\(\int \frac {(a+b x^3)^2 \cosh (c+d x)}{x^3} \, dx\) [91]

Optimal result

Integrand size = 19, antiderivative size = 141 \[ \int \frac {\left (a+b x^3\right )^2 \cosh (c+d x)}{x^3} \, dx=-\frac {6 b^2 \cosh (c+d x)}{d^4}-\frac {a^2 \cosh (c+d x)}{2 x^2}-\frac {3 b^2 x^2 \cosh (c+d x)}{d^2}+\frac {1}{2} a^2 d^2 \cosh (c) \text {Chi}(d x)+\frac {2 a b \sinh (c+d x)}{d}-\frac {a^2 d \sinh (c+d x)}{2 x}+\frac {6 b^2 x \sinh (c+d x)}{d^3}+\frac {b^2 x^3 \sinh (c+d x)}{d}+\frac {1}{2} a^2 d^2 \sinh (c) \text {Shi}(d x) \]

[Out]

1/2*a^2*d^2*Chi(d*x)*cosh(c)-6*b^2*cosh(d*x+c)/d^4-1/2*a^2*cosh(d*x+c)/x^2-3*b^2*x^2*cosh(d*x+c)/d^2+1/2*a^2*d
^2*Shi(d*x)*sinh(c)+2*a*b*sinh(d*x+c)/d-1/2*a^2*d*sinh(d*x+c)/x+6*b^2*x*sinh(d*x+c)/d^3+b^2*x^3*sinh(d*x+c)/d

Rubi [A] (verified)

Time = 0.16 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {5395, 2717, 3378, 3384, 3379, 3382, 3377, 2718} \[ \int \frac {\left (a+b x^3\right )^2 \cosh (c+d x)}{x^3} \, dx=\frac {1}{2} a^2 d^2 \cosh (c) \text {Chi}(d x)+\frac {1}{2} a^2 d^2 \sinh (c) \text {Shi}(d x)-\frac {a^2 \cosh (c+d x)}{2 x^2}-\frac {a^2 d \sinh (c+d x)}{2 x}+\frac {2 a b \sinh (c+d x)}{d}-\frac {6 b^2 \cosh (c+d x)}{d^4}+\frac {6 b^2 x \sinh (c+d x)}{d^3}-\frac {3 b^2 x^2 \cosh (c+d x)}{d^2}+\frac {b^2 x^3 \sinh (c+d x)}{d} \]

[In]

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

[Out]

(-6*b^2*Cosh[c + d*x])/d^4 - (a^2*Cosh[c + d*x])/(2*x^2) - (3*b^2*x^2*Cosh[c + d*x])/d^2 + (a^2*d^2*Cosh[c]*Co
shIntegral[d*x])/2 + (2*a*b*Sinh[c + d*x])/d - (a^2*d*Sinh[c + d*x])/(2*x) + (6*b^2*x*Sinh[c + d*x])/d^3 + (b^
2*x^3*Sinh[c + d*x])/d + (a^2*d^2*Sinh[c]*SinhIntegral[d*x])/2

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

Mathematica [A] (verified)

Time = 0.25 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.96 \[ \int \frac {\left (a+b x^3\right )^2 \cosh (c+d x)}{x^3} \, dx=\frac {1}{2} \left (-\frac {12 b^2 \cosh (c+d x)}{d^4}-\frac {a^2 \cosh (c+d x)}{x^2}-\frac {6 b^2 x^2 \cosh (c+d x)}{d^2}+a^2 d^2 \cosh (c) \text {Chi}(d x)+\frac {4 a b \sinh (c+d x)}{d}-\frac {a^2 d \sinh (c+d x)}{x}+\frac {12 b^2 x \sinh (c+d x)}{d^3}+\frac {2 b^2 x^3 \sinh (c+d x)}{d}+a^2 d^2 \sinh (c) \text {Shi}(d x)\right ) \]

[In]

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

[Out]

((-12*b^2*Cosh[c + d*x])/d^4 - (a^2*Cosh[c + d*x])/x^2 - (6*b^2*x^2*Cosh[c + d*x])/d^2 + a^2*d^2*Cosh[c]*CoshI
ntegral[d*x] + (4*a*b*Sinh[c + d*x])/d - (a^2*d*Sinh[c + d*x])/x + (12*b^2*x*Sinh[c + d*x])/d^3 + (2*b^2*x^3*S
inh[c + d*x])/d + a^2*d^2*Sinh[c]*SinhIntegral[d*x])/2

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(280\) vs. \(2(133)=266\).

Time = 0.28 (sec) , antiderivative size = 281, normalized size of antiderivative = 1.99

method	result	size
risch	\(-\frac {{\mathrm e}^{c} \operatorname {Ei}_{1}\left (-d x \right ) a^{2} d^{6} x^{2}+{\mathrm e}^{-c} \operatorname {Ei}_{1}\left (d x \right ) a^{2} d^{6} x^{2}-2 \,{\mathrm e}^{d x +c} b^{2} d^{3} x^{5}+2 \,{\mathrm e}^{-d x -c} b^{2} d^{3} x^{5}+{\mathrm e}^{d x +c} a^{2} d^{5} x +6 \,{\mathrm e}^{d x +c} b^{2} d^{2} x^{4}-{\mathrm e}^{-d x -c} a^{2} d^{5} x +6 \,{\mathrm e}^{-d x -c} b^{2} d^{2} x^{4}-4 \,{\mathrm e}^{d x +c} a b \,d^{3} x^{2}+4 \,{\mathrm e}^{-d x -c} a b \,d^{3} x^{2}+d^{4} {\mathrm e}^{d x +c} a^{2}-12 \,{\mathrm e}^{d x +c} b^{2} d \,x^{3}+d^{4} {\mathrm e}^{-d x -c} a^{2}+12 \,{\mathrm e}^{-d x -c} b^{2} d \,x^{3}+12 \,{\mathrm e}^{d x +c} b^{2} x^{2}+12 \,{\mathrm e}^{-d x -c} b^{2} x^{2}}{4 d^{4} x^{2}}\)	\(281\)
meijerg	\(\frac {8 b^{2} \cosh \left (c \right ) \sqrt {\pi }\, \left (\frac {3}{4 \sqrt {\pi }}-\frac {\left (\frac {3 x^{2} d^{2}}{2}+3\right ) \cosh \left (d x \right )}{4 \sqrt {\pi }}+\frac {d x \left (\frac {x^{2} d^{2}}{2}+3\right ) \sinh \left (d x \right )}{4 \sqrt {\pi }}\right )}{d^{4}}-\frac {8 i b^{2} \sinh \left (c \right ) \sqrt {\pi }\, \left (\frac {i x d \left (\frac {5 x^{2} d^{2}}{2}+15\right ) \cosh \left (d x \right )}{20 \sqrt {\pi }}-\frac {i \left (\frac {15 x^{2} d^{2}}{2}+15\right ) \sinh \left (d x \right )}{20 \sqrt {\pi }}\right )}{d^{4}}+\frac {2 a b \cosh \left (c \right ) \sinh \left (d x \right )}{d}-\frac {2 b a \sinh \left (c \right ) \sqrt {\pi }\, \left (\frac {1}{\sqrt {\pi }}-\frac {\cosh \left (d x \right )}{\sqrt {\pi }}\right )}{d}-\frac {a^{2} \cosh \left (c \right ) \sqrt {\pi }\, d^{2} \left (\frac {4}{\sqrt {\pi }\, x^{2} d^{2}}-\frac {2 \left (2 \gamma -3+2 \ln \left (x \right )+2 \ln \left (i d \right )\right )}{\sqrt {\pi }}-\frac {4 \left (\frac {9 x^{2} d^{2}}{2}+3\right )}{3 \sqrt {\pi }\, x^{2} d^{2}}+\frac {4 \cosh \left (d x \right )}{\sqrt {\pi }\, x^{2} d^{2}}+\frac {4 \sinh \left (d x \right )}{\sqrt {\pi }\, x d}-\frac {4 \left (\operatorname {Chi}\left (d x \right )-\ln \left (d x \right )-\gamma \right )}{\sqrt {\pi }}\right )}{8}+\frac {i a^{2} \sinh \left (c \right ) \sqrt {\pi }\, d^{2} \left (\frac {4 i \cosh \left (d x \right )}{d x \sqrt {\pi }}+\frac {4 i \sinh \left (d x \right )}{x^{2} d^{2} \sqrt {\pi }}-\frac {4 i \operatorname {Shi}\left (d x \right )}{\sqrt {\pi }}\right )}{8}\)	\(329\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.14 \[ \int \frac {\left (a+b x^3\right )^2 \cosh (c+d x)}{x^3} \, dx=-\frac {2 \, {\left (6 \, b^{2} d^{2} x^{4} + a^{2} d^{4} + 12 \, b^{2} x^{2}\right )} \cosh \left (d x + c\right ) - {\left (a^{2} d^{6} x^{2} {\rm Ei}\left (d x\right ) + a^{2} d^{6} x^{2} {\rm Ei}\left (-d x\right )\right )} \cosh \left (c\right ) - 2 \, {\left (2 \, b^{2} d^{3} x^{5} - a^{2} d^{5} x + 4 \, a b d^{3} x^{2} + 12 \, b^{2} d x^{3}\right )} \sinh \left (d x + c\right ) - {\left (a^{2} d^{6} x^{2} {\rm Ei}\left (d x\right ) - a^{2} d^{6} x^{2} {\rm Ei}\left (-d x\right )\right )} \sinh \left (c\right )}{4 \, d^{4} x^{2}} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.44 \[ \int \frac {\left (a+b x^3\right )^2 \cosh (c+d x)}{x^3} \, dx=\frac {1}{8} \, {\left (2 \, a^{2} d e^{\left (-c\right )} \Gamma \left (-1, d x\right ) + 2 \, a^{2} d e^{c} \Gamma \left (-1, -d x\right ) - \frac {8 \, {\left (d x e^{c} - e^{c}\right )} a b e^{\left (d x\right )}}{d^{2}} - \frac {8 \, {\left (d x + 1\right )} a b e^{\left (-d x - c\right )}}{d^{2}} - \frac {{\left (d^{4} x^{4} e^{c} - 4 \, d^{3} x^{3} e^{c} + 12 \, d^{2} x^{2} e^{c} - 24 \, d x e^{c} + 24 \, e^{c}\right )} b^{2} e^{\left (d x\right )}}{d^{5}} - \frac {{\left (d^{4} x^{4} + 4 \, d^{3} x^{3} + 12 \, d^{2} x^{2} + 24 \, d x + 24\right )} b^{2} e^{\left (-d x - c\right )}}{d^{5}}\right )} d + \frac {1}{4} \, {\left (b^{2} x^{4} + 8 \, a b x - \frac {2 \, a^{2}}{x^{2}}\right )} \cosh \left (d x + c\right ) \]

[In]

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

[Out]

1/8*(2*a^2*d*e^(-c)*gamma(-1, d*x) + 2*a^2*d*e^c*gamma(-1, -d*x) - 8*(d*x*e^c - e^c)*a*b*e^(d*x)/d^2 - 8*(d*x
+ 1)*a*b*e^(-d*x - c)/d^2 - (d^4*x^4*e^c - 4*d^3*x^3*e^c + 12*d^2*x^2*e^c - 24*d*x*e^c + 24*e^c)*b^2*e^(d*x)/d
^5 - (d^4*x^4 + 4*d^3*x^3 + 12*d^2*x^2 + 24*d*x + 24)*b^2*e^(-d*x - c)/d^5)*d + 1/4*(b^2*x^4 + 8*a*b*x - 2*a^2
/x^2)*cosh(d*x + c)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 280 vs. \(2 (133) = 266\).

Time = 0.26 (sec) , antiderivative size = 280, normalized size of antiderivative = 1.99 \[ \int \frac {\left (a+b x^3\right )^2 \cosh (c+d x)}{x^3} \, dx=\frac {a^{2} d^{6} x^{2} {\rm Ei}\left (-d x\right ) e^{\left (-c\right )} + a^{2} d^{6} x^{2} {\rm Ei}\left (d x\right ) e^{c} + 2 \, b^{2} d^{3} x^{5} e^{\left (d x + c\right )} - 2 \, b^{2} d^{3} x^{5} e^{\left (-d x - c\right )} - a^{2} d^{5} x e^{\left (d x + c\right )} - 6 \, b^{2} d^{2} x^{4} e^{\left (d x + c\right )} + a^{2} d^{5} x e^{\left (-d x - c\right )} - 6 \, b^{2} d^{2} x^{4} e^{\left (-d x - c\right )} + 4 \, a b d^{3} x^{2} e^{\left (d x + c\right )} - 4 \, a b d^{3} x^{2} e^{\left (-d x - c\right )} - a^{2} d^{4} e^{\left (d x + c\right )} + 12 \, b^{2} d x^{3} e^{\left (d x + c\right )} - a^{2} d^{4} e^{\left (-d x - c\right )} - 12 \, b^{2} d x^{3} e^{\left (-d x - c\right )} - 12 \, b^{2} x^{2} e^{\left (d x + c\right )} - 12 \, b^{2} x^{2} e^{\left (-d x - c\right )}}{4 \, d^{4} x^{2}} \]

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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