Integrand size = 26, antiderivative size = 385 \[ \int \frac {(e+f x) \tanh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {f \arctan (\sinh (c+d x))}{b d^2}-\frac {a^2 f \arctan (\sinh (c+d x))}{b \left (a^2+b^2\right ) d^2}+\frac {a^2 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d}-\frac {a^2 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d}+\frac {a f \log (\cosh (c+d x))}{b^2 d^2}-\frac {a^3 f \log (\cosh (c+d x))}{b^2 \left (a^2+b^2\right ) d^2}+\frac {a^2 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d^2}-\frac {a^2 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d^2}-\frac {(e+f x) \text {sech}(c+d x)}{b d}+\frac {a^2 (e+f x) \text {sech}(c+d x)}{b \left (a^2+b^2\right ) d}-\frac {a (e+f x) \tanh (c+d x)}{b^2 d}+\frac {a^3 (e+f x) \tanh (c+d x)}{b^2 \left (a^2+b^2\right ) d} \] Output:
f*arctan(sinh(d*x+c))/b/d^2-a^2*f*arctan(sinh(d*x+c))/b/(a^2+b^2)/d^2+a^2* (f*x+e)*ln(1+b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/(a^2+b^2)^(3/2)/d-a^2*(f*x+ e)*ln(1+b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/(a^2+b^2)^(3/2)/d+a*f*ln(cosh(d* x+c))/b^2/d^2-a^3*f*ln(cosh(d*x+c))/b^2/(a^2+b^2)/d^2+a^2*f*polylog(2,-b*e xp(d*x+c)/(a-(a^2+b^2)^(1/2)))/(a^2+b^2)^(3/2)/d^2-a^2*f*polylog(2,-b*exp( d*x+c)/(a+(a^2+b^2)^(1/2)))/(a^2+b^2)^(3/2)/d^2-(f*x+e)*sech(d*x+c)/b/d+a^ 2*(f*x+e)*sech(d*x+c)/b/(a^2+b^2)/d-a*(f*x+e)*tanh(d*x+c)/b^2/d+a^3*(f*x+e )*tanh(d*x+c)/b^2/(a^2+b^2)/d
Result contains complex when optimal does not.
Time = 2.76 (sec) , antiderivative size = 331, normalized size of antiderivative = 0.86 \[ \int \frac {(e+f x) \tanh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {-\frac {2 i f \arctan \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )}{a-i b}+\frac {2 i f \arctan \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )}{a+i b}+\frac {f \log (\cosh (c+d x))}{a-i b}+\frac {f \log (\cosh (c+d x))}{a+i b}+\frac {2 a^2 \left (-2 d e \text {arctanh}\left (\frac {a+b e^{c+d x}}{\sqrt {a^2+b^2}}\right )+2 c f \text {arctanh}\left (\frac {a+b e^{c+d x}}{\sqrt {a^2+b^2}}\right )+f (c+d x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )-f (c+d x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )+f \operatorname {PolyLog}\left (2,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )-f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )\right )}{\left (a^2+b^2\right )^{3/2}}-\frac {2 d (e+f x) \text {sech}(c+d x) (b+a \sinh (c+d x))}{a^2+b^2}}{2 d^2} \] Input:
Integrate[((e + f*x)*Tanh[c + d*x]^2)/(a + b*Sinh[c + d*x]),x]
Output:
(((-2*I)*f*ArcTan[Tanh[(c + d*x)/2]])/(a - I*b) + ((2*I)*f*ArcTan[Tanh[(c + d*x)/2]])/(a + I*b) + (f*Log[Cosh[c + d*x]])/(a - I*b) + (f*Log[Cosh[c + d*x]])/(a + I*b) + (2*a^2*(-2*d*e*ArcTanh[(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)*L og[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*PolyLog[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]))]))/( a^2 + b^2)^(3/2) - (2*d*(e + f*x)*Sech[c + d*x]*(b + a*Sinh[c + d*x]))/(a^ 2 + b^2))/(2*d^2)
Time = 2.68 (sec) , antiderivative size = 353, normalized size of antiderivative = 0.92, number of steps used = 23, number of rules used = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.846, Rules used = {6101, 5974, 3042, 4257, 6117, 3042, 4672, 26, 3042, 26, 3956, 6107, 3042, 3803, 25, 2694, 27, 2620, 2715, 2838, 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.
| \(\displaystyle \int \frac {(e+f x) \tanh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx\) |
| \(\Big \downarrow \) 6101 |
| \(\displaystyle \frac {\int (e+f x) \text {sech}(c+d x) \tanh (c+d x)dx}{b}-\frac {a \int \frac {(e+f x) \text {sech}(c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)}dx}{b}\) |
| \(\Big \downarrow \) 5974 |
| \(\displaystyle \frac {\frac {f \int \text {sech}(c+d x)dx}{d}-\frac {(e+f x) \text {sech}(c+d x)}{d}}{b}-\frac {a \int \frac {(e+f x) \text {sech}(c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)}dx}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {a \int \frac {(e+f x) \text {sech}(c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {-\frac {(e+f x) \text {sech}(c+d x)}{d}+\frac {f \int \csc \left (i c+i d x+\frac {\pi }{2}\right )dx}{d}}{b}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {\frac {f \arctan (\sinh (c+d x))}{d^2}-\frac {(e+f x) \text {sech}(c+d x)}{d}}{b}-\frac {a \int \frac {(e+f x) \text {sech}(c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)}dx}{b}\) |
| \(\Big \downarrow \) 6117 |
| \(\displaystyle \frac {\frac {f \arctan (\sinh (c+d x))}{d^2}-\frac {(e+f x) \text {sech}(c+d x)}{d}}{b}-\frac {a \left (\frac {\int (e+f x) \text {sech}^2(c+d x)dx}{b}-\frac {a \int \frac {(e+f x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {f \arctan (\sinh (c+d x))}{d^2}-\frac {(e+f x) \text {sech}(c+d x)}{d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\int (e+f x) \csc \left (i c+i d x+\frac {\pi }{2}\right )^2dx}{b}\right )}{b}\) |
| \(\Big \downarrow \) 4672 |
| \(\displaystyle \frac {\frac {f \arctan (\sinh (c+d x))}{d^2}-\frac {(e+f x) \text {sech}(c+d x)}{d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x) \tanh (c+d x)}{d}-\frac {i f \int -i \tanh (c+d x)dx}{d}}{b}\right )}{b}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {\frac {f \arctan (\sinh (c+d x))}{d^2}-\frac {(e+f x) \text {sech}(c+d x)}{d}}{b}-\frac {a \left (\frac {\frac {(e+f x) \tanh (c+d x)}{d}-\frac {f \int \tanh (c+d x)dx}{d}}{b}-\frac {a \int \frac {(e+f x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {f \arctan (\sinh (c+d x))}{d^2}-\frac {(e+f x) \text {sech}(c+d x)}{d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x) \tanh (c+d x)}{d}-\frac {f \int -i \tan (i c+i d x)dx}{d}}{b}\right )}{b}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {\frac {f \arctan (\sinh (c+d x))}{d^2}-\frac {(e+f x) \text {sech}(c+d x)}{d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x) \tanh (c+d x)}{d}+\frac {i f \int \tan (i c+i d x)dx}{d}}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3956 |
| \(\displaystyle \frac {\frac {f \arctan (\sinh (c+d x))}{d^2}-\frac {(e+f x) \text {sech}(c+d x)}{d}}{b}-\frac {a \left (\frac {\frac {(e+f x) \tanh (c+d x)}{d}-\frac {f \log (\cosh (c+d x))}{d^2}}{b}-\frac {a \int \frac {(e+f x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}\) |
| \(\Big \downarrow \) 6107 |
| \(\displaystyle \frac {\frac {f \arctan (\sinh (c+d x))}{d^2}-\frac {(e+f x) \text {sech}(c+d x)}{d}}{b}-\frac {a \left (\frac {\frac {(e+f x) \tanh (c+d x)}{d}-\frac {f \log (\cosh (c+d x))}{d^2}}{b}-\frac {a \left (\frac {b^2 \int \frac {e+f x}{a+b \sinh (c+d x)}dx}{a^2+b^2}+\frac {\int (e+f x) \text {sech}^2(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {f \arctan (\sinh (c+d x))}{d^2}-\frac {(e+f x) \text {sech}(c+d x)}{d}}{b}-\frac {a \left (\frac {\frac {(e+f x) \tanh (c+d x)}{d}-\frac {f \log (\cosh (c+d x))}{d^2}}{b}-\frac {a \left (\frac {\int (e+f x) \text {sech}^2(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}+\frac {b^2 \int \frac {e+f x}{a-i b \sin (i c+i d x)}dx}{a^2+b^2}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3803 |
| \(\displaystyle \frac {\frac {f \arctan (\sinh (c+d x))}{d^2}-\frac {(e+f x) \text {sech}(c+d x)}{d}}{b}-\frac {a \left (\frac {\frac {(e+f x) \tanh (c+d x)}{d}-\frac {f \log (\cosh (c+d x))}{d^2}}{b}-\frac {a \left (\frac {2 b^2 \int -\frac {e^{c+d x} (e+f x)}{-2 e^{c+d x} a-b e^{2 (c+d x)}+b}dx}{a^2+b^2}+\frac {\int (e+f x) \text {sech}^2(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {f \arctan (\sinh (c+d x))}{d^2}-\frac {(e+f x) \text {sech}(c+d x)}{d}}{b}-\frac {a \left (\frac {\frac {(e+f x) \tanh (c+d x)}{d}-\frac {f \log (\cosh (c+d x))}{d^2}}{b}-\frac {a \left (\frac {\int (e+f x) \text {sech}^2(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}-\frac {2 b^2 \int \frac {e^{c+d x} (e+f x)}{-2 e^{c+d x} a-b e^{2 (c+d x)}+b}dx}{a^2+b^2}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 2694 |
| \(\displaystyle \frac {\frac {f \arctan (\sinh (c+d x))}{d^2}-\frac {(e+f x) \text {sech}(c+d x)}{d}}{b}-\frac {a \left (\frac {\frac {(e+f x) \tanh (c+d x)}{d}-\frac {f \log (\cosh (c+d x))}{d^2}}{b}-\frac {a \left (\frac {\int (e+f x) \text {sech}^2(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}-\frac {2 b^2 \left (\frac {b \int -\frac {e^{c+d x} (e+f x)}{2 \left (a+b e^{c+d x}-\sqrt {a^2+b^2}\right )}dx}{\sqrt {a^2+b^2}}-\frac {b \int -\frac {e^{c+d x} (e+f x)}{2 \left (a+b e^{c+d x}+\sqrt {a^2+b^2}\right )}dx}{\sqrt {a^2+b^2}}\right )}{a^2+b^2}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {f \arctan (\sinh (c+d x))}{d^2}-\frac {(e+f x) \text {sech}(c+d x)}{d}}{b}-\frac {a \left (\frac {\frac {(e+f x) \tanh (c+d x)}{d}-\frac {f \log (\cosh (c+d x))}{d^2}}{b}-\frac {a \left (\frac {\int (e+f x) \text {sech}^2(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}-\frac {2 b^2 \left (\frac {b \int \frac {e^{c+d x} (e+f x)}{a+b e^{c+d x}+\sqrt {a^2+b^2}}dx}{2 \sqrt {a^2+b^2}}-\frac {b \int \frac {e^{c+d x} (e+f x)}{a+b e^{c+d x}-\sqrt {a^2+b^2}}dx}{2 \sqrt {a^2+b^2}}\right )}{a^2+b^2}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle \frac {\frac {f \arctan (\sinh (c+d x))}{d^2}-\frac {(e+f x) \text {sech}(c+d x)}{d}}{b}-\frac {a \left (\frac {\frac {(e+f x) \tanh (c+d x)}{d}-\frac {f \log (\cosh (c+d x))}{d^2}}{b}-\frac {a \left (\frac {\int (e+f x) \text {sech}^2(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}-\frac {2 b^2 \left (\frac {b \left (\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}-\frac {f \int \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right )dx}{b d}\right )}{2 \sqrt {a^2+b^2}}-\frac {b \left (\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}-\frac {f \int \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right )dx}{b d}\right )}{2 \sqrt {a^2+b^2}}\right )}{a^2+b^2}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle \frac {\frac {f \arctan (\sinh (c+d x))}{d^2}-\frac {(e+f x) \text {sech}(c+d x)}{d}}{b}-\frac {a \left (\frac {\frac {(e+f x) \tanh (c+d x)}{d}-\frac {f \log (\cosh (c+d x))}{d^2}}{b}-\frac {a \left (\frac {\int (e+f x) \text {sech}^2(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}-\frac {2 b^2 \left (\frac {b \left (\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}-\frac {f \int e^{-c-d x} \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right )de^{c+d x}}{b d^2}\right )}{2 \sqrt {a^2+b^2}}-\frac {b \left (\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}-\frac {f \int e^{-c-d x} \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right )de^{c+d x}}{b d^2}\right )}{2 \sqrt {a^2+b^2}}\right )}{a^2+b^2}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle \frac {\frac {f \arctan (\sinh (c+d x))}{d^2}-\frac {(e+f x) \text {sech}(c+d x)}{d}}{b}-\frac {a \left (\frac {\frac {(e+f x) \tanh (c+d x)}{d}-\frac {f \log (\cosh (c+d x))}{d^2}}{b}-\frac {a \left (\frac {\int (e+f x) \text {sech}^2(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}-\frac {2 b^2 \left (\frac {b \left (\frac {f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b d^2}+\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}\right )}{2 \sqrt {a^2+b^2}}-\frac {b \left (\frac {f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b d^2}+\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}\right )}{2 \sqrt {a^2+b^2}}\right )}{a^2+b^2}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {\frac {f \arctan (\sinh (c+d x))}{d^2}-\frac {(e+f x) \text {sech}(c+d x)}{d}}{b}-\frac {a \left (\frac {\frac {(e+f x) \tanh (c+d x)}{d}-\frac {f \log (\cosh (c+d x))}{d^2}}{b}-\frac {a \left (\frac {\int \left (a (e+f x) \text {sech}^2(c+d x)-b (e+f x) \text {sech}(c+d x) \tanh (c+d x)\right )dx}{a^2+b^2}-\frac {2 b^2 \left (\frac {b \left (\frac {f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b d^2}+\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}\right )}{2 \sqrt {a^2+b^2}}-\frac {b \left (\frac {f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b d^2}+\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}\right )}{2 \sqrt {a^2+b^2}}\right )}{a^2+b^2}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {f \arctan (\sinh (c+d x))}{d^2}-\frac {(e+f x) \text {sech}(c+d x)}{d}}{b}-\frac {a \left (\frac {\frac {(e+f x) \tanh (c+d x)}{d}-\frac {f \log (\cosh (c+d x))}{d^2}}{b}-\frac {a \left (\frac {-\frac {a f \log (\cosh (c+d x))}{d^2}+\frac {a (e+f x) \tanh (c+d x)}{d}-\frac {b f \arctan (\sinh (c+d x))}{d^2}+\frac {b (e+f x) \text {sech}(c+d x)}{d}}{a^2+b^2}-\frac {2 b^2 \left (\frac {b \left (\frac {f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b d^2}+\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}\right )}{2 \sqrt {a^2+b^2}}-\frac {b \left (\frac {f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b d^2}+\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}\right )}{2 \sqrt {a^2+b^2}}\right )}{a^2+b^2}\right )}{b}\right )}{b}\) |
Input:
Int[((e + f*x)*Tanh[c + d*x]^2)/(a + b*Sinh[c + d*x]),x]
Output:
((f*ArcTan[Sinh[c + d*x]])/d^2 - ((e + f*x)*Sech[c + d*x])/d)/b - (a*((-(( f*Log[Cosh[c + d*x]])/d^2) + ((e + f*x)*Tanh[c + d*x])/d)/b - (a*((-2*b^2* (-1/2*(b*(((e + f*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(b*d) + (f*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b*d^2)))/Sqrt [a^2 + b^2] + (b*(((e + f*x)*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]) ])/(b*d) + (f*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b*d^2 )))/(2*Sqrt[a^2 + b^2])))/(a^2 + b^2) + (-((b*f*ArcTan[Sinh[c + d*x]])/d^2 ) - (a*f*Log[Cosh[c + d*x]])/d^2 + (b*(e + f*x)*Sech[c + d*x])/d + (a*(e + f*x)*Tanh[c + d*x])/d)/(a^2 + b^2)))/b))/b
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[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_.)*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. \(1927\) vs. \(2(365)=730\).
Time = 1.04 (sec) , antiderivative size = 1928, normalized size of antiderivative = 5.01
Input:
int((f*x+e)*tanh(d*x+c)^2/(a+b*sinh(d*x+c)),x,method=_RETURNVERBOSE)
Output:
2*(f*x+e)*(-b*exp(d*x+c)+a)/d/(a^2+b^2)/(1+exp(2*d*x+2*c))-2/d^2/(a^2+b^2) ^(3/2)*a^4*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)))*c+2/d^2/(a^2+b^2)^(3/2)*a^2*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)))*b^2-2/d^2/(a^2+b^2)^(3/2)*a^2*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)))* b^2+2/d^2/(a^2+b^2)^(1/2)*c*a^2*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)*c*a^4*f/(2*a^2+2*b^2)*arctan h(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^2*f/(2 *a^2+2*b^2)*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))*a^2+2/d/(a^2 +b^2)^(3/2)*a^4*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)*a^4*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^2/(a^2+b^2)^(3/2)*a^4*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)))*c-2 *b^2/d/(a^2+b^2)^(3/2)*e/(2*a^2+2*b^2)*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a ^2+b^2)^(1/2))*a^2+2/d^2/(a^2+b^2)^(1/2)*b^2*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)*a^4*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)*a^4*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)))+4/d^2/(a^2+b^2)^(1/2)*a^2*f/(2*a^2+2*b^2)*arctanh(1/2 *(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))-1/d^2/(a^2+b^2)*b^2*f/(2*a^2+2*b...
Leaf count of result is larger than twice the leaf count of optimal. 1337 vs. \(2 (363) = 726\).
Time = 0.14 (sec) , antiderivative size = 1337, normalized size of antiderivative = 3.47 \[ \int \frac {(e+f x) \tanh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \] Input:
integrate((f*x+e)*tanh(d*x+c)^2/(a+b*sinh(d*x+c)),x, algorithm="fricas")
Output:
-(2*(a^3 + a*b^2)*d*f*x*cosh(d*x + c)^2 + 2*(a^3 + a*b^2)*d*f*x*sinh(d*x + c)^2 - 2*(a^3 + a*b^2)*d*e - (a^2*b*f*cosh(d*x + c)^2 + 2*a^2*b*f*cosh(d* x + c)*sinh(d*x + c) + a^2*b*f*sinh(d*x + c)^2 + a^2*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) + (a^2*b*f*cosh(d*x + c)^2 + 2*a^2*b*f*cosh(d*x + c)*sinh(d*x + c) + a^2*b*f*sinh(d*x + c)^2 + a^2*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) + (a^2*b*d*e - a^2*b*c*f + (a^2*b*d*e - a^2*b*c*f)*cosh(d*x + c)^2 + 2*(a^2*b*d*e - a^ 2*b*c*f)*cosh(d*x + c)*sinh(d*x + c) + (a^2*b*d*e - a^2*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) - (a^2*b*d*e - a^2*b*c*f + (a^2*b*d*e - a^2 *b*c*f)*cosh(d*x + c)^2 + 2*(a^2*b*d*e - a^2*b*c*f)*cosh(d*x + c)*sinh(d*x + c) + (a^2*b*d*e - a^2*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) - (a^2*b*d*f*x + a^2*b*c*f + (a^2*b*d*f*x + a^2*b*c*f)*cosh(d*x + c)^2 + 2 *(a^2*b*d*f*x + a^2*b*c*f)*cosh(d*x + c)*sinh(d*x + c) + (a^2*b*d*f*x + a^ 2*b*c*f)*sinh(d*x + c)^2)*sqrt((a^2 + b^2)/b^2)*log(-(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) + (a^2*b*d*f*x + a^2*b*c*f + (a^2*b*d*f*x + a^2*b*c*f)*cosh(d*x...
\[ \int \frac {(e+f x) \tanh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {\left (e + f x\right ) \tanh ^{2}{\left (c + d x \right )}}{a + b \sinh {\left (c + d x \right )}}\, dx \] Input:
integrate((f*x+e)*tanh(d*x+c)**2/(a+b*sinh(d*x+c)),x)
Output:
Integral((e + f*x)*tanh(c + d*x)**2/(a + b*sinh(c + d*x)), x)
\[ \int \frac {(e+f x) \tanh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )} \tanh \left (d x + c\right )^{2}}{b \sinh \left (d x + c\right ) + a} \,d x } \] Input:
integrate((f*x+e)*tanh(d*x+c)^2/(a+b*sinh(d*x+c)),x, algorithm="maxima")
Output:
(2*a^2*integrate(-x*e^(d*x + c)/(a^2*b + b^3 - (a^2*b*e^(2*c) + b^3*e^(2*c ))*e^(2*d*x) - 2*(a^3*e^c + a*b^2*e^c)*e^(d*x)), x) - 2*(b*x*e^(d*x + c) - a*x)/(a^2*d + b^2*d + (a^2*d*e^(2*c) + b^2*d*e^(2*c))*e^(2*d*x)) - 2*a*x/ ((a^2 + b^2)*d) + 2*b*arctan(e^(d*x + c))/((a^2 + b^2)*d^2) + a*log(e^(2*d *x + 2*c) + 1)/((a^2 + b^2)*d^2))*f + e*(a^2*log((b*e^(-d*x - c) - a - sqr t(a^2 + b^2))/(b*e^(-d*x - c) - a + sqrt(a^2 + b^2)))/((a^2 + b^2)^(3/2)*d ) - 2*(b*e^(-d*x - c) + a)/((a^2 + b^2 + (a^2 + b^2)*e^(-2*d*x - 2*c))*d))
Timed out. \[ \int \frac {(e+f x) \tanh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Timed out} \] Input:
integrate((f*x+e)*tanh(d*x+c)^2/(a+b*sinh(d*x+c)),x, algorithm="giac")
Output:
Timed out
Timed out. \[ \int \frac {(e+f x) \tanh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {{\mathrm {tanh}\left (c+d\,x\right )}^2\,\left (e+f\,x\right )}{a+b\,\mathrm {sinh}\left (c+d\,x\right )} \,d x \] Input:
int((tanh(c + d*x)^2*(e + f*x))/(a + b*sinh(c + d*x)),x)
Output:
int((tanh(c + d*x)^2*(e + f*x))/(a + b*sinh(c + d*x)), x)
\[ \int \frac {(e+f x) \tanh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx =\text {Too large to display} \] Input:
int((f*x+e)*tanh(d*x+c)^2/(a+b*sinh(d*x+c)),x)
Output:
(2*e**(2*c + 2*d*x)*atan(e**(c + d*x))*a**4*b*f + 4*e**(2*c + 2*d*x)*atan( e**(c + d*x))*a**2*b**3*f + 2*e**(2*c + 2*d*x)*atan(e**(c + d*x))*b**5*f + 2*atan(e**(c + d*x))*a**4*b*f + 4*atan(e**(c + d*x))*a**2*b**3*f + 2*atan (e**(c + d*x))*b**5*f + 2*e**(2*c + 2*d*x)*sqrt(a**2 + b**2)*atan((e**(c + d*x)*b*i + a*i)/sqrt(a**2 + b**2))*a**2*b**2*d*e*i + 2*sqrt(a**2 + b**2)* atan((e**(c + d*x)*b*i + a*i)/sqrt(a**2 + b**2))*a**2*b**2*d*e*i + 8*e**(5 *c + 2*d*x)*int((e**(3*d*x)*x)/(e**(6*c + 6*d*x)*b + 2*e**(5*c + 5*d*x)*a + e**(4*c + 4*d*x)*b + 4*e**(3*c + 3*d*x)*a - e**(2*c + 2*d*x)*b + 2*e**(c + d*x)*a - b),x)*a**6*d**2*f + 16*e**(5*c + 2*d*x)*int((e**(3*d*x)*x)/(e* *(6*c + 6*d*x)*b + 2*e**(5*c + 5*d*x)*a + e**(4*c + 4*d*x)*b + 4*e**(3*c + 3*d*x)*a - e**(2*c + 2*d*x)*b + 2*e**(c + d*x)*a - b),x)*a**4*b**2*d**2*f + 8*e**(5*c + 2*d*x)*int((e**(3*d*x)*x)/(e**(6*c + 6*d*x)*b + 2*e**(5*c + 5*d*x)*a + e**(4*c + 4*d*x)*b + 4*e**(3*c + 3*d*x)*a - e**(2*c + 2*d*x)*b + 2*e**(c + d*x)*a - b),x)*a**2*b**4*d**2*f + e**(2*c + 2*d*x)*log(e**(2* c + 2*d*x) + 1)*a**5*f + 2*e**(2*c + 2*d*x)*log(e**(2*c + 2*d*x) + 1)*a**3 *b**2*f + e**(2*c + 2*d*x)*log(e**(2*c + 2*d*x) + 1)*a*b**4*f - 2*e**(2*c + 2*d*x)*a**5*d*f*x - 2*e**(2*c + 2*d*x)*a**3*b**2*d*e - 4*e**(2*c + 2*d*x )*a**3*b**2*d*f*x - 2*e**(2*c + 2*d*x)*a*b**4*d*e - 2*e**(2*c + 2*d*x)*a*b **4*d*f*x - 2*e**(c + d*x)*a**4*b*d*f*x - 2*e**(c + d*x)*a**2*b**3*d*e - 4 *e**(c + d*x)*a**2*b**3*d*f*x - 2*e**(c + d*x)*b**5*d*e - 2*e**(c + d*x...