_.) + (d_.)*(x_))^(m_.)*Sech[(a_.) + (b_.)*(x_)]^(n_.)*Tanh[(a_.) + (b_.)*(x_)]^(p_.), x_Symbol] :> Simp[(-(c + d*x)^m)*(Sech[a + b*x]^n/(b*n)) , x] + Simp[d*(m/(b*n)) Int[(c + d*x)^(m - 1)*Sech[a + b*x]^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[p, 1] && GtQ[m, 0]
Int[(((e_.) + (f_.)*(x_))^(m_.)*Tanh[(c_.) + (d_.)*(x_)]^(n_.))/((a_) + (b_ .)*Sinh[(c_.) + (d_.)*(x_)]), x_Symbol] :> Simp[1/b Int[(e + f*x)^m*Sech[ c + d*x]*Tanh[c + d*x]^(n - 1), x], x] - Simp[a/b Int[(e + f*x)^m*Sech[c + d*x]*(Tanh[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]
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[(((e_.) + (f_.)*(x_))^(m_.)*Sinh[(c_.) + (d_.)*(x_)]^(p_.)*Tanh[(c_.) + (d_.)*(x_)]^(n_.))/((a_) + (b_.)*Sinh[(c_.) + (d_.)*(x_)]), x_Symbol] :> S imp[1/b Int[(e + f*x)^m*Sinh[c + d*x]^(p - 1)*Tanh[c + d*x]^n, x], x] - S imp[a/b Int[(e + f*x)^m*Sinh[c + d*x]^(p - 1)*(Tanh[c + d*x]^n/(a + b*Sin h[c + d*x])), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[ n, 0] && IGtQ[p, 0]
Int[(((e_.) + (f_.)*(x_))^(m_.)*Sech[(c_.) + (d_.)*(x_)]^(p_.)*Tanh[(c_.) + (d_.)*(x_)]^(n_.))/((a_) + (b_.)*Sinh[(c_.) + (d_.)*(x_)]), x_Symbol] :> S imp[1/b Int[(e + f*x)^m*Sech[c + d*x]^(p + 1)*Tanh[c + d*x]^(n - 1), x], x] - Simp[a/b Int[(e + f*x)^m*Sech[c + d*x]^(p + 1)*(Tanh[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]
Leaf count of result is larger than twice the leaf count of optimal. \(1896\) vs. \(2(432)=864\).
Time = 2.36 (sec) , antiderivative size = 1897, normalized size of antiderivative = 4.18
2/d^2/(a^2+b^2)*b*a^2*f/(2*a^2+2*b^2)*ln(1+exp(2*d*x+2*c))-2/d^2/(a^2+b^2) ^(5/2)*b*f*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))*a^3-2/d^2/(a^ 2+b^2)^(5/2)*b^3*f*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))*a-4/d ^2/(a^2+b^2)*b^2*f/(2*a^2+2*b^2)*a*arctan(exp(d*x+c))+2*(f*x+e)*(a*exp(d*x +c)+b)/d/(a^2+b^2)/(1+exp(2*d*x+2*c))-2/d^2/(a^2+b^2)*b*a^2*f/(2*a^2+2*b^2 )*ln(b*exp(2*d*x+2*c)+2*a*exp(d*x+c)-b)+2/d/(a^2+b^2)^(3/2)*b*a^3*e/(2*a^2 +2*b^2)*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))+1/2*f*x^2/b+2/d^ 2/(a^2+b^2)^(3/2)/b*a^5*f/(2*a^2+2*b^2)*dilog((b*exp(d*x+c)+(a^2+b^2)^(1/2 )+a)/(a+(a^2+b^2)^(1/2)))-2/d^2/(a^2+b^2)^(3/2)/b*a^5*f/(2*a^2+2*b^2)*dilo g((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))+2/d^2/(a^2+b^2)^ (3/2)*b^3*a*f/(2*a^2+2*b^2)*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/ 2))+2/d^2/(a^2+b^2)^(3/2)*b*a^3*f/(2*a^2+2*b^2)*arctanh(1/2*(2*b*exp(d*x+c )+2*a)/(a^2+b^2)^(1/2))+2/d^2/(a^2+b^2)^(3/2)*b*a^3*f/(2*a^2+2*b^2)*dilog( (b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))+2/d^2/(a^2+b^2)^(1/2 )*b*a*f/(2*a^2+2*b^2)*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))+2/ d/(a^2+b^2)^(3/2)/b*a^5*e/(2*a^2+2*b^2)*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/( a^2+b^2)^(1/2))-2/d^2/(a^2+b^2)^(3/2)*b*a^3*f/(2*a^2+2*b^2)*dilog((-b*exp( d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))-2/d/(a^2+b^2)^(3/2)/b*a^5* f/(2*a^2+2*b^2)*ln((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2))) *x+2/d/(a^2+b^2)^(3/2)/b*a^5*f/(2*a^2+2*b^2)*ln((b*exp(d*x+c)+(a^2+b^2)...
Leaf count of result is larger than twice the leaf count of optimal. 1571 vs. \(2 (430) = 860\).
Time = 0.31 (sec) , antiderivative size = 1571, normalized size of antiderivative = 3.46 \[ \int \frac {(e+f x) \sinh (c+d x) \tanh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \]
1/2*((a^4 + 2*a^2*b^2 + b^4)*d^2*f*x^2 + 2*(a^4 + 2*a^2*b^2 + b^4)*d^2*e*x + 4*(a^2*b^2 + b^4)*d*e + ((a^4 + 2*a^2*b^2 + b^4)*d^2*f*x^2 + 2*((a^4 + 2*a^2*b^2 + b^4)*d^2*e - 2*(a^2*b^2 + b^4)*d*f)*x)*cosh(d*x + c)^2 + ((a^4 + 2*a^2*b^2 + b^4)*d^2*f*x^2 + 2*((a^4 + 2*a^2*b^2 + b^4)*d^2*e - 2*(a^2* b^2 + b^4)*d*f)*x)*sinh(d*x + c)^2 - 2*(a^3*b*f*cosh(d*x + c)^2 + 2*a^3*b* f*cosh(d*x + c)*sinh(d*x + c) + a^3*b*f*sinh(d*x + c)^2 + a^3*b*f)*sqrt((a ^2 + b^2)/b^2)*dilog((a*cosh(d*x + c) + a*sinh(d*x + c) + (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2) - b)/b + 1) + 2*(a^3*b*f*cosh(d* x + c)^2 + 2*a^3*b*f*cosh(d*x + c)*sinh(d*x + c) + a^3*b*f*sinh(d*x + c)^2 + a^3*b*f)*sqrt((a^2 + b^2)/b^2)*dilog((a*cosh(d*x + c) + a*sinh(d*x + c) - (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2) - b)/b + 1) + 2*(a^3*b*d*e - a^3*b*c*f + (a^3*b*d*e - a^3*b*c*f)*cosh(d*x + c)^2 + 2*(a ^3*b*d*e - a^3*b*c*f)*cosh(d*x + c)*sinh(d*x + c) + (a^3*b*d*e - a^3*b*c*f )*sinh(d*x + c)^2)*sqrt((a^2 + b^2)/b^2)*log(2*b*cosh(d*x + c) + 2*b*sinh( d*x + c) + 2*b*sqrt((a^2 + b^2)/b^2) + 2*a) - 2*(a^3*b*d*e - a^3*b*c*f + ( a^3*b*d*e - a^3*b*c*f)*cosh(d*x + c)^2 + 2*(a^3*b*d*e - a^3*b*c*f)*cosh(d* x + c)*sinh(d*x + c) + (a^3*b*d*e - a^3*b*c*f)*sinh(d*x + c)^2)*sqrt((a^2 + b^2)/b^2)*log(2*b*cosh(d*x + c) + 2*b*sinh(d*x + c) - 2*b*sqrt((a^2 + b^ 2)/b^2) + 2*a) - 2*(a^3*b*d*f*x + a^3*b*c*f + (a^3*b*d*f*x + a^3*b*c*f)*co sh(d*x + c)^2 + 2*(a^3*b*d*f*x + a^3*b*c*f)*cosh(d*x + c)*sinh(d*x + c)...
\[ \int \frac {(e+f x) \sinh (c+d x) \tanh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {\left (e + f x\right ) \sinh {\left (c + d x \right )} \tanh ^{2}{\left (c + d x \right )}}{a + b \sinh {\left (c + d x \right )}}\, dx \]
\[ \int \frac {(e+f x) \sinh (c+d x) \tanh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )} \sinh \left (d x + c\right ) \tanh \left (d x + c\right )^{2}}{b \sinh \left (d x + c\right ) + a} \,d x } \]
-(a^3*log((b*e^(-d*x - c) - a - sqrt(a^2 + b^2))/(b*e^(-d*x - c) - a + sqr t(a^2 + b^2)))/((a^2*b + b^3)*sqrt(a^2 + b^2)*d) - 2*(a*e^(-d*x - c) - b)/ ((a^2 + b^2 + (a^2 + b^2)*e^(-2*d*x - 2*c))*d) - (d*x + c)/(b*d))*e - 1/2* (4*a^3*integrate(-x*e^(d*x + c)/(a^2*b^2 + b^4 - (a^2*b^2*e^(2*c) + b^4*e^ (2*c))*e^(2*d*x) - 2*(a^3*b*e^c + a*b^3*e^c)*e^(d*x)), x) - ((a^2*d*e^(2*c ) + b^2*d*e^(2*c))*x^2*e^(2*d*x) + 4*a*b*x*e^(d*x + c) + 4*b^2*x + (a^2*d + b^2*d)*x^2)/(a^2*b*d + b^3*d + (a^2*b*d*e^(2*c) + b^3*d*e^(2*c))*e^(2*d* x)) + 4*b*x/((a^2 + b^2)*d) + 4*a*arctan(e^(d*x + c))/((a^2 + b^2)*d^2) - 2*b*log(e^(2*d*x + 2*c) + 1)/((a^2 + b^2)*d^2))*f
Timed out. \[ \int \frac {(e+f x) \sinh (c+d x) \tanh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {(e+f x) \sinh (c+d x) \tanh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {\mathrm {sinh}\left (c+d\,x\right )\,{\mathrm {tanh}\left (c+d\,x\right )}^2\,\left (e+f\,x\right )}{a+b\,\mathrm {sinh}\left (c+d\,x\right )} \,d x \]