Integrand size = 16, antiderivative size = 79 \[ \int x \text {csch}^2(a+b x) \text {sech}(a+b x) \, dx=-\frac {2 x \arctan \left (e^{a+b x}\right )}{b}-\frac {\text {arctanh}(\cosh (a+b x))}{b^2}-\frac {x \text {csch}(a+b x)}{b}+\frac {i \operatorname {PolyLog}\left (2,-i e^{a+b x}\right )}{b^2}-\frac {i \operatorname {PolyLog}\left (2,i e^{a+b x}\right )}{b^2} \]
-2*x*arctan(exp(b*x+a))/b-arctanh(cosh(b*x+a))/b^2-x*csch(b*x+a)/b+I*polyl og(2,-I*exp(b*x+a))/b^2-I*polylog(2,I*exp(b*x+a))/b^2
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(184\) vs. \(2(79)=158\).
Time = 0.49 (sec) , antiderivative size = 184, normalized size of antiderivative = 2.33 \[ \int x \text {csch}^2(a+b x) \text {sech}(a+b x) \, dx=\frac {4 a \arctan \left (e^{a+b x}\right )-b x \coth \left (\frac {1}{2} (a+b x)\right )-2 i a \log \left (1-i e^{a+b x}\right )-2 i b x \log \left (1-i e^{a+b x}\right )+2 i a \log \left (1+i e^{a+b x}\right )+2 i b x \log \left (1+i e^{a+b x}\right )-2 \log \left (\cosh \left (\frac {1}{2} (a+b x)\right )\right )+2 \log \left (\sinh \left (\frac {1}{2} (a+b x)\right )\right )+2 i \operatorname {PolyLog}\left (2,-i e^{a+b x}\right )-2 i \operatorname {PolyLog}\left (2,i e^{a+b x}\right )+b x \tanh \left (\frac {1}{2} (a+b x)\right )}{2 b^2} \]
(4*a*ArcTan[E^(a + b*x)] - b*x*Coth[(a + b*x)/2] - (2*I)*a*Log[1 - I*E^(a + b*x)] - (2*I)*b*x*Log[1 - I*E^(a + b*x)] + (2*I)*a*Log[1 + I*E^(a + b*x) ] + (2*I)*b*x*Log[1 + I*E^(a + b*x)] - 2*Log[Cosh[(a + b*x)/2]] + 2*Log[Si nh[(a + b*x)/2]] + (2*I)*PolyLog[2, (-I)*E^(a + b*x)] - (2*I)*PolyLog[2, I *E^(a + b*x)] + b*x*Tanh[(a + b*x)/2])/(2*b^2)
Time = 0.32 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {5985, 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.
\(\displaystyle \int x \text {csch}^2(a+b x) \text {sech}(a+b x) \, dx\) |
\(\Big \downarrow \) 5985 |
\(\displaystyle -\int \left (-\frac {\arctan (\sinh (a+b x))}{b}-\frac {\text {csch}(a+b x)}{b}\right )dx-\frac {x \arctan (\sinh (a+b x))}{b}-\frac {x \text {csch}(a+b x)}{b}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {2 x \arctan \left (e^{a+b x}\right )}{b}-\frac {\text {arctanh}(\cosh (a+b x))}{b^2}+\frac {i \operatorname {PolyLog}\left (2,-i e^{a+b x}\right )}{b^2}-\frac {i \operatorname {PolyLog}\left (2,i e^{a+b x}\right )}{b^2}-\frac {x \text {csch}(a+b x)}{b}\) |
(-2*x*ArcTan[E^(a + b*x)])/b - ArcTanh[Cosh[a + b*x]]/b^2 - (x*Csch[a + b* x])/b + (I*PolyLog[2, (-I)*E^(a + b*x)])/b^2 - (I*PolyLog[2, I*E^(a + b*x) ])/b^2
3.5.90.3.1 Defintions of rubi rules used
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]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 178 vs. \(2 (72 ) = 144\).
Time = 0.99 (sec) , antiderivative size = 179, normalized size of antiderivative = 2.27
method | result | size |
risch | \(-\frac {2 x \,{\mathrm e}^{b x +a}}{b \left ({\mathrm e}^{2 b x +2 a}-1\right )}+\frac {2 a \arctan \left ({\mathrm e}^{b x +a}\right )}{b^{2}}+\frac {i \operatorname {dilog}\left (1+i {\mathrm e}^{b x +a}\right )}{b^{2}}-\frac {i \ln \left (1-i {\mathrm e}^{b x +a}\right ) x}{b}-\frac {i \ln \left (1-i {\mathrm e}^{b x +a}\right ) a}{b^{2}}+\frac {i \ln \left (1+i {\mathrm e}^{b x +a}\right ) x}{b}+\frac {i \ln \left (1+i {\mathrm e}^{b x +a}\right ) a}{b^{2}}-\frac {i \operatorname {dilog}\left (1-i {\mathrm e}^{b x +a}\right )}{b^{2}}+\frac {\ln \left ({\mathrm e}^{b x +a}-1\right )}{b^{2}}-\frac {\ln \left ({\mathrm e}^{b x +a}+1\right )}{b^{2}}\) | \(179\) |
-2/b*x*exp(b*x+a)/(exp(2*b*x+2*a)-1)+2/b^2*a*arctan(exp(b*x+a))+I/b^2*dilo g(1+I*exp(b*x+a))-I/b*ln(1-I*exp(b*x+a))*x-I/b^2*ln(1-I*exp(b*x+a))*a+I/b* ln(1+I*exp(b*x+a))*x+I/b^2*ln(1+I*exp(b*x+a))*a-I/b^2*dilog(1-I*exp(b*x+a) )+1/b^2*ln(exp(b*x+a)-1)-1/b^2*ln(exp(b*x+a)+1)
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 567 vs. \(2 (66) = 132\).
Time = 0.28 (sec) , antiderivative size = 567, normalized size of antiderivative = 7.18 \[ \int x \text {csch}^2(a+b x) \text {sech}(a+b x) \, dx=-\frac {2 \, b x \cosh \left (b x + a\right ) + 2 \, b x \sinh \left (b x + a\right ) - {\left (-i \, \cosh \left (b x + a\right )^{2} - 2 i \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) - i \, \sinh \left (b x + a\right )^{2} + i\right )} {\rm Li}_2\left (i \, \cosh \left (b x + a\right ) + i \, \sinh \left (b x + a\right )\right ) - {\left (i \, \cosh \left (b x + a\right )^{2} + 2 i \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + i \, \sinh \left (b x + a\right )^{2} - i\right )} {\rm Li}_2\left (-i \, \cosh \left (b x + a\right ) - i \, \sinh \left (b x + a\right )\right ) + {\left (\cosh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2} - 1\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + 1\right ) - {\left (i \, a \cosh \left (b x + a\right )^{2} + 2 i \, a \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + i \, a \sinh \left (b x + a\right )^{2} - i \, a\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + i\right ) - {\left (-i \, a \cosh \left (b x + a\right )^{2} - 2 i \, a \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) - i \, a \sinh \left (b x + a\right )^{2} + i \, a\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - i\right ) - {\left (\cosh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2} - 1\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - 1\right ) - {\left ({\left (i \, b x + i \, a\right )} \cosh \left (b x + a\right )^{2} - 2 \, {\left (-i \, b x - i \, a\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + {\left (i \, b x + i \, a\right )} \sinh \left (b x + a\right )^{2} - i \, b x - i \, a\right )} \log \left (i \, \cosh \left (b x + a\right ) + i \, \sinh \left (b x + a\right ) + 1\right ) - {\left ({\left (-i \, b x - i \, a\right )} \cosh \left (b x + a\right )^{2} - 2 \, {\left (i \, b x + i \, a\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + {\left (-i \, b x - i \, a\right )} \sinh \left (b x + a\right )^{2} + i \, b x + i \, a\right )} \log \left (-i \, \cosh \left (b x + a\right ) - i \, \sinh \left (b x + a\right ) + 1\right )}{b^{2} \cosh \left (b x + a\right )^{2} + 2 \, b^{2} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + b^{2} \sinh \left (b x + a\right )^{2} - b^{2}} \]
-(2*b*x*cosh(b*x + a) + 2*b*x*sinh(b*x + a) - (-I*cosh(b*x + a)^2 - 2*I*co sh(b*x + a)*sinh(b*x + a) - I*sinh(b*x + a)^2 + I)*dilog(I*cosh(b*x + a) + I*sinh(b*x + a)) - (I*cosh(b*x + a)^2 + 2*I*cosh(b*x + a)*sinh(b*x + a) + I*sinh(b*x + a)^2 - I)*dilog(-I*cosh(b*x + a) - I*sinh(b*x + a)) + (cosh( b*x + a)^2 + 2*cosh(b*x + a)*sinh(b*x + a) + sinh(b*x + a)^2 - 1)*log(cosh (b*x + a) + sinh(b*x + a) + 1) - (I*a*cosh(b*x + a)^2 + 2*I*a*cosh(b*x + a )*sinh(b*x + a) + I*a*sinh(b*x + a)^2 - I*a)*log(cosh(b*x + a) + sinh(b*x + a) + I) - (-I*a*cosh(b*x + a)^2 - 2*I*a*cosh(b*x + a)*sinh(b*x + a) - I* a*sinh(b*x + a)^2 + I*a)*log(cosh(b*x + a) + sinh(b*x + a) - I) - (cosh(b* x + a)^2 + 2*cosh(b*x + a)*sinh(b*x + a) + sinh(b*x + a)^2 - 1)*log(cosh(b *x + a) + sinh(b*x + a) - 1) - ((I*b*x + I*a)*cosh(b*x + a)^2 - 2*(-I*b*x - I*a)*cosh(b*x + a)*sinh(b*x + a) + (I*b*x + I*a)*sinh(b*x + a)^2 - I*b*x - I*a)*log(I*cosh(b*x + a) + I*sinh(b*x + a) + 1) - ((-I*b*x - I*a)*cosh( b*x + a)^2 - 2*(I*b*x + I*a)*cosh(b*x + a)*sinh(b*x + a) + (-I*b*x - I*a)* sinh(b*x + a)^2 + I*b*x + I*a)*log(-I*cosh(b*x + a) - I*sinh(b*x + a) + 1) )/(b^2*cosh(b*x + a)^2 + 2*b^2*cosh(b*x + a)*sinh(b*x + a) + b^2*sinh(b*x + a)^2 - b^2)
\[ \int x \text {csch}^2(a+b x) \text {sech}(a+b x) \, dx=\int x \operatorname {csch}^{2}{\left (a + b x \right )} \operatorname {sech}{\left (a + b x \right )}\, dx \]
\[ \int x \text {csch}^2(a+b x) \text {sech}(a+b x) \, dx=\int { x \operatorname {csch}\left (b x + a\right )^{2} \operatorname {sech}\left (b x + a\right ) \,d x } \]
-2*x*e^(b*x + a)/(b*e^(2*b*x + 2*a) - b) - log((e^(b*x + a) + 1)*e^(-a))/b ^2 + log((e^(b*x + a) - 1)*e^(-a))/b^2 - 8*integrate(1/4*x*e^(b*x + a)/(e^ (2*b*x + 2*a) + 1), x)
\[ \int x \text {csch}^2(a+b x) \text {sech}(a+b x) \, dx=\int { x \operatorname {csch}\left (b x + a\right )^{2} \operatorname {sech}\left (b x + a\right ) \,d x } \]
Timed out. \[ \int x \text {csch}^2(a+b x) \text {sech}(a+b x) \, dx=\int \frac {x}{\mathrm {cosh}\left (a+b\,x\right )\,{\mathrm {sinh}\left (a+b\,x\right )}^2} \,d x \]