3.86 \(\int \frac{(a+b x^3) \cosh (c+d x)}{x^4} \, dx\)

Optimal. Leaf size=91 \[ \frac{1}{6} a d^3 \sinh (c) \text{Chi}(d x)+\frac{1}{6} a d^3 \cosh (c) \text{Shi}(d x)-\frac{a d^2 \cosh (c+d x)}{6 x}-\frac{a d \sinh (c+d x)}{6 x^2}-\frac{a \cosh (c+d x)}{3 x^3}+b \cosh (c) \text{Chi}(d x)+b \sinh (c) \text{Shi}(d x) \]

[Out]

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

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Rubi [A]  time = 0.215468, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 5, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.294, Rules used = {5287, 3297, 3303, 3298, 3301} \[ \frac{1}{6} a d^3 \sinh (c) \text{Chi}(d x)+\frac{1}{6} a d^3 \cosh (c) \text{Shi}(d x)-\frac{a d^2 \cosh (c+d x)}{6 x}-\frac{a d \sinh (c+d x)}{6 x^2}-\frac{a \cosh (c+d x)}{3 x^3}+b \cosh (c) \text{Chi}(d x)+b \sinh (c) \text{Shi}(d x) \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x^3)*Cosh[c + d*x])/x^4,x]

[Out]

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

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]

Rubi steps

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

Mathematica [A]  time = 0.238487, size = 73, normalized size = 0.8 \[ \frac{1}{6} \left (\text{Chi}(d x) \left (a d^3 \sinh (c)+6 b \cosh (c)\right )+\text{Shi}(d x) \left (a d^3 \cosh (c)+6 b \sinh (c)\right )-\frac{a \left (\left (d^2 x^2+2\right ) \cosh (c+d x)+d x \sinh (c+d x)\right )}{x^3}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x^3)*Cosh[c + d*x])/x^4,x]

[Out]

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

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Maple [A]  time = 0.066, size = 143, normalized size = 1.6 \begin{align*}{\frac{{d}^{3}a{{\rm e}^{-c}}{\it Ei} \left ( 1,dx \right ) }{12}}-{\frac{b{{\rm e}^{-c}}{\it Ei} \left ( 1,dx \right ) }{2}}-{\frac{a{d}^{2}{{\rm e}^{-dx-c}}}{12\,x}}+{\frac{da{{\rm e}^{-dx-c}}}{12\,{x}^{2}}}-{\frac{a{{\rm e}^{-dx-c}}}{6\,{x}^{3}}}-{\frac{a{{\rm e}^{dx+c}}}{6\,{x}^{3}}}-{\frac{da{{\rm e}^{dx+c}}}{12\,{x}^{2}}}-{\frac{a{d}^{2}{{\rm e}^{dx+c}}}{12\,x}}-{\frac{{d}^{3}a{{\rm e}^{c}}{\it Ei} \left ( 1,-dx \right ) }{12}}-{\frac{b{{\rm e}^{c}}{\it Ei} \left ( 1,-dx \right ) }{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3+a)*cosh(d*x+c)/x^4,x)

[Out]

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

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Maxima [A]  time = 1.22964, size = 128, normalized size = 1.41 \begin{align*} \frac{1}{6} \,{\left ({\left (d^{2} e^{\left (-c\right )} \Gamma \left (-2, d x\right ) - d^{2} e^{c} \Gamma \left (-2, -d x\right )\right )} a - \frac{2 \, b \cosh \left (d x + c\right ) \log \left (x^{3}\right )}{d} + \frac{3 \,{\left ({\rm Ei}\left (-d x\right ) e^{\left (-c\right )} +{\rm Ei}\left (d x\right ) e^{c}\right )} b}{d}\right )} d + \frac{1}{3} \,{\left (b \log \left (x^{3}\right ) - \frac{a}{x^{3}}\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)*cosh(d*x+c)/x^4,x, algorithm="maxima")

[Out]

1/6*((d^2*e^(-c)*gamma(-2, d*x) - d^2*e^c*gamma(-2, -d*x))*a - 2*b*cosh(d*x + c)*log(x^3)/d + 3*(Ei(-d*x)*e^(-
c) + Ei(d*x)*e^c)*b/d)*d + 1/3*(b*log(x^3) - a/x^3)*cosh(d*x + c)

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Fricas [A]  time = 1.72822, size = 277, normalized size = 3.04 \begin{align*} -\frac{2 \, a d x \sinh \left (d x + c\right ) + 2 \,{\left (a d^{2} x^{2} + 2 \, a\right )} \cosh \left (d x + c\right ) -{\left ({\left (a d^{3} + 6 \, b\right )} x^{3}{\rm Ei}\left (d x\right ) -{\left (a d^{3} - 6 \, b\right )} x^{3}{\rm Ei}\left (-d x\right )\right )} \cosh \left (c\right ) -{\left ({\left (a d^{3} + 6 \, b\right )} x^{3}{\rm Ei}\left (d x\right ) +{\left (a d^{3} - 6 \, b\right )} x^{3}{\rm Ei}\left (-d x\right )\right )} \sinh \left (c\right )}{12 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)*cosh(d*x+c)/x^4,x, algorithm="fricas")

[Out]

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

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Sympy [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((b*x**3+a)*cosh(d*x+c)/x**4,x)

[Out]

Exception raised: ValueError

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Giac [A]  time = 1.27178, size = 190, normalized size = 2.09 \begin{align*} -\frac{a d^{3} x^{3}{\rm Ei}\left (-d x\right ) e^{\left (-c\right )} - a d^{3} x^{3}{\rm Ei}\left (d x\right ) e^{c} + a d^{2} x^{2} e^{\left (d x + c\right )} + a d^{2} x^{2} e^{\left (-d x - c\right )} - 6 \, b x^{3}{\rm Ei}\left (-d x\right ) e^{\left (-c\right )} - 6 \, b x^{3}{\rm Ei}\left (d x\right ) e^{c} + a d x e^{\left (d x + c\right )} - a d x e^{\left (-d x - c\right )} + 2 \, a e^{\left (d x + c\right )} + 2 \, a e^{\left (-d x - c\right )}}{12 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)*cosh(d*x+c)/x^4,x, algorithm="giac")

[Out]

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