Integrand size = 18, antiderivative size = 109 \[ \int x \text {csch}^3(a+b x) \text {sech}^2(a+b x) \, dx=\frac {\arctan (\sinh (a+b x))}{b^2}+\frac {3 x \text {arctanh}\left (e^{a+b x}\right )}{b}-\frac {\text {csch}(a+b x)}{2 b^2}+\frac {3 \operatorname {PolyLog}\left (2,-e^{a+b x}\right )}{2 b^2}-\frac {3 \operatorname {PolyLog}\left (2,e^{a+b x}\right )}{2 b^2}-\frac {3 x \text {sech}(a+b x)}{2 b}-\frac {x \text {csch}^2(a+b x) \text {sech}(a+b x)}{2 b} \]
arctan(sinh(b*x+a))/b^2+3*x*arctanh(exp(b*x+a))/b-1/2*csch(b*x+a)/b^2+3/2* polylog(2,-exp(b*x+a))/b^2-3/2*polylog(2,exp(b*x+a))/b^2-3/2*x*sech(b*x+a) /b-1/2*x*csch(b*x+a)^2*sech(b*x+a)/b
Time = 1.37 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.28 \[ \int x \text {csch}^3(a+b x) \text {sech}^2(a+b x) \, dx=-\frac {-16 \arctan \left (\tanh \left (\frac {1}{2} (a+b x)\right )\right )+2 \coth \left (\frac {1}{2} (a+b x)\right )+b x \text {csch}^2\left (\frac {1}{2} (a+b x)\right )+12 b x \log \left (1-e^{a+b x}\right )-12 b x \log \left (1+e^{a+b x}\right )-12 \operatorname {PolyLog}\left (2,-e^{a+b x}\right )+12 \operatorname {PolyLog}\left (2,e^{a+b x}\right )+b x \text {sech}^2\left (\frac {1}{2} (a+b x)\right )+8 b x \text {sech}(a+b x)-2 \tanh \left (\frac {1}{2} (a+b x)\right )}{8 b^2} \]
-1/8*(-16*ArcTan[Tanh[(a + b*x)/2]] + 2*Coth[(a + b*x)/2] + b*x*Csch[(a + b*x)/2]^2 + 12*b*x*Log[1 - E^(a + b*x)] - 12*b*x*Log[1 + E^(a + b*x)] - 12 *PolyLog[2, -E^(a + b*x)] + 12*PolyLog[2, E^(a + b*x)] + b*x*Sech[(a + b*x )/2]^2 + 8*b*x*Sech[a + b*x] - 2*Tanh[(a + b*x)/2])/b^2
Time = 0.38 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, 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.
\(\displaystyle \int x \text {csch}^3(a+b x) \text {sech}^2(a+b x) \, dx\) |
\(\Big \downarrow \) 5985 |
\(\displaystyle -\int \left (-\frac {\text {sech}(a+b x) \text {csch}^2(a+b x)}{2 b}+\frac {3 \text {arctanh}(\cosh (a+b x))}{2 b}-\frac {3 \text {sech}(a+b x)}{2 b}\right )dx+\frac {3 x \text {arctanh}(\cosh (a+b x))}{2 b}-\frac {3 x \text {sech}(a+b x)}{2 b}-\frac {x \text {csch}^2(a+b x) \text {sech}(a+b x)}{2 b}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\arctan (\sinh (a+b x))}{b^2}+\frac {3 x \text {arctanh}\left (e^{a+b x}\right )}{b}+\frac {3 \operatorname {PolyLog}\left (2,-e^{a+b x}\right )}{2 b^2}-\frac {3 \operatorname {PolyLog}\left (2,e^{a+b x}\right )}{2 b^2}-\frac {\text {csch}(a+b x)}{2 b^2}-\frac {3 x \text {sech}(a+b x)}{2 b}-\frac {x \text {csch}^2(a+b x) \text {sech}(a+b x)}{2 b}\) |
ArcTan[Sinh[a + b*x]]/b^2 + (3*x*ArcTanh[E^(a + b*x)])/b - Csch[a + b*x]/( 2*b^2) + (3*PolyLog[2, -E^(a + b*x)])/(2*b^2) - (3*PolyLog[2, E^(a + b*x)] )/(2*b^2) - (3*x*Sech[a + b*x])/(2*b) - (x*Csch[a + b*x]^2*Sech[a + b*x])/ (2*b)
3.6.17.3.1 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]
Time = 5.89 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.36
method | result | size |
risch | \(-\frac {{\mathrm e}^{b x +a} \left (3 \,{\mathrm e}^{4 b x +4 a} b x -2 \,{\mathrm e}^{2 b x +2 a} b x +{\mathrm e}^{4 b x +4 a}+3 b x -1\right )}{b^{2} \left ({\mathrm e}^{2 b x +2 a}-1\right )^{2} \left (1+{\mathrm e}^{2 b x +2 a}\right )}+\frac {2 \arctan \left ({\mathrm e}^{b x +a}\right )}{b^{2}}+\frac {3 a \ln \left ({\mathrm e}^{b x +a}-1\right )}{2 b^{2}}+\frac {3 \operatorname {dilog}\left ({\mathrm e}^{b x +a}\right )}{2 b^{2}}+\frac {3 \operatorname {dilog}\left ({\mathrm e}^{b x +a}+1\right )}{2 b^{2}}+\frac {3 \ln \left ({\mathrm e}^{b x +a}+1\right ) x}{2 b}\) | \(148\) |
-exp(b*x+a)*(3*exp(4*b*x+4*a)*b*x-2*exp(2*b*x+2*a)*b*x+exp(4*b*x+4*a)+3*b* x-1)/b^2/(exp(2*b*x+2*a)-1)^2/(1+exp(2*b*x+2*a))+2/b^2*arctan(exp(b*x+a))+ 3/2/b^2*a*ln(exp(b*x+a)-1)+3/2/b^2*dilog(exp(b*x+a))+3/2/b^2*dilog(exp(b*x +a)+1)+3/2/b*ln(exp(b*x+a)+1)*x
Leaf count of result is larger than twice the leaf count of optimal. 1694 vs. \(2 (94) = 188\).
Time = 0.28 (sec) , antiderivative size = 1694, normalized size of antiderivative = 15.54 \[ \int x \text {csch}^3(a+b x) \text {sech}^2(a+b x) \, dx=\text {Too large to display} \]
-1/2*(2*(3*b*x + 1)*cosh(b*x + a)^5 + 10*(3*b*x + 1)*cosh(b*x + a)*sinh(b* x + a)^4 + 2*(3*b*x + 1)*sinh(b*x + a)^5 - 4*b*x*cosh(b*x + a)^3 + 4*(5*(3 *b*x + 1)*cosh(b*x + a)^2 - b*x)*sinh(b*x + a)^3 + 4*(5*(3*b*x + 1)*cosh(b *x + a)^3 - 3*b*x*cosh(b*x + a))*sinh(b*x + a)^2 - 4*(cosh(b*x + a)^6 + 6* cosh(b*x + a)*sinh(b*x + a)^5 + sinh(b*x + a)^6 + (15*cosh(b*x + a)^2 - 1) *sinh(b*x + a)^4 - cosh(b*x + a)^4 + 4*(5*cosh(b*x + a)^3 - cosh(b*x + a)) *sinh(b*x + a)^3 + (15*cosh(b*x + a)^4 - 6*cosh(b*x + a)^2 - 1)*sinh(b*x + a)^2 - cosh(b*x + a)^2 + 2*(3*cosh(b*x + a)^5 - 2*cosh(b*x + a)^3 - cosh( b*x + a))*sinh(b*x + a) + 1)*arctan(cosh(b*x + a) + sinh(b*x + a)) + 2*(3* b*x - 1)*cosh(b*x + a) + 3*(cosh(b*x + a)^6 + 6*cosh(b*x + a)*sinh(b*x + a )^5 + sinh(b*x + a)^6 + (15*cosh(b*x + a)^2 - 1)*sinh(b*x + a)^4 - cosh(b* x + a)^4 + 4*(5*cosh(b*x + a)^3 - cosh(b*x + a))*sinh(b*x + a)^3 + (15*cos h(b*x + a)^4 - 6*cosh(b*x + a)^2 - 1)*sinh(b*x + a)^2 - cosh(b*x + a)^2 + 2*(3*cosh(b*x + a)^5 - 2*cosh(b*x + a)^3 - cosh(b*x + a))*sinh(b*x + a) + 1)*dilog(cosh(b*x + a) + sinh(b*x + a)) - 3*(cosh(b*x + a)^6 + 6*cosh(b*x + a)*sinh(b*x + a)^5 + sinh(b*x + a)^6 + (15*cosh(b*x + a)^2 - 1)*sinh(b*x + a)^4 - cosh(b*x + a)^4 + 4*(5*cosh(b*x + a)^3 - cosh(b*x + a))*sinh(b*x + a)^3 + (15*cosh(b*x + a)^4 - 6*cosh(b*x + a)^2 - 1)*sinh(b*x + a)^2 - c osh(b*x + a)^2 + 2*(3*cosh(b*x + a)^5 - 2*cosh(b*x + a)^3 - cosh(b*x + a)) *sinh(b*x + a) + 1)*dilog(-cosh(b*x + a) - sinh(b*x + a)) - 3*(b*x*cosh...
\[ \int x \text {csch}^3(a+b x) \text {sech}^2(a+b x) \, dx=\int x \operatorname {csch}^{3}{\left (a + b x \right )} \operatorname {sech}^{2}{\left (a + b x \right )}\, dx \]
Time = 0.31 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.52 \[ \int x \text {csch}^3(a+b x) \text {sech}^2(a+b x) \, dx=\frac {2 \, b x e^{\left (3 \, b x + 3 \, a\right )} - {\left (3 \, b x e^{\left (5 \, a\right )} + e^{\left (5 \, a\right )}\right )} e^{\left (5 \, b x\right )} - {\left (3 \, b x e^{a} - e^{a}\right )} e^{\left (b x\right )}}{b^{2} e^{\left (6 \, b x + 6 \, a\right )} - b^{2} e^{\left (4 \, b x + 4 \, a\right )} - b^{2} e^{\left (2 \, b x + 2 \, a\right )} + b^{2}} + \frac {3 \, {\left (b x \log \left (e^{\left (b x + a\right )} + 1\right ) + {\rm Li}_2\left (-e^{\left (b x + a\right )}\right )\right )}}{2 \, b^{2}} - \frac {3 \, {\left (b x \log \left (-e^{\left (b x + a\right )} + 1\right ) + {\rm Li}_2\left (e^{\left (b x + a\right )}\right )\right )}}{2 \, b^{2}} + \frac {2 \, \arctan \left (e^{\left (b x + a\right )}\right )}{b^{2}} \]
(2*b*x*e^(3*b*x + 3*a) - (3*b*x*e^(5*a) + e^(5*a))*e^(5*b*x) - (3*b*x*e^a - e^a)*e^(b*x))/(b^2*e^(6*b*x + 6*a) - b^2*e^(4*b*x + 4*a) - b^2*e^(2*b*x + 2*a) + b^2) + 3/2*(b*x*log(e^(b*x + a) + 1) + dilog(-e^(b*x + a)))/b^2 - 3/2*(b*x*log(-e^(b*x + a) + 1) + dilog(e^(b*x + a)))/b^2 + 2*arctan(e^(b* x + a))/b^2
\[ \int x \text {csch}^3(a+b x) \text {sech}^2(a+b x) \, dx=\int { x \operatorname {csch}\left (b x + a\right )^{3} \operatorname {sech}\left (b x + a\right )^{2} \,d x } \]
Timed out. \[ \int x \text {csch}^3(a+b x) \text {sech}^2(a+b x) \, dx=\int \frac {x}{{\mathrm {cosh}\left (a+b\,x\right )}^2\,{\mathrm {sinh}\left (a+b\,x\right )}^3} \,d x \]