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

Optimal. Leaf size=71 \[ \frac{b \sinh \left (a-\frac{b c}{d}\right ) \text{Chi}\left (\frac{b c}{d}+b x\right )}{d^2}+\frac{b \cosh \left (a-\frac{b c}{d}\right ) \text{Shi}\left (\frac{b c}{d}+b x\right )}{d^2}-\frac{\cosh (a+b x)}{d (c+d x)} \]

[Out]

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

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Rubi [A]  time = 0.118507, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {3297, 3303, 3298, 3301} \[ \frac{b \sinh \left (a-\frac{b c}{d}\right ) \text{Chi}\left (\frac{b c}{d}+b x\right )}{d^2}+\frac{b \cosh \left (a-\frac{b c}{d}\right ) \text{Shi}\left (\frac{b c}{d}+b x\right )}{d^2}-\frac{\cosh (a+b x)}{d (c+d x)} \]

Antiderivative was successfully verified.

[In]

Int[Cosh[a + b*x]/(c + d*x)^2,x]

[Out]

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

Mathematica [A]  time = 0.243469, size = 65, normalized size = 0.92 \[ \frac{b \sinh \left (a-\frac{b c}{d}\right ) \text{Chi}\left (b \left (\frac{c}{d}+x\right )\right )+b \cosh \left (a-\frac{b c}{d}\right ) \text{Shi}\left (b \left (\frac{c}{d}+x\right )\right )-\frac{d \cosh (a+b x)}{c+d x}}{d^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.033, size = 133, normalized size = 1.9 \begin{align*} -{\frac{b{{\rm e}^{-bx-a}}}{2\,d \left ( bdx+cb \right ) }}+{\frac{b}{2\,{d}^{2}}{{\rm e}^{-{\frac{da-cb}{d}}}}{\it Ei} \left ( 1,bx+a-{\frac{da-cb}{d}} \right ) }-{\frac{b{{\rm e}^{bx+a}}}{2\,{d}^{2}} \left ({\frac{cb}{d}}+bx \right ) ^{-1}}-{\frac{b}{2\,{d}^{2}}{{\rm e}^{{\frac{da-cb}{d}}}}{\it Ei} \left ( 1,-bx-a-{\frac{-da+cb}{d}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.35875, size = 109, normalized size = 1.54 \begin{align*} \frac{b{\left (\frac{e^{\left (-a + \frac{b c}{d}\right )} E_{1}\left (\frac{{\left (d x + c\right )} b}{d}\right )}{d} - \frac{e^{\left (a - \frac{b c}{d}\right )} E_{1}\left (-\frac{{\left (d x + c\right )} b}{d}\right )}{d}\right )}}{2 \, d} - \frac{\cosh \left (b x + a\right )}{{\left (d x + c\right )} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 1.76942, size = 316, normalized size = 4.45 \begin{align*} -\frac{2 \, d \cosh \left (b x + a\right ) -{\left ({\left (b d x + b c\right )}{\rm Ei}\left (\frac{b d x + b c}{d}\right ) -{\left (b d x + b c\right )}{\rm Ei}\left (-\frac{b d x + b c}{d}\right )\right )} \cosh \left (-\frac{b c - a d}{d}\right ) -{\left ({\left (b d x + b c\right )}{\rm Ei}\left (\frac{b d x + b c}{d}\right ) +{\left (b d x + b c\right )}{\rm Ei}\left (-\frac{b d x + b c}{d}\right )\right )} \sinh \left (-\frac{b c - a d}{d}\right )}{2 \,{\left (d^{3} x + c d^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [B]  time = 1.23058, size = 201, normalized size = 2.83 \begin{align*} \frac{b d x{\rm Ei}\left (\frac{b d x + b c}{d}\right ) e^{\left (a - \frac{b c}{d}\right )} - b d x{\rm Ei}\left (-\frac{b d x + b c}{d}\right ) e^{\left (-a + \frac{b c}{d}\right )} + b c{\rm Ei}\left (\frac{b d x + b c}{d}\right ) e^{\left (a - \frac{b c}{d}\right )} - b c{\rm Ei}\left (-\frac{b d x + b c}{d}\right ) e^{\left (-a + \frac{b c}{d}\right )} - d e^{\left (b x + a\right )} - d e^{\left (-b x - a\right )}}{2 \,{\left (d^{3} x + c d^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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