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

Optimal. Leaf size=47 \[ a d \sinh (c) \text {Chi}(d x)+a d \cosh (c) \text {Shi}(d x)-\frac {a \cosh (c+d x)}{x}+b \cosh (c) \text {Chi}(d x)+b \sinh (c) \text {Shi}(d x) \]

[Out]

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

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Rubi [A]  time = 0.23, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 9, 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} \[ a d \sinh (c) \text {Chi}(d x)+a d \cosh (c) \text {Shi}(d x)-\frac {a \cosh (c+d x)}{x}+b \cosh (c) \text {Chi}(d x)+b \sinh (c) \text {Shi}(d x) \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)*Cosh[c + d*x])/x^2,x]

[Out]

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

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

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Mathematica [A]  time = 0.14, size = 59, normalized size = 1.26 \[ a d (\sinh (c) \text {Chi}(d x)+\cosh (c) \text {Shi}(d x))-\frac {a \sinh (c) \sinh (d x)}{x}-\frac {a \cosh (c) \cosh (d x)}{x}+b \cosh (c) \text {Chi}(d x)+b \sinh (c) \text {Shi}(d x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.54, size = 76, normalized size = 1.62 \[ -\frac {2 \, a \cosh \left (d x + c\right ) - {\left ({\left (a d + b\right )} x {\rm Ei}\left (d x\right ) - {\left (a d - b\right )} x {\rm Ei}\left (-d x\right )\right )} \cosh \relax (c) - {\left ({\left (a d + b\right )} x {\rm Ei}\left (d x\right ) + {\left (a d - b\right )} x {\rm Ei}\left (-d x\right )\right )} \sinh \relax (c)}{2 \, x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.12, size = 72, normalized size = 1.53 \[ -\frac {a d x {\rm Ei}\left (-d x\right ) e^{\left (-c\right )} - a d x {\rm Ei}\left (d x\right ) e^{c} - b x {\rm Ei}\left (-d x\right ) e^{\left (-c\right )} - b x {\rm Ei}\left (d x\right ) e^{c} + a e^{\left (d x + c\right )} + a e^{\left (-d x - c\right )}}{2 \, x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.10, size = 77, normalized size = 1.64 \[ -\frac {a \,{\mathrm e}^{-d x -c}}{2 x}+\frac {d a \,{\mathrm e}^{-c} \Ei \left (1, d x \right )}{2}-\frac {b \,{\mathrm e}^{-c} \Ei \left (1, d x \right )}{2}-\frac {a \,{\mathrm e}^{d x +c}}{2 x}-\frac {d a \,{\mathrm e}^{c} \Ei \left (1, -d x \right )}{2}-\frac {b \,{\mathrm e}^{c} \Ei \left (1, -d x \right )}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*a*exp(-d*x-c)/x+1/2*d*a*exp(-c)*Ei(1,d*x)-1/2*b*exp(-c)*Ei(1,d*x)-1/2*a/x*exp(d*x+c)-1/2*d*a*exp(c)*Ei(1,
-d*x)-1/2*b*exp(c)*Ei(1,-d*x)

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maxima [A]  time = 0.41, size = 82, normalized size = 1.74 \[ -\frac {1}{2} \, {\left ({\left ({\rm Ei}\left (-d x\right ) e^{\left (-c\right )} - {\rm Ei}\left (d x\right ) e^{c}\right )} a + \frac {2 \, b \cosh \left (d x + c\right ) \log \relax (x)}{d} - \frac {{\left ({\rm Ei}\left (-d x\right ) e^{\left (-c\right )} + {\rm Ei}\left (d x\right ) e^{c}\right )} b}{d}\right )} d + {\left (b \log \relax (x) - \frac {a}{x}\right )} \cosh \left (d x + c\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\mathrm {cosh}\left (c+d\,x\right )\,\left (a+b\,x\right )}{x^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((cosh(c + d*x)*(a + b*x))/x^2,x)

[Out]

int((cosh(c + d*x)*(a + b*x))/x^2, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b x\right ) \cosh {\left (c + d x \right )}}{x^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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