\(\int \frac {x^3}{a+b \text {csch}(c+d x^2)} \, dx\) [18]

Optimal result

Integrand size = 18, antiderivative size = 225 \[ \int \frac {x^3}{a+b \text {csch}\left (c+d x^2\right )} \, dx=\frac {x^4}{4 a}-\frac {b x^2 \log \left (1+\frac {a e^{c+d x^2}}{b-\sqrt {a^2+b^2}}\right )}{2 a \sqrt {a^2+b^2} d}+\frac {b x^2 \log \left (1+\frac {a e^{c+d x^2}}{b+\sqrt {a^2+b^2}}\right )}{2 a \sqrt {a^2+b^2} d}-\frac {b \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x^2}}{b-\sqrt {a^2+b^2}}\right )}{2 a \sqrt {a^2+b^2} d^2}+\frac {b \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x^2}}{b+\sqrt {a^2+b^2}}\right )}{2 a \sqrt {a^2+b^2} d^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.35 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {5545, 4276, 3403, 2296, 2221, 2317, 2438} \[ \int \frac {x^3}{a+b \text {csch}\left (c+d x^2\right )} \, dx=-\frac {b \operatorname {PolyLog}\left (2,-\frac {a e^{d x^2+c}}{b-\sqrt {a^2+b^2}}\right )}{2 a d^2 \sqrt {a^2+b^2}}+\frac {b \operatorname {PolyLog}\left (2,-\frac {a e^{d x^2+c}}{b+\sqrt {a^2+b^2}}\right )}{2 a d^2 \sqrt {a^2+b^2}}-\frac {b x^2 \log \left (\frac {a e^{c+d x^2}}{b-\sqrt {a^2+b^2}}+1\right )}{2 a d \sqrt {a^2+b^2}}+\frac {b x^2 \log \left (\frac {a e^{c+d x^2}}{\sqrt {a^2+b^2}+b}+1\right )}{2 a d \sqrt {a^2+b^2}}+\frac {x^4}{4 a} \]

[In]

Int[x^3/(a + b*Csch[c + d*x^2]),x]

[Out]

x^4/(4*a) - (b*x^2*Log[1 + (a*E^(c + d*x^2))/(b - Sqrt[a^2 + b^2])])/(2*a*Sqrt[a^2 + b^2]*d) + (b*x^2*Log[1 +
(a*E^(c + d*x^2))/(b + Sqrt[a^2 + b^2])])/(2*a*Sqrt[a^2 + b^2]*d) - (b*PolyLog[2, -((a*E^(c + d*x^2))/(b - Sqr
t[a^2 + b^2]))])/(2*a*Sqrt[a^2 + b^2]*d^2) + (b*PolyLog[2, -((a*E^(c + d*x^2))/(b + Sqrt[a^2 + b^2]))])/(2*a*S
qrt[a^2 + b^2]*d^2)

Rule 2221

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]
- 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 2296

Int[((F_)^(u_)*((f_.) + (g_.)*(x_))^(m_.))/((a_.) + (b_.)*(F_)^(u_) + (c_.)*(F_)^(v_)), x_Symbol] :> With[{q =
Rt[b^2 - 4*a*c, 2]}, Dist[2*(c/q), Int[(f + g*x)^m*(F^u/(b - q + 2*c*F^u)), x], x] - Dist[2*(c/q), Int[(f + g
*x)^m*(F^u/(b + q + 2*c*F^u)), x], x]] /; FreeQ[{F, a, b, c, f, g}, x] && EqQ[v, 2*u] && LinearQ[u, x] && NeQ[
b^2 - 4*a*c, 0] && IGtQ[m, 0]

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 3403

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]), x_Symbol] :> Dist[2,
Int[(c + d*x)^m*(E^((-I)*e + f*fz*x)/((-I)*b + 2*a*E^((-I)*e + f*fz*x) + I*b*E^(2*((-I)*e + f*fz*x)))), x], x]
/; FreeQ[{a, b, c, d, e, f, fz}, x] && NeQ[a^2 - b^2, 0] && IGtQ[m, 0]

Rule 4276

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[
(c + d*x)^m, 1/(Sin[e + f*x]^n/(b + a*Sin[e + f*x])^n), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && ILtQ[n, 0] &
& 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*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {x}{a+b \text {csch}(c+d x)} \, dx,x,x^2\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \left (\frac {x}{a}-\frac {b x}{a (b+a \sinh (c+d x))}\right ) \, dx,x,x^2\right ) \\ & = \frac {x^4}{4 a}-\frac {b \text {Subst}\left (\int \frac {x}{b+a \sinh (c+d x)} \, dx,x,x^2\right )}{2 a} \\ & = \frac {x^4}{4 a}-\frac {b \text {Subst}\left (\int \frac {e^{c+d x} x}{-a+2 b e^{c+d x}+a e^{2 (c+d x)}} \, dx,x,x^2\right )}{a} \\ & = \frac {x^4}{4 a}-\frac {b \text {Subst}\left (\int \frac {e^{c+d x} x}{2 b-2 \sqrt {a^2+b^2}+2 a e^{c+d x}} \, dx,x,x^2\right )}{\sqrt {a^2+b^2}}+\frac {b \text {Subst}\left (\int \frac {e^{c+d x} x}{2 b+2 \sqrt {a^2+b^2}+2 a e^{c+d x}} \, dx,x,x^2\right )}{\sqrt {a^2+b^2}} \\ & = \frac {x^4}{4 a}-\frac {b x^2 \log \left (1+\frac {a e^{c+d x^2}}{b-\sqrt {a^2+b^2}}\right )}{2 a \sqrt {a^2+b^2} d}+\frac {b x^2 \log \left (1+\frac {a e^{c+d x^2}}{b+\sqrt {a^2+b^2}}\right )}{2 a \sqrt {a^2+b^2} d}+\frac {b \text {Subst}\left (\int \log \left (1+\frac {2 a e^{c+d x}}{2 b-2 \sqrt {a^2+b^2}}\right ) \, dx,x,x^2\right )}{2 a \sqrt {a^2+b^2} d}-\frac {b \text {Subst}\left (\int \log \left (1+\frac {2 a e^{c+d x}}{2 b+2 \sqrt {a^2+b^2}}\right ) \, dx,x,x^2\right )}{2 a \sqrt {a^2+b^2} d} \\ & = \frac {x^4}{4 a}-\frac {b x^2 \log \left (1+\frac {a e^{c+d x^2}}{b-\sqrt {a^2+b^2}}\right )}{2 a \sqrt {a^2+b^2} d}+\frac {b x^2 \log \left (1+\frac {a e^{c+d x^2}}{b+\sqrt {a^2+b^2}}\right )}{2 a \sqrt {a^2+b^2} d}+\frac {b \text {Subst}\left (\int \frac {\log \left (1+\frac {2 a x}{2 b-2 \sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x^2}\right )}{2 a \sqrt {a^2+b^2} d^2}-\frac {b \text {Subst}\left (\int \frac {\log \left (1+\frac {2 a x}{2 b+2 \sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x^2}\right )}{2 a \sqrt {a^2+b^2} d^2} \\ & = \frac {x^4}{4 a}-\frac {b x^2 \log \left (1+\frac {a e^{c+d x^2}}{b-\sqrt {a^2+b^2}}\right )}{2 a \sqrt {a^2+b^2} d}+\frac {b x^2 \log \left (1+\frac {a e^{c+d x^2}}{b+\sqrt {a^2+b^2}}\right )}{2 a \sqrt {a^2+b^2} d}-\frac {b \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x^2}}{b-\sqrt {a^2+b^2}}\right )}{2 a \sqrt {a^2+b^2} d^2}+\frac {b \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x^2}}{b+\sqrt {a^2+b^2}}\right )}{2 a \sqrt {a^2+b^2} d^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.78 \[ \int \frac {x^3}{a+b \text {csch}\left (c+d x^2\right )} \, dx=\frac {d x^2 \left (\sqrt {a^2+b^2} d x^2-2 b \log \left (1+\frac {a e^{c+d x^2}}{b-\sqrt {a^2+b^2}}\right )+2 b \log \left (1+\frac {a e^{c+d x^2}}{b+\sqrt {a^2+b^2}}\right )\right )-2 b \operatorname {PolyLog}\left (2,\frac {a e^{c+d x^2}}{-b+\sqrt {a^2+b^2}}\right )+2 b \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x^2}}{b+\sqrt {a^2+b^2}}\right )}{4 a \sqrt {a^2+b^2} d^2} \]

[In]

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

[Out]

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

Maple [F]

\[\int \frac {x^{3}}{a +b \,\operatorname {csch}\left (d \,x^{2}+c \right )}d x\]

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 505 vs. \(2 (193) = 386\).

Time = 0.28 (sec) , antiderivative size = 505, normalized size of antiderivative = 2.24 \[ \int \frac {x^3}{a+b \text {csch}\left (c+d x^2\right )} \, dx=\frac {{\left (a^{2} + b^{2}\right )} d^{2} x^{4} - 2 \, a b c \sqrt {\frac {a^{2} + b^{2}}{a^{2}}} \log \left (2 \, a \cosh \left (d x^{2} + c\right ) + 2 \, a \sinh \left (d x^{2} + c\right ) + 2 \, a \sqrt {\frac {a^{2} + b^{2}}{a^{2}}} + 2 \, b\right ) + 2 \, a b c \sqrt {\frac {a^{2} + b^{2}}{a^{2}}} \log \left (2 \, a \cosh \left (d x^{2} + c\right ) + 2 \, a \sinh \left (d x^{2} + c\right ) - 2 \, a \sqrt {\frac {a^{2} + b^{2}}{a^{2}}} + 2 \, b\right ) - 2 \, a b \sqrt {\frac {a^{2} + b^{2}}{a^{2}}} {\rm Li}_2\left (\frac {b \cosh \left (d x^{2} + c\right ) + b \sinh \left (d x^{2} + c\right ) + {\left (a \cosh \left (d x^{2} + c\right ) + a \sinh \left (d x^{2} + c\right )\right )} \sqrt {\frac {a^{2} + b^{2}}{a^{2}}} - a}{a} + 1\right ) + 2 \, a b \sqrt {\frac {a^{2} + b^{2}}{a^{2}}} {\rm Li}_2\left (\frac {b \cosh \left (d x^{2} + c\right ) + b \sinh \left (d x^{2} + c\right ) - {\left (a \cosh \left (d x^{2} + c\right ) + a \sinh \left (d x^{2} + c\right )\right )} \sqrt {\frac {a^{2} + b^{2}}{a^{2}}} - a}{a} + 1\right ) - 2 \, {\left (a b d x^{2} + a b c\right )} \sqrt {\frac {a^{2} + b^{2}}{a^{2}}} \log \left (-\frac {b \cosh \left (d x^{2} + c\right ) + b \sinh \left (d x^{2} + c\right ) + {\left (a \cosh \left (d x^{2} + c\right ) + a \sinh \left (d x^{2} + c\right )\right )} \sqrt {\frac {a^{2} + b^{2}}{a^{2}}} - a}{a}\right ) + 2 \, {\left (a b d x^{2} + a b c\right )} \sqrt {\frac {a^{2} + b^{2}}{a^{2}}} \log \left (-\frac {b \cosh \left (d x^{2} + c\right ) + b \sinh \left (d x^{2} + c\right ) - {\left (a \cosh \left (d x^{2} + c\right ) + a \sinh \left (d x^{2} + c\right )\right )} \sqrt {\frac {a^{2} + b^{2}}{a^{2}}} - a}{a}\right )}{4 \, {\left (a^{3} + a b^{2}\right )} d^{2}} \]

[In]

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

[Out]

1/4*((a^2 + b^2)*d^2*x^4 - 2*a*b*c*sqrt((a^2 + b^2)/a^2)*log(2*a*cosh(d*x^2 + c) + 2*a*sinh(d*x^2 + c) + 2*a*s
qrt((a^2 + b^2)/a^2) + 2*b) + 2*a*b*c*sqrt((a^2 + b^2)/a^2)*log(2*a*cosh(d*x^2 + c) + 2*a*sinh(d*x^2 + c) - 2*
a*sqrt((a^2 + b^2)/a^2) + 2*b) - 2*a*b*sqrt((a^2 + b^2)/a^2)*dilog((b*cosh(d*x^2 + c) + b*sinh(d*x^2 + c) + (a
*cosh(d*x^2 + c) + a*sinh(d*x^2 + c))*sqrt((a^2 + b^2)/a^2) - a)/a + 1) + 2*a*b*sqrt((a^2 + b^2)/a^2)*dilog((b
*cosh(d*x^2 + c) + b*sinh(d*x^2 + c) - (a*cosh(d*x^2 + c) + a*sinh(d*x^2 + c))*sqrt((a^2 + b^2)/a^2) - a)/a +
1) - 2*(a*b*d*x^2 + a*b*c)*sqrt((a^2 + b^2)/a^2)*log(-(b*cosh(d*x^2 + c) + b*sinh(d*x^2 + c) + (a*cosh(d*x^2 +
c) + a*sinh(d*x^2 + c))*sqrt((a^2 + b^2)/a^2) - a)/a) + 2*(a*b*d*x^2 + a*b*c)*sqrt((a^2 + b^2)/a^2)*log(-(b*c
osh(d*x^2 + c) + b*sinh(d*x^2 + c) - (a*cosh(d*x^2 + c) + a*sinh(d*x^2 + c))*sqrt((a^2 + b^2)/a^2) - a)/a))/((
a^3 + a*b^2)*d^2)

Sympy [F]

\[ \int \frac {x^3}{a+b \text {csch}\left (c+d x^2\right )} \, dx=\int \frac {x^{3}}{a + b \operatorname {csch}{\left (c + d x^{2} \right )}}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {x^3}{a+b \text {csch}\left (c+d x^2\right )} \, dx=\int { \frac {x^{3}}{b \operatorname {csch}\left (d x^{2} + c\right ) + a} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {x^3}{a+b \text {csch}\left (c+d x^2\right )} \, dx=\int { \frac {x^{3}}{b \operatorname {csch}\left (d x^{2} + c\right ) + a} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {x^3}{a+b \text {csch}\left (c+d x^2\right )} \, dx=\int \frac {x^3}{a+\frac {b}{\mathrm {sinh}\left (d\,x^2+c\right )}} \,d x \]

[In]

int(x^3/(a + b/sinh(c + d*x^2)),x)

[Out]

int(x^3/(a + b/sinh(c + d*x^2)), x)