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

Optimal. Leaf size=87 \[ -\frac{a}{d (c+d x)}+\frac{b f \text{Chi}\left (x f+\frac{c f}{d}\right ) \sinh \left (e-\frac{c f}{d}\right )}{d^2}+\frac{b f \cosh \left (e-\frac{c f}{d}\right ) \text{Shi}\left (x f+\frac{c f}{d}\right )}{d^2}-\frac{b \cosh (e+f x)}{d (c+d x)} \]

[Out]

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

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Rubi [A]  time = 0.151882, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.278, Rules used = {3317, 3297, 3303, 3298, 3301} \[ -\frac{a}{d (c+d x)}+\frac{b f \text{Chi}\left (x f+\frac{c f}{d}\right ) \sinh \left (e-\frac{c f}{d}\right )}{d^2}+\frac{b f \cosh \left (e-\frac{c f}{d}\right ) \text{Shi}\left (x f+\frac{c f}{d}\right )}{d^2}-\frac{b \cosh (e+f x)}{d (c+d x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 3317

Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Int[ExpandIntegrand[
(c + d*x)^m, (a + b*Sin[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[n, 0] && (EqQ[n, 1] ||
IGtQ[m, 0] || NeQ[a^2 - b^2, 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]

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

Mathematica [A]  time = 0.400004, size = 71, normalized size = 0.82 \[ \frac{-\frac{d (a+b \cosh (e+f x))}{c+d x}+b f \text{Chi}\left (f \left (\frac{c}{d}+x\right )\right ) \sinh \left (e-\frac{c f}{d}\right )+b f \cosh \left (e-\frac{c f}{d}\right ) \text{Shi}\left (f \left (\frac{c}{d}+x\right )\right )}{d^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 180., size = 0, normalized size = 0. \begin{align*} \text{hanged} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.39857, size = 117, normalized size = 1.34 \begin{align*} -\frac{1}{2} \, b{\left (\frac{e^{\left (-e + \frac{c f}{d}\right )} E_{2}\left (\frac{{\left (d x + c\right )} f}{d}\right )}{{\left (d x + c\right )} d} + \frac{e^{\left (e - \frac{c f}{d}\right )} E_{2}\left (-\frac{{\left (d x + c\right )} f}{d}\right )}{{\left (d x + c\right )} d}\right )} - \frac{a}{d^{2} x + c d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.35, size = 351, normalized size = 4.03 \begin{align*} -\frac{2 \, b d \cosh \left (f x + e\right ) + 2 \, a d -{\left ({\left (b d f x + b c f\right )}{\rm Ei}\left (\frac{d f x + c f}{d}\right ) -{\left (b d f x + b c f\right )}{\rm Ei}\left (-\frac{d f x + c f}{d}\right )\right )} \cosh \left (-\frac{d e - c f}{d}\right ) +{\left ({\left (b d f x + b c f\right )}{\rm Ei}\left (\frac{d f x + c f}{d}\right ) +{\left (b d f x + b c f\right )}{\rm Ei}\left (-\frac{d f x + c f}{d}\right )\right )} \sinh \left (-\frac{d e - c f}{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((a+b*cosh(f*x+e))/(d*x+c)^2,x, algorithm="fricas")

[Out]

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

[Out]

Timed out

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Giac [A]  time = 1.20195, size = 227, normalized size = 2.61 \begin{align*} -\frac{{\left (d f x{\rm Ei}\left (-\frac{d f x + c f}{d}\right ) e^{\left (\frac{c f}{d} - e\right )} - d f x{\rm Ei}\left (\frac{d f x + c f}{d}\right ) e^{\left (-\frac{c f}{d} + e\right )} + c f{\rm Ei}\left (-\frac{d f x + c f}{d}\right ) e^{\left (\frac{c f}{d} - e\right )} - c f{\rm Ei}\left (\frac{d f x + c f}{d}\right ) e^{\left (-\frac{c f}{d} + e\right )} + d e^{\left (f x + e\right )} + d e^{\left (-f x - e\right )}\right )} b}{2 \,{\left (d^{3} x + c d^{2}\right )}} - \frac{a}{{\left (d x + c\right )} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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