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

3.1.10.1 Optimal result

Integrand size = 18, antiderivative size = 108 \[ \int x^3 \left (a+b \text {csch}\left (c+d x^2\right )\right )^2 \, dx=\frac {a^2 x^4}{4}-\frac {2 a b x^2 \text {arctanh}\left (e^{c+d x^2}\right )}{d}-\frac {b^2 x^2 \coth \left (c+d x^2\right )}{2 d}+\frac {b^2 \log \left (\sinh \left (c+d x^2\right )\right )}{2 d^2}-\frac {a b \operatorname {PolyLog}\left (2,-e^{c+d x^2}\right )}{d^2}+\frac {a b \operatorname {PolyLog}\left (2,e^{c+d x^2}\right )}{d^2} \]

output

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

3.1.10.2 Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(276\) vs. \(2(108)=216\).

Time = 0.73 (sec) , antiderivative size = 276, normalized size of antiderivative = 2.56 \[ \int x^3 \left (a+b \text {csch}\left (c+d x^2\right )\right )^2 \, dx=\frac {\text {csch}\left (\frac {1}{2} \left (c+d x^2\right )\right ) \text {sech}\left (\frac {1}{2} \left (c+d x^2\right )\right ) \sinh (c) (\cosh (c)+\sinh (c)) \left (-2 b^2 d x^2 \cosh \left (c+d x^2\right )+2 b^2 d x^2 \sinh \left (c+d x^2\right )+a^2 d^2 x^4 \sinh \left (c+d x^2\right )+2 b^2 \log \left (1-e^{-c-d x^2}\right ) \sinh \left (c+d x^2\right )+4 a b d x^2 \log \left (1-e^{-c-d x^2}\right ) \sinh \left (c+d x^2\right )+2 b^2 \log \left (1+e^{-c-d x^2}\right ) \sinh \left (c+d x^2\right )-4 a b d x^2 \log \left (1+e^{-c-d x^2}\right ) \sinh \left (c+d x^2\right )+4 a b \operatorname {PolyLog}\left (2,-e^{-c-d x^2}\right ) \sinh \left (c+d x^2\right )-4 a b \operatorname {PolyLog}\left (2,e^{-c-d x^2}\right ) \sinh \left (c+d x^2\right )\right )}{4 d^2 \left (-1+e^{2 c}\right )} \]

input

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

output

(Csch[(c + d*x^2)/2]*Sech[(c + d*x^2)/2]*Sinh[c]*(Cosh[c] + Sinh[c])*(-2*b 
^2*d*x^2*Cosh[c + d*x^2] + 2*b^2*d*x^2*Sinh[c + d*x^2] + a^2*d^2*x^4*Sinh[ 
c + d*x^2] + 2*b^2*Log[1 - E^(-c - d*x^2)]*Sinh[c + d*x^2] + 4*a*b*d*x^2*L 
og[1 - E^(-c - d*x^2)]*Sinh[c + d*x^2] + 2*b^2*Log[1 + E^(-c - d*x^2)]*Sin 
h[c + d*x^2] - 4*a*b*d*x^2*Log[1 + E^(-c - d*x^2)]*Sinh[c + d*x^2] + 4*a*b 
*PolyLog[2, -E^(-c - d*x^2)]*Sinh[c + d*x^2] - 4*a*b*PolyLog[2, E^(-c - d* 
x^2)]*Sinh[c + d*x^2]))/(4*d^2*(-1 + E^(2*c)))
 

3.1.10.3 Rubi [A] (verified)

Time = 0.39 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {5960, 3042, 4678, 2009}

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.

\(\displaystyle \int x^3 \left (a+b \text {csch}\left (c+d x^2\right )\right )^2 \, dx\)

\(\Big \downarrow \) 5960

\(\displaystyle \frac {1}{2} \int x^2 \left (a+b \text {csch}\left (d x^2+c\right )\right )^2dx^2\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{2} \int x^2 \left (a+i b \csc \left (i d x^2+i c\right )\right )^2dx^2\)

\(\Big \downarrow \) 4678

\(\displaystyle \frac {1}{2} \int \left (a^2 x^2+b^2 \text {csch}^2\left (d x^2+c\right ) x^2+2 a b \text {csch}\left (d x^2+c\right ) x^2\right )dx^2\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{2} \left (\frac {a^2 x^4}{2}-\frac {4 a b x^2 \text {arctanh}\left (e^{c+d x^2}\right )}{d}-\frac {2 a b \operatorname {PolyLog}\left (2,-e^{d x^2+c}\right )}{d^2}+\frac {2 a b \operatorname {PolyLog}\left (2,e^{d x^2+c}\right )}{d^2}+\frac {b^2 \log \left (\sinh \left (c+d x^2\right )\right )}{d^2}-\frac {b^2 x^2 \coth \left (c+d x^2\right )}{d}\right )\)

input

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

output

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

3.1.10.3.1 Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4678

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 5960

Int[((a_.) + Csch[(c_.) + (d_.)*(x_)^(n_)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbo 
l] :> Simp[1/n   Subst[Int[x^(Simplify[(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[Simplify[(m 
 + 1)/n], 0] && IntegerQ[p]
 

3.1.10.4 Maple [F]

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

input

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

output

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

3.1.10.5 Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 683 vs. \(2 (97) = 194\).

Time = 0.28 (sec) , antiderivative size = 683, normalized size of antiderivative = 6.32 \[ \int x^3 \left (a+b \text {csch}\left (c+d x^2\right )\right )^2 \, dx=-\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 )}} \]

input

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

output

-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)*cosh(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*cosh(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 
)
 

3.1.10.6 Sympy [F]

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

input

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

output

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

3.1.10.7 Maxima [F]

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

input

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

output

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 - log((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))
 

3.1.10.8 Giac [F]

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

input

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

output

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

3.1.10.9 Mupad [F(-1)]

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

input

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

output

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