3.5.42 \(\int \frac {(e+f x) \text {csch}(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx\) [442]

3.5.42.1 Optimal result

Integrand size = 32, antiderivative size = 442 \[ \int \frac {(e+f x) \text {csch}(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {f \arctan (\sinh (c+d x))}{a d^2}+\frac {b^2 f \arctan (\sinh (c+d x))}{a \left (a^2+b^2\right ) d^2}-\frac {2 f x \text {arctanh}\left (e^{c+d x}\right )}{a d}+\frac {f x \text {arctanh}(\cosh (c+d x))}{a d}-\frac {(e+f x) \text {arctanh}(\cosh (c+d x))}{a d}-\frac {b^3 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d}+\frac {b^3 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d}+\frac {b f \log (\cosh (c+d x))}{\left (a^2+b^2\right ) d^2}-\frac {f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a d^2}+\frac {f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a d^2}-\frac {b^3 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d^2}+\frac {b^3 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d^2}+\frac {(e+f x) \text {sech}(c+d x)}{a d}-\frac {b^2 (e+f x) \text {sech}(c+d x)}{a \left (a^2+b^2\right ) d}-\frac {b (e+f x) \tanh (c+d x)}{\left (a^2+b^2\right ) d} \]

output

-f*arctan(sinh(d*x+c))/a/d^2+b^2*f*arctan(sinh(d*x+c))/a/(a^2+b^2)/d^2-2*f 
*x*arctanh(exp(d*x+c))/a/d+f*x*arctanh(cosh(d*x+c))/a/d-(f*x+e)*arctanh(co 
sh(d*x+c))/a/d+b*f*ln(cosh(d*x+c))/(a^2+b^2)/d^2-b^3*(f*x+e)*ln(1+b*exp(d* 
x+c)/(a-(a^2+b^2)^(1/2)))/a/(a^2+b^2)^(3/2)/d+b^3*(f*x+e)*ln(1+b*exp(d*x+c 
)/(a+(a^2+b^2)^(1/2)))/a/(a^2+b^2)^(3/2)/d-f*polylog(2,-exp(d*x+c))/a/d^2+ 
f*polylog(2,exp(d*x+c))/a/d^2-b^3*f*polylog(2,-b*exp(d*x+c)/(a-(a^2+b^2)^( 
1/2)))/a/(a^2+b^2)^(3/2)/d^2+b^3*f*polylog(2,-b*exp(d*x+c)/(a+(a^2+b^2)^(1 
/2)))/a/(a^2+b^2)^(3/2)/d^2+(f*x+e)*sech(d*x+c)/a/d-b^2*(f*x+e)*sech(d*x+c 
)/a/(a^2+b^2)/d-b*(f*x+e)*tanh(d*x+c)/(a^2+b^2)/d
 

3.5.42.2 Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 7.94 (sec) , antiderivative size = 428, normalized size of antiderivative = 0.97 \[ \int \frac {(e+f x) \text {csch}(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {\text {csch}(c+d x) (a+b \sinh (c+d x)) \left (-\frac {2 f \arctan \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )}{a-i b}-\frac {2 f \arctan \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )}{a+i b}-\frac {i f \log (\cosh (c+d x))}{a-i b}+\frac {i f \log (\cosh (c+d x))}{a+i b}+\frac {2 \left (d (e+f x) \left (\log \left (1-e^{c+d x}\right )-\log \left (1+e^{c+d x}\right )\right )-f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )+f \operatorname {PolyLog}\left (2,e^{c+d x}\right )\right )}{a}-\frac {2 b^3 \left (-2 d e \text {arctanh}\left (\frac {a+b e^{c+d x}}{\sqrt {a^2+b^2}}\right )+2 c f \text {arctanh}\left (\frac {a+b e^{c+d x}}{\sqrt {a^2+b^2}}\right )+f (c+d x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )-f (c+d x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )+f \operatorname {PolyLog}\left (2,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )-f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )\right )}{a \left (a^2+b^2\right )^{3/2}}+\frac {2 d (e+f x) \text {sech}(c+d x) (a-b \sinh (c+d x))}{a^2+b^2}\right )}{2 d^2 (b+a \text {csch}(c+d x))} \]

input

Integrate[((e + f*x)*Csch[c + d*x]*Sech[c + d*x]^2)/(a + b*Sinh[c + d*x]), 
x]
 

output

(Csch[c + d*x]*(a + b*Sinh[c + d*x])*((-2*f*ArcTan[Tanh[(c + d*x)/2]])/(a 
- I*b) - (2*f*ArcTan[Tanh[(c + d*x)/2]])/(a + I*b) - (I*f*Log[Cosh[c + d*x 
]])/(a - I*b) + (I*f*Log[Cosh[c + d*x]])/(a + I*b) + (2*(d*(e + f*x)*(Log[ 
1 - E^(c + d*x)] - Log[1 + E^(c + d*x)]) - f*PolyLog[2, -E^(c + d*x)] + f* 
PolyLog[2, E^(c + d*x)]))/a - (2*b^3*(-2*d*e*ArcTanh[(a + b*E^(c + d*x))/S 
qrt[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*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*(a^2 + b^2)^(3/2)) + (2*d*(e + f*x)*Sech[c + d*x]*(a - b*Sinh[ 
c + d*x]))/(a^2 + b^2)))/(2*d^2*(b + a*Csch[c + d*x]))
 

3.5.42.3 Rubi [A] (verified)

Time = 2.18 (sec) , antiderivative size = 388, normalized size of antiderivative = 0.88, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {6123, 5985, 2009, 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.

\(\displaystyle \int \frac {(e+f x) \text {csch}(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx\)

\(\Big \downarrow \) 6123

\(\displaystyle \frac {\int (e+f x) \text {csch}(c+d x) \text {sech}^2(c+d x)dx}{a}-\frac {b \int \frac {(e+f x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}\)

\(\Big \downarrow \) 5985

\(\displaystyle \frac {-f \int \left (\frac {\text {sech}(c+d x)}{d}-\frac {\text {arctanh}(\cosh (c+d x))}{d}\right )dx-\frac {(e+f x) \text {arctanh}(\cosh (c+d x))}{d}+\frac {(e+f x) \text {sech}(c+d x)}{d}}{a}-\frac {b \int \frac {(e+f x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {-f \left (\frac {\arctan (\sinh (c+d x))}{d^2}+\frac {2 x \text {arctanh}\left (e^{c+d x}\right )}{d}-\frac {x \text {arctanh}(\cosh (c+d x))}{d}+\frac {\operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {\operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )-\frac {(e+f x) \text {arctanh}(\cosh (c+d x))}{d}+\frac {(e+f x) \text {sech}(c+d x)}{d}}{a}-\frac {b \int \frac {(e+f x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}\)

\(\Big \downarrow \) 6107

\(\displaystyle \frac {-f \left (\frac {\arctan (\sinh (c+d x))}{d^2}+\frac {2 x \text {arctanh}\left (e^{c+d x}\right )}{d}-\frac {x \text {arctanh}(\cosh (c+d x))}{d}+\frac {\operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {\operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )-\frac {(e+f x) \text {arctanh}(\cosh (c+d x))}{d}+\frac {(e+f x) \text {sech}(c+d x)}{d}}{a}-\frac {b \left (\frac {b^2 \int \frac {e+f x}{a+b \sinh (c+d x)}dx}{a^2+b^2}+\frac {\int (e+f x) \text {sech}^2(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}\right )}{a}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {-f \left (\frac {\arctan (\sinh (c+d x))}{d^2}+\frac {2 x \text {arctanh}\left (e^{c+d x}\right )}{d}-\frac {x \text {arctanh}(\cosh (c+d x))}{d}+\frac {\operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {\operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )-\frac {(e+f x) \text {arctanh}(\cosh (c+d x))}{d}+\frac {(e+f x) \text {sech}(c+d x)}{d}}{a}-\frac {b \left (\frac {\int (e+f x) \text {sech}^2(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}+\frac {b^2 \int \frac {e+f x}{a-i b \sin (i c+i d x)}dx}{a^2+b^2}\right )}{a}\)

\(\Big \downarrow \) 3803

\(\displaystyle \frac {-f \left (\frac {\arctan (\sinh (c+d x))}{d^2}+\frac {2 x \text {arctanh}\left (e^{c+d x}\right )}{d}-\frac {x \text {arctanh}(\cosh (c+d x))}{d}+\frac {\operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {\operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )-\frac {(e+f x) \text {arctanh}(\cosh (c+d x))}{d}+\frac {(e+f x) \text {sech}(c+d x)}{d}}{a}-\frac {b \left (\frac {2 b^2 \int -\frac {e^{c+d x} (e+f x)}{-2 e^{c+d x} a-b e^{2 (c+d x)}+b}dx}{a^2+b^2}+\frac {\int (e+f x) \text {sech}^2(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}\right )}{a}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {-f \left (\frac {\arctan (\sinh (c+d x))}{d^2}+\frac {2 x \text {arctanh}\left (e^{c+d x}\right )}{d}-\frac {x \text {arctanh}(\cosh (c+d x))}{d}+\frac {\operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {\operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )-\frac {(e+f x) \text {arctanh}(\cosh (c+d x))}{d}+\frac {(e+f x) \text {sech}(c+d x)}{d}}{a}-\frac {b \left (\frac {\int (e+f x) \text {sech}^2(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}-\frac {2 b^2 \int \frac {e^{c+d x} (e+f x)}{-2 e^{c+d x} a-b e^{2 (c+d x)}+b}dx}{a^2+b^2}\right )}{a}\)

\(\Big \downarrow \) 2694

\(\displaystyle \frac {-f \left (\frac {\arctan (\sinh (c+d x))}{d^2}+\frac {2 x \text {arctanh}\left (e^{c+d x}\right )}{d}-\frac {x \text {arctanh}(\cosh (c+d x))}{d}+\frac {\operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {\operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )-\frac {(e+f x) \text {arctanh}(\cosh (c+d x))}{d}+\frac {(e+f x) \text {sech}(c+d x)}{d}}{a}-\frac {b \left (\frac {\int (e+f x) \text {sech}^2(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}-\frac {2 b^2 \left (\frac {b \int -\frac {e^{c+d x} (e+f x)}{2 \left (a+b e^{c+d x}-\sqrt {a^2+b^2}\right )}dx}{\sqrt {a^2+b^2}}-\frac {b \int -\frac {e^{c+d x} (e+f x)}{2 \left (a+b e^{c+d x}+\sqrt {a^2+b^2}\right )}dx}{\sqrt {a^2+b^2}}\right )}{a^2+b^2}\right )}{a}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-f \left (\frac {\arctan (\sinh (c+d x))}{d^2}+\frac {2 x \text {arctanh}\left (e^{c+d x}\right )}{d}-\frac {x \text {arctanh}(\cosh (c+d x))}{d}+\frac {\operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {\operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )-\frac {(e+f x) \text {arctanh}(\cosh (c+d x))}{d}+\frac {(e+f x) \text {sech}(c+d x)}{d}}{a}-\frac {b \left (\frac {\int (e+f x) \text {sech}^2(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}-\frac {2 b^2 \left (\frac {b \int \frac {e^{c+d x} (e+f x)}{a+b e^{c+d x}+\sqrt {a^2+b^2}}dx}{2 \sqrt {a^2+b^2}}-\frac {b \int \frac {e^{c+d x} (e+f x)}{a+b e^{c+d x}-\sqrt {a^2+b^2}}dx}{2 \sqrt {a^2+b^2}}\right )}{a^2+b^2}\right )}{a}\)

\(\Big \downarrow \) 2620

\(\displaystyle \frac {-f \left (\frac {\arctan (\sinh (c+d x))}{d^2}+\frac {2 x \text {arctanh}\left (e^{c+d x}\right )}{d}-\frac {x \text {arctanh}(\cosh (c+d x))}{d}+\frac {\operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {\operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )-\frac {(e+f x) \text {arctanh}(\cosh (c+d x))}{d}+\frac {(e+f x) \text {sech}(c+d x)}{d}}{a}-\frac {b \left (\frac {\int (e+f x) \text {sech}^2(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}-\frac {2 b^2 \left (\frac {b \left (\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}-\frac {f \int \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right )dx}{b d}\right )}{2 \sqrt {a^2+b^2}}-\frac {b \left (\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}-\frac {f \int \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right )dx}{b d}\right )}{2 \sqrt {a^2+b^2}}\right )}{a^2+b^2}\right )}{a}\)

\(\Big \downarrow \) 2715

\(\displaystyle \frac {-f \left (\frac {\arctan (\sinh (c+d x))}{d^2}+\frac {2 x \text {arctanh}\left (e^{c+d x}\right )}{d}-\frac {x \text {arctanh}(\cosh (c+d x))}{d}+\frac {\operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {\operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )-\frac {(e+f x) \text {arctanh}(\cosh (c+d x))}{d}+\frac {(e+f x) \text {sech}(c+d x)}{d}}{a}-\frac {b \left (\frac {\int (e+f x) \text {sech}^2(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}-\frac {2 b^2 \left (\frac {b \left (\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}-\frac {f \int e^{-c-d x} \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right )de^{c+d x}}{b d^2}\right )}{2 \sqrt {a^2+b^2}}-\frac {b \left (\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}-\frac {f \int e^{-c-d x} \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right )de^{c+d x}}{b d^2}\right )}{2 \sqrt {a^2+b^2}}\right )}{a^2+b^2}\right )}{a}\)

\(\Big \downarrow \) 2838

\(\displaystyle \frac {-f \left (\frac {\arctan (\sinh (c+d x))}{d^2}+\frac {2 x \text {arctanh}\left (e^{c+d x}\right )}{d}-\frac {x \text {arctanh}(\cosh (c+d x))}{d}+\frac {\operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {\operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )-\frac {(e+f x) \text {arctanh}(\cosh (c+d x))}{d}+\frac {(e+f x) \text {sech}(c+d x)}{d}}{a}-\frac {b \left (\frac {\int (e+f x) \text {sech}^2(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}-\frac {2 b^2 \left (\frac {b \left (\frac {f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b d^2}+\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}\right )}{2 \sqrt {a^2+b^2}}-\frac {b \left (\frac {f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b d^2}+\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}\right )}{2 \sqrt {a^2+b^2}}\right )}{a^2+b^2}\right )}{a}\)

\(\Big \downarrow \) 7293

\(\displaystyle \frac {-f \left (\frac {\arctan (\sinh (c+d x))}{d^2}+\frac {2 x \text {arctanh}\left (e^{c+d x}\right )}{d}-\frac {x \text {arctanh}(\cosh (c+d x))}{d}+\frac {\operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {\operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )-\frac {(e+f x) \text {arctanh}(\cosh (c+d x))}{d}+\frac {(e+f x) \text {sech}(c+d x)}{d}}{a}-\frac {b \left (\frac {\int \left (a (e+f x) \text {sech}^2(c+d x)-b (e+f x) \text {sech}(c+d x) \tanh (c+d x)\right )dx}{a^2+b^2}-\frac {2 b^2 \left (\frac {b \left (\frac {f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b d^2}+\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}\right )}{2 \sqrt {a^2+b^2}}-\frac {b \left (\frac {f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b d^2}+\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}\right )}{2 \sqrt {a^2+b^2}}\right )}{a^2+b^2}\right )}{a}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {-f \left (\frac {\arctan (\sinh (c+d x))}{d^2}+\frac {2 x \text {arctanh}\left (e^{c+d x}\right )}{d}-\frac {x \text {arctanh}(\cosh (c+d x))}{d}+\frac {\operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {\operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )-\frac {(e+f x) \text {arctanh}(\cosh (c+d x))}{d}+\frac {(e+f x) \text {sech}(c+d x)}{d}}{a}-\frac {b \left (\frac {-\frac {a f \log (\cosh (c+d x))}{d^2}+\frac {a (e+f x) \tanh (c+d x)}{d}-\frac {b f \arctan (\sinh (c+d x))}{d^2}+\frac {b (e+f x) \text {sech}(c+d x)}{d}}{a^2+b^2}-\frac {2 b^2 \left (\frac {b \left (\frac {f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b d^2}+\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}\right )}{2 \sqrt {a^2+b^2}}-\frac {b \left (\frac {f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b d^2}+\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}\right )}{2 \sqrt {a^2+b^2}}\right )}{a^2+b^2}\right )}{a}\)

input

Int[((e + f*x)*Csch[c + d*x]*Sech[c + d*x]^2)/(a + b*Sinh[c + d*x]),x]
 

output

(-(((e + f*x)*ArcTanh[Cosh[c + d*x]])/d) - f*(ArcTan[Sinh[c + d*x]]/d^2 + 
(2*x*ArcTanh[E^(c + d*x)])/d - (x*ArcTanh[Cosh[c + d*x]])/d + PolyLog[2, - 
E^(c + d*x)]/d^2 - PolyLog[2, E^(c + d*x)]/d^2) + ((e + f*x)*Sech[c + d*x] 
)/d)/a - (b*((-2*b^2*(-1/2*(b*(((e + f*x)*Log[1 + (b*E^(c + d*x))/(a - Sqr 
t[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*ArcTa 
n[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)))/a
 

3.5.42.3.1 Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2620

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]
 

rule 2694

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]
 

rule 2715

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]
 

rule 2838

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]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3803

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]
 

rule 5985

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]
 

rule 6107

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 
]
 

rule 6123

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]
 

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

3.5.42.4 Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1814\) vs. \(2(419)=838\).

Time = 10.70 (sec) , antiderivative size = 1815, normalized size of antiderivative = 4.11

method	result	size
risch	\(\text {Expression too large to display}\)	\(1815\)

input

int((f*x+e)*csch(d*x+c)*sech(d*x+c)^2/(a+b*sinh(d*x+c)),x,method=_RETURNVE 
RBOSE)
 

output

-1/(a^2+b^2)/d*f*b^2/a*ln(exp(d*x+c)+1)*x+1/(a^2+b^2)^(3/2)/d^2*a*f*b*arct 
anh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))-1/(a^2+b^2)/d^2*b^2*c*f/a*ln 
(exp(d*x+c)-1)+1/(a^2+b^2)^(3/2)/d^2*f*b^3/a*arctanh(1/2*(2*b*exp(d*x+c)+2 
*a)/(a^2+b^2)^(1/2))-1/(a^2+b^2)^(5/2)/d*f*b^5/a*ln((-b*exp(d*x+c)+(a^2+b^ 
2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))*x+1/(a^2+b^2)^(5/2)/d*f*b^5/a*ln((b*exp( 
d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))*x-1/(a^2+b^2)^(5/2)/d*a*b^3 
*f*ln((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))*x+1/(a^2+b^2 
)^(5/2)/d*a*b^3*f*ln((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2))) 
*x-1/(a^2+b^2)^(5/2)/d^2*f*b^5/a*ln((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+ 
(a^2+b^2)^(1/2)))*c+1/(a^2+b^2)^(5/2)/d^2*f*b^5/a*ln((b*exp(d*x+c)+(a^2+b^ 
2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))*c-1/(a^2+b^2)^(5/2)/d^2*a*b^3*f*ln((-b*ex 
p(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))*c+1/(a^2+b^2)^(5/2)/d^2* 
a*b^3*f*ln((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))*c-1/(a^2+ 
b^2)^(3/2)/d^2*b^3*c*f/a*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2)) 
-1/(a^2+b^2)^(5/2)/d^2*b^5*c*f/a*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2 
)^(1/2))+1/(a^2+b^2)^(5/2)/d^2*c*a^3*f*b*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/ 
(a^2+b^2)^(1/2))-1/(a^2+b^2)^(3/2)/d^2*c*a*f*b*arctanh(1/2*(2*b*exp(d*x+c) 
+2*a)/(a^2+b^2)^(1/2))-1/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+2*(f*x+e)*(a*exp(d*x+c)+b)/d/(a^2+b^...
 

3.5.42.5 Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2176 vs. \(2 (415) = 830\).

Time = 0.32 (sec) , antiderivative size = 2176, normalized size of antiderivative = 4.92 \[ \int \frac {(e+f x) \text {csch}(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \]

input

integrate((f*x+e)*csch(d*x+c)*sech(d*x+c)^2/(a+b*sinh(d*x+c)),x, algorithm 
="fricas")
 

output

-(2*(a^3*b + a*b^3)*d*f*x*cosh(d*x + c)^2 + 2*(a^3*b + a*b^3)*d*f*x*sinh(d 
*x + c)^2 - 2*(a^3*b + a*b^3)*d*e + (b^4*f*cosh(d*x + c)^2 + 2*b^4*f*cosh( 
d*x + c)*sinh(d*x + c) + b^4*f*sinh(d*x + c)^2 + b^4*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) - (b^4*f*cosh(d*x + c)^2 + 2*b^ 
4*f*cosh(d*x + c)*sinh(d*x + c) + b^4*f*sinh(d*x + c)^2 + b^4*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) - (b^4*d*e - b^4*c*f + 
 (b^4*d*e - b^4*c*f)*cosh(d*x + c)^2 + 2*(b^4*d*e - b^4*c*f)*cosh(d*x + c) 
*sinh(d*x + c) + (b^4*d*e - b^4*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) + (b^4*d*e - b^4*c*f + (b^4*d*e - b^4*c*f)*cosh(d*x + c)^2 + 2*(b^4*d 
*e - b^4*c*f)*cosh(d*x + c)*sinh(d*x + c) + (b^4*d*e - b^4*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) + (b^4*d*f*x + b^4*c*f + (b^4*d*f*x + b^4* 
c*f)*cosh(d*x + c)^2 + 2*(b^4*d*f*x + b^4*c*f)*cosh(d*x + c)*sinh(d*x + c) 
 + (b^4*d*f*x + b^4*c*f)*sinh(d*x + c)^2)*sqrt((a^2 + b^2)/b^2)*log(-(a*co 
sh(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) - (b^4*d*f*x + b^4*c*f + (b^4*d*f*x + b^4*c*f)*cos 
h(d*x + c)^2 + 2*(b^4*d*f*x + b^4*c*f)*cosh(d*x + c)*sinh(d*x + c) + (b...
 

3.5.42.6 Sympy [F(-1)]

Timed out. \[ \int \frac {(e+f x) \text {csch}(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Timed out} \]

input

integrate((f*x+e)*csch(d*x+c)*sech(d*x+c)**2/(a+b*sinh(d*x+c)),x)
 

output

Timed out
 

3.5.42.7 Maxima [F]

\[ \int \frac {(e+f x) \text {csch}(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )} \operatorname {csch}\left (d x + c\right ) \operatorname {sech}\left (d x + c\right )^{2}}{b \sinh \left (d x + c\right ) + a} \,d x } \]

input

integrate((f*x+e)*csch(d*x+c)*sech(d*x+c)^2/(a+b*sinh(d*x+c)),x, algorithm 
="maxima")
 

output

-(b^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^3 + a*b^2)*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) + log(e^(-d*x - c) + 1)/(a* 
d) - log(e^(-d*x - c) - 1)/(a*d))*e - (8*b^3*integrate(-1/4*x*e^(d*x + c)/ 
(a^3*b + a*b^3 - (a^3*b*e^(2*c) + a*b^3*e^(2*c))*e^(2*d*x) - 2*(a^4*e^c + 
a^2*b^2*e^c)*e^(d*x)), x) - 2*(a*x*e^(d*x + c) + b*x)/(a^2*d + b^2*d + (a^ 
2*d*e^(2*c) + b^2*d*e^(2*c))*e^(2*d*x)) + 2*b*x/((a^2 + b^2)*d) + 2*a*arct 
an(e^(d*x + c))/((a^2 + b^2)*d^2) - b*log(e^(2*d*x + 2*c) + 1)/((a^2 + b^2 
)*d^2) - 8*integrate(1/8*x/(a*e^(d*x + c) + a), x) - 8*integrate(1/8*x/(a* 
e^(d*x + c) - a), x))*f
 

3.5.42.8 Giac [F(-1)]

Timed out. \[ \int \frac {(e+f x) \text {csch}(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Timed out} \]

input

integrate((f*x+e)*csch(d*x+c)*sech(d*x+c)^2/(a+b*sinh(d*x+c)),x, algorithm 
="giac")
 

output

Timed out
 

3.5.42.9 Mupad [F(-1)]

Timed out. \[ \int \frac {(e+f x) \text {csch}(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {e+f\,x}{{\mathrm {cosh}\left (c+d\,x\right )}^2\,\mathrm {sinh}\left (c+d\,x\right )\,\left (a+b\,\mathrm {sinh}\left (c+d\,x\right )\right )} \,d x \]

input

int((e + f*x)/(cosh(c + d*x)^2*sinh(c + d*x)*(a + b*sinh(c + d*x))),x)
 

output

int((e + f*x)/(cosh(c + d*x)^2*sinh(c + d*x)*(a + b*sinh(c + d*x))), x)