\(\int \frac {(a+b x^3)^2 \cosh (c+d x)}{x^2} \, dx\) [90]

Optimal result

Integrand size = 19, antiderivative size = 143 \[ \int \frac {\left (a+b x^3\right )^2 \cosh (c+d x)}{x^2} \, dx=-\frac {2 a b \cosh (c+d x)}{d^2}-\frac {a^2 \cosh (c+d x)}{x}-\frac {24 b^2 x \cosh (c+d x)}{d^4}-\frac {4 b^2 x^3 \cosh (c+d x)}{d^2}+a^2 d \text {Chi}(d x) \sinh (c)+\frac {24 b^2 \sinh (c+d x)}{d^5}+\frac {2 a b x \sinh (c+d x)}{d}+\frac {12 b^2 x^2 \sinh (c+d x)}{d^3}+\frac {b^2 x^4 \sinh (c+d x)}{d}+a^2 d \cosh (c) \text {Shi}(d x) \]

[Out]

-2*a*b*cosh(d*x+c)/d^2-a^2*cosh(d*x+c)/x-24*b^2*x*cosh(d*x+c)/d^4-4*b^2*x^3*cosh(d*x+c)/d^2+a^2*d*cosh(c)*Shi(
d*x)+a^2*d*Chi(d*x)*sinh(c)+24*b^2*sinh(d*x+c)/d^5+2*a*b*x*sinh(d*x+c)/d+12*b^2*x^2*sinh(d*x+c)/d^3+b^2*x^4*si
nh(d*x+c)/d

Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {5395, 3378, 3384, 3379, 3382, 3377, 2718, 2717} \[ \int \frac {\left (a+b x^3\right )^2 \cosh (c+d x)}{x^2} \, dx=a^2 d \sinh (c) \text {Chi}(d x)+a^2 d \cosh (c) \text {Shi}(d x)-\frac {a^2 \cosh (c+d x)}{x}-\frac {2 a b \cosh (c+d x)}{d^2}+\frac {2 a b x \sinh (c+d x)}{d}+\frac {24 b^2 \sinh (c+d x)}{d^5}-\frac {24 b^2 x \cosh (c+d x)}{d^4}+\frac {12 b^2 x^2 \sinh (c+d x)}{d^3}-\frac {4 b^2 x^3 \cosh (c+d x)}{d^2}+\frac {b^2 x^4 \sinh (c+d x)}{d} \]

[In]

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

[Out]

(-2*a*b*Cosh[c + d*x])/d^2 - (a^2*Cosh[c + d*x])/x - (24*b^2*x*Cosh[c + d*x])/d^4 - (4*b^2*x^3*Cosh[c + d*x])/
d^2 + a^2*d*CoshIntegral[d*x]*Sinh[c] + (24*b^2*Sinh[c + d*x])/d^5 + (2*a*b*x*Sinh[c + d*x])/d + (12*b^2*x^2*S
inh[c + d*x])/d^3 + (b^2*x^4*Sinh[c + d*x])/d + a^2*d*Cosh[c]*SinhIntegral[d*x]

Rule 2717

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 2718

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

Rule 3377

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[(-(c + d*x)^m)*(Cos[e + f*x]/f), x]
+ Dist[d*(m/f), Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 3378

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 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 5395

Int[Cosh[(c_.) + (d_.)*(x_)]*((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegra
nd[Cosh[c + d*x], (e*x)^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && IGtQ[p, 0]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {a^2 \cosh (c+d x)}{x^2}+2 a b x \cosh (c+d x)+b^2 x^4 \cosh (c+d x)\right ) \, dx \\ & = a^2 \int \frac {\cosh (c+d x)}{x^2} \, dx+(2 a b) \int x \cosh (c+d x) \, dx+b^2 \int x^4 \cosh (c+d x) \, dx \\ & = -\frac {a^2 \cosh (c+d x)}{x}+\frac {2 a b x \sinh (c+d x)}{d}+\frac {b^2 x^4 \sinh (c+d x)}{d}-\frac {(2 a b) \int \sinh (c+d x) \, dx}{d}-\frac {\left (4 b^2\right ) \int x^3 \sinh (c+d x) \, dx}{d}+\left (a^2 d\right ) \int \frac {\sinh (c+d x)}{x} \, dx \\ & = -\frac {2 a b \cosh (c+d x)}{d^2}-\frac {a^2 \cosh (c+d x)}{x}-\frac {4 b^2 x^3 \cosh (c+d x)}{d^2}+\frac {2 a b x \sinh (c+d x)}{d}+\frac {b^2 x^4 \sinh (c+d x)}{d}+\frac {\left (12 b^2\right ) \int x^2 \cosh (c+d x) \, dx}{d^2}+\left (a^2 d \cosh (c)\right ) \int \frac {\sinh (d x)}{x} \, dx+\left (a^2 d \sinh (c)\right ) \int \frac {\cosh (d x)}{x} \, dx \\ & = -\frac {2 a b \cosh (c+d x)}{d^2}-\frac {a^2 \cosh (c+d x)}{x}-\frac {4 b^2 x^3 \cosh (c+d x)}{d^2}+a^2 d \text {Chi}(d x) \sinh (c)+\frac {2 a b x \sinh (c+d x)}{d}+\frac {12 b^2 x^2 \sinh (c+d x)}{d^3}+\frac {b^2 x^4 \sinh (c+d x)}{d}+a^2 d \cosh (c) \text {Shi}(d x)-\frac {\left (24 b^2\right ) \int x \sinh (c+d x) \, dx}{d^3} \\ & = -\frac {2 a b \cosh (c+d x)}{d^2}-\frac {a^2 \cosh (c+d x)}{x}-\frac {24 b^2 x \cosh (c+d x)}{d^4}-\frac {4 b^2 x^3 \cosh (c+d x)}{d^2}+a^2 d \text {Chi}(d x) \sinh (c)+\frac {2 a b x \sinh (c+d x)}{d}+\frac {12 b^2 x^2 \sinh (c+d x)}{d^3}+\frac {b^2 x^4 \sinh (c+d x)}{d}+a^2 d \cosh (c) \text {Shi}(d x)+\frac {\left (24 b^2\right ) \int \cosh (c+d x) \, dx}{d^4} \\ & = -\frac {2 a b \cosh (c+d x)}{d^2}-\frac {a^2 \cosh (c+d x)}{x}-\frac {24 b^2 x \cosh (c+d x)}{d^4}-\frac {4 b^2 x^3 \cosh (c+d x)}{d^2}+a^2 d \text {Chi}(d x) \sinh (c)+\frac {24 b^2 \sinh (c+d x)}{d^5}+\frac {2 a b x \sinh (c+d x)}{d}+\frac {12 b^2 x^2 \sinh (c+d x)}{d^3}+\frac {b^2 x^4 \sinh (c+d x)}{d}+a^2 d \cosh (c) \text {Shi}(d x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.26 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b x^3\right )^2 \cosh (c+d x)}{x^2} \, dx=-\frac {2 a b \cosh (c+d x)}{d^2}-\frac {a^2 \cosh (c+d x)}{x}-\frac {24 b^2 x \cosh (c+d x)}{d^4}-\frac {4 b^2 x^3 \cosh (c+d x)}{d^2}+a^2 d \text {Chi}(d x) \sinh (c)+\frac {24 b^2 \sinh (c+d x)}{d^5}+\frac {2 a b x \sinh (c+d x)}{d}+\frac {12 b^2 x^2 \sinh (c+d x)}{d^3}+\frac {b^2 x^4 \sinh (c+d x)}{d}+a^2 d \cosh (c) \text {Shi}(d x) \]

[In]

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

[Out]

(-2*a*b*Cosh[c + d*x])/d^2 - (a^2*Cosh[c + d*x])/x - (24*b^2*x*Cosh[c + d*x])/d^4 - (4*b^2*x^3*Cosh[c + d*x])/
d^2 + a^2*d*CoshIntegral[d*x]*Sinh[c] + (24*b^2*Sinh[c + d*x])/d^5 + (2*a*b*x*Sinh[c + d*x])/d + (12*b^2*x^2*S
inh[c + d*x])/d^3 + (b^2*x^4*Sinh[c + d*x])/d + a^2*d*Cosh[c]*SinhIntegral[d*x]

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.25 (sec) , antiderivative size = 306, normalized size of antiderivative = 2.14

method	result	size
meijerg	\(-\frac {16 i b^{2} \cosh \left (c \right ) \sqrt {\pi }\, \left (-\frac {i x d \left (\frac {5 x^{2} d^{2}}{2}+15\right ) \cosh \left (d x \right )}{10 \sqrt {\pi }}+\frac {i \left (\frac {5}{8} d^{4} x^{4}+\frac {15}{2} x^{2} d^{2}+15\right ) \sinh \left (d x \right )}{10 \sqrt {\pi }}\right )}{d^{5}}-\frac {16 b^{2} \sinh \left (c \right ) \sqrt {\pi }\, \left (\frac {3}{2 \sqrt {\pi }}-\frac {\left (\frac {3}{8} d^{4} x^{4}+\frac {9}{2} x^{2} d^{2}+9\right ) \cosh \left (d x \right )}{6 \sqrt {\pi }}+\frac {x d \left (\frac {3 x^{2} d^{2}}{2}+9\right ) \sinh \left (d x \right )}{6 \sqrt {\pi }}\right )}{d^{5}}-\frac {4 b a \cosh \left (c \right ) \sqrt {\pi }\, \left (-\frac {1}{2 \sqrt {\pi }}+\frac {\cosh \left (d x \right )}{2 \sqrt {\pi }}-\frac {d x \sinh \left (d x \right )}{2 \sqrt {\pi }}\right )}{d^{2}}+\frac {2 b a \sinh \left (c \right ) \left (\cosh \left (d x \right ) x d -\sinh \left (d x \right )\right )}{d^{2}}+\frac {i a^{2} \cosh \left (c \right ) \sqrt {\pi }\, d \left (\frac {4 i \cosh \left (d x \right )}{d x \sqrt {\pi }}-\frac {4 i \operatorname {Shi}\left (d x \right )}{\sqrt {\pi }}\right )}{4}+\frac {a^{2} \sinh \left (c \right ) \sqrt {\pi }\, d \left (\frac {4 \gamma -4+4 \ln \left (x \right )+4 \ln \left (i d \right )}{\sqrt {\pi }}+\frac {4}{\sqrt {\pi }}-\frac {4 \sinh \left (d x \right )}{\sqrt {\pi }\, x d}+\frac {4 \,\operatorname {Chi}\left (d x \right )-4 \ln \left (d x \right )-4 \gamma }{\sqrt {\pi }}\right )}{4}\)	\(306\)
risch	\(-\frac {{\mathrm e}^{-d x -c} b^{2} d^{4} x^{5}-{\mathrm e}^{d x +c} b^{2} d^{4} x^{5}-{\mathrm e}^{-c} \operatorname {Ei}_{1}\left (d x \right ) a^{2} d^{6} x +{\mathrm e}^{c} \operatorname {Ei}_{1}\left (-d x \right ) a^{2} d^{6} x +4 \,{\mathrm e}^{-d x -c} b^{2} d^{3} x^{4}+4 \,{\mathrm e}^{d x +c} b^{2} d^{3} x^{4}+2 \,{\mathrm e}^{-d x -c} a b \,d^{4} x^{2}-2 \,{\mathrm e}^{d x +c} a b \,d^{4} x^{2}+{\mathrm e}^{-d x -c} a^{2} d^{5}+12 d^{2} {\mathrm e}^{-d x -c} b^{2} x^{3}+{\mathrm e}^{d x +c} a^{2} d^{5}-12 d^{2} {\mathrm e}^{d x +c} b^{2} x^{3}+2 d^{3} {\mathrm e}^{-d x -c} a b x +2 d^{3} {\mathrm e}^{d x +c} a b x +24 d \,{\mathrm e}^{-d x -c} b^{2} x^{2}+24 d \,{\mathrm e}^{d x +c} b^{2} x^{2}+24 \,{\mathrm e}^{-d x -c} b^{2} x -24 \,{\mathrm e}^{d x +c} b^{2} x}{2 d^{5} x}\)	\(309\)

[In]

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

[Out]

-16*I/d^5*b^2*cosh(c)*Pi^(1/2)*(-1/10*I/Pi^(1/2)*x*d*(5/2*x^2*d^2+15)*cosh(d*x)+1/10*I/Pi^(1/2)*(5/8*d^4*x^4+1
5/2*x^2*d^2+15)*sinh(d*x))-16/d^5*b^2*sinh(c)*Pi^(1/2)*(3/2/Pi^(1/2)-1/6/Pi^(1/2)*(3/8*d^4*x^4+9/2*x^2*d^2+9)*
cosh(d*x)+1/6/Pi^(1/2)*x*d*(3/2*x^2*d^2+9)*sinh(d*x))-4*b/d^2*a*cosh(c)*Pi^(1/2)*(-1/2/Pi^(1/2)+1/2/Pi^(1/2)*c
osh(d*x)-1/2/Pi^(1/2)*d*x*sinh(d*x))+2*b/d^2*a*sinh(c)*(cosh(d*x)*x*d-sinh(d*x))+1/4*I*a^2*cosh(c)*Pi^(1/2)*d*
(4*I/d/x*cosh(d*x)/Pi^(1/2)-4*I/Pi^(1/2)*Shi(d*x))+1/4*a^2*sinh(c)*Pi^(1/2)*d*(2*(2*gamma-2+2*ln(x)+2*ln(I*d))
/Pi^(1/2)+4/Pi^(1/2)-4/Pi^(1/2)/x/d*sinh(d*x)+4/Pi^(1/2)*(Chi(d*x)-ln(d*x)-gamma))

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.12 \[ \int \frac {\left (a+b x^3\right )^2 \cosh (c+d x)}{x^2} \, dx=-\frac {2 \, {\left (4 \, b^{2} d^{3} x^{4} + a^{2} d^{5} + 2 \, a b d^{3} x + 24 \, b^{2} d x^{2}\right )} \cosh \left (d x + c\right ) - {\left (a^{2} d^{6} x {\rm Ei}\left (d x\right ) - a^{2} d^{6} x {\rm Ei}\left (-d x\right )\right )} \cosh \left (c\right ) - 2 \, {\left (b^{2} d^{4} x^{5} + 2 \, a b d^{4} x^{2} + 12 \, b^{2} d^{2} x^{3} + 24 \, b^{2} x\right )} \sinh \left (d x + c\right ) - {\left (a^{2} d^{6} x {\rm Ei}\left (d x\right ) + a^{2} d^{6} x {\rm Ei}\left (-d x\right )\right )} \sinh \left (c\right )}{2 \, d^{5} x} \]

[In]

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

[Out]

-1/2*(2*(4*b^2*d^3*x^4 + a^2*d^5 + 2*a*b*d^3*x + 24*b^2*d*x^2)*cosh(d*x + c) - (a^2*d^6*x*Ei(d*x) - a^2*d^6*x*
Ei(-d*x))*cosh(c) - 2*(b^2*d^4*x^5 + 2*a*b*d^4*x^2 + 12*b^2*d^2*x^3 + 24*b^2*x)*sinh(d*x + c) - (a^2*d^6*x*Ei(
d*x) + a^2*d^6*x*Ei(-d*x))*sinh(c))/(d^5*x)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.64 \[ \int \frac {\left (a+b x^3\right )^2 \cosh (c+d x)}{x^2} \, dx=-\frac {1}{10} \, {\left (5 \, a^{2} {\rm Ei}\left (-d x\right ) e^{\left (-c\right )} - 5 \, a^{2} {\rm Ei}\left (d x\right ) e^{c} + \frac {5 \, {\left (d^{2} x^{2} e^{c} - 2 \, d x e^{c} + 2 \, e^{c}\right )} a b e^{\left (d x\right )}}{d^{3}} + \frac {5 \, {\left (d^{2} x^{2} + 2 \, d x + 2\right )} a b e^{\left (-d x - c\right )}}{d^{3}} + \frac {{\left (d^{5} x^{5} e^{c} - 5 \, d^{4} x^{4} e^{c} + 20 \, d^{3} x^{3} e^{c} - 60 \, d^{2} x^{2} e^{c} + 120 \, d x e^{c} - 120 \, e^{c}\right )} b^{2} e^{\left (d x\right )}}{d^{6}} + \frac {{\left (d^{5} x^{5} + 5 \, d^{4} x^{4} + 20 \, d^{3} x^{3} + 60 \, d^{2} x^{2} + 120 \, d x + 120\right )} b^{2} e^{\left (-d x - c\right )}}{d^{6}}\right )} d + \frac {1}{5} \, {\left (b^{2} x^{5} + 5 \, a b x^{2} - \frac {5 \, a^{2}}{x}\right )} \cosh \left (d x + c\right ) \]

[In]

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

[Out]

-1/10*(5*a^2*Ei(-d*x)*e^(-c) - 5*a^2*Ei(d*x)*e^c + 5*(d^2*x^2*e^c - 2*d*x*e^c + 2*e^c)*a*b*e^(d*x)/d^3 + 5*(d^
2*x^2 + 2*d*x + 2)*a*b*e^(-d*x - c)/d^3 + (d^5*x^5*e^c - 5*d^4*x^4*e^c + 20*d^3*x^3*e^c - 60*d^2*x^2*e^c + 120
*d*x*e^c - 120*e^c)*b^2*e^(d*x)/d^6 + (d^5*x^5 + 5*d^4*x^4 + 20*d^3*x^3 + 60*d^2*x^2 + 120*d*x + 120)*b^2*e^(-
d*x - c)/d^6)*d + 1/5*(b^2*x^5 + 5*a*b*x^2 - 5*a^2/x)*cosh(d*x + c)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 308 vs. \(2 (143) = 286\).

Time = 0.26 (sec) , antiderivative size = 308, normalized size of antiderivative = 2.15 \[ \int \frac {\left (a+b x^3\right )^2 \cosh (c+d x)}{x^2} \, dx=\frac {b^{2} d^{4} x^{5} e^{\left (d x + c\right )} - b^{2} d^{4} x^{5} e^{\left (-d x - c\right )} - a^{2} d^{6} x {\rm Ei}\left (-d x\right ) e^{\left (-c\right )} + a^{2} d^{6} x {\rm Ei}\left (d x\right ) e^{c} - 4 \, b^{2} d^{3} x^{4} e^{\left (d x + c\right )} - 4 \, b^{2} d^{3} x^{4} e^{\left (-d x - c\right )} + 2 \, a b d^{4} x^{2} e^{\left (d x + c\right )} - 2 \, a b d^{4} x^{2} e^{\left (-d x - c\right )} - a^{2} d^{5} e^{\left (d x + c\right )} + 12 \, b^{2} d^{2} x^{3} e^{\left (d x + c\right )} - a^{2} d^{5} e^{\left (-d x - c\right )} - 12 \, b^{2} d^{2} x^{3} e^{\left (-d x - c\right )} - 2 \, a b d^{3} x e^{\left (d x + c\right )} - 2 \, a b d^{3} x e^{\left (-d x - c\right )} - 24 \, b^{2} d x^{2} e^{\left (d x + c\right )} - 24 \, b^{2} d x^{2} e^{\left (-d x - c\right )} + 24 \, b^{2} x e^{\left (d x + c\right )} - 24 \, b^{2} x e^{\left (-d x - c\right )}}{2 \, d^{5} x} \]

[In]

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

[Out]

1/2*(b^2*d^4*x^5*e^(d*x + c) - b^2*d^4*x^5*e^(-d*x - c) - a^2*d^6*x*Ei(-d*x)*e^(-c) + a^2*d^6*x*Ei(d*x)*e^c -
4*b^2*d^3*x^4*e^(d*x + c) - 4*b^2*d^3*x^4*e^(-d*x - c) + 2*a*b*d^4*x^2*e^(d*x + c) - 2*a*b*d^4*x^2*e^(-d*x - c
) - a^2*d^5*e^(d*x + c) + 12*b^2*d^2*x^3*e^(d*x + c) - a^2*d^5*e^(-d*x - c) - 12*b^2*d^2*x^3*e^(-d*x - c) - 2*
a*b*d^3*x*e^(d*x + c) - 2*a*b*d^3*x*e^(-d*x - c) - 24*b^2*d*x^2*e^(d*x + c) - 24*b^2*d*x^2*e^(-d*x - c) + 24*b
^2*x*e^(d*x + c) - 24*b^2*x*e^(-d*x - c))/(d^5*x)

Mupad [F(-1)]

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

[In]

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

[Out]

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