3.47 \(\int \frac{(a+b x^2) \cosh (c+d x)}{x^4} \, dx\)

Optimal. Leaf size=105 \[ \frac{1}{6} a d^3 \sinh (c) \text{Chi}(d x)+\frac{1}{6} a d^3 \cosh (c) \text{Shi}(d x)-\frac{a d^2 \cosh (c+d x)}{6 x}-\frac{a d \sinh (c+d x)}{6 x^2}-\frac{a \cosh (c+d x)}{3 x^3}+b d \sinh (c) \text{Chi}(d x)+b d \cosh (c) \text{Shi}(d x)-\frac{b \cosh (c+d x)}{x} \]

[Out]

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

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Rubi [A]  time = 0.233946, antiderivative size = 105, normalized size of antiderivative = 1., number of steps used = 12, 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} \[ \frac{1}{6} a d^3 \sinh (c) \text{Chi}(d x)+\frac{1}{6} a d^3 \cosh (c) \text{Shi}(d x)-\frac{a d^2 \cosh (c+d x)}{6 x}-\frac{a d \sinh (c+d x)}{6 x^2}-\frac{a \cosh (c+d x)}{3 x^3}+b d \sinh (c) \text{Chi}(d x)+b d \cosh (c) \text{Shi}(d x)-\frac{b \cosh (c+d x)}{x} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x^2)*Cosh[c + d*x])/x^4,x]

[Out]

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

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

Mathematica [A]  time = 0.22498, size = 95, normalized size = 0.9 \[ -\frac{-d x^3 \sinh (c) \left (a d^2+6 b\right ) \text{Chi}(d x)-d x^3 \cosh (c) \left (a d^2+6 b\right ) \text{Shi}(d x)+a d^2 x^2 \cosh (c+d x)+a d x \sinh (c+d x)+2 a \cosh (c+d x)+6 b x^2 \cosh (c+d x)}{6 x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x^2)*Cosh[c + d*x])/x^4,x]

[Out]

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

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Maple [A]  time = 0.059, size = 172, normalized size = 1.6 \begin{align*} -{\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{{d}^{3}a{{\rm e}^{-c}}{\it Ei} \left ( 1,dx \right ) }{12}}-{\frac{b{{\rm e}^{-dx-c}}}{2\,x}}+{\frac{db{{\rm e}^{-c}}{\it Ei} \left ( 1,dx \right ) }{2}}-{\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}^{dx+c}}}{2\,x}}-{\frac{db{{\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^2+a)*cosh(d*x+c)/x^4,x)

[Out]

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

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Maxima [A]  time = 1.21257, size = 99, normalized size = 0.94 \begin{align*} \frac{1}{6} \,{\left (a d^{2} e^{\left (-c\right )} \Gamma \left (-2, d x\right ) - a d^{2} e^{c} \Gamma \left (-2, -d x\right ) - 3 \, b{\rm Ei}\left (-d x\right ) e^{\left (-c\right )} + 3 \, b{\rm Ei}\left (d x\right ) e^{c}\right )} d - \frac{{\left (3 \, b x^{2} + a\right )} \cosh \left (d x + c\right )}{3 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)*cosh(d*x+c)/x^4,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.98027, size = 298, normalized size = 2.84 \begin{align*} -\frac{2 \, a d x \sinh \left (d x + c\right ) + 2 \,{\left ({\left (a d^{2} + 6 \, b\right )} x^{2} + 2 \, a\right )} \cosh \left (d x + c\right ) -{\left ({\left (a d^{3} + 6 \, b d\right )} x^{3}{\rm Ei}\left (d x\right ) -{\left (a d^{3} + 6 \, b d\right )} x^{3}{\rm Ei}\left (-d x\right )\right )} \cosh \left (c\right ) -{\left ({\left (a d^{3} + 6 \, b d\right )} x^{3}{\rm Ei}\left (d x\right ) +{\left (a d^{3} + 6 \, b d\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^2+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 + 6*b)*x^2 + 2*a)*cosh(d*x + c) - ((a*d^3 + 6*b*d)*x^3*Ei(d*x) - (a*d
^3 + 6*b*d)*x^3*Ei(-d*x))*cosh(c) - ((a*d^3 + 6*b*d)*x^3*Ei(d*x) + (a*d^3 + 6*b*d)*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**2+a)*cosh(d*x+c)/x**4,x)

[Out]

Exception raised: ValueError

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Giac [A]  time = 1.18282, size = 230, normalized size = 2.19 \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} + 6 \, b d x^{3}{\rm Ei}\left (-d x\right ) e^{\left (-c\right )} - 6 \, b d 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 )} + a d x e^{\left (d x + c\right )} + 6 \, b x^{2} e^{\left (d x + c\right )} - a d x e^{\left (-d x - c\right )} + 6 \, b x^{2} 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^2+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 + 6*b*d*x^3*Ei(-d*x)*e^(-c) - 6*b*d*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) + a*d*x*e^(d*x + c) + 6*b*x^2*e^(d*x + c) - a*d*x*e^(-d*x - c)
+ 6*b*x^2*e^(-d*x - c) + 2*a*e^(d*x + c) + 2*a*e^(-d*x - c))/x^3