\(\int \frac {x^5 \cosh (c+d x)}{(a+b x^3)^3} \, dx\) [107]

Optimal result

Integrand size = 19, antiderivative size = 784 \[ \int \frac {x^5 \cosh (c+d x)}{\left (a+b x^3\right )^3} \, dx=-\frac {x^3 \cosh (c+d x)}{6 b \left (a+b x^3\right )^2}-\frac {\cosh (c+d x)}{6 b^2 \left (a+b x^3\right )}-\frac {(-1)^{2/3} 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 \sqrt [3]{a} b^{8/3}}+\frac {\sqrt [3]{-1} 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 \sqrt [3]{a} b^{8/3}}-\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 \sqrt [3]{a} b^{8/3}}+\frac {2 d \text {Chi}\left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right ) \sinh \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{27 a^{2/3} b^{7/3}}-\frac {2 \sqrt [3]{-1} d \text {Chi}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right ) \sinh \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{27 a^{2/3} b^{7/3}}+\frac {2 (-1)^{2/3} d \text {Chi}\left (-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right ) \sinh \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{27 a^{2/3} b^{7/3}}-\frac {d x \sinh (c+d x)}{18 b^2 \left (a+b x^3\right )}+\frac {2 \sqrt [3]{-1} d \cosh \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^{2/3} b^{7/3}}+\frac {(-1)^{2/3} 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 \sqrt [3]{a} b^{8/3}}+\frac {2 d \cosh \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^{2/3} b^{7/3}}-\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 \sqrt [3]{a} b^{8/3}}+\frac {2 (-1)^{2/3} d \cosh \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^{2/3} b^{7/3}}+\frac {\sqrt [3]{-1} 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 \sqrt [3]{a} b^{8/3}} \]

[Out]

-1/54*d^2*Chi(a^(1/3)*d/b^(1/3)+d*x)*cosh(c-a^(1/3)*d/b^(1/3))/a^(1/3)/b^(8/3)-1/54*(-1)^(2/3)*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^(1/3)/b^(8/3)+1/54*(-1)^(1/3)*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^(1/3)/b^(8/3)-1/6*x^3*cosh(d*x+c)/b/(b*x^3
+a)^2-1/6*cosh(d*x+c)/b^2/(b*x^3+a)-2/27*(-1)^(1/3)*d*cosh(c+(-1)^(1/3)*a^(1/3)*d/b^(1/3))*Shi(-(-1)^(1/3)*a^(
1/3)*d/b^(1/3)+d*x)/a^(2/3)/b^(7/3)+2/27*d*cosh(c-a^(1/3)*d/b^(1/3))*Shi(a^(1/3)*d/b^(1/3)+d*x)/a^(2/3)/b^(7/3
)+2/27*(-1)^(2/3)*d*cosh(c-(-1)^(2/3)*a^(1/3)*d/b^(1/3))*Shi((-1)^(2/3)*a^(1/3)*d/b^(1/3)+d*x)/a^(2/3)/b^(7/3)
+2/27*d*Chi(a^(1/3)*d/b^(1/3)+d*x)*sinh(c-a^(1/3)*d/b^(1/3))/a^(2/3)/b^(7/3)-1/54*d^2*Shi(a^(1/3)*d/b^(1/3)+d*
x)*sinh(c-a^(1/3)*d/b^(1/3))/a^(1/3)/b^(8/3)-2/27*(-1)^(1/3)*d*Chi((-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^(2/3)/b^(7/3)-1/54*(-1)^(2/3)*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^(1/3)/b^(8/3)+2/27*(-1)^(2/3)*d*Chi(-(-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^(2/3)/b^(7/3)+1/54*(-1)^(1/3)*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^(1/3)/b^(8/3)-1/18*d*x*sinh(d*x+c)/b^2/(b*x^3+a)

Rubi [A] (verified)

Time = 1.28 (sec) , antiderivative size = 784, normalized size of antiderivative = 1.00, number of steps used = 36, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {5399, 5397, 5388, 3384, 3379, 3382, 5398, 5401} \[ \int \frac {x^5 \cosh (c+d x)}{\left (a+b x^3\right )^3} \, dx=\frac {2 d \sinh \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^{2/3} b^{7/3}}-\frac {2 \sqrt [3]{-1} d \sinh \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^{2/3} b^{7/3}}+\frac {2 (-1)^{2/3} d \sinh \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^{2/3} b^{7/3}}+\frac {2 \sqrt [3]{-1} d \cosh \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^{2/3} b^{7/3}}+\frac {2 d \cosh \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^{2/3} b^{7/3}}+\frac {2 (-1)^{2/3} d \cosh \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^{2/3} b^{7/3}}-\frac {(-1)^{2/3} 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 \sqrt [3]{a} b^{8/3}}+\frac {\sqrt [3]{-1} 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 \sqrt [3]{a} b^{8/3}}-\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 \sqrt [3]{a} b^{8/3}}+\frac {(-1)^{2/3} 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 \sqrt [3]{a} b^{8/3}}-\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 \sqrt [3]{a} b^{8/3}}+\frac {\sqrt [3]{-1} 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 \sqrt [3]{a} b^{8/3}}-\frac {d x \sinh (c+d x)}{18 b^2 \left (a+b x^3\right )}-\frac {\cosh (c+d x)}{6 b^2 \left (a+b x^3\right )}-\frac {x^3 \cosh (c+d x)}{6 b \left (a+b x^3\right )^2} \]

[In]

Int[(x^5*Cosh[c + d*x])/(a + b*x^3)^3,x]

[Out]

-1/6*(x^3*Cosh[c + d*x])/(b*(a + b*x^3)^2) - Cosh[c + d*x]/(6*b^2*(a + b*x^3)) - ((-1)^(2/3)*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^(1/3)*b^(8/3)) + ((-1)^(
1/3)*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
^(1/3)*b^(8/3)) - (d^2*Cosh[c - (a^(1/3)*d)/b^(1/3)]*CoshIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(54*a^(1/3)*b^(8
/3)) + (2*d*CoshIntegral[(a^(1/3)*d)/b^(1/3) + d*x]*Sinh[c - (a^(1/3)*d)/b^(1/3)])/(27*a^(2/3)*b^(7/3)) - (2*(
-1)^(1/3)*d*CoshIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x]*Sinh[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)])/(27*a
^(2/3)*b^(7/3)) + (2*(-1)^(2/3)*d*CoshIntegral[-(((-1)^(2/3)*a^(1/3)*d)/b^(1/3)) - d*x]*Sinh[c - ((-1)^(2/3)*a
^(1/3)*d)/b^(1/3)])/(27*a^(2/3)*b^(7/3)) - (d*x*Sinh[c + d*x])/(18*b^2*(a + b*x^3)) + (2*(-1)^(1/3)*d*Cosh[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^(2/3)*b^(7/3)) + ((
-1)^(2/3)*d^2*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])/(54
*a^(1/3)*b^(8/3)) + (2*d*Cosh[c - (a^(1/3)*d)/b^(1/3)]*SinhIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(27*a^(2/3)*b^
(7/3)) - (d^2*Sinh[c - (a^(1/3)*d)/b^(1/3)]*SinhIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(54*a^(1/3)*b^(8/3)) + (2
*(-1)^(2/3)*d*Cosh[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^(2/3)*b^(7/3)) + ((-1)^(1/3)*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^(1/3)*b^(8/3))

Rule 3379

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 3382

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 3384

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 5388

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*Sinh[(c_.) + (d_.)*(x_)], x_Symbol] :> Int[ExpandIntegrand[Sinh[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 5397

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

Rule 5398

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 5399

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 5401

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*} \text {integral}& = -\frac {x^3 \cosh (c+d x)}{6 b \left (a+b x^3\right )^2}+\frac {\int \frac {x^2 \cosh (c+d x)}{\left (a+b x^3\right )^2} \, dx}{2 b}+\frac {d \int \frac {x^3 \sinh (c+d x)}{\left (a+b x^3\right )^2} \, dx}{6 b} \\ & = -\frac {x^3 \cosh (c+d x)}{6 b \left (a+b x^3\right )^2}-\frac {\cosh (c+d x)}{6 b^2 \left (a+b x^3\right )}-\frac {d x \sinh (c+d x)}{18 b^2 \left (a+b x^3\right )}+\frac {d \int \frac {\sinh (c+d x)}{a+b x^3} \, dx}{18 b^2}+\frac {d \int \frac {\sinh (c+d x)}{a+b x^3} \, dx}{6 b^2}+\frac {d^2 \int \frac {x \cosh (c+d x)}{a+b x^3} \, dx}{18 b^2} \\ & = -\frac {x^3 \cosh (c+d x)}{6 b \left (a+b x^3\right )^2}-\frac {\cosh (c+d x)}{6 b^2 \left (a+b x^3\right )}-\frac {d x \sinh (c+d x)}{18 b^2 \left (a+b x^3\right )}+\frac {d \int \left (-\frac {\sinh (c+d x)}{3 a^{2/3} \left (-\sqrt [3]{a}-\sqrt [3]{b} x\right )}-\frac {\sinh (c+d x)}{3 a^{2/3} \left (-\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b} x\right )}-\frac {\sinh (c+d x)}{3 a^{2/3} \left (-\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} x\right )}\right ) \, dx}{18 b^2}+\frac {d \int \left (-\frac {\sinh (c+d x)}{3 a^{2/3} \left (-\sqrt [3]{a}-\sqrt [3]{b} x\right )}-\frac {\sinh (c+d x)}{3 a^{2/3} \left (-\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b} x\right )}-\frac {\sinh (c+d x)}{3 a^{2/3} \left (-\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} x\right )}\right ) \, dx}{6 b^2}+\frac {d^2 \int \left (-\frac {\cosh (c+d x)}{3 \sqrt [3]{a} \sqrt [3]{b} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac {(-1)^{2/3} \cosh (c+d x)}{3 \sqrt [3]{a} \sqrt [3]{b} \left (\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x\right )}+\frac {\sqrt [3]{-1} \cosh (c+d x)}{3 \sqrt [3]{a} \sqrt [3]{b} \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x\right )}\right ) \, dx}{18 b^2} \\ & = -\frac {x^3 \cosh (c+d x)}{6 b \left (a+b x^3\right )^2}-\frac {\cosh (c+d x)}{6 b^2 \left (a+b x^3\right )}-\frac {d x \sinh (c+d x)}{18 b^2 \left (a+b x^3\right )}-\frac {d \int \frac {\sinh (c+d x)}{-\sqrt [3]{a}-\sqrt [3]{b} x} \, dx}{54 a^{2/3} b^2}-\frac {d \int \frac {\sinh (c+d x)}{-\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b} x} \, dx}{54 a^{2/3} b^2}-\frac {d \int \frac {\sinh (c+d x)}{-\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} x} \, dx}{54 a^{2/3} b^2}-\frac {d \int \frac {\sinh (c+d x)}{-\sqrt [3]{a}-\sqrt [3]{b} x} \, dx}{18 a^{2/3} b^2}-\frac {d \int \frac {\sinh (c+d x)}{-\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b} x} \, dx}{18 a^{2/3} b^2}-\frac {d \int \frac {\sinh (c+d x)}{-\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} x} \, dx}{18 a^{2/3} b^2}-\frac {d^2 \int \frac {\cosh (c+d x)}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{54 \sqrt [3]{a} b^{7/3}}+\frac {\left (\sqrt [3]{-1} d^2\right ) \int \frac {\cosh (c+d x)}{\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x} \, dx}{54 \sqrt [3]{a} b^{7/3}}-\frac {\left ((-1)^{2/3} d^2\right ) \int \frac {\cosh (c+d x)}{\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x} \, dx}{54 \sqrt [3]{a} b^{7/3}} \\ & = \text {Too large to display} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 4 in optimal.

Time = 0.46 (sec) , antiderivative size = 397, normalized size of antiderivative = 0.51 \[ \int \frac {x^5 \cosh (c+d x)}{\left (a+b x^3\right )^3} \, dx=\frac {d \text {RootSum}\left [a+b \text {$\#$1}^3\&,\frac {-4 \cosh (c+d \text {$\#$1}) \text {Chi}(d (x-\text {$\#$1}))+4 \text {Chi}(d (x-\text {$\#$1})) \sinh (c+d \text {$\#$1})+4 \cosh (c+d \text {$\#$1}) \text {Shi}(d (x-\text {$\#$1}))-4 \sinh (c+d \text {$\#$1}) \text {Shi}(d (x-\text {$\#$1}))+d \cosh (c+d \text {$\#$1}) \text {Chi}(d (x-\text {$\#$1})) \text {$\#$1}-d \text {Chi}(d (x-\text {$\#$1})) \sinh (c+d \text {$\#$1}) \text {$\#$1}-d \cosh (c+d \text {$\#$1}) \text {Shi}(d (x-\text {$\#$1})) \text {$\#$1}+d \sinh (c+d \text {$\#$1}) \text {Shi}(d (x-\text {$\#$1})) \text {$\#$1}}{\text {$\#$1}^2}\&\right ]+d \text {RootSum}\left [a+b \text {$\#$1}^3\&,\frac {4 \cosh (c+d \text {$\#$1}) \text {Chi}(d (x-\text {$\#$1}))+4 \text {Chi}(d (x-\text {$\#$1})) \sinh (c+d \text {$\#$1})+4 \cosh (c+d \text {$\#$1}) \text {Shi}(d (x-\text {$\#$1}))+4 \sinh (c+d \text {$\#$1}) \text {Shi}(d (x-\text {$\#$1}))+d \cosh (c+d \text {$\#$1}) \text {Chi}(d (x-\text {$\#$1})) \text {$\#$1}+d \text {Chi}(d (x-\text {$\#$1})) \sinh (c+d \text {$\#$1}) \text {$\#$1}+d \cosh (c+d \text {$\#$1}) \text {Shi}(d (x-\text {$\#$1})) \text {$\#$1}+d \sinh (c+d \text {$\#$1}) \text {Shi}(d (x-\text {$\#$1})) \text {$\#$1}}{\text {$\#$1}^2}\&\right ]-\frac {6 b \left (3 \left (a+2 b x^3\right ) \cosh (c+d x)+d x \left (a+b x^3\right ) \sinh (c+d x)\right )}{\left (a+b x^3\right )^2}}{108 b^3} \]

[In]

Integrate[(x^5*Cosh[c + d*x])/(a + b*x^3)^3,x]

[Out]

(d*RootSum[a + b*#1^3 & , (-4*Cosh[c + d*#1]*CoshIntegral[d*(x - #1)] + 4*CoshIntegral[d*(x - #1)]*Sinh[c + d*
#1] + 4*Cosh[c + d*#1]*SinhIntegral[d*(x - #1)] - 4*Sinh[c + d*#1]*SinhIntegral[d*(x - #1)] + d*Cosh[c + d*#1]
*CoshIntegral[d*(x - #1)]*#1 - d*CoshIntegral[d*(x - #1)]*Sinh[c + d*#1]*#1 - d*Cosh[c + d*#1]*SinhIntegral[d*
(x - #1)]*#1 + d*Sinh[c + d*#1]*SinhIntegral[d*(x - #1)]*#1)/#1^2 & ] + d*RootSum[a + b*#1^3 & , (4*Cosh[c + d
*#1]*CoshIntegral[d*(x - #1)] + 4*CoshIntegral[d*(x - #1)]*Sinh[c + d*#1] + 4*Cosh[c + d*#1]*SinhIntegral[d*(x
- #1)] + 4*Sinh[c + d*#1]*SinhIntegral[d*(x - #1)] + d*Cosh[c + d*#1]*CoshIntegral[d*(x - #1)]*#1 + d*CoshInt
egral[d*(x - #1)]*Sinh[c + d*#1]*#1 + d*Cosh[c + d*#1]*SinhIntegral[d*(x - #1)]*#1 + d*Sinh[c + d*#1]*SinhInte
gral[d*(x - #1)]*#1)/#1^2 & ] - (6*b*(3*(a + 2*b*x^3)*Cosh[c + d*x] + d*x*(a + b*x^3)*Sinh[c + d*x]))/(a + b*x
^3)^2)/(108*b^3)

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 1.04 (sec) , antiderivative size = 6243, normalized size of antiderivative = 7.96

method	result	size
risch	\(\text {Expression too large to display}\)	\(6243\)

[In]

int(x^5*cosh(d*x+c)/(b*x^3+a)^3,x,method=_RETURNVERBOSE)

[Out]

result too large to display

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2980 vs. \(2 (562) = 1124\).

Time = 0.31 (sec) , antiderivative size = 2980, normalized size of antiderivative = 3.80 \[ \int \frac {x^5 \cosh (c+d x)}{\left (a+b x^3\right )^3} \, dx=\text {Too large to display} \]

[In]

integrate(x^5*cosh(d*x+c)/(b*x^3+a)^3,x, algorithm="fricas")

[Out]

1/216*(((a*d^3/b)^(2/3)*((b^2*x^6 + 2*a*b*x^3 + a^2 - sqrt(-3)*(b^2*x^6 + 2*a*b*x^3 + a^2))*cosh(d*x + c)^2 -
(b^2*x^6 + 2*a*b*x^3 + a^2 - sqrt(-3)*(b^2*x^6 + 2*a*b*x^3 + a^2))*sinh(d*x + c)^2) - 4*(a*d^3/b)^(1/3)*((b^2*
x^6 + 2*a*b*x^3 + a^2 + sqrt(-3)*(b^2*x^6 + 2*a*b*x^3 + a^2))*cosh(d*x + c)^2 - (b^2*x^6 + 2*a*b*x^3 + a^2 + s
qrt(-3)*(b^2*x^6 + 2*a*b*x^3 + a^2))*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*d^3/b)^(2/3)*((b^2*x^6 + 2*a*b*x^3 + a^2 - sqrt(-3)*(b^2*x^6 + 2*a*b
*x^3 + a^2))*cosh(d*x + c)^2 - (b^2*x^6 + 2*a*b*x^3 + a^2 - sqrt(-3)*(b^2*x^6 + 2*a*b*x^3 + a^2))*sinh(d*x + c
)^2) - 4*(-a*d^3/b)^(1/3)*((b^2*x^6 + 2*a*b*x^3 + a^2 + sqrt(-3)*(b^2*x^6 + 2*a*b*x^3 + a^2))*cosh(d*x + c)^2
- (b^2*x^6 + 2*a*b*x^3 + a^2 + sqrt(-3)*(b^2*x^6 + 2*a*b*x^3 + a^2))*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*d^3/b)^(2/3)*((b^2*x^6 + 2*a*b*x^3
+ a^2 + sqrt(-3)*(b^2*x^6 + 2*a*b*x^3 + a^2))*cosh(d*x + c)^2 - (b^2*x^6 + 2*a*b*x^3 + a^2 + sqrt(-3)*(b^2*x^
6 + 2*a*b*x^3 + a^2))*sinh(d*x + c)^2) - 4*(a*d^3/b)^(1/3)*((b^2*x^6 + 2*a*b*x^3 + a^2 - sqrt(-3)*(b^2*x^6 + 2
*a*b*x^3 + a^2))*cosh(d*x + c)^2 - (b^2*x^6 + 2*a*b*x^3 + a^2 - sqrt(-3)*(b^2*x^6 + 2*a*b*x^3 + a^2))*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*d^3
/b)^(2/3)*((b^2*x^6 + 2*a*b*x^3 + a^2 + sqrt(-3)*(b^2*x^6 + 2*a*b*x^3 + a^2))*cosh(d*x + c)^2 - (b^2*x^6 + 2*a
*b*x^3 + a^2 + sqrt(-3)*(b^2*x^6 + 2*a*b*x^3 + a^2))*sinh(d*x + c)^2) - 4*(-a*d^3/b)^(1/3)*((b^2*x^6 + 2*a*b*x
^3 + a^2 - sqrt(-3)*(b^2*x^6 + 2*a*b*x^3 + a^2))*cosh(d*x + c)^2 - (b^2*x^6 + 2*a*b*x^3 + a^2 - sqrt(-3)*(b^2*
x^6 + 2*a*b*x^3 + a^2))*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) - 2*((-a*d^3/b)^(2/3)*((b^2*x^6 + 2*a*b*x^3 + a^2)*cosh(d*x + c)^2 - (b^2*x^6 + 2*a*b
*x^3 + a^2)*sinh(d*x + c)^2) - 4*(-a*d^3/b)^(1/3)*((b^2*x^6 + 2*a*b*x^3 + a^2)*cosh(d*x + c)^2 - (b^2*x^6 + 2*
a*b*x^3 + a^2)*sinh(d*x + c)^2))*Ei(-d*x + (-a*d^3/b)^(1/3))*cosh(c + (-a*d^3/b)^(1/3)) - 2*((a*d^3/b)^(2/3)*(
(b^2*x^6 + 2*a*b*x^3 + a^2)*cosh(d*x + c)^2 - (b^2*x^6 + 2*a*b*x^3 + a^2)*sinh(d*x + c)^2) - 4*(a*d^3/b)^(1/3)
*((b^2*x^6 + 2*a*b*x^3 + a^2)*cosh(d*x + c)^2 - (b^2*x^6 + 2*a*b*x^3 + a^2)*sinh(d*x + c)^2))*Ei(d*x + (a*d^3/
b)^(1/3))*cosh(-c + (a*d^3/b)^(1/3)) + ((a*d^3/b)^(2/3)*((b^2*x^6 + 2*a*b*x^3 + a^2 - sqrt(-3)*(b^2*x^6 + 2*a*
b*x^3 + a^2))*cosh(d*x + c)^2 - (b^2*x^6 + 2*a*b*x^3 + a^2 - sqrt(-3)*(b^2*x^6 + 2*a*b*x^3 + a^2))*sinh(d*x +
c)^2) - 4*(a*d^3/b)^(1/3)*((b^2*x^6 + 2*a*b*x^3 + a^2 + sqrt(-3)*(b^2*x^6 + 2*a*b*x^3 + a^2))*cosh(d*x + c)^2
- (b^2*x^6 + 2*a*b*x^3 + a^2 + sqrt(-3)*(b^2*x^6 + 2*a*b*x^3 + a^2))*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*d^3/b)^(2/3)*((b^2*x^6 + 2*a*b*x^3 +
a^2 - sqrt(-3)*(b^2*x^6 + 2*a*b*x^3 + a^2))*cosh(d*x + c)^2 - (b^2*x^6 + 2*a*b*x^3 + a^2 - sqrt(-3)*(b^2*x^6
+ 2*a*b*x^3 + a^2))*sinh(d*x + c)^2) - 4*(-a*d^3/b)^(1/3)*((b^2*x^6 + 2*a*b*x^3 + a^2 + sqrt(-3)*(b^2*x^6 + 2*
a*b*x^3 + a^2))*cosh(d*x + c)^2 - (b^2*x^6 + 2*a*b*x^3 + a^2 + sqrt(-3)*(b^2*x^6 + 2*a*b*x^3 + a^2))*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*d^
3/b)^(2/3)*((b^2*x^6 + 2*a*b*x^3 + a^2 + sqrt(-3)*(b^2*x^6 + 2*a*b*x^3 + a^2))*cosh(d*x + c)^2 - (b^2*x^6 + 2*
a*b*x^3 + a^2 + sqrt(-3)*(b^2*x^6 + 2*a*b*x^3 + a^2))*sinh(d*x + c)^2) - 4*(a*d^3/b)^(1/3)*((b^2*x^6 + 2*a*b*x
^3 + a^2 - sqrt(-3)*(b^2*x^6 + 2*a*b*x^3 + a^2))*cosh(d*x + c)^2 - (b^2*x^6 + 2*a*b*x^3 + a^2 - sqrt(-3)*(b^2*
x^6 + 2*a*b*x^3 + a^2))*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*d^3/b)^(2/3)*((b^2*x^6 + 2*a*b*x^3 + a^2 + sqrt(-3)*(b^2*x^6 + 2*a*b*x^3 + a^2))*
cosh(d*x + c)^2 - (b^2*x^6 + 2*a*b*x^3 + a^2 + sqrt(-3)*(b^2*x^6 + 2*a*b*x^3 + a^2))*sinh(d*x + c)^2) - 4*(-a*
d^3/b)^(1/3)*((b^2*x^6 + 2*a*b*x^3 + a^2 - sqrt(-3)*(b^2*x^6 + 2*a*b*x^3 + a^2))*cosh(d*x + c)^2 - (b^2*x^6 +
2*a*b*x^3 + a^2 - sqrt(-3)*(b^2*x^6 + 2*a*b*x^3 + a^2))*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) + 2*((-a*d^3/b)^(2/3)*((b^2*x^6 + 2*a*b*x^3 + a^2)*co
sh(d*x + c)^2 - (b^2*x^6 + 2*a*b*x^3 + a^2)*sinh(d*x + c)^2) - 4*(-a*d^3/b)^(1/3)*((b^2*x^6 + 2*a*b*x^3 + a^2)
*cosh(d*x + c)^2 - (b^2*x^6 + 2*a*b*x^3 + a^2)*sinh(d*x + c)^2))*Ei(-d*x + (-a*d^3/b)^(1/3))*sinh(c + (-a*d^3/
b)^(1/3)) + 2*((a*d^3/b)^(2/3)*((b^2*x^6 + 2*a*b*x^3 + a^2)*cosh(d*x + c)^2 - (b^2*x^6 + 2*a*b*x^3 + a^2)*sinh
(d*x + c)^2) - 4*(a*d^3/b)^(1/3)*((b^2*x^6 + 2*a*b*x^3 + a^2)*cosh(d*x + c)^2 - (b^2*x^6 + 2*a*b*x^3 + a^2)*si
nh(d*x + c)^2))*Ei(d*x + (a*d^3/b)^(1/3))*sinh(-c + (a*d^3/b)^(1/3)) - 36*(2*a*b*x^3 + a^2)*cosh(d*x + c) - 12
*(a*b*d*x^4 + a^2*d*x)*sinh(d*x + c))/((a*b^4*x^6 + 2*a^2*b^3*x^3 + a^3*b^2)*cosh(d*x + c)^2 - (a*b^4*x^6 + 2*
a^2*b^3*x^3 + a^3*b^2)*sinh(d*x + c)^2)

Sympy [F(-1)]

Timed out. \[ \int \frac {x^5 \cosh (c+d x)}{\left (a+b x^3\right )^3} \, dx=\text {Timed out} \]

[In]

integrate(x**5*cosh(d*x+c)/(b*x**3+a)**3,x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {x^5 \cosh (c+d x)}{\left (a+b x^3\right )^3} \, dx=\int { \frac {x^{5} \cosh \left (d x + c\right )}{{\left (b x^{3} + a\right )}^{3}} \,d x } \]

[In]

integrate(x^5*cosh(d*x+c)/(b*x^3+a)^3,x, algorithm="maxima")

[Out]

1/2*((b*d^4*x^5*e^(2*c) + 4*b*d^3*x^4*e^(2*c) + 20*b*d^2*x^3*e^(2*c) + 120*b*d*x^2*e^(2*c) - 3*(3*a*d^3*e^(2*c
) - 280*b*e^(2*c))*x)*e^(d*x) - (b*d^4*x^5 - 4*b*d^3*x^4 + 20*b*d^2*x^3 - 120*b*d*x^2 + 3*(3*a*d^3 + 280*b)*x)
*e^(-d*x))/(b^4*d^5*x^9*e^c + 3*a*b^3*d^5*x^6*e^c + 3*a^2*b^2*d^5*x^3*e^c + a^3*b*d^5*e^c) - 1/2*integrate(3*(
60*a*b*d^2*x^2*e^c - 3*a^2*d^3*e^c + 4*(9*a*b*d^3*e^c - 560*b^2*e^c)*x^3 + 280*a*b*e^c - 3*(a^2*d^4*e^c - 120*
a*b*d*e^c)*x)*e^(d*x)/(b^5*d^5*x^12 + 4*a*b^4*d^5*x^9 + 6*a^2*b^3*d^5*x^6 + 4*a^3*b^2*d^5*x^3 + a^4*b*d^5), x)
- 1/2*integrate(-3*(60*a*b*d^2*x^2 + 3*a^2*d^3 - 4*(9*a*b*d^3 + 560*b^2)*x^3 + 280*a*b - 3*(a^2*d^4 + 120*a*b
*d)*x)*e^(-d*x)/(b^5*d^5*x^12*e^c + 4*a*b^4*d^5*x^9*e^c + 6*a^2*b^3*d^5*x^6*e^c + 4*a^3*b^2*d^5*x^3*e^c + a^4*
b*d^5*e^c), x)

Giac [F]

\[ \int \frac {x^5 \cosh (c+d x)}{\left (a+b x^3\right )^3} \, dx=\int { \frac {x^{5} \cosh \left (d x + c\right )}{{\left (b x^{3} + a\right )}^{3}} \,d x } \]

[In]

integrate(x^5*cosh(d*x+c)/(b*x^3+a)^3,x, algorithm="giac")

[Out]

integrate(x^5*cosh(d*x + c)/(b*x^3 + a)^3, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {x^5 \cosh (c+d x)}{\left (a+b x^3\right )^3} \, dx=\int \frac {x^5\,\mathrm {cosh}\left (c+d\,x\right )}{{\left (b\,x^3+a\right )}^3} \,d x \]

[In]

int((x^5*cosh(c + d*x))/(a + b*x^3)^3,x)

[Out]

int((x^5*cosh(c + d*x))/(a + b*x^3)^3, x)