\(\int \frac {(a+b x^3)^2 \cosh (c+d x)}{x} \, dx\) [89]

Optimal result

Integrand size = 19, antiderivative size = 160 \[ \int \frac {\left (a+b x^3\right )^2 \cosh (c+d x)}{x} \, dx=-\frac {120 b^2 \cosh (c+d x)}{d^6}-\frac {4 a b x \cosh (c+d x)}{d^2}-\frac {60 b^2 x^2 \cosh (c+d x)}{d^4}-\frac {5 b^2 x^4 \cosh (c+d x)}{d^2}+a^2 \cosh (c) \text {Chi}(d x)+\frac {4 a b \sinh (c+d x)}{d^3}+\frac {120 b^2 x \sinh (c+d x)}{d^5}+\frac {2 a b x^2 \sinh (c+d x)}{d}+\frac {20 b^2 x^3 \sinh (c+d x)}{d^3}+\frac {b^2 x^5 \sinh (c+d x)}{d}+a^2 \sinh (c) \text {Shi}(d x) \]

[Out]

a^2*Chi(d*x)*cosh(c)-120*b^2*cosh(d*x+c)/d^6-4*a*b*x*cosh(d*x+c)/d^2-60*b^2*x^2*cosh(d*x+c)/d^4-5*b^2*x^4*cosh
(d*x+c)/d^2+a^2*Shi(d*x)*sinh(c)+4*a*b*sinh(d*x+c)/d^3+120*b^2*x*sinh(d*x+c)/d^5+2*a*b*x^2*sinh(d*x+c)/d+20*b^
2*x^3*sinh(d*x+c)/d^3+b^2*x^5*sinh(d*x+c)/d

Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {5395, 3384, 3379, 3382, 3377, 2717, 2718} \[ \int \frac {\left (a+b x^3\right )^2 \cosh (c+d x)}{x} \, dx=a^2 \cosh (c) \text {Chi}(d x)+a^2 \sinh (c) \text {Shi}(d x)+\frac {4 a b \sinh (c+d x)}{d^3}-\frac {4 a b x \cosh (c+d x)}{d^2}+\frac {2 a b x^2 \sinh (c+d x)}{d}-\frac {120 b^2 \cosh (c+d x)}{d^6}+\frac {120 b^2 x \sinh (c+d x)}{d^5}-\frac {60 b^2 x^2 \cosh (c+d x)}{d^4}+\frac {20 b^2 x^3 \sinh (c+d x)}{d^3}-\frac {5 b^2 x^4 \cosh (c+d x)}{d^2}+\frac {b^2 x^5 \sinh (c+d x)}{d} \]

[In]

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

[Out]

(-120*b^2*Cosh[c + d*x])/d^6 - (4*a*b*x*Cosh[c + d*x])/d^2 - (60*b^2*x^2*Cosh[c + d*x])/d^4 - (5*b^2*x^4*Cosh[
c + d*x])/d^2 + a^2*Cosh[c]*CoshIntegral[d*x] + (4*a*b*Sinh[c + d*x])/d^3 + (120*b^2*x*Sinh[c + d*x])/d^5 + (2
*a*b*x^2*Sinh[c + d*x])/d + (20*b^2*x^3*Sinh[c + d*x])/d^3 + (b^2*x^5*Sinh[c + d*x])/d + a^2*Sinh[c]*SinhInteg
ral[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 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 a b x^2 \cosh (c+d x)+b^2 x^5 \cosh (c+d x)\right ) \, dx \\ & = a^2 \int \frac {\cosh (c+d x)}{x} \, dx+(2 a b) \int x^2 \cosh (c+d x) \, dx+b^2 \int x^5 \cosh (c+d x) \, dx \\ & = \frac {2 a b x^2 \sinh (c+d x)}{d}+\frac {b^2 x^5 \sinh (c+d x)}{d}-\frac {(4 a b) \int x \sinh (c+d x) \, dx}{d}-\frac {\left (5 b^2\right ) \int x^4 \sinh (c+d x) \, dx}{d}+\left (a^2 \cosh (c)\right ) \int \frac {\cosh (d x)}{x} \, dx+\left (a^2 \sinh (c)\right ) \int \frac {\sinh (d x)}{x} \, dx \\ & = -\frac {4 a b x \cosh (c+d x)}{d^2}-\frac {5 b^2 x^4 \cosh (c+d x)}{d^2}+a^2 \cosh (c) \text {Chi}(d x)+\frac {2 a b x^2 \sinh (c+d x)}{d}+\frac {b^2 x^5 \sinh (c+d x)}{d}+a^2 \sinh (c) \text {Shi}(d x)+\frac {(4 a b) \int \cosh (c+d x) \, dx}{d^2}+\frac {\left (20 b^2\right ) \int x^3 \cosh (c+d x) \, dx}{d^2} \\ & = -\frac {4 a b x \cosh (c+d x)}{d^2}-\frac {5 b^2 x^4 \cosh (c+d x)}{d^2}+a^2 \cosh (c) \text {Chi}(d x)+\frac {4 a b \sinh (c+d x)}{d^3}+\frac {2 a b x^2 \sinh (c+d x)}{d}+\frac {20 b^2 x^3 \sinh (c+d x)}{d^3}+\frac {b^2 x^5 \sinh (c+d x)}{d}+a^2 \sinh (c) \text {Shi}(d x)-\frac {\left (60 b^2\right ) \int x^2 \sinh (c+d x) \, dx}{d^3} \\ & = -\frac {4 a b x \cosh (c+d x)}{d^2}-\frac {60 b^2 x^2 \cosh (c+d x)}{d^4}-\frac {5 b^2 x^4 \cosh (c+d x)}{d^2}+a^2 \cosh (c) \text {Chi}(d x)+\frac {4 a b \sinh (c+d x)}{d^3}+\frac {2 a b x^2 \sinh (c+d x)}{d}+\frac {20 b^2 x^3 \sinh (c+d x)}{d^3}+\frac {b^2 x^5 \sinh (c+d x)}{d}+a^2 \sinh (c) \text {Shi}(d x)+\frac {\left (120 b^2\right ) \int x \cosh (c+d x) \, dx}{d^4} \\ & = -\frac {4 a b x \cosh (c+d x)}{d^2}-\frac {60 b^2 x^2 \cosh (c+d x)}{d^4}-\frac {5 b^2 x^4 \cosh (c+d x)}{d^2}+a^2 \cosh (c) \text {Chi}(d x)+\frac {4 a b \sinh (c+d x)}{d^3}+\frac {120 b^2 x \sinh (c+d x)}{d^5}+\frac {2 a b x^2 \sinh (c+d x)}{d}+\frac {20 b^2 x^3 \sinh (c+d x)}{d^3}+\frac {b^2 x^5 \sinh (c+d x)}{d}+a^2 \sinh (c) \text {Shi}(d x)-\frac {\left (120 b^2\right ) \int \sinh (c+d x) \, dx}{d^5} \\ & = -\frac {120 b^2 \cosh (c+d x)}{d^6}-\frac {4 a b x \cosh (c+d x)}{d^2}-\frac {60 b^2 x^2 \cosh (c+d x)}{d^4}-\frac {5 b^2 x^4 \cosh (c+d x)}{d^2}+a^2 \cosh (c) \text {Chi}(d x)+\frac {4 a b \sinh (c+d x)}{d^3}+\frac {120 b^2 x \sinh (c+d x)}{d^5}+\frac {2 a b x^2 \sinh (c+d x)}{d}+\frac {20 b^2 x^3 \sinh (c+d x)}{d^3}+\frac {b^2 x^5 \sinh (c+d x)}{d}+a^2 \sinh (c) \text {Shi}(d x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.31 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.68 \[ \int \frac {\left (a+b x^3\right )^2 \cosh (c+d x)}{x} \, dx=-\frac {b \left (4 a d^4 x+5 b \left (24+12 d^2 x^2+d^4 x^4\right )\right ) \cosh (c+d x)}{d^6}+a^2 \cosh (c) \text {Chi}(d x)+\frac {b \left (2 a d^2 \left (2+d^2 x^2\right )+b x \left (120+20 d^2 x^2+d^4 x^4\right )\right ) \sinh (c+d x)}{d^5}+a^2 \sinh (c) \text {Shi}(d x) \]

[In]

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

[Out]

-((b*(4*a*d^4*x + 5*b*(24 + 12*d^2*x^2 + d^4*x^4))*Cosh[c + d*x])/d^6) + a^2*Cosh[c]*CoshIntegral[d*x] + (b*(2
*a*d^2*(2 + d^2*x^2) + b*x*(120 + 20*d^2*x^2 + d^4*x^4))*Sinh[c + d*x])/d^5 + a^2*Sinh[c]*SinhIntegral[d*x]

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.24 (sec) , antiderivative size = 303, normalized size of antiderivative = 1.89

method	result	size
meijerg	\(-\frac {32 b^{2} \cosh \left (c \right ) \sqrt {\pi }\, \left (-\frac {15}{4 \sqrt {\pi }}+\frac {\left (\frac {15}{8} d^{4} x^{4}+\frac {45}{2} x^{2} d^{2}+45\right ) \cosh \left (d x \right )}{12 \sqrt {\pi }}-\frac {x d \left (\frac {3}{8} d^{4} x^{4}+\frac {15}{2} x^{2} d^{2}+45\right ) \sinh \left (d x \right )}{12 \sqrt {\pi }}\right )}{d^{6}}+\frac {32 i b^{2} \sinh \left (c \right ) \sqrt {\pi }\, \left (-\frac {i x d \left (\frac {7}{8} d^{4} x^{4}+\frac {35}{2} x^{2} d^{2}+105\right ) \cosh \left (d x \right )}{28 \sqrt {\pi }}+\frac {i \left (\frac {35}{8} d^{4} x^{4}+\frac {105}{2} x^{2} d^{2}+105\right ) \sinh \left (d x \right )}{28 \sqrt {\pi }}\right )}{d^{6}}+\frac {8 i a b \cosh \left (c \right ) \sqrt {\pi }\, \left (\frac {i x d \cosh \left (d x \right )}{2 \sqrt {\pi }}-\frac {i \left (\frac {3 x^{2} d^{2}}{2}+3\right ) \sinh \left (d x \right )}{6 \sqrt {\pi }}\right )}{d^{3}}+\frac {8 b a \sinh \left (c \right ) \sqrt {\pi }\, \left (-\frac {1}{2 \sqrt {\pi }}+\frac {\left (\frac {x^{2} d^{2}}{2}+1\right ) \cosh \left (d x \right )}{2 \sqrt {\pi }}-\frac {d x \sinh \left (d x \right )}{2 \sqrt {\pi }}\right )}{d^{3}}+\frac {a^{2} \cosh \left (c \right ) \sqrt {\pi }\, \left (\frac {2 \gamma +2 \ln \left (x \right )+2 \ln \left (i d \right )}{\sqrt {\pi }}+\frac {2 \,\operatorname {Chi}\left (d x \right )-2 \ln \left (d x \right )-2 \gamma }{\sqrt {\pi }}\right )}{2}+a^{2} \operatorname {Shi}\left (d x \right ) \sinh \left (c \right )\)	\(303\)
risch	\(-\frac {{\mathrm e}^{-d x -c} b^{2} x^{5}}{2 d}+\frac {{\mathrm e}^{d x +c} b^{2} x^{5}}{2 d}-\frac {5 \,{\mathrm e}^{-d x -c} b^{2} x^{4}}{2 d^{2}}-\frac {5 \,{\mathrm e}^{d x +c} b^{2} x^{4}}{2 d^{2}}-\frac {a^{2} {\mathrm e}^{c} \operatorname {Ei}_{1}\left (-d x \right )}{2}-\frac {{\mathrm e}^{-d x -c} a b \,x^{2}}{d}-\frac {a^{2} {\mathrm e}^{-c} \operatorname {Ei}_{1}\left (d x \right )}{2}+\frac {{\mathrm e}^{d x +c} a b \,x^{2}}{d}-\frac {10 \,{\mathrm e}^{-d x -c} b^{2} x^{3}}{d^{3}}+\frac {10 \,{\mathrm e}^{d x +c} b^{2} x^{3}}{d^{3}}-\frac {2 \,{\mathrm e}^{-d x -c} a b x}{d^{2}}-\frac {2 \,{\mathrm e}^{d x +c} a b x}{d^{2}}-\frac {30 \,{\mathrm e}^{-d x -c} b^{2} x^{2}}{d^{4}}-\frac {30 \,{\mathrm e}^{d x +c} b^{2} x^{2}}{d^{4}}-\frac {2 \,{\mathrm e}^{-d x -c} a b}{d^{3}}+\frac {2 \,{\mathrm e}^{d x +c} a b}{d^{3}}-\frac {60 \,{\mathrm e}^{-d x -c} b^{2} x}{d^{5}}+\frac {60 \,{\mathrm e}^{d x +c} b^{2} x}{d^{5}}-\frac {60 \,{\mathrm e}^{-d x -c} b^{2}}{d^{6}}-\frac {60 \,{\mathrm e}^{d x +c} b^{2}}{d^{6}}\)	\(335\)

[In]

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

[Out]

-32/d^6*b^2*cosh(c)*Pi^(1/2)*(-15/4/Pi^(1/2)+1/12/Pi^(1/2)*(15/8*d^4*x^4+45/2*x^2*d^2+45)*cosh(d*x)-1/12/Pi^(1
/2)*x*d*(3/8*d^4*x^4+15/2*x^2*d^2+45)*sinh(d*x))+32*I/d^6*b^2*sinh(c)*Pi^(1/2)*(-1/28*I/Pi^(1/2)*x*d*(7/8*d^4*
x^4+35/2*x^2*d^2+105)*cosh(d*x)+1/28*I/Pi^(1/2)*(35/8*d^4*x^4+105/2*x^2*d^2+105)*sinh(d*x))+8*I/d^3*a*b*cosh(c
)*Pi^(1/2)*(1/2*I/Pi^(1/2)*x*d*cosh(d*x)-1/6*I/Pi^(1/2)*(3/2*x^2*d^2+3)*sinh(d*x))+8/d^3*b*a*sinh(c)*Pi^(1/2)*
(-1/2/Pi^(1/2)+1/2/Pi^(1/2)*(1/2*x^2*d^2+1)*cosh(d*x)-1/2/Pi^(1/2)*d*x*sinh(d*x))+1/2*a^2*cosh(c)*Pi^(1/2)*((2
*gamma+2*ln(x)+2*ln(I*d))/Pi^(1/2)+2/Pi^(1/2)*(Chi(d*x)-ln(d*x)-gamma))+a^2*Shi(d*x)*sinh(c)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.01 \[ \int \frac {\left (a+b x^3\right )^2 \cosh (c+d x)}{x} \, dx=-\frac {2 \, {\left (5 \, b^{2} d^{4} x^{4} + 4 \, a b d^{4} x + 60 \, b^{2} d^{2} x^{2} + 120 \, b^{2}\right )} \cosh \left (d x + c\right ) - {\left (a^{2} d^{6} {\rm Ei}\left (d x\right ) + a^{2} d^{6} {\rm Ei}\left (-d x\right )\right )} \cosh \left (c\right ) - 2 \, {\left (b^{2} d^{5} x^{5} + 2 \, a b d^{5} x^{2} + 20 \, b^{2} d^{3} x^{3} + 4 \, a b d^{3} + 120 \, b^{2} d x\right )} \sinh \left (d x + c\right ) - {\left (a^{2} d^{6} {\rm Ei}\left (d x\right ) - a^{2} d^{6} {\rm Ei}\left (-d x\right )\right )} \sinh \left (c\right )}{2 \, d^{6}} \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 2.39 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.05 \[ \int \frac {\left (a+b x^3\right )^2 \cosh (c+d x)}{x} \, dx=a^{2} \sinh {\left (c \right )} \operatorname {Shi}{\left (d x \right )} + a^{2} \cosh {\left (c \right )} \operatorname {Chi}\left (d x\right ) + 2 a b \left (\begin {cases} \frac {x^{2} \sinh {\left (c + d x \right )}}{d} - \frac {2 x \cosh {\left (c + d x \right )}}{d^{2}} + \frac {2 \sinh {\left (c + d x \right )}}{d^{3}} & \text {for}\: d \neq 0 \\\frac {x^{3} \cosh {\left (c \right )}}{3} & \text {otherwise} \end {cases}\right ) + b^{2} \left (\begin {cases} \frac {x^{5} \sinh {\left (c + d x \right )}}{d} - \frac {5 x^{4} \cosh {\left (c + d x \right )}}{d^{2}} + \frac {20 x^{3} \sinh {\left (c + d x \right )}}{d^{3}} - \frac {60 x^{2} \cosh {\left (c + d x \right )}}{d^{4}} + \frac {120 x \sinh {\left (c + d x \right )}}{d^{5}} - \frac {120 \cosh {\left (c + d x \right )}}{d^{6}} & \text {for}\: d \neq 0 \\\frac {x^{6} \cosh {\left (c \right )}}{6} & \text {otherwise} \end {cases}\right ) \]

[In]

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

[Out]

a**2*sinh(c)*Shi(d*x) + a**2*cosh(c)*Chi(d*x) + 2*a*b*Piecewise((x**2*sinh(c + d*x)/d - 2*x*cosh(c + d*x)/d**2
+ 2*sinh(c + d*x)/d**3, Ne(d, 0)), (x**3*cosh(c)/3, True)) + b**2*Piecewise((x**5*sinh(c + d*x)/d - 5*x**4*co
sh(c + d*x)/d**2 + 20*x**3*sinh(c + d*x)/d**3 - 60*x**2*cosh(c + d*x)/d**4 + 120*x*sinh(c + d*x)/d**5 - 120*co
sh(c + d*x)/d**6, Ne(d, 0)), (x**6*cosh(c)/6, True))

Maxima [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 289, normalized size of antiderivative = 1.81 \[ \int \frac {\left (a+b x^3\right )^2 \cosh (c+d x)}{x} \, dx=-\frac {1}{12} \, {\left (4 \, a b {\left (\frac {{\left (d^{3} x^{3} e^{c} - 3 \, d^{2} x^{2} e^{c} + 6 \, d x e^{c} - 6 \, e^{c}\right )} e^{\left (d x\right )}}{d^{4}} + \frac {{\left (d^{3} x^{3} + 3 \, d^{2} x^{2} + 6 \, d x + 6\right )} e^{\left (-d x - c\right )}}{d^{4}}\right )} + b^{2} {\left (\frac {{\left (d^{6} x^{6} e^{c} - 6 \, d^{5} x^{5} e^{c} + 30 \, d^{4} x^{4} e^{c} - 120 \, d^{3} x^{3} e^{c} + 360 \, d^{2} x^{2} e^{c} - 720 \, d x e^{c} + 720 \, e^{c}\right )} e^{\left (d x\right )}}{d^{7}} + \frac {{\left (d^{6} x^{6} + 6 \, d^{5} x^{5} + 30 \, d^{4} x^{4} + 120 \, d^{3} x^{3} + 360 \, d^{2} x^{2} + 720 \, d x + 720\right )} e^{\left (-d x - c\right )}}{d^{7}}\right )} + \frac {4 \, a^{2} \cosh \left (d x + c\right ) \log \left (x^{3}\right )}{d} - \frac {6 \, {\left ({\rm Ei}\left (-d x\right ) e^{\left (-c\right )} + {\rm Ei}\left (d x\right ) e^{c}\right )} a^{2}}{d}\right )} d + \frac {1}{6} \, {\left (b^{2} x^{6} + 4 \, a b x^{3} + 2 \, a^{2} \log \left (x^{3}\right )\right )} \cosh \left (d x + c\right ) \]

[In]

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

[Out]

-1/12*(4*a*b*((d^3*x^3*e^c - 3*d^2*x^2*e^c + 6*d*x*e^c - 6*e^c)*e^(d*x)/d^4 + (d^3*x^3 + 3*d^2*x^2 + 6*d*x + 6
)*e^(-d*x - c)/d^4) + b^2*((d^6*x^6*e^c - 6*d^5*x^5*e^c + 30*d^4*x^4*e^c - 120*d^3*x^3*e^c + 360*d^2*x^2*e^c -
720*d*x*e^c + 720*e^c)*e^(d*x)/d^7 + (d^6*x^6 + 6*d^5*x^5 + 30*d^4*x^4 + 120*d^3*x^3 + 360*d^2*x^2 + 720*d*x
+ 720)*e^(-d*x - c)/d^7) + 4*a^2*cosh(d*x + c)*log(x^3)/d - 6*(Ei(-d*x)*e^(-c) + Ei(d*x)*e^c)*a^2/d)*d + 1/6*(
b^2*x^6 + 4*a*b*x^3 + 2*a^2*log(x^3))*cosh(d*x + c)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 331 vs. \(2 (160) = 320\).

Time = 0.27 (sec) , antiderivative size = 331, normalized size of antiderivative = 2.07 \[ \int \frac {\left (a+b x^3\right )^2 \cosh (c+d x)}{x} \, dx=\frac {b^{2} d^{5} x^{5} e^{\left (d x + c\right )} - b^{2} d^{5} x^{5} e^{\left (-d x - c\right )} - 5 \, b^{2} d^{4} x^{4} e^{\left (d x + c\right )} - 5 \, b^{2} d^{4} x^{4} e^{\left (-d x - c\right )} + 2 \, a b d^{5} x^{2} e^{\left (d x + c\right )} - 2 \, a b d^{5} x^{2} e^{\left (-d x - c\right )} + a^{2} d^{6} {\rm Ei}\left (-d x\right ) e^{\left (-c\right )} + a^{2} d^{6} {\rm Ei}\left (d x\right ) e^{c} + 20 \, b^{2} d^{3} x^{3} e^{\left (d x + c\right )} - 20 \, b^{2} d^{3} x^{3} e^{\left (-d x - c\right )} - 4 \, a b d^{4} x e^{\left (d x + c\right )} - 4 \, a b d^{4} x e^{\left (-d x - c\right )} - 60 \, b^{2} d^{2} x^{2} e^{\left (d x + c\right )} - 60 \, b^{2} d^{2} x^{2} e^{\left (-d x - c\right )} + 4 \, a b d^{3} e^{\left (d x + c\right )} - 4 \, a b d^{3} e^{\left (-d x - c\right )} + 120 \, b^{2} d x e^{\left (d x + c\right )} - 120 \, b^{2} d x e^{\left (-d x - c\right )} - 120 \, b^{2} e^{\left (d x + c\right )} - 120 \, b^{2} e^{\left (-d x - c\right )}}{2 \, d^{6}} \]

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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