3.93 \(\int \frac{(a+b x^3)^2 \cosh (c+d x)}{x^5} \, dx\)

Optimal. Leaf size=167 \[ \frac{1}{24} a^2 d^4 \cosh (c) \text{Chi}(d x)+\frac{1}{24} a^2 d^4 \sinh (c) \text{Shi}(d x)-\frac{a^2 d^2 \cosh (c+d x)}{24 x^2}-\frac{a^2 d^3 \sinh (c+d x)}{24 x}-\frac{a^2 d \sinh (c+d x)}{12 x^3}-\frac{a^2 \cosh (c+d x)}{4 x^4}+2 a b d \sinh (c) \text{Chi}(d x)+2 a b d \cosh (c) \text{Shi}(d x)-\frac{2 a b \cosh (c+d x)}{x}-\frac{b^2 \cosh (c+d x)}{d^2}+\frac{b^2 x \sinh (c+d x)}{d} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.30798, antiderivative size = 167, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 7, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.368, Rules used = {5287, 3297, 3303, 3298, 3301, 3296, 2638} \[ \frac{1}{24} a^2 d^4 \cosh (c) \text{Chi}(d x)+\frac{1}{24} a^2 d^4 \sinh (c) \text{Shi}(d x)-\frac{a^2 d^2 \cosh (c+d x)}{24 x^2}-\frac{a^2 d^3 \sinh (c+d x)}{24 x}-\frac{a^2 d \sinh (c+d x)}{12 x^3}-\frac{a^2 \cosh (c+d x)}{4 x^4}+2 a b d \sinh (c) \text{Chi}(d x)+2 a b d \cosh (c) \text{Shi}(d x)-\frac{2 a b \cosh (c+d x)}{x}-\frac{b^2 \cosh (c+d x)}{d^2}+\frac{b^2 x \sinh (c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x^3)^2*Cosh[c + d*x])/x^5,x]

[Out]

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

Rule 5287

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]

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 3296

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 2638

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

Rubi steps

\begin{align*} \int \frac{\left (a+b x^3\right )^2 \cosh (c+d x)}{x^5} \, dx &=\int \left (\frac{a^2 \cosh (c+d x)}{x^5}+\frac{2 a b \cosh (c+d x)}{x^2}+b^2 x \cosh (c+d x)\right ) \, dx\\ &=a^2 \int \frac{\cosh (c+d x)}{x^5} \, dx+(2 a b) \int \frac{\cosh (c+d x)}{x^2} \, dx+b^2 \int x \cosh (c+d x) \, dx\\ &=-\frac{a^2 \cosh (c+d x)}{4 x^4}-\frac{2 a b \cosh (c+d x)}{x}+\frac{b^2 x \sinh (c+d x)}{d}-\frac{b^2 \int \sinh (c+d x) \, dx}{d}+\frac{1}{4} \left (a^2 d\right ) \int \frac{\sinh (c+d x)}{x^4} \, dx+(2 a b d) \int \frac{\sinh (c+d x)}{x} \, dx\\ &=-\frac{b^2 \cosh (c+d x)}{d^2}-\frac{a^2 \cosh (c+d x)}{4 x^4}-\frac{2 a b \cosh (c+d x)}{x}-\frac{a^2 d \sinh (c+d x)}{12 x^3}+\frac{b^2 x \sinh (c+d x)}{d}+\frac{1}{12} \left (a^2 d^2\right ) \int \frac{\cosh (c+d x)}{x^3} \, dx+(2 a b d \cosh (c)) \int \frac{\sinh (d x)}{x} \, dx+(2 a b d \sinh (c)) \int \frac{\cosh (d x)}{x} \, dx\\ &=-\frac{b^2 \cosh (c+d x)}{d^2}-\frac{a^2 \cosh (c+d x)}{4 x^4}-\frac{a^2 d^2 \cosh (c+d x)}{24 x^2}-\frac{2 a b \cosh (c+d x)}{x}+2 a b d \text{Chi}(d x) \sinh (c)-\frac{a^2 d \sinh (c+d x)}{12 x^3}+\frac{b^2 x \sinh (c+d x)}{d}+2 a b d \cosh (c) \text{Shi}(d x)+\frac{1}{24} \left (a^2 d^3\right ) \int \frac{\sinh (c+d x)}{x^2} \, dx\\ &=-\frac{b^2 \cosh (c+d x)}{d^2}-\frac{a^2 \cosh (c+d x)}{4 x^4}-\frac{a^2 d^2 \cosh (c+d x)}{24 x^2}-\frac{2 a b \cosh (c+d x)}{x}+2 a b d \text{Chi}(d x) \sinh (c)-\frac{a^2 d \sinh (c+d x)}{12 x^3}-\frac{a^2 d^3 \sinh (c+d x)}{24 x}+\frac{b^2 x \sinh (c+d x)}{d}+2 a b d \cosh (c) \text{Shi}(d x)+\frac{1}{24} \left (a^2 d^4\right ) \int \frac{\cosh (c+d x)}{x} \, dx\\ &=-\frac{b^2 \cosh (c+d x)}{d^2}-\frac{a^2 \cosh (c+d x)}{4 x^4}-\frac{a^2 d^2 \cosh (c+d x)}{24 x^2}-\frac{2 a b \cosh (c+d x)}{x}+2 a b d \text{Chi}(d x) \sinh (c)-\frac{a^2 d \sinh (c+d x)}{12 x^3}-\frac{a^2 d^3 \sinh (c+d x)}{24 x}+\frac{b^2 x \sinh (c+d x)}{d}+2 a b d \cosh (c) \text{Shi}(d x)+\frac{1}{24} \left (a^2 d^4 \cosh (c)\right ) \int \frac{\cosh (d x)}{x} \, dx+\frac{1}{24} \left (a^2 d^4 \sinh (c)\right ) \int \frac{\sinh (d x)}{x} \, dx\\ &=-\frac{b^2 \cosh (c+d x)}{d^2}-\frac{a^2 \cosh (c+d x)}{4 x^4}-\frac{a^2 d^2 \cosh (c+d x)}{24 x^2}-\frac{2 a b \cosh (c+d x)}{x}+\frac{1}{24} a^2 d^4 \cosh (c) \text{Chi}(d x)+2 a b d \text{Chi}(d x) \sinh (c)-\frac{a^2 d \sinh (c+d x)}{12 x^3}-\frac{a^2 d^3 \sinh (c+d x)}{24 x}+\frac{b^2 x \sinh (c+d x)}{d}+2 a b d \cosh (c) \text{Shi}(d x)+\frac{1}{24} a^2 d^4 \sinh (c) \text{Shi}(d x)\\ \end{align*}

Mathematica [A]  time = 0.539467, size = 150, normalized size = 0.9 \[ \frac{1}{24} \left (-\frac{a^2 d^2 \cosh (c+d x)}{x^2}-\frac{a^2 d^3 \sinh (c+d x)}{x}-\frac{2 a^2 d \sinh (c+d x)}{x^3}-\frac{6 a^2 \cosh (c+d x)}{x^4}+a d \text{Chi}(d x) \left (a d^3 \cosh (c)+48 b \sinh (c)\right )+a d \text{Shi}(d x) \left (a d^3 \sinh (c)+48 b \cosh (c)\right )-\frac{48 a b \cosh (c+d x)}{x}-\frac{24 b^2 \cosh (c+d x)}{d^2}+\frac{24 b^2 x \sinh (c+d x)}{d}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x^3)^2*Cosh[c + d*x])/x^5,x]

[Out]

((-24*b^2*Cosh[c + d*x])/d^2 - (6*a^2*Cosh[c + d*x])/x^4 - (a^2*d^2*Cosh[c + d*x])/x^2 - (48*a*b*Cosh[c + d*x]
)/x + a*d*CoshIntegral[d*x]*(a*d^3*Cosh[c] + 48*b*Sinh[c]) - (2*a^2*d*Sinh[c + d*x])/x^3 - (a^2*d^3*Sinh[c + d
*x])/x + (24*b^2*x*Sinh[c + d*x])/d + a*d*(48*b*Cosh[c] + a*d^3*Sinh[c])*SinhIntegral[d*x])/24

________________________________________________________________________________________

Maple [A]  time = 0.174, size = 292, normalized size = 1.8 \begin{align*} -{\frac{{b}^{2}{{\rm e}^{-dx-c}}x}{2\,d}}-{\frac{{d}^{4}{a}^{2}{{\rm e}^{-c}}{\it Ei} \left ( 1,dx \right ) }{48}}+{\frac{{d}^{3}{a}^{2}{{\rm e}^{-dx-c}}}{48\,x}}-{\frac{{b}^{2}{{\rm e}^{-dx-c}}}{2\,{d}^{2}}}-{\frac{{a}^{2}{d}^{2}{{\rm e}^{-dx-c}}}{48\,{x}^{2}}}+{\frac{d{a}^{2}{{\rm e}^{-dx-c}}}{24\,{x}^{3}}}-{\frac{{a}^{2}{{\rm e}^{-dx-c}}}{8\,{x}^{4}}}-{\frac{ab{{\rm e}^{-dx-c}}}{x}}+dab{{\rm e}^{-c}}{\it Ei} \left ( 1,dx \right ) -{\frac{{d}^{4}{a}^{2}{{\rm e}^{c}}{\it Ei} \left ( 1,-dx \right ) }{48}}-{\frac{{{\rm e}^{dx+c}}{b}^{2}}{2\,{d}^{2}}}-{\frac{{a}^{2}{d}^{2}{{\rm e}^{dx+c}}}{48\,{x}^{2}}}-{\frac{{d}^{3}{a}^{2}{{\rm e}^{dx+c}}}{48\,x}}-{\frac{{{\rm e}^{dx+c}}{a}^{2}}{8\,{x}^{4}}}-{\frac{d{a}^{2}{{\rm e}^{dx+c}}}{24\,{x}^{3}}}-dab{{\rm e}^{c}}{\it Ei} \left ( 1,-dx \right ) +{\frac{{{\rm e}^{dx+c}}{b}^{2}x}{2\,d}}-{\frac{ab{{\rm e}^{dx+c}}}{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3+a)^2*cosh(d*x+c)/x^5,x)

[Out]

-1/2*b^2/d*exp(-d*x-c)*x-1/48*d^4*a^2*exp(-c)*Ei(1,d*x)+1/48*d^3*a^2*exp(-d*x-c)/x-1/2*b^2/d^2*exp(-d*x-c)-1/4
8*d^2*a^2*exp(-d*x-c)/x^2+1/24*d*a^2*exp(-d*x-c)/x^3-1/8*a^2*exp(-d*x-c)/x^4-a*b*exp(-d*x-c)/x+d*a*b*exp(-c)*E
i(1,d*x)-1/48*d^4*a^2*exp(c)*Ei(1,-d*x)-1/2*b^2/d^2*exp(d*x+c)-1/48*d^2*a^2/x^2*exp(d*x+c)-1/48*d^3*a^2/x*exp(
d*x+c)-1/8*a^2/x^4*exp(d*x+c)-1/24*d*a^2/x^3*exp(d*x+c)-d*a*b*exp(c)*Ei(1,-d*x)+1/2*b^2/d*exp(d*x+c)*x-a*b/x*e
xp(d*x+c)

________________________________________________________________________________________

Maxima [A]  time = 1.24745, size = 208, normalized size = 1.25 \begin{align*} \frac{1}{8} \,{\left (a^{2} d^{3} e^{\left (-c\right )} \Gamma \left (-3, d x\right ) + a^{2} d^{3} e^{c} \Gamma \left (-3, -d x\right ) - 8 \, a b{\rm Ei}\left (-d x\right ) e^{\left (-c\right )} + 8 \, a b{\rm Ei}\left (d x\right ) e^{c} - \frac{2 \,{\left (d^{2} x^{2} e^{c} - 2 \, d x e^{c} + 2 \, e^{c}\right )} b^{2} e^{\left (d x\right )}}{d^{3}} - \frac{2 \,{\left (d^{2} x^{2} + 2 \, d x + 2\right )} b^{2} e^{\left (-d x - c\right )}}{d^{3}}\right )} d + \frac{1}{4} \,{\left (2 \, b^{2} x^{2} - \frac{8 \, a b x^{3} + a^{2}}{x^{4}}\right )} \cosh \left (d x + c\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^2*cosh(d*x+c)/x^5,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.80379, size = 435, normalized size = 2.6 \begin{align*} -\frac{2 \,{\left (a^{2} d^{4} x^{2} + 48 \, a b d^{2} x^{3} + 24 \, b^{2} x^{4} + 6 \, a^{2} d^{2}\right )} \cosh \left (d x + c\right ) -{\left ({\left (a^{2} d^{6} + 48 \, a b d^{3}\right )} x^{4}{\rm Ei}\left (d x\right ) +{\left (a^{2} d^{6} - 48 \, a b d^{3}\right )} x^{4}{\rm Ei}\left (-d x\right )\right )} \cosh \left (c\right ) + 2 \,{\left (a^{2} d^{5} x^{3} - 24 \, b^{2} d x^{5} + 2 \, a^{2} d^{3} x\right )} \sinh \left (d x + c\right ) -{\left ({\left (a^{2} d^{6} + 48 \, a b d^{3}\right )} x^{4}{\rm Ei}\left (d x\right ) -{\left (a^{2} d^{6} - 48 \, a b d^{3}\right )} x^{4}{\rm Ei}\left (-d x\right )\right )} \sinh \left (c\right )}{48 \, d^{2} x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^2*cosh(d*x+c)/x^5,x, algorithm="fricas")

[Out]

-1/48*(2*(a^2*d^4*x^2 + 48*a*b*d^2*x^3 + 24*b^2*x^4 + 6*a^2*d^2)*cosh(d*x + c) - ((a^2*d^6 + 48*a*b*d^3)*x^4*E
i(d*x) + (a^2*d^6 - 48*a*b*d^3)*x^4*Ei(-d*x))*cosh(c) + 2*(a^2*d^5*x^3 - 24*b^2*d*x^5 + 2*a^2*d^3*x)*sinh(d*x
+ c) - ((a^2*d^6 + 48*a*b*d^3)*x^4*Ei(d*x) - (a^2*d^6 - 48*a*b*d^3)*x^4*Ei(-d*x))*sinh(c))/(d^2*x^4)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b x^{3}\right )^{2} \cosh{\left (c + d x \right )}}{x^{5}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**3+a)**2*cosh(d*x+c)/x**5,x)

[Out]

Integral((a + b*x**3)**2*cosh(c + d*x)/x**5, x)

________________________________________________________________________________________

Giac [B]  time = 1.33292, size = 425, normalized size = 2.54 \begin{align*} \frac{a^{2} d^{6} x^{4}{\rm Ei}\left (-d x\right ) e^{\left (-c\right )} + a^{2} d^{6} x^{4}{\rm Ei}\left (d x\right ) e^{c} - a^{2} d^{5} x^{3} e^{\left (d x + c\right )} + a^{2} d^{5} x^{3} e^{\left (-d x - c\right )} - 48 \, a b d^{3} x^{4}{\rm Ei}\left (-d x\right ) e^{\left (-c\right )} + 48 \, a b d^{3} x^{4}{\rm Ei}\left (d x\right ) e^{c} - a^{2} d^{4} x^{2} e^{\left (d x + c\right )} + 24 \, b^{2} d x^{5} e^{\left (d x + c\right )} - a^{2} d^{4} x^{2} e^{\left (-d x - c\right )} - 24 \, b^{2} d x^{5} e^{\left (-d x - c\right )} - 48 \, a b d^{2} x^{3} e^{\left (d x + c\right )} - 48 \, a b d^{2} x^{3} e^{\left (-d x - c\right )} - 2 \, a^{2} d^{3} x e^{\left (d x + c\right )} - 24 \, b^{2} x^{4} e^{\left (d x + c\right )} + 2 \, a^{2} d^{3} x e^{\left (-d x - c\right )} - 24 \, b^{2} x^{4} e^{\left (-d x - c\right )} - 6 \, a^{2} d^{2} e^{\left (d x + c\right )} - 6 \, a^{2} d^{2} e^{\left (-d x - c\right )}}{48 \, d^{2} x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^2*cosh(d*x+c)/x^5,x, algorithm="giac")

[Out]

1/48*(a^2*d^6*x^4*Ei(-d*x)*e^(-c) + a^2*d^6*x^4*Ei(d*x)*e^c - a^2*d^5*x^3*e^(d*x + c) + a^2*d^5*x^3*e^(-d*x -
c) - 48*a*b*d^3*x^4*Ei(-d*x)*e^(-c) + 48*a*b*d^3*x^4*Ei(d*x)*e^c - a^2*d^4*x^2*e^(d*x + c) + 24*b^2*d*x^5*e^(d
*x + c) - a^2*d^4*x^2*e^(-d*x - c) - 24*b^2*d*x^5*e^(-d*x - c) - 48*a*b*d^2*x^3*e^(d*x + c) - 48*a*b*d^2*x^3*e
^(-d*x - c) - 2*a^2*d^3*x*e^(d*x + c) - 24*b^2*x^4*e^(d*x + c) + 2*a^2*d^3*x*e^(-d*x - c) - 24*b^2*x^4*e^(-d*x
- c) - 6*a^2*d^2*e^(d*x + c) - 6*a^2*d^2*e^(-d*x - c))/(d^2*x^4)