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

Optimal. Leaf size=150 \[ -\frac{\cosh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (x d+\frac{a d}{b}\right )}{a^2}-\frac{\sinh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (x d+\frac{a d}{b}\right )}{a^2}+\frac{\cosh (c) \text{Chi}(d x)}{a^2}+\frac{\sinh (c) \text{Shi}(d x)}{a^2}-\frac{d \sinh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (x d+\frac{a d}{b}\right )}{a b}-\frac{d \cosh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (x d+\frac{a d}{b}\right )}{a b}+\frac{\cosh (c+d x)}{a (a+b x)} \]

[Out]

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

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Rubi [A]  time = 0.400274, antiderivative size = 150, 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 = {6742, 3303, 3298, 3301, 3297} \[ -\frac{\cosh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (x d+\frac{a d}{b}\right )}{a^2}-\frac{\sinh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (x d+\frac{a d}{b}\right )}{a^2}+\frac{\cosh (c) \text{Chi}(d x)}{a^2}+\frac{\sinh (c) \text{Shi}(d x)}{a^2}-\frac{d \sinh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (x d+\frac{a d}{b}\right )}{a b}-\frac{d \cosh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (x d+\frac{a d}{b}\right )}{a b}+\frac{\cosh (c+d x)}{a (a+b x)} \]

Antiderivative was successfully verified.

[In]

Int[Cosh[c + d*x]/(x*(a + b*x)^2),x]

[Out]

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

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]

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]

Rubi steps

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

Mathematica [A]  time = 1.05038, size = 241, normalized size = 1.61 \[ -\frac{a^2 d \sinh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (\frac{d (a+b x)}{b}\right )+a^2 d \cosh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (\frac{d (a+b x)}{b}\right )+b^2 x \sinh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (d \left (\frac{a}{b}+x\right )\right )+a b d x \sinh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (\frac{d (a+b x)}{b}\right )-b \cosh (c) (a+b x) \text{Chi}(d x)+b (a+b x) \cosh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (d \left (\frac{a}{b}+x\right )\right )-a b \sinh (c) \text{Shi}(d x)+a b \sinh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (d \left (\frac{a}{b}+x\right )\right )+a b d x \cosh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (\frac{d (a+b x)}{b}\right )-a b \cosh (c+d x)-b^2 x \sinh (c) \text{Shi}(d x)}{a^2 b (a+b x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((-(a*b*Cosh[c + d*x]) - b*(a + b*x)*Cosh[c]*CoshIntegral[d*x] + b*(a + b*x)*Cosh[c - (a*d)/b]*CoshIntegral[d
*(a/b + x)] + a^2*d*CoshIntegral[(d*(a + b*x))/b]*Sinh[c - (a*d)/b] + a*b*d*x*CoshIntegral[(d*(a + b*x))/b]*Si
nh[c - (a*d)/b] - a*b*Sinh[c]*SinhIntegral[d*x] - b^2*x*Sinh[c]*SinhIntegral[d*x] + a*b*Sinh[c - (a*d)/b]*Sinh
Integral[d*(a/b + x)] + b^2*x*Sinh[c - (a*d)/b]*SinhIntegral[d*(a/b + x)] + a^2*d*Cosh[c - (a*d)/b]*SinhIntegr
al[(d*(a + b*x))/b] + a*b*d*x*Cosh[c - (a*d)/b]*SinhIntegral[(d*(a + b*x))/b])/(a^2*b*(a + b*x)))

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Maple [A]  time = 0.05, size = 254, normalized size = 1.7 \begin{align*}{\frac{{{\rm e}^{-dx-c}}d}{2\,a \left ( \left ( dx+c \right ) b+da-cb \right ) }}-{\frac{{{\rm e}^{-c}}{\it Ei} \left ( 1,dx \right ) }{2\,{a}^{2}}}-{\frac{d}{2\,ab}{{\rm e}^{{\frac{da-cb}{b}}}}{\it Ei} \left ( 1,dx+c+{\frac{da-cb}{b}} \right ) }+{\frac{1}{2\,{a}^{2}}{{\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}^{2}}}+{\frac{d{{\rm e}^{dx+c}}}{2\,ab} \left ({\frac{da}{b}}+dx \right ) ^{-1}}+{\frac{d}{2\,ab}{{\rm e}^{-{\frac{da-cb}{b}}}}{\it Ei} \left ( 1,-dx-c-{\frac{da-cb}{b}} \right ) }+{\frac{1}{2\,{a}^{2}}{{\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)^2,x)

[Out]

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

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Maxima [A]  time = 1.43188, size = 306, normalized size = 2.04 \begin{align*} -\frac{1}{2} \, d{\left (\frac{e^{\left (-c + \frac{a d}{b}\right )} E_{1}\left (\frac{{\left (b x + a\right )} d}{b}\right )}{a b} - \frac{e^{\left (c - \frac{a d}{b}\right )} E_{1}\left (-\frac{{\left (b x + a\right )} d}{b}\right )}{a b} - \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^{2} d} - \frac{2 \, \cosh \left (d x + c\right ) \log \left (b x + a\right )}{a^{2} d} + \frac{2 \, \cosh \left (d x + c\right ) \log \left (x\right )}{a^{2} d} - \frac{{\rm Ei}\left (-d x\right ) e^{\left (-c\right )} +{\rm Ei}\left (d x\right ) e^{c}}{a^{2} d}\right )} +{\left (\frac{1}{a b x + a^{2}} - \frac{\log \left (b x + a\right )}{a^{2}} + \frac{\log \left (x\right )}{a^{2}}\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)^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 2.15082, size = 579, normalized size = 3.86 \begin{align*} \frac{2 \, a b \cosh \left (d x + c\right ) +{\left ({\left (b^{2} x + a b\right )}{\rm Ei}\left (d x\right ) +{\left (b^{2} x + a b\right )}{\rm Ei}\left (-d x\right )\right )} \cosh \left (c\right ) -{\left ({\left (a^{2} d + a b +{\left (a b d + b^{2}\right )} x\right )}{\rm Ei}\left (\frac{b d x + a d}{b}\right ) -{\left (a^{2} d - a b +{\left (a b d - b^{2}\right )} x\right )}{\rm Ei}\left (-\frac{b d x + a d}{b}\right )\right )} \cosh \left (-\frac{b c - a d}{b}\right ) +{\left ({\left (b^{2} x + a b\right )}{\rm Ei}\left (d x\right ) -{\left (b^{2} x + a b\right )}{\rm Ei}\left (-d x\right )\right )} \sinh \left (c\right ) +{\left ({\left (a^{2} d + a b +{\left (a b d + b^{2}\right )} x\right )}{\rm Ei}\left (\frac{b d x + a d}{b}\right ) +{\left (a^{2} d - a b +{\left (a b d - b^{2}\right )} x\right )}{\rm Ei}\left (-\frac{b d x + a d}{b}\right )\right )} \sinh \left (-\frac{b c - a d}{b}\right )}{2 \,{\left (a^{2} b^{2} x + a^{3} b\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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 )^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.21112, size = 439, normalized size = 2.93 \begin{align*} -\frac{{\left (a b d x{\rm Ei}\left (\frac{b d x + a d}{b}\right ) e^{\left (c - \frac{a d}{b}\right )} - a b d x{\rm Ei}\left (-\frac{b d x + a d}{b}\right ) e^{\left (-c + \frac{a d}{b}\right )} - b^{2} x{\rm Ei}\left (-d x\right ) e^{\left (-c\right )} + a^{2} d{\rm Ei}\left (\frac{b d x + a d}{b}\right ) e^{\left (c - \frac{a d}{b}\right )} + b^{2} x{\rm Ei}\left (\frac{b d x + a d}{b}\right ) e^{\left (c - \frac{a d}{b}\right )} - b^{2} x{\rm Ei}\left (d x\right ) e^{c} - a^{2} d{\rm Ei}\left (-\frac{b d x + a d}{b}\right ) e^{\left (-c + \frac{a d}{b}\right )} + b^{2} x{\rm Ei}\left (-\frac{b d x + a d}{b}\right ) e^{\left (-c + \frac{a d}{b}\right )} - a b{\rm Ei}\left (-d x\right ) e^{\left (-c\right )} + a b{\rm Ei}\left (\frac{b d x + a d}{b}\right ) e^{\left (c - \frac{a d}{b}\right )} - a b{\rm Ei}\left (d x\right ) e^{c} + a b{\rm Ei}\left (-\frac{b d x + a d}{b}\right ) e^{\left (-c + \frac{a d}{b}\right )} - a b e^{\left (d x + c\right )} - a b e^{\left (-d x - c\right )}\right )} b}{2 \,{\left (a^{2} b^{3} x + a^{3} b^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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