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

Optimal. Leaf size=183 \[ -\frac{a^2}{d (c+d x)}+\frac{2 a b f \text{Chi}\left (x f+\frac{c f}{d}\right ) \sinh \left (e-\frac{c f}{d}\right )}{d^2}+\frac{2 a b f \cosh \left (e-\frac{c f}{d}\right ) \text{Shi}\left (x f+\frac{c f}{d}\right )}{d^2}-\frac{2 a b \cosh (e+f x)}{d (c+d x)}+\frac{b^2 f \text{Chi}\left (2 x f+\frac{2 c f}{d}\right ) \sinh \left (2 e-\frac{2 c f}{d}\right )}{d^2}+\frac{b^2 f \cosh \left (2 e-\frac{2 c f}{d}\right ) \text{Shi}\left (2 x f+\frac{2 c f}{d}\right )}{d^2}-\frac{b^2 \cosh ^2(e+f x)}{d (c+d x)} \]

[Out]

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

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Rubi [A]  time = 0.344979, antiderivative size = 183, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 7, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.35, Rules used = {3317, 3297, 3303, 3298, 3301, 3313, 12} \[ -\frac{a^2}{d (c+d x)}+\frac{2 a b f \text{Chi}\left (x f+\frac{c f}{d}\right ) \sinh \left (e-\frac{c f}{d}\right )}{d^2}+\frac{2 a b f \cosh \left (e-\frac{c f}{d}\right ) \text{Shi}\left (x f+\frac{c f}{d}\right )}{d^2}-\frac{2 a b \cosh (e+f x)}{d (c+d x)}+\frac{b^2 f \text{Chi}\left (2 x f+\frac{2 c f}{d}\right ) \sinh \left (2 e-\frac{2 c f}{d}\right )}{d^2}+\frac{b^2 f \cosh \left (2 e-\frac{2 c f}{d}\right ) \text{Shi}\left (2 x f+\frac{2 c f}{d}\right )}{d^2}-\frac{b^2 \cosh ^2(e+f x)}{d (c+d x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rubi steps

\begin{align*} \int \frac{(a+b \cosh (e+f x))^2}{(c+d x)^2} \, dx &=\int \left (\frac{a^2}{(c+d x)^2}+\frac{2 a b \cosh (e+f x)}{(c+d x)^2}+\frac{b^2 \cosh ^2(e+f x)}{(c+d x)^2}\right ) \, dx\\ &=-\frac{a^2}{d (c+d x)}+(2 a b) \int \frac{\cosh (e+f x)}{(c+d x)^2} \, dx+b^2 \int \frac{\cosh ^2(e+f x)}{(c+d x)^2} \, dx\\ &=-\frac{a^2}{d (c+d x)}-\frac{2 a b \cosh (e+f x)}{d (c+d x)}-\frac{b^2 \cosh ^2(e+f x)}{d (c+d x)}+\frac{(2 a b f) \int \frac{\sinh (e+f x)}{c+d x} \, dx}{d}+\frac{\left (2 i b^2 f\right ) \int -\frac{i \sinh (2 e+2 f x)}{2 (c+d x)} \, dx}{d}\\ &=-\frac{a^2}{d (c+d x)}-\frac{2 a b \cosh (e+f x)}{d (c+d x)}-\frac{b^2 \cosh ^2(e+f x)}{d (c+d x)}+\frac{\left (b^2 f\right ) \int \frac{\sinh (2 e+2 f x)}{c+d x} \, dx}{d}+\frac{\left (2 a 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 (2 a 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^2}{d (c+d x)}-\frac{2 a b \cosh (e+f x)}{d (c+d x)}-\frac{b^2 \cosh ^2(e+f x)}{d (c+d x)}+\frac{2 a b f \text{Chi}\left (\frac{c f}{d}+f x\right ) \sinh \left (e-\frac{c f}{d}\right )}{d^2}+\frac{2 a b f \cosh \left (e-\frac{c f}{d}\right ) \text{Shi}\left (\frac{c f}{d}+f x\right )}{d^2}+\frac{\left (b^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 (b^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{a^2}{d (c+d x)}-\frac{2 a b \cosh (e+f x)}{d (c+d x)}-\frac{b^2 \cosh ^2(e+f x)}{d (c+d x)}+\frac{b^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 b f \text{Chi}\left (\frac{c f}{d}+f x\right ) \sinh \left (e-\frac{c f}{d}\right )}{d^2}+\frac{2 a b f \cosh \left (e-\frac{c f}{d}\right ) \text{Shi}\left (\frac{c f}{d}+f x\right )}{d^2}+\frac{b^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 = 0.760369, size = 233, normalized size = 1.27 \[ \frac{-2 a^2 d+4 a b f (c+d x) \text{Chi}\left (f \left (\frac{c}{d}+x\right )\right ) \sinh \left (e-\frac{c f}{d}\right )+4 a b c f \cosh \left (e-\frac{c f}{d}\right ) \text{Shi}\left (f \left (\frac{c}{d}+x\right )\right )+4 a b d f x \cosh \left (e-\frac{c f}{d}\right ) \text{Shi}\left (f \left (\frac{c}{d}+x\right )\right )-4 a b d \cosh (e+f x)+2 b^2 f (c+d x) \text{Chi}\left (\frac{2 f (c+d x)}{d}\right ) \sinh \left (2 e-\frac{2 c f}{d}\right )+2 b^2 c f \cosh \left (2 e-\frac{2 c f}{d}\right ) \text{Shi}\left (\frac{2 f (c+d x)}{d}\right )+2 b^2 d f x \cosh \left (2 e-\frac{2 c f}{d}\right ) \text{Shi}\left (\frac{2 f (c+d x)}{d}\right )-b^2 d \cosh (2 (e+f x))-b^2 d}{2 d^2 (c+d x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*a^2*d - b^2*d - 4*a*b*d*Cosh[e + f*x] - b^2*d*Cosh[2*(e + f*x)] + 2*b^2*f*(c + d*x)*CoshIntegral[(2*f*(c +
d*x))/d]*Sinh[2*e - (2*c*f)/d] + 4*a*b*f*(c + d*x)*CoshIntegral[f*(c/d + x)]*Sinh[e - (c*f)/d] + 4*a*b*c*f*Co
sh[e - (c*f)/d]*SinhIntegral[f*(c/d + x)] + 4*a*b*d*f*x*Cosh[e - (c*f)/d]*SinhIntegral[f*(c/d + x)] + 2*b^2*c*
f*Cosh[2*e - (2*c*f)/d]*SinhIntegral[(2*f*(c + d*x))/d] + 2*b^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 = 0.12, size = 319, normalized size = 1.7 \begin{align*} -{\frac{abf{{\rm e}^{-fx-e}}}{d \left ( dfx+cf \right ) }}+{\frac{abf}{{d}^{2}}{{\rm e}^{{\frac{cf-de}{d}}}}{\it Ei} \left ( 1,fx+e+{\frac{cf-de}{d}} \right ) }-{\frac{abf{{\rm e}^{fx+e}}}{{d}^{2}} \left ({\frac{cf}{d}}+fx \right ) ^{-1}}-{\frac{abf}{{d}^{2}}{{\rm e}^{-{\frac{cf-de}{d}}}}{\it Ei} \left ( 1,-fx-e-{\frac{cf-de}{d}} \right ) }-{\frac{{a}^{2}}{d \left ( dx+c \right ) }}-{\frac{{b}^{2}}{2\,d \left ( dx+c \right ) }}-{\frac{f{b}^{2}{{\rm e}^{-2\,fx-2\,e}}}{4\,d \left ( dfx+cf \right ) }}+{\frac{f{b}^{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{b}^{2}{{\rm e}^{2\,fx+2\,e}}}{4\,{d}^{2}} \left ({\frac{cf}{d}}+fx \right ) ^{-1}}-{\frac{f{b}^{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+b*cosh(f*x+e))^2/(d*x+c)^2,x)

[Out]

-a*b*f*exp(-f*x-e)/d/(d*f*x+c*f)+a*b*f/d^2*exp((c*f-d*e)/d)*Ei(1,f*x+e+(c*f-d*e)/d)-a*b*f/d^2*exp(f*x+e)/(c*f/
d+f*x)-a*b*f/d^2*exp(-(c*f-d*e)/d)*Ei(1,-f*x-e-(c*f-d*e)/d)-a^2/d/(d*x+c)-1/2*b^2/d/(d*x+c)-1/4*b^2*f*exp(-2*f
*x-2*e)/d/(d*f*x+c*f)+1/2*b^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*b^2/d^2*exp(2*f*x+2
*e)/(c*f/d+f*x)-1/2*f*b^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 = 1.3857, size = 244, normalized size = 1.33 \begin{align*} -\frac{1}{4} \, b^{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 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^{2}}{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))^2/(d*x+c)^2,x, algorithm="maxima")

[Out]

-1/4*b^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*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^2/(d^2*x + c*d)

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Fricas [A]  time = 2.15579, size = 778, normalized size = 4.25 \begin{align*} -\frac{b^{2} d \cosh \left (f x + e\right )^{2} + b^{2} d \sinh \left (f x + e\right )^{2} + 4 \, a b d \cosh \left (f x + e\right ) +{\left (2 \, a^{2} + b^{2}\right )} d - 2 \,{\left ({\left (a b d f x + a b c f\right )}{\rm Ei}\left (\frac{d f x + c f}{d}\right ) -{\left (a b d f x + a 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^{2} d f x + b^{2} c f\right )}{\rm Ei}\left (\frac{2 \,{\left (d f x + c f\right )}}{d}\right ) -{\left (b^{2} d f x + b^{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 b d f x + a b c f\right )}{\rm Ei}\left (\frac{d f x + c f}{d}\right ) +{\left (a b d f x + a 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 ) +{\left ({\left (b^{2} d f x + b^{2} c f\right )}{\rm Ei}\left (\frac{2 \,{\left (d f x + c f\right )}}{d}\right ) +{\left (b^{2} d f x + b^{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+b*cosh(f*x+e))^2/(d*x+c)^2,x, algorithm="fricas")

[Out]

-1/2*(b^2*d*cosh(f*x + e)^2 + b^2*d*sinh(f*x + e)^2 + 4*a*b*d*cosh(f*x + e) + (2*a^2 + b^2)*d - 2*((a*b*d*f*x
+ a*b*c*f)*Ei((d*f*x + c*f)/d) - (a*b*d*f*x + a*b*c*f)*Ei(-(d*f*x + c*f)/d))*cosh(-(d*e - c*f)/d) - ((b^2*d*f*
x + b^2*c*f)*Ei(2*(d*f*x + c*f)/d) - (b^2*d*f*x + b^2*c*f)*Ei(-2*(d*f*x + c*f)/d))*cosh(-2*(d*e - c*f)/d) + 2*
((a*b*d*f*x + a*b*c*f)*Ei((d*f*x + c*f)/d) + (a*b*d*f*x + a*b*c*f)*Ei(-(d*f*x + c*f)/d))*sinh(-(d*e - c*f)/d)
+ ((b^2*d*f*x + b^2*c*f)*Ei(2*(d*f*x + c*f)/d) + (b^2*d*f*x + b^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*} \int \frac{\left (a + b \cosh{\left (e + f x \right )}\right )^{2}}{\left (c + d x\right )^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.92411, size = 485, normalized size = 2.65 \begin{align*} -\frac{2 \, b^{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 b d f x{\rm Ei}\left (-\frac{d f x + c f}{d}\right ) e^{\left (\frac{c f}{d} - e\right )} - 4 \, a b d f x{\rm Ei}\left (\frac{d f x + c f}{d}\right ) e^{\left (-\frac{c f}{d} + e\right )} - 2 \, b^{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 \, b^{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 b c f{\rm Ei}\left (-\frac{d f x + c f}{d}\right ) e^{\left (\frac{c f}{d} - e\right )} - 4 \, a b c f{\rm Ei}\left (\frac{d f x + c f}{d}\right ) e^{\left (-\frac{c f}{d} + e\right )} - 2 \, b^{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 )} + b^{2} d e^{\left (2 \, f x + 2 \, e\right )} + 4 \, a b d e^{\left (f x + e\right )} + 4 \, a b d e^{\left (-f x - e\right )} + b^{2} d e^{\left (-2 \, f x - 2 \, e\right )}}{4 \,{\left (d^{3} x + c d^{2}\right )}} - \frac{2 \, a^{2} + b^{2}}{2 \,{\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))^2/(d*x+c)^2,x, algorithm="giac")

[Out]

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