Integrand size = 20, antiderivative size = 64 \[ \int x^2 \text {csch}^2(a+b x) \text {sech}^2(a+b x) \, dx=-\frac {2 x^2}{b}-\frac {2 x^2 \coth (2 a+2 b x)}{b}+\frac {2 x \log \left (1-e^{4 (a+b x)}\right )}{b^2}+\frac {\operatorname {PolyLog}\left (2,e^{4 (a+b x)}\right )}{2 b^3} \] Output:
-2*x^2/b-2*x^2*coth(2*b*x+2*a)/b+2*x*ln(1-exp(4*b*x+4*a))/b^2+1/2*polylog( 2,exp(4*b*x+4*a))/b^3
Leaf count is larger than twice the leaf count of optimal. \(216\) vs. \(2(64)=128\).
Time = 0.90 (sec) , antiderivative size = 216, normalized size of antiderivative = 3.38 \[ \int x^2 \text {csch}^2(a+b x) \text {sech}^2(a+b x) \, dx=4 \left (-\frac {e^{4 a} \left (4 b^2 e^{-4 a} x^2-2 b \left (1-e^{-4 a}\right ) x \log \left (1-e^{-a-b x}\right )-2 b \left (1-e^{-4 a}\right ) x \log \left (1+e^{-a-b x}\right )-2 b \left (1-e^{-4 a}\right ) x \log \left (1+e^{-2 (a+b x)}\right )+2 \left (1-e^{-4 a}\right ) \operatorname {PolyLog}\left (2,-e^{-a-b x}\right )+2 \left (1-e^{-4 a}\right ) \operatorname {PolyLog}\left (2,e^{-a-b x}\right )+\left (1-e^{-4 a}\right ) \operatorname {PolyLog}\left (2,-e^{-2 (a+b x)}\right )\right )}{4 b^3 \left (-1+e^{4 a}\right )}+\frac {x^2 \text {csch}(2 a) \text {csch}(2 a+2 b x) \sinh (2 b x)}{2 b}\right ) \] Input:
Integrate[x^2*Csch[a + b*x]^2*Sech[a + b*x]^2,x]
Output:
4*(-1/4*(E^(4*a)*((4*b^2*x^2)/E^(4*a) - 2*b*(1 - E^(-4*a))*x*Log[1 - E^(-a - b*x)] - 2*b*(1 - E^(-4*a))*x*Log[1 + E^(-a - b*x)] - 2*b*(1 - E^(-4*a)) *x*Log[1 + E^(-2*(a + b*x))] + 2*(1 - E^(-4*a))*PolyLog[2, -E^(-a - b*x)] + 2*(1 - E^(-4*a))*PolyLog[2, E^(-a - b*x)] + (1 - E^(-4*a))*PolyLog[2, -E ^(-2*(a + b*x))]))/(b^3*(-1 + E^(4*a))) + (x^2*Csch[2*a]*Csch[2*a + 2*b*x] *Sinh[2*b*x])/(2*b))
Result contains complex when optimal does not.
Time = 0.54 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.50, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.550, Rules used = {5984, 3042, 25, 4672, 26, 3042, 26, 4201, 2620, 2715, 2838}
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.
| \(\displaystyle \int x^2 \text {csch}^2(a+b x) \text {sech}^2(a+b x) \, dx\) |
| \(\Big \downarrow \) 5984 |
| \(\displaystyle 4 \int x^2 \text {csch}^2(2 a+2 b x)dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle 4 \int -x^2 \csc (2 i a+2 i b x)^2dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -4 \int x^2 \csc (2 i a+2 i b x)^2dx\) |
| \(\Big \downarrow \) 4672 |
| \(\displaystyle -4 \left (\frac {x^2 \coth (2 a+2 b x)}{2 b}-\frac {i \int -i x \coth (2 a+2 b x)dx}{b}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -4 \left (\frac {x^2 \coth (2 a+2 b x)}{2 b}-\frac {\int x \coth (2 a+2 b x)dx}{b}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -4 \left (\frac {x^2 \coth (2 a+2 b x)}{2 b}-\frac {\int -i x \tan \left (2 i a+2 i b x+\frac {\pi }{2}\right )dx}{b}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -4 \left (\frac {x^2 \coth (2 a+2 b x)}{2 b}+\frac {i \int x \tan \left (\frac {1}{2} (4 i a+\pi )+2 i b x\right )dx}{b}\right )\) |
| \(\Big \downarrow \) 4201 |
| \(\displaystyle -4 \left (\frac {x^2 \coth (2 a+2 b x)}{2 b}+\frac {i \left (2 i \int \frac {e^{4 a+4 b x-i \pi } x}{1+e^{4 a+4 b x-i \pi }}dx-\frac {i x^2}{2}\right )}{b}\right )\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle -4 \left (\frac {x^2 \coth (2 a+2 b x)}{2 b}+\frac {i \left (2 i \left (\frac {x \log \left (1+e^{4 a+4 b x-i \pi }\right )}{4 b}-\frac {\int \log \left (1+e^{4 a+4 b x-i \pi }\right )dx}{4 b}\right )-\frac {i x^2}{2}\right )}{b}\right )\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle -4 \left (\frac {x^2 \coth (2 a+2 b x)}{2 b}+\frac {i \left (2 i \left (\frac {x \log \left (1+e^{4 a+4 b x-i \pi }\right )}{4 b}-\frac {\int e^{-4 a-4 b x+i \pi } \log \left (1+e^{4 a+4 b x-i \pi }\right )de^{4 a+4 b x-i \pi }}{16 b^2}\right )-\frac {i x^2}{2}\right )}{b}\right )\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle -4 \left (\frac {x^2 \coth (2 a+2 b x)}{2 b}+\frac {i \left (2 i \left (\frac {\operatorname {PolyLog}\left (2,-e^{4 a+4 b x-i \pi }\right )}{16 b^2}+\frac {x \log \left (1+e^{4 a+4 b x-i \pi }\right )}{4 b}\right )-\frac {i x^2}{2}\right )}{b}\right )\) |
Input:
Int[x^2*Csch[a + b*x]^2*Sech[a + b*x]^2,x]
Output:
-4*((x^2*Coth[2*a + 2*b*x])/(2*b) + (I*((-1/2*I)*x^2 + (2*I)*((x*Log[1 + E ^(4*a - I*Pi + 4*b*x)])/(4*b) + PolyLog[2, -E^(4*a - I*Pi + 4*b*x)]/(16*b^ 2))))/b)
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
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]
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]
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]
Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (Complex[0, fz_])*(f_.)*(x_)], x _Symbol] :> Simp[(-I)*((c + d*x)^(m + 1)/(d*(m + 1))), x] + Simp[2*I Int[ (c + d*x)^m*(E^(2*((-I)*e + f*fz*x))/(1 + E^(2*((-I)*e + f*fz*x)))), x], x] /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[m, 0]
Int[csc[(e_.) + (f_.)*(x_)]^2*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp [(-(c + d*x)^m)*(Cot[e + f*x]/f), x] + Simp[d*(m/f) Int[(c + d*x)^(m - 1) *Cot[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]
Int[Csch[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.)*Sech[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Simp[2^n Int[(c + d*x)^m*Csch[2*a + 2*b*x ]^n, x], x] /; FreeQ[{a, b, c, d}, x] && RationalQ[m] && IntegerQ[n]
Leaf count of result is larger than twice the leaf count of optimal. \(198\) vs. \(2(62)=124\).
Time = 5.30 (sec) , antiderivative size = 199, normalized size of antiderivative = 3.11
| method | result | size |
| risch | \(-\frac {4 x^{2}}{b \left ({\mathrm e}^{2 b x +2 a}-1\right ) \left ({\mathrm e}^{2 b x +2 a}+1\right )}-\frac {4 x^{2}}{b}-\frac {8 a x}{b^{2}}-\frac {4 a^{2}}{b^{3}}+\frac {2 \ln \left (1-{\mathrm e}^{b x +a}\right ) x}{b^{2}}+\frac {2 \ln \left (1-{\mathrm e}^{b x +a}\right ) a}{b^{3}}+\frac {2 \operatorname {polylog}\left (2, {\mathrm e}^{b x +a}\right )}{b^{3}}+\frac {2 \ln \left ({\mathrm e}^{b x +a}+1\right ) x}{b^{2}}+\frac {2 \operatorname {polylog}\left (2, -{\mathrm e}^{b x +a}\right )}{b^{3}}+\frac {2 x \ln \left ({\mathrm e}^{2 b x +2 a}+1\right )}{b^{2}}+\frac {\operatorname {polylog}\left (2, -{\mathrm e}^{2 b x +2 a}\right )}{b^{3}}-\frac {2 a \ln \left ({\mathrm e}^{b x +a}-1\right )}{b^{3}}+\frac {8 a \ln \left ({\mathrm e}^{b x +a}\right )}{b^{3}}\) | \(199\) |
Input:
int(x^2*csch(b*x+a)^2*sech(b*x+a)^2,x,method=_RETURNVERBOSE)
Output:
-4*x^2/b/(exp(2*b*x+2*a)-1)/(exp(2*b*x+2*a)+1)-4/b*x^2-8*a*x/b^2-4/b^3*a^2 +2/b^2*ln(1-exp(b*x+a))*x+2/b^3*ln(1-exp(b*x+a))*a+2*polylog(2,exp(b*x+a)) /b^3+2/b^2*ln(exp(b*x+a)+1)*x+2*polylog(2,-exp(b*x+a))/b^3+2*x*ln(exp(2*b* x+2*a)+1)/b^2+polylog(2,-exp(2*b*x+2*a))/b^3-2/b^3*a*ln(exp(b*x+a)-1)+8/b^ 3*a*ln(exp(b*x+a))
Result contains complex when optimal does not.
Time = 0.10 (sec) , antiderivative size = 1327, normalized size of antiderivative = 20.73 \[ \int x^2 \text {csch}^2(a+b x) \text {sech}^2(a+b x) \, dx=\text {Too large to display} \] Input:
integrate(x^2*csch(b*x+a)^2*sech(b*x+a)^2,x, algorithm="fricas")
Output:
-2*(2*(b^2*x^2 - a^2)*cosh(b*x + a)^4 + 8*(b^2*x^2 - a^2)*cosh(b*x + a)^3* sinh(b*x + a) + 12*(b^2*x^2 - a^2)*cosh(b*x + a)^2*sinh(b*x + a)^2 + 8*(b^ 2*x^2 - a^2)*cosh(b*x + a)*sinh(b*x + a)^3 + 2*(b^2*x^2 - a^2)*sinh(b*x + a)^4 + 2*a^2 - (cosh(b*x + a)^4 + 4*cosh(b*x + a)^3*sinh(b*x + a) + 6*cosh (b*x + a)^2*sinh(b*x + a)^2 + 4*cosh(b*x + a)*sinh(b*x + a)^3 + sinh(b*x + a)^4 - 1)*dilog(cosh(b*x + a) + sinh(b*x + a)) - (cosh(b*x + a)^4 + 4*cos h(b*x + a)^3*sinh(b*x + a) + 6*cosh(b*x + a)^2*sinh(b*x + a)^2 + 4*cosh(b* x + a)*sinh(b*x + a)^3 + sinh(b*x + a)^4 - 1)*dilog(I*cosh(b*x + a) + I*si nh(b*x + a)) - (cosh(b*x + a)^4 + 4*cosh(b*x + a)^3*sinh(b*x + a) + 6*cosh (b*x + a)^2*sinh(b*x + a)^2 + 4*cosh(b*x + a)*sinh(b*x + a)^3 + sinh(b*x + a)^4 - 1)*dilog(-I*cosh(b*x + a) - I*sinh(b*x + a)) - (cosh(b*x + a)^4 + 4*cosh(b*x + a)^3*sinh(b*x + a) + 6*cosh(b*x + a)^2*sinh(b*x + a)^2 + 4*co sh(b*x + a)*sinh(b*x + a)^3 + sinh(b*x + a)^4 - 1)*dilog(-cosh(b*x + a) - sinh(b*x + a)) - (b*x*cosh(b*x + a)^4 + 4*b*x*cosh(b*x + a)^3*sinh(b*x + a ) + 6*b*x*cosh(b*x + a)^2*sinh(b*x + a)^2 + 4*b*x*cosh(b*x + a)*sinh(b*x + a)^3 + b*x*sinh(b*x + a)^4 - b*x)*log(cosh(b*x + a) + sinh(b*x + a) + 1) + (a*cosh(b*x + a)^4 + 4*a*cosh(b*x + a)^3*sinh(b*x + a) + 6*a*cosh(b*x + a)^2*sinh(b*x + a)^2 + 4*a*cosh(b*x + a)*sinh(b*x + a)^3 + a*sinh(b*x + a) ^4 - a)*log(cosh(b*x + a) + sinh(b*x + a) + I) + (a*cosh(b*x + a)^4 + 4*a* cosh(b*x + a)^3*sinh(b*x + a) + 6*a*cosh(b*x + a)^2*sinh(b*x + a)^2 + 4...
\[ \int x^2 \text {csch}^2(a+b x) \text {sech}^2(a+b x) \, dx=\int x^{2} \operatorname {csch}^{2}{\left (a + b x \right )} \operatorname {sech}^{2}{\left (a + b x \right )}\, dx \] Input:
integrate(x**2*csch(b*x+a)**2*sech(b*x+a)**2,x)
Output:
Integral(x**2*csch(a + b*x)**2*sech(a + b*x)**2, x)
Time = 0.06 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.84 \[ \int x^2 \text {csch}^2(a+b x) \text {sech}^2(a+b x) \, dx=-\frac {4 \, x^{2}}{b e^{\left (4 \, b x + 4 \, a\right )} - b} - \frac {4 \, x^{2}}{b} + \frac {2 \, b x \log \left (e^{\left (2 \, b x + 2 \, a\right )} + 1\right ) + {\rm Li}_2\left (-e^{\left (2 \, b x + 2 \, a\right )}\right )}{b^{3}} + \frac {2 \, {\left (b x \log \left (e^{\left (b x + a\right )} + 1\right ) + {\rm Li}_2\left (-e^{\left (b x + a\right )}\right )\right )}}{b^{3}} + \frac {2 \, {\left (b x \log \left (-e^{\left (b x + a\right )} + 1\right ) + {\rm Li}_2\left (e^{\left (b x + a\right )}\right )\right )}}{b^{3}} \] Input:
integrate(x^2*csch(b*x+a)^2*sech(b*x+a)^2,x, algorithm="maxima")
Output:
-4*x^2/(b*e^(4*b*x + 4*a) - b) - 4*x^2/b + (2*b*x*log(e^(2*b*x + 2*a) + 1) + dilog(-e^(2*b*x + 2*a)))/b^3 + 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
\[ \int x^2 \text {csch}^2(a+b x) \text {sech}^2(a+b x) \, dx=\int { x^{2} \operatorname {csch}\left (b x + a\right )^{2} \operatorname {sech}\left (b x + a\right )^{2} \,d x } \] Input:
integrate(x^2*csch(b*x+a)^2*sech(b*x+a)^2,x, algorithm="giac")
Output:
integrate(x^2*csch(b*x + a)^2*sech(b*x + a)^2, x)
Timed out. \[ \int x^2 \text {csch}^2(a+b x) \text {sech}^2(a+b x) \, dx=\int \frac {x^2}{{\mathrm {cosh}\left (a+b\,x\right )}^2\,{\mathrm {sinh}\left (a+b\,x\right )}^2} \,d x \] Input:
int(x^2/(cosh(a + b*x)^2*sinh(a + b*x)^2),x)
Output:
int(x^2/(cosh(a + b*x)^2*sinh(a + b*x)^2), x)
\[ \int x^2 \text {csch}^2(a+b x) \text {sech}^2(a+b x) \, dx=\frac {-16 e^{4 b x +4 a} \left (\int \frac {x}{e^{8 b x +8 a}-2 e^{4 b x +4 a}+1}d x \right ) b^{2}+e^{4 b x +4 a} \mathrm {log}\left (e^{2 b x +2 a}+1\right )+e^{4 b x +4 a} \mathrm {log}\left (e^{b x +a}-1\right )+e^{4 b x +4 a} \mathrm {log}\left (e^{b x +a}+1\right )-4 e^{4 b x +4 a} b x +16 \left (\int \frac {x}{e^{8 b x +8 a}-2 e^{4 b x +4 a}+1}d x \right ) b^{2}-\mathrm {log}\left (e^{2 b x +2 a}+1\right )-\mathrm {log}\left (e^{b x +a}-1\right )-\mathrm {log}\left (e^{b x +a}+1\right )-8 b^{2} x^{2}}{2 b^{3} \left (e^{4 b x +4 a}-1\right )} \] Input:
int(x^2*csch(b*x+a)^2*sech(b*x+a)^2,x)
Output:
( - 16*e**(4*a + 4*b*x)*int(x/(e**(8*a + 8*b*x) - 2*e**(4*a + 4*b*x) + 1), x)*b**2 + e**(4*a + 4*b*x)*log(e**(2*a + 2*b*x) + 1) + e**(4*a + 4*b*x)*lo g(e**(a + b*x) - 1) + e**(4*a + 4*b*x)*log(e**(a + b*x) + 1) - 4*e**(4*a + 4*b*x)*b*x + 16*int(x/(e**(8*a + 8*b*x) - 2*e**(4*a + 4*b*x) + 1),x)*b**2 - log(e**(2*a + 2*b*x) + 1) - log(e**(a + b*x) - 1) - log(e**(a + b*x) + 1) - 8*b**2*x**2)/(2*b**3*(e**(4*a + 4*b*x) - 1))