\(\int \frac {(e+f x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx\) [315]

Optimal result

Integrand size = 26, antiderivative size = 560 \[ \int \frac {(e+f x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {2 a b^2 (e+f x) \arctan \left (e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d}+\frac {a (e+f x) \arctan \left (e^{c+d x}\right )}{\left (a^2+b^2\right ) d}+\frac {b^3 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d}+\frac {b^3 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d}-\frac {b^3 (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{\left (a^2+b^2\right )^2 d}-\frac {i a b^2 f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^2}-\frac {i a f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{2 \left (a^2+b^2\right ) d^2}+\frac {i a b^2 f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac {i a f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{2 \left (a^2+b^2\right ) d^2}+\frac {b^3 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac {b^3 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d^2}-\frac {b^3 f \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 \left (a^2+b^2\right )^2 d^2}+\frac {a f \text {sech}(c+d x)}{2 \left (a^2+b^2\right ) d^2}+\frac {b (e+f x) \text {sech}^2(c+d x)}{2 \left (a^2+b^2\right ) d}-\frac {b f \tanh (c+d x)}{2 \left (a^2+b^2\right ) d^2}+\frac {a (e+f x) \text {sech}(c+d x) \tanh (c+d x)}{2 \left (a^2+b^2\right ) d} \] Output:

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

Mathematica [A] (warning: unable to verify)

Time = 8.48 (sec) , antiderivative size = 832, normalized size of antiderivative = 1.49 \[ \int \frac {(e+f x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {b^3 \left (-2 d e (c+d x)+2 c f (c+d x)-f (c+d x)^2+\frac {4 a \sqrt {a^2+b^2} d e \arctan \left (\frac {a+b e^{c+d x}}{\sqrt {-a^2-b^2}}\right )}{\sqrt {-\left (a^2+b^2\right )^2}}-\frac {4 a \sqrt {-\left (a^2+b^2\right )^2} d e \text {arctanh}\left (\frac {a+b e^{c+d x}}{\sqrt {a^2+b^2}}\right )}{\left (-a^2-b^2\right )^{3/2}}+2 f (c+d x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )+2 f (c+d x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )-2 c f \log \left (b-2 a e^{c+d x}-b e^{2 (c+d x)}\right )+2 d e \log \left (2 a e^{c+d x}+b \left (-1+e^{2 (c+d x)}\right )\right )+2 f \operatorname {PolyLog}\left (2,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )+2 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )\right )}{2 \left (a^2+b^2\right )^2 d^2}-\frac {-2 b^3 d e (c+d x)+2 b^3 c f (c+d x)-b^3 f (c+d x)^2-2 a^3 d e \arctan \left (e^{c+d x}\right )-6 a b^2 d e \arctan \left (e^{c+d x}\right )+2 a^3 c f \arctan \left (e^{c+d x}\right )+6 a b^2 c f \arctan \left (e^{c+d x}\right )-i a^3 f (c+d x) \log \left (1-i e^{c+d x}\right )-3 i a b^2 f (c+d x) \log \left (1-i e^{c+d x}\right )+i a^3 f (c+d x) \log \left (1+i e^{c+d x}\right )+3 i a b^2 f (c+d x) \log \left (1+i e^{c+d x}\right )+2 b^3 d e \log \left (1+e^{2 (c+d x)}\right )-2 b^3 c f \log \left (1+e^{2 (c+d x)}\right )+2 b^3 f (c+d x) \log \left (1+e^{2 (c+d x)}\right )+i a \left (a^2+3 b^2\right ) f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )-i a \left (a^2+3 b^2\right ) f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )+b^3 f \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 \left (a^2+b^2\right )^2 d^2}+\frac {\text {sech}(c+d x) (a f-b f \sinh (c+d x))}{2 \left (a^2+b^2\right ) d^2}+\frac {\text {sech}^2(c+d x) (b d e-b c f+b f (c+d x)+a d e \sinh (c+d x)-a c f \sinh (c+d x)+a f (c+d x) \sinh (c+d x))}{2 \left (a^2+b^2\right ) d^2} \] Input:

Integrate[((e + f*x)*Sech[c + d*x]^3)/(a + b*Sinh[c + d*x]),x]
 

Output:

(b^3*(-2*d*e*(c + d*x) + 2*c*f*(c + d*x) - f*(c + d*x)^2 + (4*a*Sqrt[a^2 + 
 b^2]*d*e*ArcTan[(a + b*E^(c + d*x))/Sqrt[-a^2 - b^2]])/Sqrt[-(a^2 + b^2)^ 
2] - (4*a*Sqrt[-(a^2 + b^2)^2]*d*e*ArcTanh[(a + b*E^(c + d*x))/Sqrt[a^2 + 
b^2]])/(-a^2 - b^2)^(3/2) + 2*f*(c + d*x)*Log[1 + (b*E^(c + d*x))/(a - Sqr 
t[a^2 + b^2])] + 2*f*(c + d*x)*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2 
])] - 2*c*f*Log[b - 2*a*E^(c + d*x) - b*E^(2*(c + d*x))] + 2*d*e*Log[2*a*E 
^(c + d*x) + b*(-1 + E^(2*(c + d*x)))] + 2*f*PolyLog[2, (b*E^(c + d*x))/(- 
a + Sqrt[a^2 + b^2])] + 2*f*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b 
^2]))]))/(2*(a^2 + b^2)^2*d^2) - (-2*b^3*d*e*(c + d*x) + 2*b^3*c*f*(c + d* 
x) - b^3*f*(c + d*x)^2 - 2*a^3*d*e*ArcTan[E^(c + d*x)] - 6*a*b^2*d*e*ArcTa 
n[E^(c + d*x)] + 2*a^3*c*f*ArcTan[E^(c + d*x)] + 6*a*b^2*c*f*ArcTan[E^(c + 
 d*x)] - I*a^3*f*(c + d*x)*Log[1 - I*E^(c + d*x)] - (3*I)*a*b^2*f*(c + d*x 
)*Log[1 - I*E^(c + d*x)] + I*a^3*f*(c + d*x)*Log[1 + I*E^(c + d*x)] + (3*I 
)*a*b^2*f*(c + d*x)*Log[1 + I*E^(c + d*x)] + 2*b^3*d*e*Log[1 + E^(2*(c + d 
*x))] - 2*b^3*c*f*Log[1 + E^(2*(c + d*x))] + 2*b^3*f*(c + d*x)*Log[1 + E^( 
2*(c + d*x))] + I*a*(a^2 + 3*b^2)*f*PolyLog[2, (-I)*E^(c + d*x)] - I*a*(a^ 
2 + 3*b^2)*f*PolyLog[2, I*E^(c + d*x)] + b^3*f*PolyLog[2, -E^(2*(c + d*x)) 
])/(2*(a^2 + b^2)^2*d^2) + (Sech[c + d*x]*(a*f - b*f*Sinh[c + d*x]))/(2*(a 
^2 + b^2)*d^2) + (Sech[c + d*x]^2*(b*d*e - b*c*f + b*f*(c + d*x) + a*d*e*S 
inh[c + d*x] - a*c*f*Sinh[c + d*x] + a*f*(c + d*x)*Sinh[c + d*x]))/(2*(...
 

Rubi [A] (verified)

Time = 1.99 (sec) , antiderivative size = 485, normalized size of antiderivative = 0.87, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {6107, 6107, 6095, 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.

\(\displaystyle \int \frac {(e+f x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx\)

\(\Big \downarrow \) 6107

\(\displaystyle \frac {\int (e+f x) \text {sech}^3(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}+\frac {b^2 \int \frac {(e+f x) \text {sech}(c+d x)}{a+b \sinh (c+d x)}dx}{a^2+b^2}\)

\(\Big \downarrow \) 6107

\(\displaystyle \frac {\int (e+f x) \text {sech}^3(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}+\frac {b^2 \left (\frac {b^2 \int \frac {(e+f x) \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{a^2+b^2}+\frac {\int (e+f x) \text {sech}(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}\right )}{a^2+b^2}\)

\(\Big \downarrow \) 6095

\(\displaystyle \frac {\int (e+f x) \text {sech}^3(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}+\frac {b^2 \left (\frac {b^2 \left (\int \frac {e^{c+d x} (e+f x)}{a+b e^{c+d x}-\sqrt {a^2+b^2}}dx+\int \frac {e^{c+d x} (e+f x)}{a+b e^{c+d x}+\sqrt {a^2+b^2}}dx-\frac {(e+f x)^2}{2 b f}\right )}{a^2+b^2}+\frac {\int (e+f x) \text {sech}(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}\right )}{a^2+b^2}\)

\(\Big \downarrow \) 2620

\(\displaystyle \frac {\int (e+f x) \text {sech}^3(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}+\frac {b^2 \left (\frac {b^2 \left (-\frac {f \int \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right )dx}{b d}-\frac {f \int \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right )dx}{b d}+\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}+\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}-\frac {(e+f x)^2}{2 b f}\right )}{a^2+b^2}+\frac {\int (e+f x) \text {sech}(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}\right )}{a^2+b^2}\)

\(\Big \downarrow \) 2715

\(\displaystyle \frac {b^2 \left (\frac {b^2 \left (-\frac {f \int e^{-c-d x} \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right )de^{c+d x}}{b d^2}-\frac {f \int e^{-c-d x} \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right )de^{c+d x}}{b d^2}+\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}+\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}-\frac {(e+f x)^2}{2 b f}\right )}{a^2+b^2}+\frac {\int (e+f x) \text {sech}(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}\right )}{a^2+b^2}+\frac {\int (e+f x) \text {sech}^3(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}\)

\(\Big \downarrow \) 2838

\(\displaystyle \frac {b^2 \left (\frac {\int (e+f x) \text {sech}(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}+\frac {b^2 \left (\frac {f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b d^2}+\frac {f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b d^2}+\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}+\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}-\frac {(e+f x)^2}{2 b f}\right )}{a^2+b^2}\right )}{a^2+b^2}+\frac {\int (e+f x) \text {sech}^3(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}\)

\(\Big \downarrow \) 7293

\(\displaystyle \frac {b^2 \left (\frac {\int (a (e+f x) \text {sech}(c+d x)-b (e+f x) \tanh (c+d x))dx}{a^2+b^2}+\frac {b^2 \left (\frac {f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b d^2}+\frac {f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b d^2}+\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}+\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}-\frac {(e+f x)^2}{2 b f}\right )}{a^2+b^2}\right )}{a^2+b^2}+\frac {\int \left (a (e+f x) \text {sech}^3(c+d x)-b (e+f x) \text {sech}^2(c+d x) \tanh (c+d x)\right )dx}{a^2+b^2}\)

\(\Big \downarrow \) 2009

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

Input:

Int[((e + f*x)*Sech[c + d*x]^3)/(a + b*Sinh[c + d*x]),x]
 

Output:

(b^2*((b^2*(-1/2*(e + f*x)^2/(b*f) + ((e + f*x)*Log[1 + (b*E^(c + d*x))/(a 
 - Sqrt[a^2 + b^2])])/(b*d) + ((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) + (f*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/( 
b*d^2)))/(a^2 + b^2) + ((b*(e + f*x)^2)/(2*f) + (2*a*(e + f*x)*ArcTan[E^(c 
 + d*x)])/d - (b*(e + f*x)*Log[1 + E^(2*(c + d*x))])/d - (I*a*f*PolyLog[2, 
 (-I)*E^(c + d*x)])/d^2 + (I*a*f*PolyLog[2, I*E^(c + d*x)])/d^2 - (b*f*Pol 
yLog[2, -E^(2*(c + d*x))])/(2*d^2))/(a^2 + b^2)))/(a^2 + b^2) + ((a*(e + f 
*x)*ArcTan[E^(c + d*x)])/d - ((I/2)*a*f*PolyLog[2, (-I)*E^(c + d*x)])/d^2 
+ ((I/2)*a*f*PolyLog[2, I*E^(c + d*x)])/d^2 + (a*f*Sech[c + d*x])/(2*d^2) 
+ (b*(e + f*x)*Sech[c + d*x]^2)/(2*d) - (b*f*Tanh[c + d*x])/(2*d^2) + (a*( 
e + f*x)*Sech[c + d*x]*Tanh[c + d*x])/(2*d))/(a^2 + b^2)
 

Defintions of rubi rules used

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 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 6095

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]
 

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 7293

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

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 2050 vs. \(2 (520 ) = 1040\).

Time = 31.58 (sec) , antiderivative size = 2051, normalized size of antiderivative = 3.66

method	result	size
risch	\(\text {Expression too large to display}\)	\(2051\)

Input:

int((f*x+e)*sech(d*x+c)^3/(a+b*sinh(d*x+c)),x,method=_RETURNVERBOSE)
 

Output:

-1/d^2/(a^2+b^2)^(1/2)*c*a*b*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)*a^3*e/(2*a^2+2*b^2)*arctan(exp(d*x+c))+ 
(a*d*f*x*exp(3*d*x+3*c)+a*d*e*exp(3*d*x+3*c)+2*b*d*f*x*exp(2*d*x+2*c)-a*d* 
f*x*exp(d*x+c)+a*f*exp(3*d*x+3*c)+2*b*d*e*exp(2*d*x+2*c)-a*d*e*exp(d*x+c)+ 
b*f*exp(2*d*x+2*c)+a*f*exp(d*x+c)+b*f)/d^2/(a^2+b^2)/(1+exp(2*d*x+2*c))^2+ 
2/d/(a^2+b^2)*b^3*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)*b^3*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/(a^2+b^2)*b^3*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)*b^3*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/(a^2+b^2)*b^3*f/(2*a^2+2*b^2)*ln(1+I*exp(d*x+c))*x-2/d^2/(a^ 
2+b^2)*b^3*f/(2*a^2+2*b^2)*ln(1+I*exp(d*x+c))*c-2/d/(a^2+b^2)*b^3*f/(2*a^2 
+2*b^2)*ln(1-I*exp(d*x+c))*x-2/d^2/(a^2+b^2)*b^3*f/(2*a^2+2*b^2)*ln(1-I*ex 
p(d*x+c))*c+6/d/(a^2+b^2)*a*b^2*e/(2*a^2+2*b^2)*arctan(exp(d*x+c))-2/d^2/( 
a^2+b^2)*c*a^3*f/(2*a^2+2*b^2)*arctan(exp(d*x+c))+I/d^2/(a^2+b^2)*a^3*f/(2 
*a^2+2*b^2)*dilog(1-I*exp(d*x+c))+2/d^2/(a^2+b^2)*c*b^3*f/(2*a^2+2*b^2)*ln 
(1+exp(2*d*x+2*c))-2/d^2/(a^2+b^2)*c*b^3*f/(2*a^2+2*b^2)*ln(b*exp(2*d*x+2* 
c)+2*a*exp(d*x+c)-b)-I/d^2/(a^2+b^2)*a^3*f/(2*a^2+2*b^2)*dilog(1+I*exp(d*x 
+c))+I/d/(a^2+b^2)*a^3*f/(2*a^2+2*b^2)*ln(1-I*exp(d*x+c))*x+I/d^2/(a^2+b^2 
)*a^3*f/(2*a^2+2*b^2)*ln(1-I*exp(d*x+c))*c+1/d/(a^2+b^2)^(1/2)*a*b*e/(2...
 

Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 4729 vs. \(2 (505) = 1010\).

Time = 0.20 (sec) , antiderivative size = 4729, normalized size of antiderivative = 8.44 \[ \int \frac {(e+f x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \] Input:

integrate((f*x+e)*sech(d*x+c)^3/(a+b*sinh(d*x+c)),x, algorithm="fricas")
 

Output:

Too large to include
 

Sympy [F]

\[ \int \frac {(e+f x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {\left (e + f x\right ) \operatorname {sech}^{3}{\left (c + d x \right )}}{a + b \sinh {\left (c + d x \right )}}\, dx \] Input:

integrate((f*x+e)*sech(d*x+c)**3/(a+b*sinh(d*x+c)),x)
 

Output:

Integral((e + f*x)*sech(c + d*x)**3/(a + b*sinh(c + d*x)), x)
 

Maxima [F]

\[ \int \frac {(e+f x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )} \operatorname {sech}\left (d x + c\right )^{3}}{b \sinh \left (d x + c\right ) + a} \,d x } \] Input:

integrate((f*x+e)*sech(d*x+c)^3/(a+b*sinh(d*x+c)),x, algorithm="maxima")
 

Output:

(b^3*log(-2*a*e^(-d*x - c) + b*e^(-2*d*x - 2*c) - b)/((a^4 + 2*a^2*b^2 + b 
^4)*d) - b^3*log(e^(-2*d*x - 2*c) + 1)/((a^4 + 2*a^2*b^2 + b^4)*d) - (a^3 
+ 3*a*b^2)*arctan(e^(-d*x - c))/((a^4 + 2*a^2*b^2 + b^4)*d) + (a*e^(-d*x - 
 c) + 2*b*e^(-2*d*x - 2*c) - a*e^(-3*d*x - 3*c))/((a^2 + b^2 + 2*(a^2 + b^ 
2)*e^(-2*d*x - 2*c) + (a^2 + b^2)*e^(-4*d*x - 4*c))*d))*e + f*(((a*d*x*e^( 
3*c) + a*e^(3*c))*e^(3*d*x) + (2*b*d*x*e^(2*c) + b*e^(2*c))*e^(2*d*x) - (a 
*d*x*e^c - a*e^c)*e^(d*x) + b)/(a^2*d^2 + b^2*d^2 + (a^2*d^2*e^(4*c) + b^2 
*d^2*e^(4*c))*e^(4*d*x) + 2*(a^2*d^2*e^(2*c) + b^2*d^2*e^(2*c))*e^(2*d*x)) 
 - 8*integrate(-1/4*(a*b^3*x*e^(d*x + c) - b^4*x)/(a^4*b + 2*a^2*b^3 + b^5 
 - (a^4*b*e^(2*c) + 2*a^2*b^3*e^(2*c) + b^5*e^(2*c))*e^(2*d*x) - 2*(a^5*e^ 
c + 2*a^3*b^2*e^c + a*b^4*e^c)*e^(d*x)), x) + 8*integrate(1/8*(2*b^3*x + ( 
a^3*e^c + 3*a*b^2*e^c)*x*e^(d*x))/(a^4 + 2*a^2*b^2 + b^4 + (a^4*e^(2*c) + 
2*a^2*b^2*e^(2*c) + b^4*e^(2*c))*e^(2*d*x)), x))
 

Giac [F(-1)]

Timed out. \[ \int \frac {(e+f x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Timed out} \] Input:

integrate((f*x+e)*sech(d*x+c)^3/(a+b*sinh(d*x+c)),x, algorithm="giac")
 

Output:

Timed out
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(e+f x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {e+f\,x}{{\mathrm {cosh}\left (c+d\,x\right )}^3\,\left (a+b\,\mathrm {sinh}\left (c+d\,x\right )\right )} \,d x \] Input:

int((e + f*x)/(cosh(c + d*x)^3*(a + b*sinh(c + d*x))),x)
                                                                                    
                                                                                    
 

Output:

int((e + f*x)/(cosh(c + d*x)^3*(a + b*sinh(c + d*x))), x)
 

Reduce [F]

\[ \int \frac {(e+f x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {too large to display} \] Input:

int((f*x+e)*sech(d*x+c)^3/(a+b*sinh(d*x+c)),x)
 

Output:

(e**(4*c + 4*d*x)*atan(e**(c + d*x))*a**3*b*d*e + 3*e**(4*c + 4*d*x)*atan( 
e**(c + d*x))*a*b**3*d*e + 2*e**(2*c + 2*d*x)*atan(e**(c + d*x))*a**3*b*d* 
e + 6*e**(2*c + 2*d*x)*atan(e**(c + d*x))*a*b**3*d*e + atan(e**(c + d*x))* 
a**3*b*d*e + 3*atan(e**(c + d*x))*a*b**3*d*e - 32*e**(7*c + 4*d*x)*int((e* 
*(3*d*x)*x)/(e**(8*c + 8*d*x)*b + 2*e**(7*c + 7*d*x)*a + 2*e**(6*c + 6*d*x 
)*b + 6*e**(5*c + 5*d*x)*a + 6*e**(3*c + 3*d*x)*a - 2*e**(2*c + 2*d*x)*b + 
 2*e**(c + d*x)*a - b),x)*a**5*d**2*f - 64*e**(7*c + 4*d*x)*int((e**(3*d*x 
)*x)/(e**(8*c + 8*d*x)*b + 2*e**(7*c + 7*d*x)*a + 2*e**(6*c + 6*d*x)*b + 6 
*e**(5*c + 5*d*x)*a + 6*e**(3*c + 3*d*x)*a - 2*e**(2*c + 2*d*x)*b + 2*e**( 
c + d*x)*a - b),x)*a**3*b**2*d**2*f - 32*e**(7*c + 4*d*x)*int((e**(3*d*x)* 
x)/(e**(8*c + 8*d*x)*b + 2*e**(7*c + 7*d*x)*a + 2*e**(6*c + 6*d*x)*b + 6*e 
**(5*c + 5*d*x)*a + 6*e**(3*c + 3*d*x)*a - 2*e**(2*c + 2*d*x)*b + 2*e**(c 
+ d*x)*a - b),x)*a*b**4*d**2*f + 16*e**(6*c + 4*d*x)*int((e**(2*d*x)*x)/(e 
**(8*c + 8*d*x)*b + 2*e**(7*c + 7*d*x)*a + 2*e**(6*c + 6*d*x)*b + 6*e**(5* 
c + 5*d*x)*a + 6*e**(3*c + 3*d*x)*a - 2*e**(2*c + 2*d*x)*b + 2*e**(c + d*x 
)*a - b),x)*a**4*b*d**2*f + 32*e**(6*c + 4*d*x)*int((e**(2*d*x)*x)/(e**(8* 
c + 8*d*x)*b + 2*e**(7*c + 7*d*x)*a + 2*e**(6*c + 6*d*x)*b + 6*e**(5*c + 5 
*d*x)*a + 6*e**(3*c + 3*d*x)*a - 2*e**(2*c + 2*d*x)*b + 2*e**(c + d*x)*a - 
 b),x)*a**2*b**3*d**2*f + 16*e**(6*c + 4*d*x)*int((e**(2*d*x)*x)/(e**(8*c 
+ 8*d*x)*b + 2*e**(7*c + 7*d*x)*a + 2*e**(6*c + 6*d*x)*b + 6*e**(5*c + ...