3.109 \(\int \frac {x^3 \cosh (c+d x)}{(a+b x^3)^3} \, dx\)

Optimal. Leaf size=776 \[ -\frac {\sqrt [3]{-1} \cosh \left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}+c\right ) \text {Chi}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{27 a^{5/3} b^{4/3}}+\frac {(-1)^{2/3} \cosh \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Chi}\left (-x d-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{27 a^{5/3} b^{4/3}}+\frac {\cosh \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Chi}\left (x d+\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{27 a^{5/3} b^{4/3}}+\frac {\sqrt [3]{-1} \sinh \left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}+c\right ) \text {Shi}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{27 a^{5/3} b^{4/3}}+\frac {\sinh \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Shi}\left (x d+\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{27 a^{5/3} b^{4/3}}+\frac {(-1)^{2/3} \sinh \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Shi}\left (x d+\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{27 a^{5/3} b^{4/3}}-\frac {d^2 \cosh \left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}+c\right ) \text {Chi}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{54 a b^2}-\frac {d^2 \cosh \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Chi}\left (-x d-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{54 a b^2}-\frac {d^2 \cosh \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Chi}\left (x d+\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{54 a b^2}+\frac {d^2 \sinh \left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}+c\right ) \text {Shi}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{54 a b^2}-\frac {d^2 \sinh \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Shi}\left (x d+\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{54 a b^2}-\frac {d^2 \sinh \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Shi}\left (x d+\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{54 a b^2}-\frac {d \sinh (c+d x)}{18 b^2 x \left (a+b x^3\right )}+\frac {\cosh (c+d x)}{18 a b^2 x^2}-\frac {\cosh (c+d x)}{18 b^2 x^2 \left (a+b x^3\right )}+\frac {d \sinh (c+d x)}{18 a b^2 x}-\frac {x \cosh (c+d x)}{6 b \left (a+b x^3\right )^2} \]

[Out]

1/27*Chi(a^(1/3)*d/b^(1/3)+d*x)*cosh(c-a^(1/3)*d/b^(1/3))/a^(5/3)/b^(4/3)-1/54*d^2*Chi(a^(1/3)*d/b^(1/3)+d*x)*
cosh(c-a^(1/3)*d/b^(1/3))/a/b^2-1/27*(-1)^(1/3)*Chi((-1)^(1/3)*a^(1/3)*d/b^(1/3)-d*x)*cosh(c+(-1)^(1/3)*a^(1/3
)*d/b^(1/3))/a^(5/3)/b^(4/3)-1/54*d^2*Chi((-1)^(1/3)*a^(1/3)*d/b^(1/3)-d*x)*cosh(c+(-1)^(1/3)*a^(1/3)*d/b^(1/3
))/a/b^2+1/27*(-1)^(2/3)*Chi(-(-1)^(2/3)*a^(1/3)*d/b^(1/3)-d*x)*cosh(c-(-1)^(2/3)*a^(1/3)*d/b^(1/3))/a^(5/3)/b
^(4/3)-1/54*d^2*Chi(-(-1)^(2/3)*a^(1/3)*d/b^(1/3)-d*x)*cosh(c-(-1)^(2/3)*a^(1/3)*d/b^(1/3))/a/b^2+1/18*cosh(d*
x+c)/a/b^2/x^2-1/6*x*cosh(d*x+c)/b/(b*x^3+a)^2-1/18*cosh(d*x+c)/b^2/x^2/(b*x^3+a)+1/27*Shi(a^(1/3)*d/b^(1/3)+d
*x)*sinh(c-a^(1/3)*d/b^(1/3))/a^(5/3)/b^(4/3)-1/54*d^2*Shi(a^(1/3)*d/b^(1/3)+d*x)*sinh(c-a^(1/3)*d/b^(1/3))/a/
b^2-1/27*(-1)^(1/3)*Shi(-(-1)^(1/3)*a^(1/3)*d/b^(1/3)+d*x)*sinh(c+(-1)^(1/3)*a^(1/3)*d/b^(1/3))/a^(5/3)/b^(4/3
)-1/54*d^2*Shi(-(-1)^(1/3)*a^(1/3)*d/b^(1/3)+d*x)*sinh(c+(-1)^(1/3)*a^(1/3)*d/b^(1/3))/a/b^2+1/27*(-1)^(2/3)*S
hi((-1)^(2/3)*a^(1/3)*d/b^(1/3)+d*x)*sinh(c-(-1)^(2/3)*a^(1/3)*d/b^(1/3))/a^(5/3)/b^(4/3)-1/54*d^2*Shi((-1)^(2
/3)*a^(1/3)*d/b^(1/3)+d*x)*sinh(c-(-1)^(2/3)*a^(1/3)*d/b^(1/3))/a/b^2+1/18*d*sinh(d*x+c)/a/b^2/x-1/18*d*sinh(d
*x+c)/b^2/x/(b*x^3+a)

________________________________________________________________________________________

Rubi [A]  time = 2.66, antiderivative size = 776, normalized size of antiderivative = 1.00, number of steps used = 71, number of rules used = 10, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.526, Rules used = {5291, 5279, 5293, 3297, 3303, 3298, 3301, 5281, 5292, 5290} \[ -\frac {\sqrt [3]{-1} \cosh \left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}+c\right ) \text {Chi}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{27 a^{5/3} b^{4/3}}+\frac {(-1)^{2/3} \cosh \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Chi}\left (-x d-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{27 a^{5/3} b^{4/3}}+\frac {\cosh \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Chi}\left (x d+\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{27 a^{5/3} b^{4/3}}+\frac {\sqrt [3]{-1} \sinh \left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}+c\right ) \text {Shi}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{27 a^{5/3} b^{4/3}}+\frac {\sinh \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Shi}\left (x d+\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{27 a^{5/3} b^{4/3}}+\frac {(-1)^{2/3} \sinh \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Shi}\left (x d+\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{27 a^{5/3} b^{4/3}}-\frac {d^2 \cosh \left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}+c\right ) \text {Chi}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{54 a b^2}-\frac {d^2 \cosh \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Chi}\left (-x d-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{54 a b^2}-\frac {d^2 \cosh \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Chi}\left (x d+\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{54 a b^2}+\frac {d^2 \sinh \left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}+c\right ) \text {Shi}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{54 a b^2}-\frac {d^2 \sinh \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Shi}\left (x d+\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{54 a b^2}-\frac {d^2 \sinh \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Shi}\left (x d+\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{54 a b^2}-\frac {d \sinh (c+d x)}{18 b^2 x \left (a+b x^3\right )}-\frac {\cosh (c+d x)}{18 b^2 x^2 \left (a+b x^3\right )}+\frac {\cosh (c+d x)}{18 a b^2 x^2}+\frac {d \sinh (c+d x)}{18 a b^2 x}-\frac {x \cosh (c+d x)}{6 b \left (a+b x^3\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[(x^3*Cosh[c + d*x])/(a + b*x^3)^3,x]

[Out]

Cosh[c + d*x]/(18*a*b^2*x^2) - (x*Cosh[c + d*x])/(6*b*(a + b*x^3)^2) - Cosh[c + d*x]/(18*b^2*x^2*(a + b*x^3))
- ((-1)^(1/3)*Cosh[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*CoshIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(27
*a^(5/3)*b^(4/3)) - (d^2*Cosh[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*CoshIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3)
- d*x])/(54*a*b^2) + ((-1)^(2/3)*Cosh[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*CoshIntegral[-(((-1)^(2/3)*a^(1/3)*d
)/b^(1/3)) - d*x])/(27*a^(5/3)*b^(4/3)) - (d^2*Cosh[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*CoshIntegral[-(((-1)^(
2/3)*a^(1/3)*d)/b^(1/3)) - d*x])/(54*a*b^2) + (Cosh[c - (a^(1/3)*d)/b^(1/3)]*CoshIntegral[(a^(1/3)*d)/b^(1/3)
+ d*x])/(27*a^(5/3)*b^(4/3)) - (d^2*Cosh[c - (a^(1/3)*d)/b^(1/3)]*CoshIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(54
*a*b^2) + (d*Sinh[c + d*x])/(18*a*b^2*x) - (d*Sinh[c + d*x])/(18*b^2*x*(a + b*x^3)) + ((-1)^(1/3)*Sinh[c + ((-
1)^(1/3)*a^(1/3)*d)/b^(1/3)]*SinhIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(27*a^(5/3)*b^(4/3)) + (d^2*S
inh[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*SinhIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(54*a*b^2) + (Sinh
[c - (a^(1/3)*d)/b^(1/3)]*SinhIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(27*a^(5/3)*b^(4/3)) - (d^2*Sinh[c - (a^(1/
3)*d)/b^(1/3)]*SinhIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(54*a*b^2) + ((-1)^(2/3)*Sinh[c - ((-1)^(2/3)*a^(1/3)*
d)/b^(1/3)]*SinhIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x])/(27*a^(5/3)*b^(4/3)) - (d^2*Sinh[c - ((-1)^(2/
3)*a^(1/3)*d)/b^(1/3)]*SinhIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x])/(54*a*b^2)

Rule 3297

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[((c + d*x)^(m + 1)*Sin[e + f*x])/(d*(
m + 1)), x] - Dist[f/(d*(m + 1)), Int[(c + d*x)^(m + 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && LtQ[
m, -1]

Rule 3298

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(I*SinhIntegral[(c*f*fz)
/d + f*fz*x])/d, x] /; FreeQ[{c, d, e, f, fz}, x] && EqQ[d*e - c*f*fz*I, 0]

Rule 3301

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CoshIntegral[(c*f*fz)/d
+ f*fz*x]/d, x] /; FreeQ[{c, d, e, f, fz}, x] && EqQ[d*(e - Pi/2) - c*f*fz*I, 0]

Rule 3303

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Dist[Cos[(d*e - c*f)/d], Int[Sin[(c*f)/d + f*x]
/(c + d*x), x], x] + Dist[Sin[(d*e - c*f)/d], Int[Cos[(c*f)/d + f*x]/(c + d*x), x], x] /; FreeQ[{c, d, e, f},
x] && NeQ[d*e - c*f, 0]

Rule 5279

Int[Cosh[(c_.) + (d_.)*(x_)]*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x^(-n + 1)*(a + b*x^n)^(p + 1)*
Cosh[c + d*x])/(b*n*(p + 1)), x] + (-Dist[(-n + 1)/(b*n*(p + 1)), Int[((a + b*x^n)^(p + 1)*Cosh[c + d*x])/x^n,
x], x] - Dist[d/(b*n*(p + 1)), Int[x^(-n + 1)*(a + b*x^n)^(p + 1)*Sinh[c + d*x], x], x]) /; FreeQ[{a, b, c, d
}, x] && IntegerQ[p] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[n, 2]

Rule 5281

Int[Cosh[(c_.) + (d_.)*(x_)]*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[Cosh[c + d*x], (a
+ b*x^n)^p, x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[p, 0] && IGtQ[n, 0] && (EqQ[n, 2] || EqQ[p, -1])

Rule 5290

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*Sinh[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[(x^(m - n + 1)*(a + b
*x^n)^(p + 1)*Sinh[c + d*x])/(b*n*(p + 1)), x] + (-Dist[(m - n + 1)/(b*n*(p + 1)), Int[x^(m - n)*(a + b*x^n)^(
p + 1)*Sinh[c + d*x], x], x] - Dist[d/(b*n*(p + 1)), Int[x^(m - n + 1)*(a + b*x^n)^(p + 1)*Cosh[c + d*x], x],
x]) /; FreeQ[{a, b, c, d}, x] && ILtQ[p, -1] && IGtQ[n, 0] && RationalQ[m] && (GtQ[m - n + 1, 0] || GtQ[n, 2])

Rule 5291

Int[Cosh[(c_.) + (d_.)*(x_)]*(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x^(m - n + 1)*(a + b
*x^n)^(p + 1)*Cosh[c + d*x])/(b*n*(p + 1)), x] + (-Dist[(m - n + 1)/(b*n*(p + 1)), Int[x^(m - n)*(a + b*x^n)^(
p + 1)*Cosh[c + d*x], x], x] - Dist[d/(b*n*(p + 1)), Int[x^(m - n + 1)*(a + b*x^n)^(p + 1)*Sinh[c + d*x], x],
x]) /; FreeQ[{a, b, c, d}, x] && ILtQ[p, -1] && IGtQ[n, 0] && RationalQ[m] && (GtQ[m - n + 1, 0] || GtQ[n, 2])

Rule 5292

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*Sinh[(c_.) + (d_.)*(x_)], x_Symbol] :> Int[ExpandIntegrand[Sinh[c
+ d*x], x^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[p, 0] && IntegerQ[m] && IGtQ[n, 0] && (Eq
Q[n, 2] || EqQ[p, -1])

Rule 5293

Int[Cosh[(c_.) + (d_.)*(x_)]*(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[Cosh[c
+ d*x], x^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[p, 0] && IntegerQ[m] && IGtQ[n, 0] && (Eq
Q[n, 2] || EqQ[p, -1])

Rubi steps

\begin {align*} \int \frac {x^3 \cosh (c+d x)}{\left (a+b x^3\right )^3} \, dx &=-\frac {x \cosh (c+d x)}{6 b \left (a+b x^3\right )^2}+\frac {\int \frac {\cosh (c+d x)}{\left (a+b x^3\right )^2} \, dx}{6 b}+\frac {d \int \frac {x \sinh (c+d x)}{\left (a+b x^3\right )^2} \, dx}{6 b}\\ &=-\frac {x \cosh (c+d x)}{6 b \left (a+b x^3\right )^2}-\frac {\cosh (c+d x)}{18 b^2 x^2 \left (a+b x^3\right )}-\frac {d \sinh (c+d x)}{18 b^2 x \left (a+b x^3\right )}-\frac {\int \frac {\cosh (c+d x)}{x^3 \left (a+b x^3\right )} \, dx}{9 b^2}+\frac {d^2 \int \frac {\cosh (c+d x)}{x \left (a+b x^3\right )} \, dx}{18 b^2}\\ &=-\frac {x \cosh (c+d x)}{6 b \left (a+b x^3\right )^2}-\frac {\cosh (c+d x)}{18 b^2 x^2 \left (a+b x^3\right )}-\frac {d \sinh (c+d x)}{18 b^2 x \left (a+b x^3\right )}-\frac {\int \left (\frac {\cosh (c+d x)}{a x^3}-\frac {b \cosh (c+d x)}{a \left (a+b x^3\right )}\right ) \, dx}{9 b^2}+\frac {d^2 \int \left (\frac {\cosh (c+d x)}{a x}-\frac {b x^2 \cosh (c+d x)}{a \left (a+b x^3\right )}\right ) \, dx}{18 b^2}\\ &=-\frac {x \cosh (c+d x)}{6 b \left (a+b x^3\right )^2}-\frac {\cosh (c+d x)}{18 b^2 x^2 \left (a+b x^3\right )}-\frac {d \sinh (c+d x)}{18 b^2 x \left (a+b x^3\right )}-\frac {\int \frac {\cosh (c+d x)}{x^3} \, dx}{9 a b^2}+\frac {\int \frac {\cosh (c+d x)}{a+b x^3} \, dx}{9 a b}+\frac {d^2 \int \frac {\cosh (c+d x)}{x} \, dx}{18 a b^2}-\frac {d^2 \int \frac {x^2 \cosh (c+d x)}{a+b x^3} \, dx}{18 a b}\\ &=\frac {\cosh (c+d x)}{18 a b^2 x^2}-\frac {x \cosh (c+d x)}{6 b \left (a+b x^3\right )^2}-\frac {\cosh (c+d x)}{18 b^2 x^2 \left (a+b x^3\right )}-\frac {d \sinh (c+d x)}{18 b^2 x \left (a+b x^3\right )}+\frac {\int \left (-\frac {\cosh (c+d x)}{3 a^{2/3} \left (-\sqrt [3]{a}-\sqrt [3]{b} x\right )}-\frac {\cosh (c+d x)}{3 a^{2/3} \left (-\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b} x\right )}-\frac {\cosh (c+d x)}{3 a^{2/3} \left (-\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} x\right )}\right ) \, dx}{9 a b}-\frac {d \int \frac {\sinh (c+d x)}{x^2} \, dx}{18 a b^2}-\frac {d^2 \int \left (\frac {\cosh (c+d x)}{3 b^{2/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac {\cosh (c+d x)}{3 b^{2/3} \left (-\sqrt [3]{-1} \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac {\cosh (c+d x)}{3 b^{2/3} \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right )}\right ) \, dx}{18 a b}+\frac {\left (d^2 \cosh (c)\right ) \int \frac {\cosh (d x)}{x} \, dx}{18 a b^2}+\frac {\left (d^2 \sinh (c)\right ) \int \frac {\sinh (d x)}{x} \, dx}{18 a b^2}\\ &=\frac {\cosh (c+d x)}{18 a b^2 x^2}-\frac {x \cosh (c+d x)}{6 b \left (a+b x^3\right )^2}-\frac {\cosh (c+d x)}{18 b^2 x^2 \left (a+b x^3\right )}+\frac {d^2 \cosh (c) \text {Chi}(d x)}{18 a b^2}+\frac {d \sinh (c+d x)}{18 a b^2 x}-\frac {d \sinh (c+d x)}{18 b^2 x \left (a+b x^3\right )}+\frac {d^2 \sinh (c) \text {Shi}(d x)}{18 a b^2}-\frac {\int \frac {\cosh (c+d x)}{-\sqrt [3]{a}-\sqrt [3]{b} x} \, dx}{27 a^{5/3} b}-\frac {\int \frac {\cosh (c+d x)}{-\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b} x} \, dx}{27 a^{5/3} b}-\frac {\int \frac {\cosh (c+d x)}{-\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} x} \, dx}{27 a^{5/3} b}-\frac {d^2 \int \frac {\cosh (c+d x)}{x} \, dx}{18 a b^2}-\frac {d^2 \int \frac {\cosh (c+d x)}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{54 a b^{5/3}}-\frac {d^2 \int \frac {\cosh (c+d x)}{-\sqrt [3]{-1} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{54 a b^{5/3}}-\frac {d^2 \int \frac {\cosh (c+d x)}{(-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{54 a b^{5/3}}\\ &=\frac {\cosh (c+d x)}{18 a b^2 x^2}-\frac {x \cosh (c+d x)}{6 b \left (a+b x^3\right )^2}-\frac {\cosh (c+d x)}{18 b^2 x^2 \left (a+b x^3\right )}+\frac {d^2 \cosh (c) \text {Chi}(d x)}{18 a b^2}+\frac {d \sinh (c+d x)}{18 a b^2 x}-\frac {d \sinh (c+d x)}{18 b^2 x \left (a+b x^3\right )}+\frac {d^2 \sinh (c) \text {Shi}(d x)}{18 a b^2}-\frac {\left (d^2 \cosh (c)\right ) \int \frac {\cosh (d x)}{x} \, dx}{18 a b^2}-\frac {\cosh \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \int \frac {\cosh \left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{-\sqrt [3]{a}-\sqrt [3]{b} x} \, dx}{27 a^{5/3} b}-\frac {\left (d^2 \cosh \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac {\cosh \left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{54 a b^{5/3}}-\frac {\cosh \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \int \frac {\cos \left (\frac {(-1)^{5/6} \sqrt [3]{a} d}{\sqrt [3]{b}}-i d x\right )}{-\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} x} \, dx}{27 a^{5/3} b}-\frac {\left (d^2 \cosh \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac {\cos \left (\frac {(-1)^{5/6} \sqrt [3]{a} d}{\sqrt [3]{b}}-i d x\right )}{-\sqrt [3]{-1} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{54 a b^{5/3}}-\frac {\cosh \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \int \frac {\cos \left (\frac {\sqrt [6]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-i d x\right )}{-\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b} x} \, dx}{27 a^{5/3} b}-\frac {\left (d^2 \cosh \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac {\cos \left (\frac {\sqrt [6]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-i d x\right )}{(-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{54 a b^{5/3}}-\frac {\left (d^2 \sinh (c)\right ) \int \frac {\sinh (d x)}{x} \, dx}{18 a b^2}-\frac {\sinh \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \int \frac {\sinh \left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{-\sqrt [3]{a}-\sqrt [3]{b} x} \, dx}{27 a^{5/3} b}-\frac {\left (d^2 \sinh \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac {\sinh \left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{54 a b^{5/3}}-\frac {\left (i \sinh \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac {\sin \left (\frac {(-1)^{5/6} \sqrt [3]{a} d}{\sqrt [3]{b}}-i d x\right )}{-\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} x} \, dx}{27 a^{5/3} b}-\frac {\left (i d^2 \sinh \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac {\sin \left (\frac {(-1)^{5/6} \sqrt [3]{a} d}{\sqrt [3]{b}}-i d x\right )}{-\sqrt [3]{-1} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{54 a b^{5/3}}-\frac {\left (i \sinh \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac {\sin \left (\frac {\sqrt [6]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-i d x\right )}{-\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b} x} \, dx}{27 a^{5/3} b}-\frac {\left (i d^2 \sinh \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac {\sin \left (\frac {\sqrt [6]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-i d x\right )}{(-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{54 a b^{5/3}}\\ &=\frac {\cosh (c+d x)}{18 a b^2 x^2}-\frac {x \cosh (c+d x)}{6 b \left (a+b x^3\right )^2}-\frac {\cosh (c+d x)}{18 b^2 x^2 \left (a+b x^3\right )}-\frac {\sqrt [3]{-1} \cosh \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Chi}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{27 a^{5/3} b^{4/3}}-\frac {d^2 \cosh \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Chi}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{54 a b^2}+\frac {(-1)^{2/3} \cosh \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Chi}\left (-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{27 a^{5/3} b^{4/3}}-\frac {d^2 \cosh \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Chi}\left (-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{54 a b^2}+\frac {\cosh \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Chi}\left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{27 a^{5/3} b^{4/3}}-\frac {d^2 \cosh \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Chi}\left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{54 a b^2}+\frac {d \sinh (c+d x)}{18 a b^2 x}-\frac {d \sinh (c+d x)}{18 b^2 x \left (a+b x^3\right )}+\frac {\sqrt [3]{-1} \sinh \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Shi}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{27 a^{5/3} b^{4/3}}+\frac {d^2 \sinh \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Shi}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{54 a b^2}+\frac {\sinh \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Shi}\left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{27 a^{5/3} b^{4/3}}-\frac {d^2 \sinh \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Shi}\left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{54 a b^2}+\frac {(-1)^{2/3} \sinh \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Shi}\left (\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{27 a^{5/3} b^{4/3}}-\frac {d^2 \sinh \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Shi}\left (\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{54 a b^2}\\ \end {align*}

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Mathematica [C]  time = 0.71, size = 429, normalized size = 0.55 \[ -\frac {\text {RootSum}\left [\text {$\#$1}^3 b+a\& ,\frac {-\text {$\#$1}^2 d^2 \sinh (\text {$\#$1} d+c) \text {Chi}(d (x-\text {$\#$1}))+\text {$\#$1}^2 d^2 \cosh (\text {$\#$1} d+c) \text {Chi}(d (x-\text {$\#$1}))+\text {$\#$1}^2 d^2 \sinh (\text {$\#$1} d+c) \text {Shi}(d (x-\text {$\#$1}))-\text {$\#$1}^2 d^2 \cosh (\text {$\#$1} d+c) \text {Shi}(d (x-\text {$\#$1}))+2 \sinh (\text {$\#$1} d+c) \text {Chi}(d (x-\text {$\#$1}))-2 \cosh (\text {$\#$1} d+c) \text {Chi}(d (x-\text {$\#$1}))-2 \sinh (\text {$\#$1} d+c) \text {Shi}(d (x-\text {$\#$1}))+2 \cosh (\text {$\#$1} d+c) \text {Shi}(d (x-\text {$\#$1}))}{\text {$\#$1}^2}\& \right ]+\text {RootSum}\left [\text {$\#$1}^3 b+a\& ,\frac {\text {$\#$1}^2 d^2 \sinh (\text {$\#$1} d+c) \text {Chi}(d (x-\text {$\#$1}))+\text {$\#$1}^2 d^2 \cosh (\text {$\#$1} d+c) \text {Chi}(d (x-\text {$\#$1}))+\text {$\#$1}^2 d^2 \sinh (\text {$\#$1} d+c) \text {Shi}(d (x-\text {$\#$1}))+\text {$\#$1}^2 d^2 \cosh (\text {$\#$1} d+c) \text {Shi}(d (x-\text {$\#$1}))-2 \sinh (\text {$\#$1} d+c) \text {Chi}(d (x-\text {$\#$1}))-2 \cosh (\text {$\#$1} d+c) \text {Chi}(d (x-\text {$\#$1}))-2 \sinh (\text {$\#$1} d+c) \text {Shi}(d (x-\text {$\#$1}))-2 \cosh (\text {$\#$1} d+c) \text {Shi}(d (x-\text {$\#$1}))}{\text {$\#$1}^2}\& \right ]-\frac {6 b x \left (d x \left (a+b x^3\right ) \sinh (c+d x)+\left (b x^3-2 a\right ) \cosh (c+d x)\right )}{\left (a+b x^3\right )^2}}{108 a b^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/108*(RootSum[a + b*#1^3 & , (-2*Cosh[c + d*#1]*CoshIntegral[d*(x - #1)] + 2*CoshIntegral[d*(x - #1)]*Sinh[c
+ d*#1] + 2*Cosh[c + d*#1]*SinhIntegral[d*(x - #1)] - 2*Sinh[c + d*#1]*SinhIntegral[d*(x - #1)] + d^2*Cosh[c
+ d*#1]*CoshIntegral[d*(x - #1)]*#1^2 - d^2*CoshIntegral[d*(x - #1)]*Sinh[c + d*#1]*#1^2 - d^2*Cosh[c + d*#1]*
SinhIntegral[d*(x - #1)]*#1^2 + d^2*Sinh[c + d*#1]*SinhIntegral[d*(x - #1)]*#1^2)/#1^2 & ] + RootSum[a + b*#1^
3 & , (-2*Cosh[c + d*#1]*CoshIntegral[d*(x - #1)] - 2*CoshIntegral[d*(x - #1)]*Sinh[c + d*#1] - 2*Cosh[c + d*#
1]*SinhIntegral[d*(x - #1)] - 2*Sinh[c + d*#1]*SinhIntegral[d*(x - #1)] + d^2*Cosh[c + d*#1]*CoshIntegral[d*(x
- #1)]*#1^2 + d^2*CoshIntegral[d*(x - #1)]*Sinh[c + d*#1]*#1^2 + d^2*Cosh[c + d*#1]*SinhIntegral[d*(x - #1)]*
#1^2 + d^2*Sinh[c + d*#1]*SinhIntegral[d*(x - #1)]*#1^2)/#1^2 & ] - (6*b*x*((-2*a + b*x^3)*Cosh[c + d*x] + d*x
*(a + b*x^3)*Sinh[c + d*x]))/(a + b*x^3)^2)/(a*b^2)

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fricas [B]  time = 0.65, size = 2962, normalized size = 3.82 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/108*(((a*b^2*d^3*x^6 + 2*a^2*b*d^3*x^3 + a^3*d^3)*cosh(d*x + c)^2 - (a*b^2*d^3*x^6 + 2*a^2*b*d^3*x^3 + a^3*
d^3)*sinh(d*x + c)^2 + (a*d^3/b)^(1/3)*((b^3*x^6 + 2*a*b^2*x^3 + a^2*b + sqrt(-3)*(b^3*x^6 + 2*a*b^2*x^3 + a^2
*b))*cosh(d*x + c)^2 - (b^3*x^6 + 2*a*b^2*x^3 + a^2*b + sqrt(-3)*(b^3*x^6 + 2*a*b^2*x^3 + a^2*b))*sinh(d*x + c
)^2))*Ei(d*x - 1/2*(a*d^3/b)^(1/3)*(sqrt(-3) + 1))*cosh(1/2*(a*d^3/b)^(1/3)*(sqrt(-3) + 1) + c) + ((a*b^2*d^3*
x^6 + 2*a^2*b*d^3*x^3 + a^3*d^3)*cosh(d*x + c)^2 - (a*b^2*d^3*x^6 + 2*a^2*b*d^3*x^3 + a^3*d^3)*sinh(d*x + c)^2
- (-a*d^3/b)^(1/3)*((b^3*x^6 + 2*a*b^2*x^3 + a^2*b + sqrt(-3)*(b^3*x^6 + 2*a*b^2*x^3 + a^2*b))*cosh(d*x + c)^
2 - (b^3*x^6 + 2*a*b^2*x^3 + a^2*b + sqrt(-3)*(b^3*x^6 + 2*a*b^2*x^3 + a^2*b))*sinh(d*x + c)^2))*Ei(-d*x - 1/2
*(-a*d^3/b)^(1/3)*(sqrt(-3) + 1))*cosh(1/2*(-a*d^3/b)^(1/3)*(sqrt(-3) + 1) - c) + ((a*b^2*d^3*x^6 + 2*a^2*b*d^
3*x^3 + a^3*d^3)*cosh(d*x + c)^2 - (a*b^2*d^3*x^6 + 2*a^2*b*d^3*x^3 + a^3*d^3)*sinh(d*x + c)^2 + (a*d^3/b)^(1/
3)*((b^3*x^6 + 2*a*b^2*x^3 + a^2*b - sqrt(-3)*(b^3*x^6 + 2*a*b^2*x^3 + a^2*b))*cosh(d*x + c)^2 - (b^3*x^6 + 2*
a*b^2*x^3 + a^2*b - sqrt(-3)*(b^3*x^6 + 2*a*b^2*x^3 + a^2*b))*sinh(d*x + c)^2))*Ei(d*x + 1/2*(a*d^3/b)^(1/3)*(
sqrt(-3) - 1))*cosh(1/2*(a*d^3/b)^(1/3)*(sqrt(-3) - 1) - c) + ((a*b^2*d^3*x^6 + 2*a^2*b*d^3*x^3 + a^3*d^3)*cos
h(d*x + c)^2 - (a*b^2*d^3*x^6 + 2*a^2*b*d^3*x^3 + a^3*d^3)*sinh(d*x + c)^2 - (-a*d^3/b)^(1/3)*((b^3*x^6 + 2*a*
b^2*x^3 + a^2*b - sqrt(-3)*(b^3*x^6 + 2*a*b^2*x^3 + a^2*b))*cosh(d*x + c)^2 - (b^3*x^6 + 2*a*b^2*x^3 + a^2*b -
sqrt(-3)*(b^3*x^6 + 2*a*b^2*x^3 + a^2*b))*sinh(d*x + c)^2))*Ei(-d*x + 1/2*(-a*d^3/b)^(1/3)*(sqrt(-3) - 1))*co
sh(1/2*(-a*d^3/b)^(1/3)*(sqrt(-3) - 1) + c) + ((a*b^2*d^3*x^6 + 2*a^2*b*d^3*x^3 + a^3*d^3)*cosh(d*x + c)^2 - (
a*b^2*d^3*x^6 + 2*a^2*b*d^3*x^3 + a^3*d^3)*sinh(d*x + c)^2 + 2*(-a*d^3/b)^(1/3)*((b^3*x^6 + 2*a*b^2*x^3 + a^2*
b)*cosh(d*x + c)^2 - (b^3*x^6 + 2*a*b^2*x^3 + a^2*b)*sinh(d*x + c)^2))*Ei(-d*x + (-a*d^3/b)^(1/3))*cosh(c + (-
a*d^3/b)^(1/3)) + ((a*b^2*d^3*x^6 + 2*a^2*b*d^3*x^3 + a^3*d^3)*cosh(d*x + c)^2 - (a*b^2*d^3*x^6 + 2*a^2*b*d^3*
x^3 + a^3*d^3)*sinh(d*x + c)^2 - 2*(a*d^3/b)^(1/3)*((b^3*x^6 + 2*a*b^2*x^3 + a^2*b)*cosh(d*x + c)^2 - (b^3*x^6
+ 2*a*b^2*x^3 + a^2*b)*sinh(d*x + c)^2))*Ei(d*x + (a*d^3/b)^(1/3))*cosh(-c + (a*d^3/b)^(1/3)) + ((a*b^2*d^3*x
^6 + 2*a^2*b*d^3*x^3 + a^3*d^3)*cosh(d*x + c)^2 - (a*b^2*d^3*x^6 + 2*a^2*b*d^3*x^3 + a^3*d^3)*sinh(d*x + c)^2
+ (a*d^3/b)^(1/3)*((b^3*x^6 + 2*a*b^2*x^3 + a^2*b + sqrt(-3)*(b^3*x^6 + 2*a*b^2*x^3 + a^2*b))*cosh(d*x + c)^2
- (b^3*x^6 + 2*a*b^2*x^3 + a^2*b + sqrt(-3)*(b^3*x^6 + 2*a*b^2*x^3 + a^2*b))*sinh(d*x + c)^2))*Ei(d*x - 1/2*(a
*d^3/b)^(1/3)*(sqrt(-3) + 1))*sinh(1/2*(a*d^3/b)^(1/3)*(sqrt(-3) + 1) + c) + ((a*b^2*d^3*x^6 + 2*a^2*b*d^3*x^3
+ a^3*d^3)*cosh(d*x + c)^2 - (a*b^2*d^3*x^6 + 2*a^2*b*d^3*x^3 + a^3*d^3)*sinh(d*x + c)^2 - (-a*d^3/b)^(1/3)*(
(b^3*x^6 + 2*a*b^2*x^3 + a^2*b + sqrt(-3)*(b^3*x^6 + 2*a*b^2*x^3 + a^2*b))*cosh(d*x + c)^2 - (b^3*x^6 + 2*a*b^
2*x^3 + a^2*b + sqrt(-3)*(b^3*x^6 + 2*a*b^2*x^3 + a^2*b))*sinh(d*x + c)^2))*Ei(-d*x - 1/2*(-a*d^3/b)^(1/3)*(sq
rt(-3) + 1))*sinh(1/2*(-a*d^3/b)^(1/3)*(sqrt(-3) + 1) - c) - ((a*b^2*d^3*x^6 + 2*a^2*b*d^3*x^3 + a^3*d^3)*cosh
(d*x + c)^2 - (a*b^2*d^3*x^6 + 2*a^2*b*d^3*x^3 + a^3*d^3)*sinh(d*x + c)^2 + (a*d^3/b)^(1/3)*((b^3*x^6 + 2*a*b^
2*x^3 + a^2*b - sqrt(-3)*(b^3*x^6 + 2*a*b^2*x^3 + a^2*b))*cosh(d*x + c)^2 - (b^3*x^6 + 2*a*b^2*x^3 + a^2*b - s
qrt(-3)*(b^3*x^6 + 2*a*b^2*x^3 + a^2*b))*sinh(d*x + c)^2))*Ei(d*x + 1/2*(a*d^3/b)^(1/3)*(sqrt(-3) - 1))*sinh(1
/2*(a*d^3/b)^(1/3)*(sqrt(-3) - 1) - c) - ((a*b^2*d^3*x^6 + 2*a^2*b*d^3*x^3 + a^3*d^3)*cosh(d*x + c)^2 - (a*b^2
*d^3*x^6 + 2*a^2*b*d^3*x^3 + a^3*d^3)*sinh(d*x + c)^2 - (-a*d^3/b)^(1/3)*((b^3*x^6 + 2*a*b^2*x^3 + a^2*b - sqr
t(-3)*(b^3*x^6 + 2*a*b^2*x^3 + a^2*b))*cosh(d*x + c)^2 - (b^3*x^6 + 2*a*b^2*x^3 + a^2*b - sqrt(-3)*(b^3*x^6 +
2*a*b^2*x^3 + a^2*b))*sinh(d*x + c)^2))*Ei(-d*x + 1/2*(-a*d^3/b)^(1/3)*(sqrt(-3) - 1))*sinh(1/2*(-a*d^3/b)^(1/
3)*(sqrt(-3) - 1) + c) - ((a*b^2*d^3*x^6 + 2*a^2*b*d^3*x^3 + a^3*d^3)*cosh(d*x + c)^2 - (a*b^2*d^3*x^6 + 2*a^2
*b*d^3*x^3 + a^3*d^3)*sinh(d*x + c)^2 + 2*(-a*d^3/b)^(1/3)*((b^3*x^6 + 2*a*b^2*x^3 + a^2*b)*cosh(d*x + c)^2 -
(b^3*x^6 + 2*a*b^2*x^3 + a^2*b)*sinh(d*x + c)^2))*Ei(-d*x + (-a*d^3/b)^(1/3))*sinh(c + (-a*d^3/b)^(1/3)) - ((a
*b^2*d^3*x^6 + 2*a^2*b*d^3*x^3 + a^3*d^3)*cosh(d*x + c)^2 - (a*b^2*d^3*x^6 + 2*a^2*b*d^3*x^3 + a^3*d^3)*sinh(d
*x + c)^2 - 2*(a*d^3/b)^(1/3)*((b^3*x^6 + 2*a*b^2*x^3 + a^2*b)*cosh(d*x + c)^2 - (b^3*x^6 + 2*a*b^2*x^3 + a^2*
b)*sinh(d*x + c)^2))*Ei(d*x + (a*d^3/b)^(1/3))*sinh(-c + (a*d^3/b)^(1/3)) - 6*(a*b^2*d*x^4 - 2*a^2*b*d*x)*cosh
(d*x + c) - 6*(a*b^2*d^2*x^5 + a^2*b*d^2*x^2)*sinh(d*x + c))/((a^2*b^4*d*x^6 + 2*a^3*b^3*d*x^3 + a^4*b^2*d)*co
sh(d*x + c)^2 - (a^2*b^4*d*x^6 + 2*a^3*b^3*d*x^3 + a^4*b^2*d)*sinh(d*x + c)^2)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{3} \cosh \left (d x + c\right )}{{\left (b x^{3} + a\right )}^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 0.40, size = 1456, normalized size = 1.88 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/36*d^6*exp(-d*x-c)/a/(b^2*d^6*x^6+2*a*b*d^6*x^3+a^2*d^6)*x^4-1/18*d^6*exp(-d*x-c)/b/(b^2*d^6*x^6+2*a*b*d^6*x
^3+a^2*d^6)*x-1/36*d^7*exp(-d*x-c)/a/(b^2*d^6*x^6+2*a*b*d^6*x^3+a^2*d^6)*x^5-1/36*d^7*exp(-d*x-c)/b/(b^2*d^6*x
^6+2*a*b*d^6*x^3+a^2*d^6)*x^2+1/108/d*c^3/a^2/b*sum((_R1^2-2*_R1*c+c^2+6*_R1-6*c+10)/(_R1^2-2*_R1*c+c^2)*exp(-
_R1)*Ei(1,d*x-_R1+c),_R1=RootOf(_Z^3*b-3*_Z^2*b*c+3*_Z*b*c^2+a*d^3-b*c^3))+1/36/d*c/a^2/b^2*sum((_R1^2*b*c^2-_
R1*a*d^3-2*_R1*b*c^3-a*c*d^3+b*c^4+8*_R1^2*b*c-10*_R1*b*c^2-2*a*d^3+2*b*c^3+8*_R1*b*c+2*b*c^2)/(_R1^2-2*_R1*c+
c^2)*exp(-_R1)*Ei(1,d*x-_R1+c),_R1=RootOf(_Z^3*b-3*_Z^2*b*c+3*_Z*b*c^2+a*d^3-b*c^3))-1/36/d*c^2/a^2/b^2*sum((_
R1^2*b*c-2*_R1*b*c^2-a*d^3+b*c^3+4*_R1^2*b-2*_R1*b*c-2*b*c^2+4*_R1*b+6*b*c)/(_R1^2-2*_R1*c+c^2)*exp(-_R1)*Ei(1
,d*x-_R1+c),_R1=RootOf(_Z^3*b-3*_Z^2*b*c+3*_Z*b*c^2+a*d^3-b*c^3))+1/108/d/a^2/b^2*sum((_R1^2*a*d^3-_R1^2*b*c^3
+_R1*a*c*d^3+2*_R1*b*c^4+a*c^2*d^3-b*c^5-12*_R1^2*b*c^2+18*_R1*b*c^3+6*a*c*d^3-6*b*c^4-12*_R1*b*c^2-2*a*d^3+2*
b*c^3)/(_R1^2-2*_R1*c+c^2)*exp(-_R1)*Ei(1,d*x-_R1+c),_R1=RootOf(_Z^3*b-3*_Z^2*b*c+3*_Z*b*c^2+a*d^3-b*c^3))+1/3
6*d^7*exp(d*x+c)/a/(b^2*d^6*x^6+2*a*b*d^6*x^3+a^2*d^6)*x^5+1/36*d^7*exp(d*x+c)/b/(b^2*d^6*x^6+2*a*b*d^6*x^3+a^
2*d^6)*x^2+1/108/d/a^2/b^2*sum((_R1^2*a*d^3-_R1^2*b*c^3+_R1*a*c*d^3+2*_R1*b*c^4+a*c^2*d^3-b*c^5+12*_R1^2*b*c^2
-18*_R1*b*c^3-6*a*c*d^3+6*b*c^4-12*_R1*b*c^2-2*a*d^3+2*b*c^3)/(_R1^2-2*_R1*c+c^2)*exp(_R1)*Ei(1,-d*x+_R1-c),_R
1=RootOf(_Z^3*b-3*_Z^2*b*c+3*_Z*b*c^2+a*d^3-b*c^3))+1/108/d*c^3/a^2/b*sum((_R1^2-2*_R1*c+c^2-6*_R1+6*c+10)/(_R
1^2-2*_R1*c+c^2)*exp(_R1)*Ei(1,-d*x+_R1-c),_R1=RootOf(_Z^3*b-3*_Z^2*b*c+3*_Z*b*c^2+a*d^3-b*c^3))-1/36/d*c^2/a^
2/b^2*sum((_R1^2*b*c-2*_R1*b*c^2-a*d^3+b*c^3-4*_R1^2*b+2*_R1*b*c+2*b*c^2+4*_R1*b+6*b*c)/(_R1^2-2*_R1*c+c^2)*ex
p(_R1)*Ei(1,-d*x+_R1-c),_R1=RootOf(_Z^3*b-3*_Z^2*b*c+3*_Z*b*c^2+a*d^3-b*c^3))+1/36*d^6*exp(d*x+c)/a/(b^2*d^6*x
^6+2*a*b*d^6*x^3+a^2*d^6)*x^4-1/18*d^6*exp(d*x+c)/b/(b^2*d^6*x^6+2*a*b*d^6*x^3+a^2*d^6)*x+1/36/d*c/a^2/b^2*sum
((_R1^2*b*c^2-_R1*a*d^3-2*_R1*b*c^3-a*c*d^3+b*c^4-8*_R1^2*b*c+10*_R1*b*c^2+2*a*d^3-2*b*c^3+8*_R1*b*c+2*b*c^2)/
(_R1^2-2*_R1*c+c^2)*exp(_R1)*Ei(1,-d*x+_R1-c),_R1=RootOf(_Z^3*b-3*_Z^2*b*c+3*_Z*b*c^2+a*d^3-b*c^3))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {{\left (d^{2} x^{3} e^{\left (2 \, c\right )} + 6 \, d x^{2} e^{\left (2 \, c\right )} + 42 \, x e^{\left (2 \, c\right )}\right )} e^{\left (d x\right )} - {\left (d^{2} x^{3} - 6 \, d x^{2} + 42 \, x\right )} e^{\left (-d x\right )}}{2 \, {\left (b^{3} d^{3} x^{9} e^{c} + 3 \, a b^{2} d^{3} x^{6} e^{c} + 3 \, a^{2} b d^{3} x^{3} e^{c} + a^{3} d^{3} e^{c}\right )}} + \frac {1}{2} \, \int -\frac {3 \, {\left (3 \, a d^{2} x^{2} e^{c} - 112 \, b x^{3} e^{c} + 18 \, a d x e^{c} + 14 \, a e^{c}\right )} e^{\left (d x\right )}}{b^{4} d^{3} x^{12} + 4 \, a b^{3} d^{3} x^{9} + 6 \, a^{2} b^{2} d^{3} x^{6} + 4 \, a^{3} b d^{3} x^{3} + a^{4} d^{3}}\,{d x} - \frac {1}{2} \, \int -\frac {3 \, {\left (3 \, a d^{2} x^{2} - 112 \, b x^{3} - 18 \, a d x + 14 \, a\right )} e^{\left (-d x\right )}}{b^{4} d^{3} x^{12} e^{c} + 4 \, a b^{3} d^{3} x^{9} e^{c} + 6 \, a^{2} b^{2} d^{3} x^{6} e^{c} + 4 \, a^{3} b d^{3} x^{3} e^{c} + a^{4} d^{3} e^{c}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*((d^2*x^3*e^(2*c) + 6*d*x^2*e^(2*c) + 42*x*e^(2*c))*e^(d*x) - (d^2*x^3 - 6*d*x^2 + 42*x)*e^(-d*x))/(b^3*d^
3*x^9*e^c + 3*a*b^2*d^3*x^6*e^c + 3*a^2*b*d^3*x^3*e^c + a^3*d^3*e^c) + 1/2*integrate(-3*(3*a*d^2*x^2*e^c - 112
*b*x^3*e^c + 18*a*d*x*e^c + 14*a*e^c)*e^(d*x)/(b^4*d^3*x^12 + 4*a*b^3*d^3*x^9 + 6*a^2*b^2*d^3*x^6 + 4*a^3*b*d^
3*x^3 + a^4*d^3), x) - 1/2*integrate(-3*(3*a*d^2*x^2 - 112*b*x^3 - 18*a*d*x + 14*a)*e^(-d*x)/(b^4*d^3*x^12*e^c
+ 4*a*b^3*d^3*x^9*e^c + 6*a^2*b^2*d^3*x^6*e^c + 4*a^3*b*d^3*x^3*e^c + a^4*d^3*e^c), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^3\,\mathrm {cosh}\left (c+d\,x\right )}{{\left (b\,x^3+a\right )}^3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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