3.1.10 \(\int x^3 (a+b \text {csch}(c+d x^2))^2 \, dx\) [10]

Optimal. Leaf size=108 \[ \frac {a^2 x^4}{4}-\frac {2 a b x^2 \tanh ^{-1}\left (e^{c+d x^2}\right )}{d}-\frac {b^2 x^2 \coth \left (c+d x^2\right )}{2 d}+\frac {b^2 \log \left (\sinh \left (c+d x^2\right )\right )}{2 d^2}-\frac {a b \text {PolyLog}\left (2,-e^{c+d x^2}\right )}{d^2}+\frac {a b \text {PolyLog}\left (2,e^{c+d x^2}\right )}{d^2} \]

[Out]

1/4*a^2*x^4-2*a*b*x^2*arctanh(exp(d*x^2+c))/d-1/2*b^2*x^2*coth(d*x^2+c)/d+1/2*b^2*ln(sinh(d*x^2+c))/d^2-a*b*po
lylog(2,-exp(d*x^2+c))/d^2+a*b*polylog(2,exp(d*x^2+c))/d^2

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Rubi [A]
time = 0.12, antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {5545, 4275, 4267, 2317, 2438, 4269, 3556} \begin {gather*} \frac {a^2 x^4}{4}-\frac {a b \text {Li}_2\left (-e^{d x^2+c}\right )}{d^2}+\frac {a b \text {Li}_2\left (e^{d x^2+c}\right )}{d^2}-\frac {2 a b x^2 \tanh ^{-1}\left (e^{c+d x^2}\right )}{d}+\frac {b^2 \log \left (\sinh \left (c+d x^2\right )\right )}{2 d^2}-\frac {b^2 x^2 \coth \left (c+d x^2\right )}{2 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^3*(a + b*Csch[c + d*x^2])^2,x]

[Out]

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

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 3556

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{c, d}, x]

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 4269

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 4275

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

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(260\) vs. \(2(108)=216\).
time = 2.90, size = 260, normalized size = 2.41 \begin {gather*} \frac {4 b^2 d x^2 \coth (c)+2 d x^2 \left (a^2 d x^2-2 b^2 \coth (c)\right )-4 b^2 \left (d x^2 \coth (c)-\log \left (\sinh \left (c+d x^2\right )\right )\right )+8 a b \left (2 \tanh ^{-1}(\tanh (c)) \tanh ^{-1}\left (\cosh (c)+\sinh (c) \tanh \left (\frac {d x^2}{2}\right )\right )+\frac {\left (\left (d x^2+\tanh ^{-1}(\tanh (c))\right ) \left (\log \left (1-e^{-d x^2-\tanh ^{-1}(\tanh (c))}\right )-\log \left (1+e^{-d x^2-\tanh ^{-1}(\tanh (c))}\right )\right )+\text {PolyLog}\left (2,-e^{-d x^2-\tanh ^{-1}(\tanh (c))}\right )-\text {PolyLog}\left (2,e^{-d x^2-\tanh ^{-1}(\tanh (c))}\right )\right ) \text {sech}(c)}{\sqrt {\text {sech}^2(c)}}\right )+2 b^2 d x^2 \text {csch}\left (\frac {c}{2}\right ) \text {csch}\left (\frac {1}{2} \left (c+d x^2\right )\right ) \sinh \left (\frac {d x^2}{2}\right )-2 b^2 d x^2 \text {sech}\left (\frac {c}{2}\right ) \text {sech}\left (\frac {1}{2} \left (c+d x^2\right )\right ) \sinh \left (\frac {d x^2}{2}\right )}{8 d^2} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(4*b^2*d*x^2*Coth[c] + 2*d*x^2*(a^2*d*x^2 - 2*b^2*Coth[c]) - 4*b^2*(d*x^2*Coth[c] - Log[Sinh[c + d*x^2]]) + 8*
a*b*(2*ArcTanh[Tanh[c]]*ArcTanh[Cosh[c] + Sinh[c]*Tanh[(d*x^2)/2]] + (((d*x^2 + ArcTanh[Tanh[c]])*(Log[1 - E^(
-(d*x^2) - ArcTanh[Tanh[c]])] - Log[1 + E^(-(d*x^2) - ArcTanh[Tanh[c]])]) + PolyLog[2, -E^(-(d*x^2) - ArcTanh[
Tanh[c]])] - PolyLog[2, E^(-(d*x^2) - ArcTanh[Tanh[c]])])*Sech[c])/Sqrt[Sech[c]^2]) + 2*b^2*d*x^2*Csch[c/2]*Cs
ch[(c + d*x^2)/2]*Sinh[(d*x^2)/2] - 2*b^2*d*x^2*Sech[c/2]*Sech[(c + d*x^2)/2]*Sinh[(d*x^2)/2])/(8*d^2)

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Maple [F]
time = 1.38, size = 0, normalized size = 0.00 \[\int x^{3} \left (a +b \,\mathrm {csch}\left (d \,x^{2}+c \right )\right )^{2}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 683 vs. \(2 (97) = 194\).
time = 0.52, size = 683, normalized size = 6.32 \begin {gather*} -\frac {a^{2} d^{2} x^{4} - 4 \, b^{2} c - {\left (a^{2} d^{2} x^{4} - 4 \, b^{2} d x^{2} - 4 \, b^{2} c\right )} \cosh \left (d x^{2} + c\right )^{2} - 2 \, {\left (a^{2} d^{2} x^{4} - 4 \, b^{2} d x^{2} - 4 \, b^{2} c\right )} \cosh \left (d x^{2} + c\right ) \sinh \left (d x^{2} + c\right ) - {\left (a^{2} d^{2} x^{4} - 4 \, b^{2} d x^{2} - 4 \, b^{2} c\right )} \sinh \left (d x^{2} + c\right )^{2} - 4 \, {\left (a b \cosh \left (d x^{2} + c\right )^{2} + 2 \, a b \cosh \left (d x^{2} + c\right ) \sinh \left (d x^{2} + c\right ) + a b \sinh \left (d x^{2} + c\right )^{2} - a b\right )} {\rm Li}_2\left (\cosh \left (d x^{2} + c\right ) + \sinh \left (d x^{2} + c\right )\right ) + 4 \, {\left (a b \cosh \left (d x^{2} + c\right )^{2} + 2 \, a b \cosh \left (d x^{2} + c\right ) \sinh \left (d x^{2} + c\right ) + a b \sinh \left (d x^{2} + c\right )^{2} - a b\right )} {\rm Li}_2\left (-\cosh \left (d x^{2} + c\right ) - \sinh \left (d x^{2} + c\right )\right ) - 2 \, {\left (2 \, a b d x^{2} - {\left (2 \, a b d x^{2} - b^{2}\right )} \cosh \left (d x^{2} + c\right )^{2} - 2 \, {\left (2 \, a b d x^{2} - b^{2}\right )} \cosh \left (d x^{2} + c\right ) \sinh \left (d x^{2} + c\right ) - {\left (2 \, a b d x^{2} - b^{2}\right )} \sinh \left (d x^{2} + c\right )^{2} - b^{2}\right )} \log \left (\cosh \left (d x^{2} + c\right ) + \sinh \left (d x^{2} + c\right ) + 1\right ) - 2 \, {\left (2 \, a b c - {\left (2 \, a b c - b^{2}\right )} \cosh \left (d x^{2} + c\right )^{2} - 2 \, {\left (2 \, a b c - b^{2}\right )} \cosh \left (d x^{2} + c\right ) \sinh \left (d x^{2} + c\right ) - {\left (2 \, a b c - b^{2}\right )} \sinh \left (d x^{2} + c\right )^{2} - b^{2}\right )} \log \left (\cosh \left (d x^{2} + c\right ) + \sinh \left (d x^{2} + c\right ) - 1\right ) + 4 \, {\left (a b d x^{2} + a b c - {\left (a b d x^{2} + a b c\right )} \cosh \left (d x^{2} + c\right )^{2} - 2 \, {\left (a b d x^{2} + a b c\right )} \cosh \left (d x^{2} + c\right ) \sinh \left (d x^{2} + c\right ) - {\left (a b d x^{2} + a b c\right )} \sinh \left (d x^{2} + c\right )^{2}\right )} \log \left (-\cosh \left (d x^{2} + c\right ) - \sinh \left (d x^{2} + c\right ) + 1\right )}{4 \, {\left (d^{2} \cosh \left (d x^{2} + c\right )^{2} + 2 \, d^{2} \cosh \left (d x^{2} + c\right ) \sinh \left (d x^{2} + c\right ) + d^{2} \sinh \left (d x^{2} + c\right )^{2} - d^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(a^2*d^2*x^4 - 4*b^2*c - (a^2*d^2*x^4 - 4*b^2*d*x^2 - 4*b^2*c)*cosh(d*x^2 + c)^2 - 2*(a^2*d^2*x^4 - 4*b^2
*d*x^2 - 4*b^2*c)*cosh(d*x^2 + c)*sinh(d*x^2 + c) - (a^2*d^2*x^4 - 4*b^2*d*x^2 - 4*b^2*c)*sinh(d*x^2 + c)^2 -
4*(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)*dilog(cosh(d*x
^2 + c) + sinh(d*x^2 + c)) + 4*(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)*dilog(-cosh(d*x^2 + c) - sinh(d*x^2 + c)) - 2*(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)*log
(cosh(d*x^2 + c) + sinh(d*x^2 + c) + 1) - 2*(2*a*b*c - (2*a*b*c - b^2)*cosh(d*x^2 + c)^2 - 2*(2*a*b*c - b^2)*c
osh(d*x^2 + c)*sinh(d*x^2 + c) - (2*a*b*c - b^2)*sinh(d*x^2 + c)^2 - b^2)*log(cosh(d*x^2 + c) + sinh(d*x^2 + c
) - 1) + 4*(a*b*d*x^2 + a*b*c - (a*b*d*x^2 + a*b*c)*cosh(d*x^2 + c)^2 - 2*(a*b*d*x^2 + a*b*c)*cosh(d*x^2 + c)*
sinh(d*x^2 + c) - (a*b*d*x^2 + a*b*c)*sinh(d*x^2 + c)^2)*log(-cosh(d*x^2 + c) - sinh(d*x^2 + c) + 1))/(d^2*cos
h(d*x^2 + c)^2 + 2*d^2*cosh(d*x^2 + c)*sinh(d*x^2 + c) + d^2*sinh(d*x^2 + c)^2 - 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 )^{2}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

[Out]

int(x^3*(a + b/sinh(c + d*x^2))^2, x)

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