Integrand size = 20, antiderivative size = 85 \[ \int x^3 \text {csch}^2(a+b x) \text {sech}^2(a+b x) \, dx=-\frac {2 x^3}{b}-\frac {2 x^3 \coth (2 a+2 b x)}{b}+\frac {3 x^2 \log \left (1-e^{4 (a+b x)}\right )}{b^2}+\frac {3 x \operatorname {PolyLog}\left (2,e^{4 (a+b x)}\right )}{2 b^3}-\frac {3 \operatorname {PolyLog}\left (3,e^{4 (a+b x)}\right )}{8 b^4} \]
-2*x^3/b-2*x^3*coth(2*b*x+2*a)/b+3*x^2*ln(1-exp(4*b*x+4*a))/b^2+3/2*x*poly log(2,exp(4*b*x+4*a))/b^3-3/8*polylog(3,exp(4*b*x+4*a))/b^4
Leaf count is larger than twice the leaf count of optimal. \(307\) vs. \(2(85)=170\).
Time = 1.48 (sec) , antiderivative size = 307, normalized size of antiderivative = 3.61 \[ \int x^3 \text {csch}^2(a+b x) \text {sech}^2(a+b x) \, dx=4 \left (-\frac {e^{4 a} \left (8 b^3 e^{-4 a} x^3-6 b^2 \left (1-e^{-4 a}\right ) x^2 \log \left (1-e^{-a-b x}\right )-6 b^2 \left (1-e^{-4 a}\right ) x^2 \log \left (1+e^{-a-b x}\right )-6 b^2 \left (1-e^{-4 a}\right ) x^2 \log \left (1+e^{-2 (a+b x)}\right )+12 b \left (1-e^{-4 a}\right ) x \operatorname {PolyLog}\left (2,-e^{-a-b x}\right )+12 b \left (1-e^{-4 a}\right ) x \operatorname {PolyLog}\left (2,e^{-a-b x}\right )+6 b \left (1-e^{-4 a}\right ) x \operatorname {PolyLog}\left (2,-e^{-2 (a+b x)}\right )+12 \left (1-e^{-4 a}\right ) \operatorname {PolyLog}\left (3,-e^{-a-b x}\right )+12 \left (1-e^{-4 a}\right ) \operatorname {PolyLog}\left (3,e^{-a-b x}\right )+3 \left (1-e^{-4 a}\right ) \operatorname {PolyLog}\left (3,-e^{-2 (a+b x)}\right )\right )}{8 b^4 \left (-1+e^{4 a}\right )}+\frac {x^3 \text {csch}(2 a) \text {csch}(2 a+2 b x) \sinh (2 b x)}{2 b}\right ) \]
4*(-1/8*(E^(4*a)*((8*b^3*x^3)/E^(4*a) - 6*b^2*(1 - E^(-4*a))*x^2*Log[1 - E ^(-a - b*x)] - 6*b^2*(1 - E^(-4*a))*x^2*Log[1 + E^(-a - b*x)] - 6*b^2*(1 - E^(-4*a))*x^2*Log[1 + E^(-2*(a + b*x))] + 12*b*(1 - E^(-4*a))*x*PolyLog[2 , -E^(-a - b*x)] + 12*b*(1 - E^(-4*a))*x*PolyLog[2, E^(-a - b*x)] + 6*b*(1 - E^(-4*a))*x*PolyLog[2, -E^(-2*(a + b*x))] + 12*(1 - E^(-4*a))*PolyLog[3 , -E^(-a - b*x)] + 12*(1 - E^(-4*a))*PolyLog[3, E^(-a - b*x)] + 3*(1 - E^( -4*a))*PolyLog[3, -E^(-2*(a + b*x))]))/(b^4*(-1 + E^(4*a))) + (x^3*Csch[2* a]*Csch[2*a + 2*b*x]*Sinh[2*b*x])/(2*b))
Result contains complex when optimal does not.
Time = 0.72 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.59, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {5984, 3042, 25, 4672, 26, 3042, 26, 4201, 2620, 3011, 2720, 7143}
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.
| \(\displaystyle \int x^3 \text {csch}^2(a+b x) \text {sech}^2(a+b x) \, dx\) |
| \(\Big \downarrow \) 5984 |
| \(\displaystyle 4 \int x^3 \text {csch}^2(2 a+2 b x)dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle 4 \int -x^3 \csc (2 i a+2 i b x)^2dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -4 \int x^3 \csc (2 i a+2 i b x)^2dx\) |
| \(\Big \downarrow \) 4672 |
| \(\displaystyle -4 \left (\frac {x^3 \coth (2 a+2 b x)}{2 b}-\frac {3 i \int -i x^2 \coth (2 a+2 b x)dx}{2 b}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -4 \left (\frac {x^3 \coth (2 a+2 b x)}{2 b}-\frac {3 \int x^2 \coth (2 a+2 b x)dx}{2 b}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -4 \left (\frac {x^3 \coth (2 a+2 b x)}{2 b}-\frac {3 \int -i x^2 \tan \left (2 i a+2 i b x+\frac {\pi }{2}\right )dx}{2 b}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -4 \left (\frac {x^3 \coth (2 a+2 b x)}{2 b}+\frac {3 i \int x^2 \tan \left (\frac {1}{2} (4 i a+\pi )+2 i b x\right )dx}{2 b}\right )\) |
| \(\Big \downarrow \) 4201 |
| \(\displaystyle -4 \left (\frac {x^3 \coth (2 a+2 b x)}{2 b}+\frac {3 i \left (2 i \int \frac {e^{4 a+4 b x-i \pi } x^2}{1+e^{4 a+4 b x-i \pi }}dx-\frac {i x^3}{3}\right )}{2 b}\right )\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle -4 \left (\frac {x^3 \coth (2 a+2 b x)}{2 b}+\frac {3 i \left (2 i \left (\frac {x^2 \log \left (1+e^{4 a+4 b x-i \pi }\right )}{4 b}-\frac {\int x \log \left (1+e^{4 a+4 b x-i \pi }\right )dx}{2 b}\right )-\frac {i x^3}{3}\right )}{2 b}\right )\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle -4 \left (\frac {x^3 \coth (2 a+2 b x)}{2 b}+\frac {3 i \left (2 i \left (\frac {x^2 \log \left (1+e^{4 a+4 b x-i \pi }\right )}{4 b}-\frac {\frac {\int \operatorname {PolyLog}\left (2,-e^{4 a+4 b x-i \pi }\right )dx}{4 b}-\frac {x \operatorname {PolyLog}\left (2,-e^{4 a+4 b x-i \pi }\right )}{4 b}}{2 b}\right )-\frac {i x^3}{3}\right )}{2 b}\right )\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle -4 \left (\frac {x^3 \coth (2 a+2 b x)}{2 b}+\frac {3 i \left (2 i \left (\frac {x^2 \log \left (1+e^{4 a+4 b x-i \pi }\right )}{4 b}-\frac {\frac {\int e^{-4 a-4 b x+i \pi } \operatorname {PolyLog}\left (2,-e^{4 a+4 b x-i \pi }\right )de^{4 a+4 b x-i \pi }}{16 b^2}-\frac {x \operatorname {PolyLog}\left (2,-e^{4 a+4 b x-i \pi }\right )}{4 b}}{2 b}\right )-\frac {i x^3}{3}\right )}{2 b}\right )\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle -4 \left (\frac {x^3 \coth (2 a+2 b x)}{2 b}+\frac {3 i \left (2 i \left (\frac {x^2 \log \left (1+e^{4 a+4 b x-i \pi }\right )}{4 b}-\frac {\frac {\operatorname {PolyLog}\left (3,-e^{4 a+4 b x-i \pi }\right )}{16 b^2}-\frac {x \operatorname {PolyLog}\left (2,-e^{4 a+4 b x-i \pi }\right )}{4 b}}{2 b}\right )-\frac {i x^3}{3}\right )}{2 b}\right )\) |
-4*((x^3*Coth[2*a + 2*b*x])/(2*b) + (((3*I)/2)*((-1/3*I)*x^3 + (2*I)*((x^2 *Log[1 + E^(4*a - I*Pi + 4*b*x)])/(4*b) - (-1/4*(x*PolyLog[2, -E^(4*a - I* Pi + 4*b*x)])/b + PolyLog[3, -E^(4*a - I*Pi + 4*b*x)]/(16*b^2))/(2*b))))/b )
3.5.95.3.1 Defintions of rubi rules used
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[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct ionOfExponentialQ[u, x] && !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ [{a, m, n}, x] && IntegerQ[m*n]] && !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) *(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.) *(x_))^(m_.), x_Symbol] :> Simp[(-(f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Simp[g*(m/(b*c*n*Log[F])) Int[(f + g*x)^( m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e , f, g, n}, x] && GtQ[m, 0]
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]
Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d , e, n, p}, x] && EqQ[b*d, a*e]
Leaf count of result is larger than twice the leaf count of optimal. \(262\) vs. \(2(81)=162\).
Time = 6.98 (sec) , antiderivative size = 263, normalized size of antiderivative = 3.09
| method | result | size |
| risch | \(-\frac {4 x^{3}}{b \left (1+{\mathrm e}^{2 b x +2 a}\right ) \left ({\mathrm e}^{2 b x +2 a}-1\right )}+\frac {8 a^{3}}{b^{4}}+\frac {12 x \,a^{2}}{b^{3}}-\frac {12 a^{2} \ln \left ({\mathrm e}^{b x +a}\right )}{b^{4}}+\frac {3 a^{2} \ln \left ({\mathrm e}^{b x +a}-1\right )}{b^{4}}-\frac {3 \ln \left (1-{\mathrm e}^{b x +a}\right ) a^{2}}{b^{4}}-\frac {4 x^{3}}{b}-\frac {6 \operatorname {polylog}\left (3, -{\mathrm e}^{b x +a}\right )}{b^{4}}-\frac {6 \operatorname {polylog}\left (3, {\mathrm e}^{b x +a}\right )}{b^{4}}-\frac {3 \operatorname {polylog}\left (3, -{\mathrm e}^{2 b x +2 a}\right )}{2 b^{4}}+\frac {3 x^{2} \ln \left (1+{\mathrm e}^{2 b x +2 a}\right )}{b^{2}}+\frac {3 x \operatorname {polylog}\left (2, -{\mathrm e}^{2 b x +2 a}\right )}{b^{3}}+\frac {3 \ln \left ({\mathrm e}^{b x +a}+1\right ) x^{2}}{b^{2}}+\frac {6 x \operatorname {polylog}\left (2, -{\mathrm e}^{b x +a}\right )}{b^{3}}+\frac {3 \ln \left (1-{\mathrm e}^{b x +a}\right ) x^{2}}{b^{2}}+\frac {6 x \operatorname {polylog}\left (2, {\mathrm e}^{b x +a}\right )}{b^{3}}\) | \(263\) |
-4*x^3/b/(1+exp(2*b*x+2*a))/(exp(2*b*x+2*a)-1)+8/b^4*a^3+12*x/b^3*a^2-12/b ^4*a^2*ln(exp(b*x+a))+3/b^4*a^2*ln(exp(b*x+a)-1)-3/b^4*ln(1-exp(b*x+a))*a^ 2-4*x^3/b-6*polylog(3,-exp(b*x+a))/b^4-6*polylog(3,exp(b*x+a))/b^4-3/2*pol ylog(3,-exp(2*b*x+2*a))/b^4+3*x^2*ln(1+exp(2*b*x+2*a))/b^2+3*x*polylog(2,- exp(2*b*x+2*a))/b^3+3/b^2*ln(exp(b*x+a)+1)*x^2+6*x*polylog(2,-exp(b*x+a))/ b^3+3/b^2*ln(1-exp(b*x+a))*x^2+6*x*polylog(2,exp(b*x+a))/b^3
Result contains complex when optimal does not.
Time = 0.29 (sec) , antiderivative size = 1924, normalized size of antiderivative = 22.64 \[ \int x^3 \text {csch}^2(a+b x) \text {sech}^2(a+b x) \, dx=\text {Too large to display} \]
-(4*(b^3*x^3 + a^3)*cosh(b*x + a)^4 + 16*(b^3*x^3 + a^3)*cosh(b*x + a)^3*s inh(b*x + a) + 24*(b^3*x^3 + a^3)*cosh(b*x + a)^2*sinh(b*x + a)^2 + 16*(b^ 3*x^3 + a^3)*cosh(b*x + a)*sinh(b*x + a)^3 + 4*(b^3*x^3 + a^3)*sinh(b*x + a)^4 - 4*a^3 - 6*(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)*dilog(cosh(b*x + a) + sinh(b*x + a)) - 6*(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*sin h(b*x + a)^4 - b*x)*dilog(I*cosh(b*x + a) + I*sinh(b*x + a)) - 6*(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)*dilog(-I*cosh(b*x + a) - I*sinh(b*x + a)) - 6*(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)*d ilog(-cosh(b*x + a) - sinh(b*x + a)) - 3*(b^2*x^2*cosh(b*x + a)^4 + 4*b^2* x^2*cosh(b*x + a)^3*sinh(b*x + a) + 6*b^2*x^2*cosh(b*x + a)^2*sinh(b*x + a )^2 + 4*b^2*x^2*cosh(b*x + a)*sinh(b*x + a)^3 + b^2*x^2*sinh(b*x + a)^4 - b^2*x^2)*log(cosh(b*x + a) + sinh(b*x + a) + 1) - 3*(a^2*cosh(b*x + a)^4 + 4*a^2*cosh(b*x + a)^3*sinh(b*x + a) + 6*a^2*cosh(b*x + a)^2*sinh(b*x + a) ^2 + 4*a^2*cosh(b*x + a)*sinh(b*x + a)^3 + a^2*sinh(b*x + a)^4 - a^2)*l...
\[ \int x^3 \text {csch}^2(a+b x) \text {sech}^2(a+b x) \, dx=\int x^{3} \operatorname {csch}^{2}{\left (a + b x \right )} \operatorname {sech}^{2}{\left (a + b x \right )}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 180 vs. \(2 (80) = 160\).
Time = 0.21 (sec) , antiderivative size = 180, normalized size of antiderivative = 2.12 \[ \int x^3 \text {csch}^2(a+b x) \text {sech}^2(a+b x) \, dx=-\frac {4 \, x^{3}}{b e^{\left (4 \, b x + 4 \, a\right )} - b} - \frac {4 \, x^{3}}{b} + \frac {3 \, {\left (2 \, b^{2} x^{2} \log \left (e^{\left (2 \, b x + 2 \, a\right )} + 1\right ) + 2 \, b x {\rm Li}_2\left (-e^{\left (2 \, b x + 2 \, a\right )}\right ) - {\rm Li}_{3}(-e^{\left (2 \, b x + 2 \, a\right )})\right )}}{2 \, b^{4}} + \frac {3 \, {\left (b^{2} x^{2} \log \left (e^{\left (b x + a\right )} + 1\right ) + 2 \, b x {\rm Li}_2\left (-e^{\left (b x + a\right )}\right ) - 2 \, {\rm Li}_{3}(-e^{\left (b x + a\right )})\right )}}{b^{4}} + \frac {3 \, {\left (b^{2} x^{2} \log \left (-e^{\left (b x + a\right )} + 1\right ) + 2 \, b x {\rm Li}_2\left (e^{\left (b x + a\right )}\right ) - 2 \, {\rm Li}_{3}(e^{\left (b x + a\right )})\right )}}{b^{4}} \]
-4*x^3/(b*e^(4*b*x + 4*a) - b) - 4*x^3/b + 3/2*(2*b^2*x^2*log(e^(2*b*x + 2 *a) + 1) + 2*b*x*dilog(-e^(2*b*x + 2*a)) - polylog(3, -e^(2*b*x + 2*a)))/b ^4 + 3*(b^2*x^2*log(e^(b*x + a) + 1) + 2*b*x*dilog(-e^(b*x + a)) - 2*polyl og(3, -e^(b*x + a)))/b^4 + 3*(b^2*x^2*log(-e^(b*x + a) + 1) + 2*b*x*dilog( e^(b*x + a)) - 2*polylog(3, e^(b*x + a)))/b^4
\[ \int x^3 \text {csch}^2(a+b x) \text {sech}^2(a+b x) \, dx=\int { x^{3} \operatorname {csch}\left (b x + a\right )^{2} \operatorname {sech}\left (b x + a\right )^{2} \,d x } \]
Timed out. \[ \int x^3 \text {csch}^2(a+b x) \text {sech}^2(a+b x) \, dx=\int \frac {x^3}{{\mathrm {cosh}\left (a+b\,x\right )}^2\,{\mathrm {sinh}\left (a+b\,x\right )}^2} \,d x \]