3.106 \(\int \frac{\cosh (c+d x)}{x (a+b x^3)^2} \, dx\)

Optimal. Leaf size=697 \[ \text{result too large to display} \]

[Out]

Cosh[c + d*x]/(3*a*b*x^3) - Cosh[c + d*x]/(3*b*x^3*(a + b*x^3)) + (Cosh[c]*CoshIntegral[d*x])/a^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])/(3*a^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])/(3*a^2) - (Cosh[c - (a^(1/3)*
d)/b^(1/3)]*CoshIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(3*a^2) - (d*CoshIntegral[(a^(1/3)*d)/b^(1/3) + d*x]*Sinh
[c - (a^(1/3)*d)/b^(1/3)])/(9*a^(5/3)*b^(1/3)) + ((-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)])/(9*a^(5/3)*b^(1/3)) - ((-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)])/(9*a^(5/3)*b^(1/3)) + (Sinh[c]*SinhIntegra
l[d*x])/a^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])/(9*a^(5/3)*b^(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])/(3*a^2) - (d*Cosh[c - (a^(1/3)*d)/b^(1/3)]*SinhIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(9*a^(5/3
)*b^(1/3)) - (Sinh[c - (a^(1/3)*d)/b^(1/3)]*SinhIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(3*a^2) - ((-1)^(2/3)*d*C
osh[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*SinhIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x])/(9*a^(5/3)*b^(1/3)
) - (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])/(3*a^2)

________________________________________________________________________________________

Rubi [A]  time = 1.47159, antiderivative size = 697, normalized size of antiderivative = 1., number of steps used = 41, number of rules used = 8, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.421, Rules used = {5291, 5293, 3297, 3303, 3298, 3301, 5292, 5280} \[ -\frac{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 )}{9 a^{5/3} \sqrt [3]{b}}+\frac{\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 )}{9 a^{5/3} \sqrt [3]{b}}-\frac{(-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 )}{9 a^{5/3} \sqrt [3]{b}}-\frac{\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 )}{3 a^2}-\frac{\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 )}{3 a^2}-\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 )}{3 a^2}+\frac{\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 )}{3 a^2}-\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 )}{3 a^2}-\frac{\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 )}{3 a^2}-\frac{\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 )}{9 a^{5/3} \sqrt [3]{b}}-\frac{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 )}{9 a^{5/3} \sqrt [3]{b}}-\frac{(-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 )}{9 a^{5/3} \sqrt [3]{b}}+\frac{\cosh (c) \text{Chi}(d x)}{a^2}+\frac{\sinh (c) \text{Shi}(d x)}{a^2}-\frac{\cosh (c+d x)}{3 b x^3 \left (a+b x^3\right )}+\frac{\cosh (c+d x)}{3 a b x^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

Cosh[c + d*x]/(3*a*b*x^3) - Cosh[c + d*x]/(3*b*x^3*(a + b*x^3)) + (Cosh[c]*CoshIntegral[d*x])/a^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])/(3*a^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])/(3*a^2) - (Cosh[c - (a^(1/3)*
d)/b^(1/3)]*CoshIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(3*a^2) - (d*CoshIntegral[(a^(1/3)*d)/b^(1/3) + d*x]*Sinh
[c - (a^(1/3)*d)/b^(1/3)])/(9*a^(5/3)*b^(1/3)) + ((-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)])/(9*a^(5/3)*b^(1/3)) - ((-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)])/(9*a^(5/3)*b^(1/3)) + (Sinh[c]*SinhIntegra
l[d*x])/a^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])/(9*a^(5/3)*b^(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])/(3*a^2) - (d*Cosh[c - (a^(1/3)*d)/b^(1/3)]*SinhIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(9*a^(5/3
)*b^(1/3)) - (Sinh[c - (a^(1/3)*d)/b^(1/3)]*SinhIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(3*a^2) - ((-1)^(2/3)*d*C
osh[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*SinhIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x])/(9*a^(5/3)*b^(1/3)
) - (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])/(3*a^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 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])

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 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 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 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 5280

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])

Rubi steps

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

Mathematica [C]  time = 8.61079, size = 5530, normalized size = 7.93 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

Result too large to show

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Maple [C]  time = 0.072, size = 338, normalized size = 0.5 \begin{align*}{\frac{{{\rm e}^{-dx-c}}{d}^{3}}{6\,a \left ( \left ( dx+c \right ) ^{3}b-3\, \left ( dx+c \right ) ^{2}bc+3\, \left ( dx+c \right ) b{c}^{2}+a{d}^{3}-b{c}^{3} \right ) }}-{\frac{{{\rm e}^{-c}}{\it Ei} \left ( 1,dx \right ) }{2\,{a}^{2}}}+{\frac{1}{18\,{a}^{2}b}\sum _{{\it \_R1}={\it RootOf} \left ( b{{\it \_Z}}^{3}-3\,{{\it \_Z}}^{2}bc+3\,{\it \_Z}\,b{c}^{2}+a{d}^{3}-b{c}^{3} \right ) }{\frac{ \left ( -a{d}^{3}+3\,{{\it \_R1}}^{2}b-6\,{\it \_R1}\,bc+3\,b{c}^{2} \right ){{\rm e}^{-{\it \_R1}}}{\it Ei} \left ( 1,dx-{\it \_R1}+c \right ) }{{{\it \_R1}}^{2}-2\,{\it \_R1}\,c+{c}^{2}}}}+{\frac{{{\rm e}^{dx+c}}{d}^{3}}{6\,a \left ( \left ( dx+c \right ) ^{3}b-3\, \left ( dx+c \right ) ^{2}bc+3\, \left ( dx+c \right ) b{c}^{2}+a{d}^{3}-b{c}^{3} \right ) }}-{\frac{{{\rm e}^{c}}{\it Ei} \left ( 1,-dx \right ) }{2\,{a}^{2}}}+{\frac{1}{18\,{a}^{2}b}\sum _{{\it \_R1}={\it RootOf} \left ( b{{\it \_Z}}^{3}-3\,{{\it \_Z}}^{2}bc+3\,{\it \_Z}\,b{c}^{2}+a{d}^{3}-b{c}^{3} \right ) }{\frac{ \left ( a{d}^{3}+3\,{{\it \_R1}}^{2}b-6\,{\it \_R1}\,bc+3\,b{c}^{2} \right ){{\rm e}^{{\it \_R1}}}{\it Ei} \left ( 1,-dx+{\it \_R1}-c \right ) }{{{\it \_R1}}^{2}-2\,{\it \_R1}\,c+{c}^{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*exp(-d*x-c)*d^3/a/((d*x+c)^3*b-3*(d*x+c)^2*b*c+3*(d*x+c)*b*c^2+a*d^3-b*c^3)-1/2/a^2*exp(-c)*Ei(1,d*x)+1/18
/a^2/b*sum((-a*d^3+3*_R1^2*b-6*_R1*b*c+3*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))+1/6*exp(d*x+c)*d^3/a/((d*x+c)^3*b-3*(d*x+c)^2*b*c+3*(d*x+c)*b*c^2+a*d^3-
b*c^3)-1/2/a^2*exp(c)*Ei(1,-d*x)+1/18/a^2/b*sum((a*d^3+3*_R1^2*b-6*_R1*b*c+3*b*c^2)/(_R1^2-2*_R1*c+c^2)*exp(_R
1)*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(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.32726, size = 4420, normalized size = 6.34 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/36*((6*(b*x^3 + a)*cosh(d*x + c)^2 - 6*(b*x^3 + a)*sinh(d*x + c)^2 - (a*d^3/b)^(1/3)*((b*x^3 + sqrt(-3)*(b*
x^3 + a) + a)*cosh(d*x + c)^2 - (b*x^3 + sqrt(-3)*(b*x^3 + a) + a)*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) + (6*(b*x^3 + a)*cosh(d*x + c)^2 - 6*(b*x^3 +
a)*sinh(d*x + c)^2 - (-a*d^3/b)^(1/3)*((b*x^3 + sqrt(-3)*(b*x^3 + a) + a)*cosh(d*x + c)^2 - (b*x^3 + sqrt(-3)
*(b*x^3 + a) + a)*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) + (6*(b*x^3 + a)*cosh(d*x + c)^2 - 6*(b*x^3 + a)*sinh(d*x + c)^2 - (a*d^3/b)^(1/3)*((b*x^3
- sqrt(-3)*(b*x^3 + a) + a)*cosh(d*x + c)^2 - (b*x^3 - sqrt(-3)*(b*x^3 + a) + a)*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) + (6*(b*x^3 + a)*cosh(d*x + c)^
2 - 6*(b*x^3 + a)*sinh(d*x + c)^2 - (-a*d^3/b)^(1/3)*((b*x^3 - sqrt(-3)*(b*x^3 + a) + a)*cosh(d*x + c)^2 - (b*
x^3 - sqrt(-3)*(b*x^3 + a) + a)*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*(3*(b*x^3 + a)*cosh(d*x + c)^2 - 3*(b*x^3 + a)*sinh(d*x + c)^2 + (-a*d^3/
b)^(1/3)*((b*x^3 + a)*cosh(d*x + c)^2 - (b*x^3 + a)*sinh(d*x + c)^2))*Ei(-d*x + (-a*d^3/b)^(1/3))*cosh(c + (-a
*d^3/b)^(1/3)) + 2*(3*(b*x^3 + a)*cosh(d*x + c)^2 - 3*(b*x^3 + a)*sinh(d*x + c)^2 + (a*d^3/b)^(1/3)*((b*x^3 +
a)*cosh(d*x + c)^2 - (b*x^3 + a)*sinh(d*x + c)^2))*Ei(d*x + (a*d^3/b)^(1/3))*cosh(-c + (a*d^3/b)^(1/3)) + (6*(
b*x^3 + a)*cosh(d*x + c)^2 - 6*(b*x^3 + a)*sinh(d*x + c)^2 - (a*d^3/b)^(1/3)*((b*x^3 + sqrt(-3)*(b*x^3 + a) +
a)*cosh(d*x + c)^2 - (b*x^3 + sqrt(-3)*(b*x^3 + a) + a)*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) + (6*(b*x^3 + a)*cosh(d*x + c)^2 - 6*(b*x^3 + a)*sinh(d*
x + c)^2 - (-a*d^3/b)^(1/3)*((b*x^3 + sqrt(-3)*(b*x^3 + a) + a)*cosh(d*x + c)^2 - (b*x^3 + sqrt(-3)*(b*x^3 + a
) + a)*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) - (6*(b*x^3 + a)*cosh(d*x + c)^2 - 6*(b*x^3 + a)*sinh(d*x + c)^2 - (a*d^3/b)^(1/3)*((b*x^3 - sqrt(-3)*
(b*x^3 + a) + a)*cosh(d*x + c)^2 - (b*x^3 - sqrt(-3)*(b*x^3 + a) + a)*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) - (6*(b*x^3 + a)*cosh(d*x + c)^2 - 6*(b*x^
3 + a)*sinh(d*x + c)^2 - (-a*d^3/b)^(1/3)*((b*x^3 - sqrt(-3)*(b*x^3 + a) + a)*cosh(d*x + c)^2 - (b*x^3 - sqrt(
-3)*(b*x^3 + a) + a)*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*(3*(b*x^3 + a)*cosh(d*x + c)^2 - 3*(b*x^3 + a)*sinh(d*x + c)^2 + (-a*d^3/b)^(1/3)*((
b*x^3 + a)*cosh(d*x + c)^2 - (b*x^3 + a)*sinh(d*x + c)^2))*Ei(-d*x + (-a*d^3/b)^(1/3))*sinh(c + (-a*d^3/b)^(1/
3)) - 2*(3*(b*x^3 + a)*cosh(d*x + c)^2 - 3*(b*x^3 + a)*sinh(d*x + c)^2 + (a*d^3/b)^(1/3)*((b*x^3 + a)*cosh(d*x
+ c)^2 - (b*x^3 + a)*sinh(d*x + c)^2))*Ei(d*x + (a*d^3/b)^(1/3))*sinh(-c + (a*d^3/b)^(1/3)) - 12*a*cosh(d*x +
c) - 18*((b*x^3 + a)*Ei(d*x) + (b*x^3 + a)*Ei(-d*x))*cosh(c) - 18*((b*x^3 + a)*Ei(d*x) - (b*x^3 + a)*Ei(-d*x)
)*sinh(c))/((a^2*b*x^3 + a^3)*cosh(d*x + c)^2 - (a^2*b*x^3 + a^3)*sinh(d*x + c)^2)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cosh \left (d x + c\right )}{{\left (b x^{3} + a\right )}^{2} x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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