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

Optimal. Leaf size=157 \[ \frac{a^2 f \text{Chi}\left (2 x f+\frac{2 c f}{d}\right ) \sinh \left (2 e-\frac{2 c f}{d}\right )}{d^2}+\frac{2 a^2 f \text{Chi}\left (x f+\frac{c f}{d}\right ) \sinh \left (e-\frac{c f}{d}\right )}{d^2}+\frac{2 a^2 f \cosh \left (e-\frac{c f}{d}\right ) \text{Shi}\left (x f+\frac{c f}{d}\right )}{d^2}+\frac{a^2 f \cosh \left (2 e-\frac{2 c f}{d}\right ) \text{Shi}\left (2 x f+\frac{2 c f}{d}\right )}{d^2}-\frac{4 a^2 \cosh ^4\left (\frac{e}{2}+\frac{f x}{2}\right )}{d (c+d x)} \]

[Out]

(-4*a^2*Cosh[e/2 + (f*x)/2]^4)/(d*(c + d*x)) + (a^2*f*CoshIntegral[(2*c*f)/d + 2*f*x]*Sinh[2*e - (2*c*f)/d])/d
^2 + (2*a^2*f*CoshIntegral[(c*f)/d + f*x]*Sinh[e - (c*f)/d])/d^2 + (2*a^2*f*Cosh[e - (c*f)/d]*SinhIntegral[(c*
f)/d + f*x])/d^2 + (a^2*f*Cosh[2*e - (2*c*f)/d]*SinhIntegral[(2*c*f)/d + 2*f*x])/d^2

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Rubi [A]  time = 0.332978, antiderivative size = 157, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3318, 3313, 3303, 3298, 3301} \[ \frac{a^2 f \text{Chi}\left (2 x f+\frac{2 c f}{d}\right ) \sinh \left (2 e-\frac{2 c f}{d}\right )}{d^2}+\frac{2 a^2 f \text{Chi}\left (x f+\frac{c f}{d}\right ) \sinh \left (e-\frac{c f}{d}\right )}{d^2}+\frac{2 a^2 f \cosh \left (e-\frac{c f}{d}\right ) \text{Shi}\left (x f+\frac{c f}{d}\right )}{d^2}+\frac{a^2 f \cosh \left (2 e-\frac{2 c f}{d}\right ) \text{Shi}\left (2 x f+\frac{2 c f}{d}\right )}{d^2}-\frac{4 a^2 \cosh ^4\left (\frac{e}{2}+\frac{f x}{2}\right )}{d (c+d x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-4*a^2*Cosh[e/2 + (f*x)/2]^4)/(d*(c + d*x)) + (a^2*f*CoshIntegral[(2*c*f)/d + 2*f*x]*Sinh[2*e - (2*c*f)/d])/d
^2 + (2*a^2*f*CoshIntegral[(c*f)/d + f*x]*Sinh[e - (c*f)/d])/d^2 + (2*a^2*f*Cosh[e - (c*f)/d]*SinhIntegral[(c*
f)/d + f*x])/d^2 + (a^2*f*Cosh[2*e - (2*c*f)/d]*SinhIntegral[(2*c*f)/d + 2*f*x])/d^2

Rule 3318

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

Rule 3313

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Simp[((c + d*x)^(m + 1)*Sin[e + f*x]^
n)/(d*(m + 1)), x] - Dist[(f*n)/(d*(m + 1)), Int[ExpandTrigReduce[(c + d*x)^(m + 1), Cos[e + f*x]*Sin[e + f*x]
^(n - 1), x], x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && GeQ[m, -2] && 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+a \cosh (e+f x))^2}{(c+d x)^2} \, dx &=\left (4 a^2\right ) \int \frac{\sin ^4\left (\frac{1}{2} (i e+\pi )+\frac{i f x}{2}\right )}{(c+d x)^2} \, dx\\ &=-\frac{4 a^2 \cosh ^4\left (\frac{e}{2}+\frac{f x}{2}\right )}{d (c+d x)}+\frac{\left (8 i a^2 f\right ) \int \left (-\frac{i \sinh (e+f x)}{4 (c+d x)}-\frac{i \sinh (2 e+2 f x)}{8 (c+d x)}\right ) \, dx}{d}\\ &=-\frac{4 a^2 \cosh ^4\left (\frac{e}{2}+\frac{f x}{2}\right )}{d (c+d x)}+\frac{\left (a^2 f\right ) \int \frac{\sinh (2 e+2 f x)}{c+d x} \, dx}{d}+\frac{\left (2 a^2 f\right ) \int \frac{\sinh (e+f x)}{c+d x} \, dx}{d}\\ &=-\frac{4 a^2 \cosh ^4\left (\frac{e}{2}+\frac{f x}{2}\right )}{d (c+d x)}+\frac{\left (a^2 f \cosh \left (2 e-\frac{2 c f}{d}\right )\right ) \int \frac{\sinh \left (\frac{2 c f}{d}+2 f x\right )}{c+d x} \, dx}{d}+\frac{\left (2 a^2 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 (a^2 f \sinh \left (2 e-\frac{2 c f}{d}\right )\right ) \int \frac{\cosh \left (\frac{2 c f}{d}+2 f x\right )}{c+d x} \, dx}{d}+\frac{\left (2 a^2 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{4 a^2 \cosh ^4\left (\frac{e}{2}+\frac{f x}{2}\right )}{d (c+d x)}+\frac{a^2 f \text{Chi}\left (\frac{2 c f}{d}+2 f x\right ) \sinh \left (2 e-\frac{2 c f}{d}\right )}{d^2}+\frac{2 a^2 f \text{Chi}\left (\frac{c f}{d}+f x\right ) \sinh \left (e-\frac{c f}{d}\right )}{d^2}+\frac{2 a^2 f \cosh \left (e-\frac{c f}{d}\right ) \text{Shi}\left (\frac{c f}{d}+f x\right )}{d^2}+\frac{a^2 f \cosh \left (2 e-\frac{2 c f}{d}\right ) \text{Shi}\left (\frac{2 c f}{d}+2 f x\right )}{d^2}\\ \end{align*}

Mathematica [A]  time = 0.630069, size = 207, normalized size = 1.32 \[ \frac{a^2 \left (2 f (c+d x) \text{Chi}\left (\frac{2 f (c+d x)}{d}\right ) \sinh \left (2 e-\frac{2 c f}{d}\right )+4 f (c+d x) \text{Chi}\left (f \left (\frac{c}{d}+x\right )\right ) \sinh \left (e-\frac{c f}{d}\right )+4 d f x \cosh \left (e-\frac{c f}{d}\right ) \text{Shi}\left (f \left (\frac{c}{d}+x\right )\right )+2 d f x \cosh \left (2 e-\frac{2 c f}{d}\right ) \text{Shi}\left (\frac{2 f (c+d x)}{d}\right )+4 c f \cosh \left (e-\frac{c f}{d}\right ) \text{Shi}\left (f \left (\frac{c}{d}+x\right )\right )+2 c f \cosh \left (2 e-\frac{2 c f}{d}\right ) \text{Shi}\left (\frac{2 f (c+d x)}{d}\right )-4 d \cosh (e+f x)-d \cosh (2 (e+f x))-3 d\right )}{2 d^2 (c+d x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^2*(-3*d - 4*d*Cosh[e + f*x] - d*Cosh[2*(e + f*x)] + 2*f*(c + d*x)*CoshIntegral[(2*f*(c + d*x))/d]*Sinh[2*e
- (2*c*f)/d] + 4*f*(c + d*x)*CoshIntegral[f*(c/d + x)]*Sinh[e - (c*f)/d] + 4*c*f*Cosh[e - (c*f)/d]*SinhIntegra
l[f*(c/d + x)] + 4*d*f*x*Cosh[e - (c*f)/d]*SinhIntegral[f*(c/d + x)] + 2*c*f*Cosh[2*e - (2*c*f)/d]*SinhIntegra
l[(2*f*(c + d*x))/d] + 2*d*f*x*Cosh[2*e - (2*c*f)/d]*SinhIntegral[(2*f*(c + d*x))/d]))/(2*d^2*(c + d*x))

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Maple [A]  time = 0.123, size = 308, normalized size = 2. \begin{align*} -{\frac{f{a}^{2}{{\rm e}^{-fx-e}}}{d \left ( dfx+cf \right ) }}+{\frac{f{a}^{2}}{{d}^{2}}{{\rm e}^{{\frac{cf-de}{d}}}}{\it Ei} \left ( 1,fx+e+{\frac{cf-de}{d}} \right ) }-{\frac{f{a}^{2}{{\rm e}^{fx+e}}}{{d}^{2}} \left ({\frac{cf}{d}}+fx \right ) ^{-1}}-{\frac{f{a}^{2}}{{d}^{2}}{{\rm e}^{-{\frac{cf-de}{d}}}}{\it Ei} \left ( 1,-fx-e-{\frac{cf-de}{d}} \right ) }-{\frac{3\,{a}^{2}}{2\,d \left ( dx+c \right ) }}-{\frac{f{a}^{2}{{\rm e}^{-2\,fx-2\,e}}}{4\,d \left ( dfx+cf \right ) }}+{\frac{f{a}^{2}}{2\,{d}^{2}}{{\rm e}^{2\,{\frac{cf-de}{d}}}}{\it Ei} \left ( 1,2\,fx+2\,e+2\,{\frac{cf-de}{d}} \right ) }-{\frac{f{a}^{2}{{\rm e}^{2\,fx+2\,e}}}{4\,{d}^{2}} \left ({\frac{cf}{d}}+fx \right ) ^{-1}}-{\frac{f{a}^{2}}{2\,{d}^{2}}{{\rm e}^{-2\,{\frac{cf-de}{d}}}}{\it Ei} \left ( 1,-2\,fx-2\,e-2\,{\frac{cf-de}{d}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a^2*f*exp(-f*x-e)/d/(d*f*x+c*f)+a^2*f/d^2*exp((c*f-d*e)/d)*Ei(1,f*x+e+(c*f-d*e)/d)-f*a^2/d^2*exp(f*x+e)/(c*f/
d+f*x)-f*a^2/d^2*exp(-(c*f-d*e)/d)*Ei(1,-f*x-e-(c*f-d*e)/d)-3/2*a^2/d/(d*x+c)-1/4*a^2*f*exp(-2*f*x-2*e)/d/(d*f
*x+c*f)+1/2*a^2*f/d^2*exp(2*(c*f-d*e)/d)*Ei(1,2*f*x+2*e+2*(c*f-d*e)/d)-1/4*f*a^2/d^2*exp(2*f*x+2*e)/(c*f/d+f*x
)-1/2*f*a^2/d^2*exp(-2*(c*f-d*e)/d)*Ei(1,-2*f*x-2*e-2*(c*f-d*e)/d)

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Maxima [A]  time = 1.32004, size = 246, normalized size = 1.57 \begin{align*} -\frac{1}{4} \, a^{2}{\left (\frac{e^{\left (-2 \, e + \frac{2 \, c f}{d}\right )} E_{2}\left (\frac{2 \,{\left (d x + c\right )} f}{d}\right )}{{\left (d x + c\right )} d} + \frac{e^{\left (2 \, e - \frac{2 \, c f}{d}\right )} E_{2}\left (-\frac{2 \,{\left (d x + c\right )} f}{d}\right )}{{\left (d x + c\right )} d} + \frac{2}{d^{2} x + c d}\right )} - a^{2}{\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^{2}}{d^{2} x + c d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*a^2*(e^(-2*e + 2*c*f/d)*exp_integral_e(2, 2*(d*x + c)*f/d)/((d*x + c)*d) + e^(2*e - 2*c*f/d)*exp_integral
_e(2, -2*(d*x + c)*f/d)/((d*x + c)*d) + 2/(d^2*x + c*d)) - a^2*(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^2/(d^2*x + c*d)

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Fricas [B]  time = 2.16994, size = 767, normalized size = 4.89 \begin{align*} -\frac{a^{2} d \cosh \left (f x + e\right )^{2} + a^{2} d \sinh \left (f x + e\right )^{2} + 4 \, a^{2} d \cosh \left (f x + e\right ) + 3 \, a^{2} d - 2 \,{\left ({\left (a^{2} d f x + a^{2} c f\right )}{\rm Ei}\left (\frac{d f x + c f}{d}\right ) -{\left (a^{2} d f x + a^{2} 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 (a^{2} d f x + a^{2} c f\right )}{\rm Ei}\left (\frac{2 \,{\left (d f x + c f\right )}}{d}\right ) -{\left (a^{2} d f x + a^{2} c f\right )}{\rm Ei}\left (-\frac{2 \,{\left (d f x + c f\right )}}{d}\right )\right )} \cosh \left (-\frac{2 \,{\left (d e - c f\right )}}{d}\right ) + 2 \,{\left ({\left (a^{2} d f x + a^{2} c f\right )}{\rm Ei}\left (\frac{d f x + c f}{d}\right ) +{\left (a^{2} d f x + a^{2} c f\right )}{\rm Ei}\left (-\frac{d f x + c f}{d}\right )\right )} \sinh \left (-\frac{d e - c f}{d}\right ) +{\left ({\left (a^{2} d f x + a^{2} c f\right )}{\rm Ei}\left (\frac{2 \,{\left (d f x + c f\right )}}{d}\right ) +{\left (a^{2} d f x + a^{2} c f\right )}{\rm Ei}\left (-\frac{2 \,{\left (d f x + c f\right )}}{d}\right )\right )} \sinh \left (-\frac{2 \,{\left (d e - c f\right )}}{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+a*cosh(f*x+e))^2/(d*x+c)^2,x, algorithm="fricas")

[Out]

-1/2*(a^2*d*cosh(f*x + e)^2 + a^2*d*sinh(f*x + e)^2 + 4*a^2*d*cosh(f*x + e) + 3*a^2*d - 2*((a^2*d*f*x + a^2*c*
f)*Ei((d*f*x + c*f)/d) - (a^2*d*f*x + a^2*c*f)*Ei(-(d*f*x + c*f)/d))*cosh(-(d*e - c*f)/d) - ((a^2*d*f*x + a^2*
c*f)*Ei(2*(d*f*x + c*f)/d) - (a^2*d*f*x + a^2*c*f)*Ei(-2*(d*f*x + c*f)/d))*cosh(-2*(d*e - c*f)/d) + 2*((a^2*d*
f*x + a^2*c*f)*Ei((d*f*x + c*f)/d) + (a^2*d*f*x + a^2*c*f)*Ei(-(d*f*x + c*f)/d))*sinh(-(d*e - c*f)/d) + ((a^2*
d*f*x + a^2*c*f)*Ei(2*(d*f*x + c*f)/d) + (a^2*d*f*x + a^2*c*f)*Ei(-2*(d*f*x + c*f)/d))*sinh(-2*(d*e - c*f)/d))
/(d^3*x + c*d^2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} a^{2} \left (\int \frac{2 \cosh{\left (e + f x \right )}}{c^{2} + 2 c d x + d^{2} x^{2}}\, dx + \int \frac{\cosh ^{2}{\left (e + f x \right )}}{c^{2} + 2 c d x + d^{2} x^{2}}\, dx + \int \frac{1}{c^{2} + 2 c d x + d^{2} x^{2}}\, dx\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**2*(Integral(2*cosh(e + f*x)/(c**2 + 2*c*d*x + d**2*x**2), x) + Integral(cosh(e + f*x)**2/(c**2 + 2*c*d*x +
d**2*x**2), x) + Integral(1/(c**2 + 2*c*d*x + d**2*x**2), x))

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Giac [B]  time = 2.31396, size = 485, normalized size = 3.09 \begin{align*} -\frac{2 \, a^{2} d f x{\rm Ei}\left (-\frac{2 \,{\left (d f x + c f\right )}}{d}\right ) e^{\left (\frac{2 \, c f}{d} - 2 \, e\right )} + 4 \, a^{2} d f x{\rm Ei}\left (-\frac{d f x + c f}{d}\right ) e^{\left (\frac{c f}{d} - e\right )} - 4 \, a^{2} d f x{\rm Ei}\left (\frac{d f x + c f}{d}\right ) e^{\left (-\frac{c f}{d} + e\right )} - 2 \, a^{2} d f x{\rm Ei}\left (\frac{2 \,{\left (d f x + c f\right )}}{d}\right ) e^{\left (-\frac{2 \, c f}{d} + 2 \, e\right )} + 2 \, a^{2} c f{\rm Ei}\left (-\frac{2 \,{\left (d f x + c f\right )}}{d}\right ) e^{\left (\frac{2 \, c f}{d} - 2 \, e\right )} + 4 \, a^{2} c f{\rm Ei}\left (-\frac{d f x + c f}{d}\right ) e^{\left (\frac{c f}{d} - e\right )} - 4 \, a^{2} c f{\rm Ei}\left (\frac{d f x + c f}{d}\right ) e^{\left (-\frac{c f}{d} + e\right )} - 2 \, a^{2} c f{\rm Ei}\left (\frac{2 \,{\left (d f x + c f\right )}}{d}\right ) e^{\left (-\frac{2 \, c f}{d} + 2 \, e\right )} + a^{2} d e^{\left (2 \, f x + 2 \, e\right )} + 4 \, a^{2} d e^{\left (f x + e\right )} + 4 \, a^{2} d e^{\left (-f x - e\right )} + a^{2} d e^{\left (-2 \, f x - 2 \, e\right )}}{4 \,{\left (d^{3} x + c d^{2}\right )}} - \frac{3 \, a^{2}}{2 \,{\left (d x + c\right )} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(2*a^2*d*f*x*Ei(-2*(d*f*x + c*f)/d)*e^(2*c*f/d - 2*e) + 4*a^2*d*f*x*Ei(-(d*f*x + c*f)/d)*e^(c*f/d - e) -
4*a^2*d*f*x*Ei((d*f*x + c*f)/d)*e^(-c*f/d + e) - 2*a^2*d*f*x*Ei(2*(d*f*x + c*f)/d)*e^(-2*c*f/d + 2*e) + 2*a^2*
c*f*Ei(-2*(d*f*x + c*f)/d)*e^(2*c*f/d - 2*e) + 4*a^2*c*f*Ei(-(d*f*x + c*f)/d)*e^(c*f/d - e) - 4*a^2*c*f*Ei((d*
f*x + c*f)/d)*e^(-c*f/d + e) - 2*a^2*c*f*Ei(2*(d*f*x + c*f)/d)*e^(-2*c*f/d + 2*e) + a^2*d*e^(2*f*x + 2*e) + 4*
a^2*d*e^(f*x + e) + 4*a^2*d*e^(-f*x - e) + a^2*d*e^(-2*f*x - 2*e))/(d^3*x + c*d^2) - 3/2*a^2/((d*x + c)*d)