3.1.3 \(\int x^3 (a+b \text {csch}(c+d x^2)) \, dx\) [3]

Optimal. Leaf size=68 \[ \frac {a x^4}{4}-\frac {b x^2 \tanh ^{-1}\left (e^{c+d x^2}\right )}{d}-\frac {b \text {PolyLog}\left (2,-e^{c+d x^2}\right )}{2 d^2}+\frac {b \text {PolyLog}\left (2,e^{c+d x^2}\right )}{2 d^2} \]

[Out]

1/4*a*x^4-b*x^2*arctanh(exp(d*x^2+c))/d-1/2*b*polylog(2,-exp(d*x^2+c))/d^2+1/2*b*polylog(2,exp(d*x^2+c))/d^2

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Rubi [A]
time = 0.06, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {14, 5545, 4267, 2317, 2438} \begin {gather*} \frac {a x^4}{4}-\frac {b \text {Li}_2\left (-e^{d x^2+c}\right )}{2 d^2}+\frac {b \text {Li}_2\left (e^{d x^2+c}\right )}{2 d^2}-\frac {b x^2 \tanh ^{-1}\left (e^{c+d x^2}\right )}{d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^3*(a + b*Csch[c + d*x^2]),x]

[Out]

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

Rule 14

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]]

Rule 2317

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 2438

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 4267

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 5545

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]

Rubi steps

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

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Mathematica [A]
time = 0.10, size = 108, normalized size = 1.59 \begin {gather*} \frac {1}{4} \left (a x^4+\frac {2 b \left (\left (c+d x^2\right ) \left (\log \left (1-e^{-c-d x^2}\right )-\log \left (1+e^{-c-d x^2}\right )\right )-c \log \left (\tanh \left (\frac {1}{2} \left (c+d x^2\right )\right )\right )+\text {PolyLog}\left (2,-e^{-c-d x^2}\right )-\text {PolyLog}\left (2,e^{-c-d x^2}\right )\right )}{d^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^3*(a + b*Csch[c + d*x^2]),x]

[Out]

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

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Maple [F]
time = 0.76, size = 0, normalized size = 0.00 \[\int x^{3} \left (a +b \,\mathrm {csch}\left (d \,x^{2}+c \right )\right )\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(a+b*csch(d*x^2+c)),x)

[Out]

int(x^3*(a+b*csch(d*x^2+c)),x)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(a+b*csch(d*x^2+c)),x, algorithm="maxima")

[Out]

1/4*a*x^4 + 2*b*integrate(x^3/(e^(d*x^2 + c) - e^(-d*x^2 - c)), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 144 vs. \(2 (57) = 114\).
time = 0.40, size = 144, normalized size = 2.12 \begin {gather*} \frac {a d^{2} x^{4} - 2 \, b d x^{2} \log \left (\cosh \left (d x^{2} + c\right ) + \sinh \left (d x^{2} + c\right ) + 1\right ) - 2 \, b c \log \left (\cosh \left (d x^{2} + c\right ) + \sinh \left (d x^{2} + c\right ) - 1\right ) + 2 \, b {\rm Li}_2\left (\cosh \left (d x^{2} + c\right ) + \sinh \left (d x^{2} + c\right )\right ) - 2 \, b {\rm Li}_2\left (-\cosh \left (d x^{2} + c\right ) - \sinh \left (d x^{2} + c\right )\right ) + 2 \, {\left (b d x^{2} + b c\right )} \log \left (-\cosh \left (d x^{2} + c\right ) - \sinh \left (d x^{2} + c\right ) + 1\right )}{4 \, d^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(a+b*csch(d*x^2+c)),x, algorithm="fricas")

[Out]

1/4*(a*d^2*x^4 - 2*b*d*x^2*log(cosh(d*x^2 + c) + sinh(d*x^2 + c) + 1) - 2*b*c*log(cosh(d*x^2 + c) + sinh(d*x^2
+ c) - 1) + 2*b*dilog(cosh(d*x^2 + c) + sinh(d*x^2 + c)) - 2*b*dilog(-cosh(d*x^2 + c) - sinh(d*x^2 + c)) + 2*
(b*d*x^2 + b*c)*log(-cosh(d*x^2 + c) - sinh(d*x^2 + c) + 1))/d^2

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{3} \left (a + b \operatorname {csch}{\left (c + d x^{2} \right )}\right )\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(a+b*csch(d*x**2+c)),x)

[Out]

Integral(x**3*(a + b*csch(c + d*x**2)), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(a+b*csch(d*x^2+c)),x, algorithm="giac")

[Out]

integrate((b*csch(d*x^2 + c) + a)*x^3, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int x^3\,\left (a+\frac {b}{\mathrm {sinh}\left (d\,x^2+c\right )}\right ) \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(a + b/sinh(c + d*x^2)),x)

[Out]

int(x^3*(a + b/sinh(c + d*x^2)), x)

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