Integrand size = 20, antiderivative size = 202 \[ \int x^2 \text {csch}^2(a+b x) \text {sech}^3(a+b x) \, dx=-\frac {3 x^2 \arctan \left (e^{a+b x}\right )}{b}+\frac {\arctan (\sinh (a+b x))}{b^3}-\frac {4 x \text {arctanh}\left (e^{a+b x}\right )}{b^2}-\frac {x^2 \text {csch}(a+b x)}{b}-\frac {2 \operatorname {PolyLog}\left (2,-e^{a+b x}\right )}{b^3}+\frac {3 i x \operatorname {PolyLog}\left (2,-i e^{a+b x}\right )}{b^2}-\frac {3 i x \operatorname {PolyLog}\left (2,i e^{a+b x}\right )}{b^2}+\frac {2 \operatorname {PolyLog}\left (2,e^{a+b x}\right )}{b^3}-\frac {3 i \operatorname {PolyLog}\left (3,-i e^{a+b x}\right )}{b^3}+\frac {3 i \operatorname {PolyLog}\left (3,i e^{a+b x}\right )}{b^3}-\frac {x \text {sech}(a+b x)}{b^2}-\frac {x^2 \text {sech}(a+b x) \tanh (a+b x)}{2 b} \] Output:
-3*x^2*arctan(exp(b*x+a))/b+arctan(sinh(b*x+a))/b^3-4*x*arctanh(exp(b*x+a) )/b^2-x^2*csch(b*x+a)/b-2*polylog(2,-exp(b*x+a))/b^3+3*I*x*polylog(2,-I*ex p(b*x+a))/b^2-3*I*x*polylog(2,I*exp(b*x+a))/b^2+2*polylog(2,exp(b*x+a))/b^ 3-3*I*polylog(3,-I*exp(b*x+a))/b^3+3*I*polylog(3,I*exp(b*x+a))/b^3-x*sech( b*x+a)/b^2-1/2*x^2*sech(b*x+a)*tanh(b*x+a)/b
Time = 4.53 (sec) , antiderivative size = 310, normalized size of antiderivative = 1.53 \[ \int x^2 \text {csch}^2(a+b x) \text {sech}^3(a+b x) \, dx=\frac {4 \arctan \left (e^{a+b x}\right )-2 b^2 x^2 \text {csch}(a)+4 b x \log \left (1-e^{a+b x}\right )-3 i b^2 x^2 \log \left (1-i e^{a+b x}\right )+3 i b^2 x^2 \log \left (1+i e^{a+b x}\right )-4 b x \log \left (1+e^{a+b x}\right )-4 \operatorname {PolyLog}\left (2,-e^{a+b x}\right )+6 i b x \operatorname {PolyLog}\left (2,-i e^{a+b x}\right )-6 i b x \operatorname {PolyLog}\left (2,i e^{a+b x}\right )+4 \operatorname {PolyLog}\left (2,e^{a+b x}\right )-6 i \operatorname {PolyLog}\left (3,-i e^{a+b x}\right )+6 i \operatorname {PolyLog}\left (3,i e^{a+b x}\right )-2 b x \text {sech}(a+b x)+b^2 x^2 \text {csch}\left (\frac {a}{2}\right ) \text {csch}\left (\frac {1}{2} (a+b x)\right ) \sinh \left (\frac {b x}{2}\right )+b^2 x^2 \text {sech}\left (\frac {a}{2}\right ) \text {sech}\left (\frac {1}{2} (a+b x)\right ) \sinh \left (\frac {b x}{2}\right )-b^2 x^2 \text {sech}(a) \text {sech}^2(a+b x) \sinh (b x)-b^2 x^2 \text {sech}(a+b x) \tanh (a)}{2 b^3} \] Input:
Integrate[x^2*Csch[a + b*x]^2*Sech[a + b*x]^3,x]
Output:
(4*ArcTan[E^(a + b*x)] - 2*b^2*x^2*Csch[a] + 4*b*x*Log[1 - E^(a + b*x)] - (3*I)*b^2*x^2*Log[1 - I*E^(a + b*x)] + (3*I)*b^2*x^2*Log[1 + I*E^(a + b*x) ] - 4*b*x*Log[1 + E^(a + b*x)] - 4*PolyLog[2, -E^(a + b*x)] + (6*I)*b*x*Po lyLog[2, (-I)*E^(a + b*x)] - (6*I)*b*x*PolyLog[2, I*E^(a + b*x)] + 4*PolyL og[2, E^(a + b*x)] - (6*I)*PolyLog[3, (-I)*E^(a + b*x)] + (6*I)*PolyLog[3, I*E^(a + b*x)] - 2*b*x*Sech[a + b*x] + b^2*x^2*Csch[a/2]*Csch[(a + b*x)/2 ]*Sinh[(b*x)/2] + b^2*x^2*Sech[a/2]*Sech[(a + b*x)/2]*Sinh[(b*x)/2] - b^2* x^2*Sech[a]*Sech[a + b*x]^2*Sinh[b*x] - b^2*x^2*Sech[a + b*x]*Tanh[a])/(2* b^3)
Time = 0.71 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.02, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {5985, 27, 2010, 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.
\(\displaystyle \int x^2 \text {csch}^2(a+b x) \text {sech}^3(a+b x) \, dx\) |
\(\Big \downarrow \) 5985 |
\(\displaystyle -2 \int -\frac {1}{2} x \left (-\frac {\text {csch}(a+b x) \text {sech}^2(a+b x)}{b}+\frac {3 \arctan (\sinh (a+b x))}{b}+\frac {3 \text {csch}(a+b x)}{b}\right )dx-\frac {3 x^2 \arctan (\sinh (a+b x))}{2 b}-\frac {3 x^2 \text {csch}(a+b x)}{2 b}+\frac {x^2 \text {csch}(a+b x) \text {sech}^2(a+b x)}{2 b}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \int x \left (-\frac {\text {csch}(a+b x) \text {sech}^2(a+b x)}{b}+\frac {3 \arctan (\sinh (a+b x))}{b}+\frac {3 \text {csch}(a+b x)}{b}\right )dx-\frac {3 x^2 \arctan (\sinh (a+b x))}{2 b}-\frac {3 x^2 \text {csch}(a+b x)}{2 b}+\frac {x^2 \text {csch}(a+b x) \text {sech}^2(a+b x)}{2 b}\) |
\(\Big \downarrow \) 2010 |
\(\displaystyle \int \left (\frac {3 x (\arctan (\sinh (a+b x))+\text {csch}(a+b x))}{b}-\frac {x \text {csch}(a+b x) \text {sech}^2(a+b x)}{b}\right )dx-\frac {3 x^2 \arctan (\sinh (a+b x))}{2 b}-\frac {3 x^2 \text {csch}(a+b x)}{2 b}+\frac {x^2 \text {csch}(a+b x) \text {sech}^2(a+b x)}{2 b}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\arctan (\sinh (a+b x))}{b^3}-\frac {3 x^2 \arctan \left (e^{a+b x}\right )}{b}-\frac {4 x \text {arctanh}\left (e^{a+b x}\right )}{b^2}-\frac {2 \operatorname {PolyLog}\left (2,-e^{a+b x}\right )}{b^3}+\frac {2 \operatorname {PolyLog}\left (2,e^{a+b x}\right )}{b^3}-\frac {3 i \operatorname {PolyLog}\left (3,-i e^{a+b x}\right )}{b^3}+\frac {3 i \operatorname {PolyLog}\left (3,i e^{a+b x}\right )}{b^3}+\frac {3 i x \operatorname {PolyLog}\left (2,-i e^{a+b x}\right )}{b^2}-\frac {3 i x \operatorname {PolyLog}\left (2,i e^{a+b x}\right )}{b^2}-\frac {x \text {sech}(a+b x)}{b^2}-\frac {3 x^2 \text {csch}(a+b x)}{2 b}+\frac {x^2 \text {csch}(a+b x) \text {sech}^2(a+b x)}{2 b}\) |
Input:
Int[x^2*Csch[a + b*x]^2*Sech[a + b*x]^3,x]
Output:
(-3*x^2*ArcTan[E^(a + b*x)])/b + ArcTan[Sinh[a + b*x]]/b^3 - (4*x*ArcTanh[ E^(a + b*x)])/b^2 - (3*x^2*Csch[a + b*x])/(2*b) - (2*PolyLog[2, -E^(a + b* x)])/b^3 + ((3*I)*x*PolyLog[2, (-I)*E^(a + b*x)])/b^2 - ((3*I)*x*PolyLog[2 , I*E^(a + b*x)])/b^2 + (2*PolyLog[2, E^(a + b*x)])/b^3 - ((3*I)*PolyLog[3 , (-I)*E^(a + b*x)])/b^3 + ((3*I)*PolyLog[3, I*E^(a + b*x)])/b^3 - (x*Sech [a + b*x])/b^2 + (x^2*Csch[a + b*x]*Sech[a + b*x]^2)/(2*b)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x] , x] /; FreeQ[{c, m}, x] && SumQ[u] && !LinearQ[u, x] && !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]
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]
\[\int x^{2} \operatorname {csch}\left (b x +a \right )^{2} \operatorname {sech}\left (b x +a \right )^{3}d x\]
Input:
int(x^2*csch(b*x+a)^2*sech(b*x+a)^3,x)
Output:
int(x^2*csch(b*x+a)^2*sech(b*x+a)^3,x)
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 3825 vs. \(2 (172) = 344\).
Time = 0.17 (sec) , antiderivative size = 3825, normalized size of antiderivative = 18.94 \[ \int x^2 \text {csch}^2(a+b x) \text {sech}^3(a+b x) \, dx=\text {Too large to display} \] Input:
integrate(x^2*csch(b*x+a)^2*sech(b*x+a)^3,x, algorithm="fricas")
Output:
Too large to include
\[ \int x^2 \text {csch}^2(a+b x) \text {sech}^3(a+b x) \, dx=\int x^{2} \operatorname {csch}^{2}{\left (a + b x \right )} \operatorname {sech}^{3}{\left (a + b x \right )}\, dx \] Input:
integrate(x**2*csch(b*x+a)**2*sech(b*x+a)**3,x)
Output:
Integral(x**2*csch(a + b*x)**2*sech(a + b*x)**3, x)
\[ \int x^2 \text {csch}^2(a+b x) \text {sech}^3(a+b x) \, dx=\int { x^{2} \operatorname {csch}\left (b x + a\right )^{2} \operatorname {sech}\left (b x + a\right )^{3} \,d x } \] Input:
integrate(x^2*csch(b*x+a)^2*sech(b*x+a)^3,x, algorithm="maxima")
Output:
-96*b^2*integrate(1/32*x^2*e^(b*x + a)/(b^2*e^(2*b*x + 2*a) + b^2), x) - ( 2*b*x^2*e^(3*b*x + 3*a) + (3*b*x^2*e^(5*a) + 2*x*e^(5*a))*e^(5*b*x) + (3*b *x^2*e^a - 2*x*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) - 2*(b*x*log(e^(b*x + a) + 1) + dilog(-e^(b*x + a)))/b^3 + 2*(b*x*log(-e^(b*x + a) + 1) + dilog(e^(b*x + a)))/b^3 + 2*arc tan(e^(b*x + a))/b^3
\[ \int x^2 \text {csch}^2(a+b x) \text {sech}^3(a+b x) \, dx=\int { x^{2} \operatorname {csch}\left (b x + a\right )^{2} \operatorname {sech}\left (b x + a\right )^{3} \,d x } \] Input:
integrate(x^2*csch(b*x+a)^2*sech(b*x+a)^3,x, algorithm="giac")
Output:
integrate(x^2*csch(b*x + a)^2*sech(b*x + a)^3, x)
Timed out. \[ \int x^2 \text {csch}^2(a+b x) \text {sech}^3(a+b x) \, dx=\int \frac {x^2}{{\mathrm {cosh}\left (a+b\,x\right )}^3\,{\mathrm {sinh}\left (a+b\,x\right )}^2} \,d x \] Input:
int(x^2/(cosh(a + b*x)^3*sinh(a + b*x)^2),x)
Output:
int(x^2/(cosh(a + b*x)^3*sinh(a + b*x)^2), x)
\[ \int x^2 \text {csch}^2(a+b x) \text {sech}^3(a+b x) \, dx=\text {too large to display} \] Input:
int(x^2*csch(b*x+a)^2*sech(b*x+a)^3,x)
Output:
(8*( - 4*e**(6*a + 6*b*x)*atan(e**(a + b*x)) - 4*e**(4*a + 4*b*x)*atan(e** (a + b*x)) + 4*e**(2*a + 2*b*x)*atan(e**(a + b*x)) + 4*atan(e**(a + b*x)) + 40*e**(9*a + 6*b*x)*int((e**(3*b*x)*x**2)/(e**(10*a + 10*b*x) + e**(8*a + 8*b*x) - 2*e**(6*a + 6*b*x) - 2*e**(4*a + 4*b*x) + e**(2*a + 2*b*x) + 1) ,x)*b**3 + 16*e**(9*a + 6*b*x)*int((e**(3*b*x)*x)/(e**(10*a + 10*b*x) + e* *(8*a + 8*b*x) - 2*e**(6*a + 6*b*x) - 2*e**(4*a + 4*b*x) + e**(2*a + 2*b*x ) + 1),x)*b**2 - 20*e**(7*a + 6*b*x)*int((e**(b*x)*x**2)/(e**(10*a + 10*b* x) + e**(8*a + 8*b*x) - 2*e**(6*a + 6*b*x) - 2*e**(4*a + 4*b*x) + e**(2*a + 2*b*x) + 1),x)*b**3 - 48*e**(7*a + 6*b*x)*int((e**(b*x)*x)/(e**(10*a + 1 0*b*x) + e**(8*a + 8*b*x) - 2*e**(6*a + 6*b*x) - 2*e**(4*a + 4*b*x) + e**( 2*a + 2*b*x) + 1),x)*b**2 + e**(6*a + 6*b*x)*log(e**(a + b*x) - 1) - e**(6 *a + 6*b*x)*log(e**(a + b*x) + 1) - 2*e**(5*a + 5*b*x) + 40*e**(7*a + 4*b* x)*int((e**(3*b*x)*x**2)/(e**(10*a + 10*b*x) + e**(8*a + 8*b*x) - 2*e**(6* a + 6*b*x) - 2*e**(4*a + 4*b*x) + e**(2*a + 2*b*x) + 1),x)*b**3 + 16*e**(7 *a + 4*b*x)*int((e**(3*b*x)*x)/(e**(10*a + 10*b*x) + e**(8*a + 8*b*x) - 2* e**(6*a + 6*b*x) - 2*e**(4*a + 4*b*x) + e**(2*a + 2*b*x) + 1),x)*b**2 - 20 *e**(5*a + 4*b*x)*int((e**(b*x)*x**2)/(e**(10*a + 10*b*x) + e**(8*a + 8*b* x) - 2*e**(6*a + 6*b*x) - 2*e**(4*a + 4*b*x) + e**(2*a + 2*b*x) + 1),x)*b* *3 - 48*e**(5*a + 4*b*x)*int((e**(b*x)*x)/(e**(10*a + 10*b*x) + e**(8*a + 8*b*x) - 2*e**(6*a + 6*b*x) - 2*e**(4*a + 4*b*x) + e**(2*a + 2*b*x) + 1...