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

Optimal. Leaf size=88 \[ \frac {1}{2} a d^2 \cosh (c) \text {Chi}(d x)+\frac {1}{2} a d^2 \sinh (c) \text {Shi}(d x)-\frac {a \cosh (c+d x)}{2 x^2}-\frac {a d \sinh (c+d x)}{2 x}+b d \sinh (c) \text {Chi}(d x)+b d \cosh (c) \text {Shi}(d x)-\frac {b \cosh (c+d x)}{x} \]

[Out]

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

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Rubi [A]  time = 0.28, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 5, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6742, 3297, 3303, 3298, 3301} \[ \frac {1}{2} a d^2 \cosh (c) \text {Chi}(d x)+\frac {1}{2} a d^2 \sinh (c) \text {Shi}(d x)-\frac {a \cosh (c+d x)}{2 x^2}-\frac {a d \sinh (c+d x)}{2 x}+b d \sinh (c) \text {Chi}(d x)+b d \cosh (c) \text {Shi}(d x)-\frac {b \cosh (c+d x)}{x} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)*Cosh[c + d*x])/x^3,x]

[Out]

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

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 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 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 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

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

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Mathematica [A]  time = 0.17, size = 78, normalized size = 0.89 \[ \frac {d x^2 \text {Chi}(d x) (a d \cosh (c)+2 b \sinh (c))+d x^2 \text {Shi}(d x) (a d \sinh (c)+2 b \cosh (c))-a d x \sinh (c+d x)-a \cosh (c+d x)-2 b x \cosh (c+d x)}{2 x^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.52, size = 116, normalized size = 1.32 \[ -\frac {2 \, a d x \sinh \left (d x + c\right ) + 2 \, {\left (2 \, b x + a\right )} \cosh \left (d x + c\right ) - {\left ({\left (a d^{2} + 2 \, b d\right )} x^{2} {\rm Ei}\left (d x\right ) + {\left (a d^{2} - 2 \, b d\right )} x^{2} {\rm Ei}\left (-d x\right )\right )} \cosh \relax (c) - {\left ({\left (a d^{2} + 2 \, b d\right )} x^{2} {\rm Ei}\left (d x\right ) - {\left (a d^{2} - 2 \, b d\right )} x^{2} {\rm Ei}\left (-d x\right )\right )} \sinh \relax (c)}{4 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(2*a*d*x*sinh(d*x + c) + 2*(2*b*x + a)*cosh(d*x + c) - ((a*d^2 + 2*b*d)*x^2*Ei(d*x) + (a*d^2 - 2*b*d)*x^2
*Ei(-d*x))*cosh(c) - ((a*d^2 + 2*b*d)*x^2*Ei(d*x) - (a*d^2 - 2*b*d)*x^2*Ei(-d*x))*sinh(c))/x^2

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giac [A]  time = 0.12, size = 134, normalized size = 1.52 \[ \frac {a d^{2} x^{2} {\rm Ei}\left (-d x\right ) e^{\left (-c\right )} + a d^{2} x^{2} {\rm Ei}\left (d x\right ) e^{c} - 2 \, b d x^{2} {\rm Ei}\left (-d x\right ) e^{\left (-c\right )} + 2 \, b d x^{2} {\rm Ei}\left (d x\right ) e^{c} - a d x e^{\left (d x + c\right )} + a d x e^{\left (-d x - c\right )} - 2 \, b x e^{\left (d x + c\right )} - 2 \, b x e^{\left (-d x - c\right )} - a e^{\left (d x + c\right )} - a e^{\left (-d x - c\right )}}{4 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.10, size = 139, normalized size = 1.58 \[ \frac {d a \,{\mathrm e}^{-d x -c}}{4 x}-\frac {a \,{\mathrm e}^{-d x -c}}{4 x^{2}}-\frac {d^{2} a \,{\mathrm e}^{-c} \Ei \left (1, d x \right )}{4}-\frac {b \,{\mathrm e}^{-d x -c}}{2 x}+\frac {d b \,{\mathrm e}^{-c} \Ei \left (1, d x \right )}{2}-\frac {a \,{\mathrm e}^{d x +c}}{4 x^{2}}-\frac {d a \,{\mathrm e}^{d x +c}}{4 x}-\frac {d^{2} a \,{\mathrm e}^{c} \Ei \left (1, -d x \right )}{4}-\frac {b \,{\mathrm e}^{d x +c}}{2 x}-\frac {d b \,{\mathrm e}^{c} \Ei \left (1, -d x \right )}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*d*a*exp(-d*x-c)/x-1/4*a*exp(-d*x-c)/x^2-1/4*d^2*a*exp(-c)*Ei(1,d*x)-1/2*b*exp(-d*x-c)/x+1/2*d*b*exp(-c)*Ei
(1,d*x)-1/4*a/x^2*exp(d*x+c)-1/4*d*a/x*exp(d*x+c)-1/4*d^2*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 = 0.43, size = 66, normalized size = 0.75 \[ \frac {1}{4} \, {\left (a d e^{\left (-c\right )} \Gamma \left (-1, d x\right ) + a d e^{c} \Gamma \left (-1, -d x\right ) - 2 \, b {\rm Ei}\left (-d x\right ) e^{\left (-c\right )} + 2 \, b {\rm Ei}\left (d x\right ) e^{c}\right )} d - \frac {{\left (2 \, b x + a\right )} \cosh \left (d x + c\right )}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\mathrm {cosh}\left (c+d\,x\right )\,\left (a+b\,x\right )}{x^3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((cosh(c + d*x)*(a + b*x))/x^3,x)

[Out]

int((cosh(c + d*x)*(a + b*x))/x^3, x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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