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

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

[Out]

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

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Rubi [A]  time = 0.109993, 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} \[ \frac{d \sinh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (x d+\frac{a d}{b}\right )}{b^2}+\frac{d \cosh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (x d+\frac{a d}{b}\right )}{b^2}-\frac{\cosh (c+d x)}{b (a+b x)} \]

Antiderivative was successfully verified.

[In]

Int[Cosh[c + d*x]/(a + b*x)^2,x]

[Out]

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

Mathematica [A]  time = 0.201944, size = 65, normalized size = 0.92 \[ \frac{d \sinh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (d \left (\frac{a}{b}+x\right )\right )+d \cosh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (d \left (\frac{a}{b}+x\right )\right )-\frac{b \cosh (c+d x)}{a+b x}}{b^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.025, size = 132, normalized size = 1.9 \begin{align*} -{\frac{d{{\rm e}^{-dx-c}}}{2\,b \left ( bdx+da \right ) }}+{\frac{d}{2\,{b}^{2}}{{\rm e}^{{\frac{da-cb}{b}}}}{\it Ei} \left ( 1,dx+c+{\frac{da-cb}{b}} \right ) }-{\frac{d{{\rm e}^{dx+c}}}{2\,{b}^{2}} \left ({\frac{da}{b}}+dx \right ) ^{-1}}-{\frac{d}{2\,{b}^{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)/(b*x+a)^2,x)

[Out]

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

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Maxima [A]  time = 1.20947, size = 109, normalized size = 1.54 \begin{align*} \frac{d{\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 )}}{2 \, b} - \frac{\cosh \left (d x + c\right )}{{\left (b x + a\right )} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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Fricas [B]  time = 2.03666, size = 316, normalized size = 4.45 \begin{align*} -\frac{2 \, b \cosh \left (d x + c\right ) -{\left ({\left (b d x + a d\right )}{\rm Ei}\left (\frac{b d x + a d}{b}\right ) -{\left (b d x + a d\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 d x + a d\right )}{\rm Ei}\left (\frac{b d x + a d}{b}\right ) +{\left (b d x + a d\right )}{\rm Ei}\left (-\frac{b d x + a d}{b}\right )\right )} \sinh \left (-\frac{b c - a d}{b}\right )}{2 \,{\left (b^{3} x + a b^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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Giac [B]  time = 1.21064, size = 201, normalized size = 2.83 \begin{align*} \frac{b d x{\rm Ei}\left (\frac{b d x + a d}{b}\right ) e^{\left (c - \frac{a d}{b}\right )} - b d x{\rm Ei}\left (-\frac{b d x + a d}{b}\right ) e^{\left (-c + \frac{a d}{b}\right )} + a d{\rm Ei}\left (\frac{b d x + a d}{b}\right ) e^{\left (c - \frac{a d}{b}\right )} - a d{\rm Ei}\left (-\frac{b d x + a d}{b}\right ) e^{\left (-c + \frac{a d}{b}\right )} - b e^{\left (d x + c\right )} - b e^{\left (-d x - c\right )}}{2 \,{\left (b^{3} x + a b^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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