3.8 \(\int x^5 (a+b \text {csch}(c+d x^2))^2 \, dx\)

Optimal. Leaf size=196 \[ \frac {a^2 x^6}{6}+\frac {2 a b \text {Li}_3\left (-e^{d x^2+c}\right )}{d^3}-\frac {2 a b \text {Li}_3\left (e^{d x^2+c}\right )}{d^3}-\frac {2 a b x^2 \text {Li}_2\left (-e^{d x^2+c}\right )}{d^2}+\frac {2 a b x^2 \text {Li}_2\left (e^{d x^2+c}\right )}{d^2}-\frac {2 a b x^4 \tanh ^{-1}\left (e^{c+d x^2}\right )}{d}+\frac {b^2 \text {Li}_2\left (e^{2 \left (d x^2+c\right )}\right )}{2 d^3}+\frac {b^2 x^2 \log \left (1-e^{2 \left (c+d x^2\right )}\right )}{d^2}-\frac {b^2 x^4 \coth \left (c+d x^2\right )}{2 d}-\frac {b^2 x^4}{2 d} \]

[Out]

-1/2*b^2*x^4/d+1/6*a^2*x^6-2*a*b*x^4*arctanh(exp(d*x^2+c))/d-1/2*b^2*x^4*coth(d*x^2+c)/d+b^2*x^2*ln(1-exp(2*d*
x^2+2*c))/d^2-2*a*b*x^2*polylog(2,-exp(d*x^2+c))/d^2+2*a*b*x^2*polylog(2,exp(d*x^2+c))/d^2+1/2*b^2*polylog(2,e
xp(2*d*x^2+2*c))/d^3+2*a*b*polylog(3,-exp(d*x^2+c))/d^3-2*a*b*polylog(3,exp(d*x^2+c))/d^3

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Rubi [A]  time = 0.38, antiderivative size = 196, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 11, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.611, Rules used = {5437, 4190, 4182, 2531, 2282, 6589, 4184, 3716, 2190, 2279, 2391} \[ -\frac {2 a b x^2 \text {PolyLog}\left (2,-e^{c+d x^2}\right )}{d^2}+\frac {2 a b x^2 \text {PolyLog}\left (2,e^{c+d x^2}\right )}{d^2}+\frac {2 a b \text {PolyLog}\left (3,-e^{c+d x^2}\right )}{d^3}-\frac {2 a b \text {PolyLog}\left (3,e^{c+d x^2}\right )}{d^3}+\frac {b^2 \text {PolyLog}\left (2,e^{2 \left (c+d x^2\right )}\right )}{2 d^3}+\frac {a^2 x^6}{6}-\frac {2 a b x^4 \tanh ^{-1}\left (e^{c+d x^2}\right )}{d}+\frac {b^2 x^2 \log \left (1-e^{2 \left (c+d x^2\right )}\right )}{d^2}-\frac {b^2 x^4 \coth \left (c+d x^2\right )}{2 d}-\frac {b^2 x^4}{2 d} \]

Antiderivative was successfully verified.

[In]

Int[x^5*(a + b*Csch[c + d*x^2])^2,x]

[Out]

-(b^2*x^4)/(2*d) + (a^2*x^6)/6 - (2*a*b*x^4*ArcTanh[E^(c + d*x^2)])/d - (b^2*x^4*Coth[c + d*x^2])/(2*d) + (b^2
*x^2*Log[1 - E^(2*(c + d*x^2))])/d^2 - (2*a*b*x^2*PolyLog[2, -E^(c + d*x^2)])/d^2 + (2*a*b*x^2*PolyLog[2, E^(c
+ d*x^2)])/d^2 + (b^2*PolyLog[2, E^(2*(c + d*x^2))])/(2*d^3) + (2*a*b*PolyLog[3, -E^(c + d*x^2)])/d^3 - (2*a*
b*PolyLog[3, E^(c + d*x^2)])/d^3

Rule 2190

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*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), x]
- Dist[(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]

Rule 2279

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[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]

Rule 2282

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 2391

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]

Rule 2531

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] + Dist[(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]

Rule 3716

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)], x_Symbol] :> -Simp[(I*(c
+ d*x)^(m + 1))/(d*(m + 1)), x] + Dist[2*I, Int[((c + d*x)^m*E^(2*(-(I*e) + f*fz*x)))/(E^(2*I*k*Pi)*(1 + E^(2*
(-(I*e) + f*fz*x))/E^(2*I*k*Pi))), x], x] /; FreeQ[{c, d, e, f, fz}, x] && IntegerQ[4*k] && IGtQ[m, 0]

Rule 4182

Int[csc[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(-2*(c + d*x)^m*Ar
cTanh[E^(-(I*e) + f*fz*x)])/(f*fz*I), x] + (-Dist[(d*m)/(f*fz*I), Int[(c + d*x)^(m - 1)*Log[1 - E^(-(I*e) + f*
fz*x)], x], x] + Dist[(d*m)/(f*fz*I), Int[(c + d*x)^(m - 1)*Log[1 + E^(-(I*e) + f*fz*x)], x], x]) /; FreeQ[{c,
d, e, f, fz}, x] && IGtQ[m, 0]

Rule 4184

Int[csc[(e_.) + (f_.)*(x_)]^2*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> -Simp[((c + d*x)^m*Cot[e + f*x])/f, x]
+ Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Cot[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 4190

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[
(c + d*x)^m, (a + b*Csc[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[m, 0] && IGtQ[n, 0]

Rule 5437

Int[((a_.) + Csch[(c_.) + (d_.)*(x_)^(n_)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simpli
fy[(m + 1)/n] - 1)*(a + b*Csch[c + d*x])^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && IGtQ[Simplif
y[(m + 1)/n], 0] && IntegerQ[p]

Rule 6589

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> 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]

Rubi steps

\begin {align*} \int x^5 \left (a+b \text {csch}\left (c+d x^2\right )\right )^2 \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int x^2 (a+b \text {csch}(c+d x))^2 \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (a^2 x^2+2 a b x^2 \text {csch}(c+d x)+b^2 x^2 \text {csch}^2(c+d x)\right ) \, dx,x,x^2\right )\\ &=\frac {a^2 x^6}{6}+(a b) \operatorname {Subst}\left (\int x^2 \text {csch}(c+d x) \, dx,x,x^2\right )+\frac {1}{2} b^2 \operatorname {Subst}\left (\int x^2 \text {csch}^2(c+d x) \, dx,x,x^2\right )\\ &=\frac {a^2 x^6}{6}-\frac {2 a b x^4 \tanh ^{-1}\left (e^{c+d x^2}\right )}{d}-\frac {b^2 x^4 \coth \left (c+d x^2\right )}{2 d}-\frac {(2 a b) \operatorname {Subst}\left (\int x \log \left (1-e^{c+d x}\right ) \, dx,x,x^2\right )}{d}+\frac {(2 a b) \operatorname {Subst}\left (\int x \log \left (1+e^{c+d x}\right ) \, dx,x,x^2\right )}{d}+\frac {b^2 \operatorname {Subst}\left (\int x \coth (c+d x) \, dx,x,x^2\right )}{d}\\ &=-\frac {b^2 x^4}{2 d}+\frac {a^2 x^6}{6}-\frac {2 a b x^4 \tanh ^{-1}\left (e^{c+d x^2}\right )}{d}-\frac {b^2 x^4 \coth \left (c+d x^2\right )}{2 d}-\frac {2 a b x^2 \text {Li}_2\left (-e^{c+d x^2}\right )}{d^2}+\frac {2 a b x^2 \text {Li}_2\left (e^{c+d x^2}\right )}{d^2}+\frac {(2 a b) \operatorname {Subst}\left (\int \text {Li}_2\left (-e^{c+d x}\right ) \, dx,x,x^2\right )}{d^2}-\frac {(2 a b) \operatorname {Subst}\left (\int \text {Li}_2\left (e^{c+d x}\right ) \, dx,x,x^2\right )}{d^2}-\frac {\left (2 b^2\right ) \operatorname {Subst}\left (\int \frac {e^{2 (c+d x)} x}{1-e^{2 (c+d x)}} \, dx,x,x^2\right )}{d}\\ &=-\frac {b^2 x^4}{2 d}+\frac {a^2 x^6}{6}-\frac {2 a b x^4 \tanh ^{-1}\left (e^{c+d x^2}\right )}{d}-\frac {b^2 x^4 \coth \left (c+d x^2\right )}{2 d}+\frac {b^2 x^2 \log \left (1-e^{2 \left (c+d x^2\right )}\right )}{d^2}-\frac {2 a b x^2 \text {Li}_2\left (-e^{c+d x^2}\right )}{d^2}+\frac {2 a b x^2 \text {Li}_2\left (e^{c+d x^2}\right )}{d^2}+\frac {(2 a b) \operatorname {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{c+d x^2}\right )}{d^3}-\frac {(2 a b) \operatorname {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{c+d x^2}\right )}{d^3}-\frac {b^2 \operatorname {Subst}\left (\int \log \left (1-e^{2 (c+d x)}\right ) \, dx,x,x^2\right )}{d^2}\\ &=-\frac {b^2 x^4}{2 d}+\frac {a^2 x^6}{6}-\frac {2 a b x^4 \tanh ^{-1}\left (e^{c+d x^2}\right )}{d}-\frac {b^2 x^4 \coth \left (c+d x^2\right )}{2 d}+\frac {b^2 x^2 \log \left (1-e^{2 \left (c+d x^2\right )}\right )}{d^2}-\frac {2 a b x^2 \text {Li}_2\left (-e^{c+d x^2}\right )}{d^2}+\frac {2 a b x^2 \text {Li}_2\left (e^{c+d x^2}\right )}{d^2}+\frac {2 a b \text {Li}_3\left (-e^{c+d x^2}\right )}{d^3}-\frac {2 a b \text {Li}_3\left (e^{c+d x^2}\right )}{d^3}-\frac {b^2 \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 \left (c+d x^2\right )}\right )}{2 d^3}\\ &=-\frac {b^2 x^4}{2 d}+\frac {a^2 x^6}{6}-\frac {2 a b x^4 \tanh ^{-1}\left (e^{c+d x^2}\right )}{d}-\frac {b^2 x^4 \coth \left (c+d x^2\right )}{2 d}+\frac {b^2 x^2 \log \left (1-e^{2 \left (c+d x^2\right )}\right )}{d^2}-\frac {2 a b x^2 \text {Li}_2\left (-e^{c+d x^2}\right )}{d^2}+\frac {2 a b x^2 \text {Li}_2\left (e^{c+d x^2}\right )}{d^2}+\frac {b^2 \text {Li}_2\left (e^{2 \left (c+d x^2\right )}\right )}{2 d^3}+\frac {2 a b \text {Li}_3\left (-e^{c+d x^2}\right )}{d^3}-\frac {2 a b \text {Li}_3\left (e^{c+d x^2}\right )}{d^3}\\ \end {align*}

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Mathematica [B]  time = 6.27, size = 595, normalized size = 3.04 \[ -\frac {-2 a^2 e^{2 c} d^3 x^6+2 a^2 d^3 x^6-12 a b e^{2 c} d^2 x^4 \log \left (1-e^{-c-d x^2}\right )+12 a b d^2 x^4 \log \left (1-e^{-c-d x^2}\right )+12 a b e^{2 c} d^2 x^4 \log \left (e^{-c-d x^2}+1\right )-12 a b d^2 x^4 \log \left (e^{-c-d x^2}+1\right )+12 b \left (e^{2 c}-1\right ) \left (b-2 a d x^2\right ) \text {Li}_2\left (-e^{-d x^2-c}\right )+12 b \left (e^{2 c}-1\right ) \left (2 a d x^2+b\right ) \text {Li}_2\left (e^{-d x^2-c}\right )-24 a b e^{2 c} \text {Li}_3\left (-e^{-d x^2-c}\right )+24 a b \text {Li}_3\left (-e^{-d x^2-c}\right )+24 a b e^{2 c} \text {Li}_3\left (e^{-d x^2-c}\right )-24 a b \text {Li}_3\left (e^{-d x^2-c}\right )-3 b^2 e^{2 c} d^2 x^4 \text {csch}\left (\frac {c}{2}\right ) \sinh \left (\frac {d x^2}{2}\right ) \text {csch}\left (\frac {1}{2} \left (c+d x^2\right )\right )+3 b^2 d^2 x^4 \text {csch}\left (\frac {c}{2}\right ) \sinh \left (\frac {d x^2}{2}\right ) \text {csch}\left (\frac {1}{2} \left (c+d x^2\right )\right )+3 b^2 e^{2 c} d^2 x^4 \text {sech}\left (\frac {c}{2}\right ) \sinh \left (\frac {d x^2}{2}\right ) \text {sech}\left (\frac {1}{2} \left (c+d x^2\right )\right )-3 b^2 d^2 x^4 \text {sech}\left (\frac {c}{2}\right ) \sinh \left (\frac {d x^2}{2}\right ) \text {sech}\left (\frac {1}{2} \left (c+d x^2\right )\right )-12 b^2 e^{2 c} d x^2 \log \left (1-e^{-c-d x^2}\right )+12 b^2 d x^2 \log \left (1-e^{-c-d x^2}\right )-12 b^2 e^{2 c} d x^2 \log \left (e^{-c-d x^2}+1\right )+12 b^2 d x^2 \log \left (e^{-c-d x^2}+1\right )+12 b^2 d^2 x^4}{12 \left (e^{2 c}-1\right ) d^3} \]

Antiderivative was successfully verified.

[In]

Integrate[x^5*(a + b*Csch[c + d*x^2])^2,x]

[Out]

-1/12*(12*b^2*d^2*x^4 + 2*a^2*d^3*x^6 - 2*a^2*d^3*E^(2*c)*x^6 + 12*b^2*d*x^2*Log[1 - E^(-c - d*x^2)] - 12*b^2*
d*E^(2*c)*x^2*Log[1 - E^(-c - d*x^2)] + 12*a*b*d^2*x^4*Log[1 - E^(-c - d*x^2)] - 12*a*b*d^2*E^(2*c)*x^4*Log[1
- E^(-c - d*x^2)] + 12*b^2*d*x^2*Log[1 + E^(-c - d*x^2)] - 12*b^2*d*E^(2*c)*x^2*Log[1 + E^(-c - d*x^2)] - 12*a
*b*d^2*x^4*Log[1 + E^(-c - d*x^2)] + 12*a*b*d^2*E^(2*c)*x^4*Log[1 + E^(-c - d*x^2)] + 12*b*(-1 + E^(2*c))*(b -
2*a*d*x^2)*PolyLog[2, -E^(-c - d*x^2)] + 12*b*(-1 + E^(2*c))*(b + 2*a*d*x^2)*PolyLog[2, E^(-c - d*x^2)] + 24*
a*b*PolyLog[3, -E^(-c - d*x^2)] - 24*a*b*E^(2*c)*PolyLog[3, -E^(-c - d*x^2)] - 24*a*b*PolyLog[3, E^(-c - d*x^2
)] + 24*a*b*E^(2*c)*PolyLog[3, E^(-c - d*x^2)] + 3*b^2*d^2*x^4*Csch[c/2]*Csch[(c + d*x^2)/2]*Sinh[(d*x^2)/2] -
3*b^2*d^2*E^(2*c)*x^4*Csch[c/2]*Csch[(c + d*x^2)/2]*Sinh[(d*x^2)/2] - 3*b^2*d^2*x^4*Sech[c/2]*Sech[(c + d*x^2
)/2]*Sinh[(d*x^2)/2] + 3*b^2*d^2*E^(2*c)*x^4*Sech[c/2]*Sech[(c + d*x^2)/2]*Sinh[(d*x^2)/2])/(d^3*(-1 + E^(2*c)
))

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fricas [C]  time = 0.50, size = 1031, normalized size = 5.26 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(a+b*csch(d*x^2+c))^2,x, algorithm="fricas")

[Out]

-1/6*(a^2*d^3*x^6 + 6*b^2*c^2 - (a^2*d^3*x^6 - 6*b^2*d^2*x^4 + 6*b^2*c^2)*cosh(d*x^2 + c)^2 - 2*(a^2*d^3*x^6 -
6*b^2*d^2*x^4 + 6*b^2*c^2)*cosh(d*x^2 + c)*sinh(d*x^2 + c) - (a^2*d^3*x^6 - 6*b^2*d^2*x^4 + 6*b^2*c^2)*sinh(d
*x^2 + c)^2 + 6*(2*a*b*d*x^2 - (2*a*b*d*x^2 + b^2)*cosh(d*x^2 + c)^2 - 2*(2*a*b*d*x^2 + b^2)*cosh(d*x^2 + c)*s
inh(d*x^2 + c) - (2*a*b*d*x^2 + b^2)*sinh(d*x^2 + c)^2 + b^2)*dilog(cosh(d*x^2 + c) + sinh(d*x^2 + c)) - 6*(2*
a*b*d*x^2 - (2*a*b*d*x^2 - b^2)*cosh(d*x^2 + c)^2 - 2*(2*a*b*d*x^2 - b^2)*cosh(d*x^2 + c)*sinh(d*x^2 + c) - (2
*a*b*d*x^2 - b^2)*sinh(d*x^2 + c)^2 - b^2)*dilog(-cosh(d*x^2 + c) - sinh(d*x^2 + c)) - 6*(a*b*d^2*x^4 - b^2*d*
x^2 - (a*b*d^2*x^4 - b^2*d*x^2)*cosh(d*x^2 + c)^2 - 2*(a*b*d^2*x^4 - b^2*d*x^2)*cosh(d*x^2 + c)*sinh(d*x^2 + c
) - (a*b*d^2*x^4 - b^2*d*x^2)*sinh(d*x^2 + c)^2)*log(cosh(d*x^2 + c) + sinh(d*x^2 + c) + 1) + 6*(a*b*c^2 - b^2
*c - (a*b*c^2 - b^2*c)*cosh(d*x^2 + c)^2 - 2*(a*b*c^2 - b^2*c)*cosh(d*x^2 + c)*sinh(d*x^2 + c) - (a*b*c^2 - b^
2*c)*sinh(d*x^2 + c)^2)*log(cosh(d*x^2 + c) + sinh(d*x^2 + c) - 1) + 6*(a*b*d^2*x^4 + b^2*d*x^2 - a*b*c^2 + b^
2*c - (a*b*d^2*x^4 + b^2*d*x^2 - a*b*c^2 + b^2*c)*cosh(d*x^2 + c)^2 - 2*(a*b*d^2*x^4 + b^2*d*x^2 - a*b*c^2 + b
^2*c)*cosh(d*x^2 + c)*sinh(d*x^2 + c) - (a*b*d^2*x^4 + b^2*d*x^2 - a*b*c^2 + b^2*c)*sinh(d*x^2 + c)^2)*log(-co
sh(d*x^2 + c) - sinh(d*x^2 + c) + 1) + 12*(a*b*cosh(d*x^2 + c)^2 + 2*a*b*cosh(d*x^2 + c)*sinh(d*x^2 + c) + a*b
*sinh(d*x^2 + c)^2 - a*b)*polylog(3, cosh(d*x^2 + c) + sinh(d*x^2 + c)) - 12*(a*b*cosh(d*x^2 + c)^2 + 2*a*b*co
sh(d*x^2 + c)*sinh(d*x^2 + c) + a*b*sinh(d*x^2 + c)^2 - a*b)*polylog(3, -cosh(d*x^2 + c) - sinh(d*x^2 + c)))/(
d^3*cosh(d*x^2 + c)^2 + 2*d^3*cosh(d*x^2 + c)*sinh(d*x^2 + c) + d^3*sinh(d*x^2 + c)^2 - d^3)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \operatorname {csch}\left (d x^{2} + c\right ) + a\right )}^{2} x^{5}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(a+b*csch(d*x^2+c))^2,x, algorithm="giac")

[Out]

integrate((b*csch(d*x^2 + c) + a)^2*x^5, x)

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maple [F]  time = 0.43, size = 0, normalized size = 0.00 \[ \int x^{5} \left (a +b \,\mathrm {csch}\left (d \,x^{2}+c \right )\right )^{2}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(a+b*csch(d*x^2+c))^2,x)

[Out]

int(x^5*(a+b*csch(d*x^2+c))^2,x)

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maxima [A]  time = 0.79, size = 271, normalized size = 1.38 \[ \frac {1}{6} \, a^{2} x^{6} - \frac {b^{2} x^{4}}{d e^{\left (2 \, d x^{2} + 2 \, c\right )} - d} - \frac {{\left (d^{2} x^{4} \log \left (e^{\left (d x^{2} + c\right )} + 1\right ) + 2 \, d x^{2} {\rm Li}_2\left (-e^{\left (d x^{2} + c\right )}\right ) - 2 \, {\rm Li}_{3}(-e^{\left (d x^{2} + c\right )})\right )} a b}{d^{3}} + \frac {{\left (d^{2} x^{4} \log \left (-e^{\left (d x^{2} + c\right )} + 1\right ) + 2 \, d x^{2} {\rm Li}_2\left (e^{\left (d x^{2} + c\right )}\right ) - 2 \, {\rm Li}_{3}(e^{\left (d x^{2} + c\right )})\right )} a b}{d^{3}} + \frac {{\left (d x^{2} \log \left (e^{\left (d x^{2} + c\right )} + 1\right ) + {\rm Li}_2\left (-e^{\left (d x^{2} + c\right )}\right )\right )} b^{2}}{d^{3}} + \frac {{\left (d x^{2} \log \left (-e^{\left (d x^{2} + c\right )} + 1\right ) + {\rm Li}_2\left (e^{\left (d x^{2} + c\right )}\right )\right )} b^{2}}{d^{3}} - \frac {2 \, a b d^{3} x^{6} + 3 \, b^{2} d^{2} x^{4}}{6 \, d^{3}} + \frac {2 \, a b d^{3} x^{6} - 3 \, b^{2} d^{2} x^{4}}{6 \, d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(a+b*csch(d*x^2+c))^2,x, algorithm="maxima")

[Out]

1/6*a^2*x^6 - b^2*x^4/(d*e^(2*d*x^2 + 2*c) - d) - (d^2*x^4*log(e^(d*x^2 + c) + 1) + 2*d*x^2*dilog(-e^(d*x^2 +
c)) - 2*polylog(3, -e^(d*x^2 + c)))*a*b/d^3 + (d^2*x^4*log(-e^(d*x^2 + c) + 1) + 2*d*x^2*dilog(e^(d*x^2 + c))
- 2*polylog(3, e^(d*x^2 + c)))*a*b/d^3 + (d*x^2*log(e^(d*x^2 + c) + 1) + dilog(-e^(d*x^2 + c)))*b^2/d^3 + (d*x
^2*log(-e^(d*x^2 + c) + 1) + dilog(e^(d*x^2 + c)))*b^2/d^3 - 1/6*(2*a*b*d^3*x^6 + 3*b^2*d^2*x^4)/d^3 + 1/6*(2*
a*b*d^3*x^6 - 3*b^2*d^2*x^4)/d^3

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int x^5\,{\left (a+\frac {b}{\mathrm {sinh}\left (d\,x^2+c\right )}\right )}^2 \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(a + b/sinh(c + d*x^2))^2,x)

[Out]

int(x^5*(a + b/sinh(c + d*x^2))^2, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{5} \left (a + b \operatorname {csch}{\left (c + d x^{2} \right )}\right )^{2}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5*(a+b*csch(d*x**2+c))**2,x)

[Out]

Integral(x**5*(a + b*csch(c + d*x**2))**2, x)

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