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

Optimal. Leaf size=73 \[ -\frac{\cosh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (x d+\frac{a d}{b}\right )}{a}-\frac{\sinh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (x d+\frac{a d}{b}\right )}{a}+\frac{\cosh (c) \text{Chi}(d x)}{a}+\frac{\sinh (c) \text{Shi}(d x)}{a} \]

[Out]

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

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Rubi [A]  time = 0.258935, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235, Rules used = {6742, 3303, 3298, 3301} \[ -\frac{\cosh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (x d+\frac{a d}{b}\right )}{a}-\frac{\sinh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (x d+\frac{a d}{b}\right )}{a}+\frac{\cosh (c) \text{Chi}(d x)}{a}+\frac{\sinh (c) \text{Shi}(d x)}{a} \]

Antiderivative was successfully verified.

[In]

Int[Cosh[c + d*x]/(x*(a + b*x)),x]

[Out]

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

Rule 6742

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

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

Mathematica [A]  time = 0.140752, size = 63, normalized size = 0.86 \[ \frac{-\cosh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (d \left (\frac{a}{b}+x\right )\right )-\sinh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (d \left (\frac{a}{b}+x\right )\right )+\cosh (c) \text{Chi}(d x)+\sinh (c) \text{Shi}(d x)}{a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.031, size = 108, normalized size = 1.5 \begin{align*} -{\frac{{{\rm e}^{-c}}{\it Ei} \left ( 1,dx \right ) }{2\,a}}+{\frac{1}{2\,a}{{\rm e}^{{\frac{da-cb}{b}}}}{\it Ei} \left ( 1,dx+c+{\frac{da-cb}{b}} \right ) }-{\frac{{{\rm e}^{c}}{\it Ei} \left ( 1,-dx \right ) }{2\,a}}+{\frac{1}{2\,a}{{\rm e}^{-{\frac{da-cb}{b}}}}{\it Ei} \left ( 1,-dx-c-{\frac{da-cb}{b}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.43734, size = 209, normalized size = 2.86 \begin{align*} \frac{1}{2} \, d{\left (\frac{b{\left (\frac{e^{\left (-c + \frac{a d}{b}\right )} E_{1}\left (\frac{{\left (b x + a\right )} d}{b}\right )}{b} + \frac{e^{\left (c - \frac{a d}{b}\right )} E_{1}\left (-\frac{{\left (b x + a\right )} d}{b}\right )}{b}\right )}}{a d} + \frac{2 \, \cosh \left (d x + c\right ) \log \left (b x + a\right )}{a d} - \frac{2 \, \cosh \left (d x + c\right ) \log \left (x\right )}{a d} + \frac{{\rm Ei}\left (-d x\right ) e^{\left (-c\right )} +{\rm Ei}\left (d x\right ) e^{c}}{a d}\right )} -{\left (\frac{\log \left (b x + a\right )}{a} - \frac{\log \left (x\right )}{a}\right )} \cosh \left (d x + c\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.06579, size = 277, normalized size = 3.79 \begin{align*} \frac{{\left ({\rm Ei}\left (d x\right ) +{\rm Ei}\left (-d x\right )\right )} \cosh \left (c\right ) -{\left ({\rm Ei}\left (\frac{b d x + a d}{b}\right ) +{\rm Ei}\left (-\frac{b d x + a d}{b}\right )\right )} \cosh \left (-\frac{b c - a d}{b}\right ) +{\left ({\rm Ei}\left (d x\right ) -{\rm Ei}\left (-d x\right )\right )} \sinh \left (c\right ) +{\left ({\rm Ei}\left (\frac{b d x + a d}{b}\right ) -{\rm Ei}\left (-\frac{b d x + a d}{b}\right )\right )} \sinh \left (-\frac{b c - a d}{b}\right )}{2 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.15917, size = 101, normalized size = 1.38 \begin{align*} \frac{{\rm Ei}\left (-d x\right ) e^{\left (-c\right )} -{\rm Ei}\left (\frac{b d x + a d}{b}\right ) e^{\left (c - \frac{a d}{b}\right )} +{\rm Ei}\left (d x\right ) e^{c} -{\rm Ei}\left (-\frac{b d x + a d}{b}\right ) e^{\left (-c + \frac{a d}{b}\right )}}{2 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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