\(\int \frac {(a+a \cosh (e+f x))^2}{(c+d x)^2} \, dx\) [109]

Optimal result

Integrand size = 20, antiderivative size = 157 \[ \int \frac {(a+a \cosh (e+f x))^2}{(c+d x)^2} \, dx=-\frac {4 a^2 \cosh ^4\left (\frac {e}{2}+\frac {f x}{2}\right )}{d (c+d x)}+\frac {a^2 f \text {Chi}\left (\frac {2 c f}{d}+2 f x\right ) \sinh \left (2 e-\frac {2 c f}{d}\right )}{d^2}+\frac {2 a^2 f \text {Chi}\left (\frac {c f}{d}+f x\right ) \sinh \left (e-\frac {c f}{d}\right )}{d^2}+\frac {2 a^2 f \cosh \left (e-\frac {c f}{d}\right ) \text {Shi}\left (\frac {c f}{d}+f x\right )}{d^2}+\frac {a^2 f \cosh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (\frac {2 c f}{d}+2 f x\right )}{d^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.24 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3399, 3394, 3384, 3379, 3382} \[ \int \frac {(a+a \cosh (e+f x))^2}{(c+d x)^2} \, dx=\frac {a^2 f \text {Chi}\left (2 x f+\frac {2 c f}{d}\right ) \sinh \left (2 e-\frac {2 c f}{d}\right )}{d^2}+\frac {2 a^2 f \text {Chi}\left (x f+\frac {c f}{d}\right ) \sinh \left (e-\frac {c f}{d}\right )}{d^2}+\frac {2 a^2 f \cosh \left (e-\frac {c f}{d}\right ) \text {Shi}\left (x f+\frac {c f}{d}\right )}{d^2}+\frac {a^2 f \cosh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (2 x f+\frac {2 c f}{d}\right )}{d^2}-\frac {4 a^2 \cosh ^4\left (\frac {e}{2}+\frac {f x}{2}\right )}{d (c+d x)} \]

[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 3379

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 3382

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 3384

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 3394

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 3399

Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[(2*a)^n, Int[(c
+ d*x)^m*Sin[(1/2)*(e + Pi*(a/(2*b))) + 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])

Rubi steps \begin{align*} \text {integral}& = \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] (verified)

Time = 0.63 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.32 \[ \int \frac {(a+a \cosh (e+f x))^2}{(c+d x)^2} \, dx=\frac {a^2 \left (-3 d-4 d \cosh (e+f x)-d \cosh (2 (e+f x))+2 f (c+d x) \text {Chi}\left (\frac {2 f (c+d x)}{d}\right ) \sinh \left (2 e-\frac {2 c f}{d}\right )+4 f (c+d x) \text {Chi}\left (f \left (\frac {c}{d}+x\right )\right ) \sinh \left (e-\frac {c f}{d}\right )+4 c f \cosh \left (e-\frac {c f}{d}\right ) \text {Shi}\left (f \left (\frac {c}{d}+x\right )\right )+4 d f x \cosh \left (e-\frac {c f}{d}\right ) \text {Shi}\left (f \left (\frac {c}{d}+x\right )\right )+2 c f \cosh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (\frac {2 f (c+d x)}{d}\right )+2 d f x \cosh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (\frac {2 f (c+d x)}{d}\right )\right )}{2 d^2 (c+d x)} \]

[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))

Maple [A] (verified)

Time = 0.51 (sec) , antiderivative size = 308, normalized size of antiderivative = 1.96

method	result	size
risch	\(-\frac {f \,a^{2} {\mathrm e}^{-f x -e}}{d \left (d x f +c f \right )}+\frac {f \,a^{2} {\mathrm e}^{\frac {c f -d e}{d}} \operatorname {Ei}_{1}\left (f x +e +\frac {c f -d e}{d}\right )}{d^{2}}-\frac {f \,a^{2} {\mathrm e}^{f x +e}}{d^{2} \left (\frac {c f}{d}+f x \right )}-\frac {f \,a^{2} {\mathrm e}^{-\frac {c f -d e}{d}} \operatorname {Ei}_{1}\left (-f x -e -\frac {c f -d e}{d}\right )}{d^{2}}-\frac {3 a^{2}}{2 d \left (d x +c \right )}-\frac {f \,a^{2} {\mathrm e}^{-2 f x -2 e}}{4 d \left (d x f +c f \right )}+\frac {f \,a^{2} {\mathrm e}^{\frac {2 c f -2 d e}{d}} \operatorname {Ei}_{1}\left (2 f x +2 e +\frac {2 c f -2 d e}{d}\right )}{2 d^{2}}-\frac {f \,a^{2} {\mathrm e}^{2 f x +2 e}}{4 d^{2} \left (\frac {c f}{d}+f x \right )}-\frac {f \,a^{2} {\mathrm e}^{-\frac {2 \left (c f -d e \right )}{d}} \operatorname {Ei}_{1}\left (-2 f x -2 e -\frac {2 \left (c f -d e \right )}{d}\right )}{2 d^{2}}\)	\(308\)

[In]

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

[Out]

-f*a^2*exp(-f*x-e)/d/(d*f*x+c*f)+f*a^2/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*f*a^2*exp(-2*f*x-2*e)/d/(d*f
*x+c*f)+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)-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)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 359 vs. \(2 (156) = 312\).

Time = 0.25 (sec) , antiderivative size = 359, normalized size of antiderivative = 2.29 \[ \int \frac {(a+a \cosh (e+f x))^2}{(c+d x)^2} \, dx=-\frac {a^{2} d \cosh \left (f x + e\right )^{2} + a^{2} d \sinh \left (f x + e\right )^{2} + 4 \, a^{2} d \cosh \left (f x + e\right ) + 3 \, a^{2} d - 2 \, {\left ({\left (a^{2} d f x + a^{2} c f\right )} {\rm Ei}\left (\frac {d f x + c f}{d}\right ) - {\left (a^{2} d f x + a^{2} c f\right )} {\rm Ei}\left (-\frac {d f x + c f}{d}\right )\right )} \cosh \left (-\frac {d e - c f}{d}\right ) - {\left ({\left (a^{2} d f x + a^{2} c f\right )} {\rm Ei}\left (\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) - {\left (a^{2} d f x + a^{2} c f\right )} {\rm Ei}\left (-\frac {2 \, {\left (d f x + c f\right )}}{d}\right )\right )} \cosh \left (-\frac {2 \, {\left (d e - c f\right )}}{d}\right ) + 2 \, {\left ({\left (a^{2} d f x + a^{2} c f\right )} {\rm Ei}\left (\frac {d f x + c f}{d}\right ) + {\left (a^{2} d f x + a^{2} c f\right )} {\rm Ei}\left (-\frac {d f x + c f}{d}\right )\right )} \sinh \left (-\frac {d e - c f}{d}\right ) + {\left ({\left (a^{2} d f x + a^{2} c f\right )} {\rm Ei}\left (\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) + {\left (a^{2} d f x + a^{2} c f\right )} {\rm Ei}\left (-\frac {2 \, {\left (d f x + c f\right )}}{d}\right )\right )} \sinh \left (-\frac {2 \, {\left (d e - c f\right )}}{d}\right )}{2 \, {\left (d^{3} x + c d^{2}\right )}} \]

[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)

Sympy [F]

\[ \int \frac {(a+a \cosh (e+f x))^2}{(c+d x)^2} \, dx=a^{2} \left (\int \frac {2 \cosh {\left (e + f x \right )}}{c^{2} + 2 c d x + d^{2} x^{2}}\, dx + \int \frac {\cosh ^{2}{\left (e + f x \right )}}{c^{2} + 2 c d x + d^{2} x^{2}}\, dx + \int \frac {1}{c^{2} + 2 c d x + d^{2} x^{2}}\, dx\right ) \]

[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))

Maxima [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.16 \[ \int \frac {(a+a \cosh (e+f x))^2}{(c+d x)^2} \, dx=-\frac {1}{4} \, a^{2} {\left (\frac {e^{\left (-2 \, e + \frac {2 \, c f}{d}\right )} E_{2}\left (\frac {2 \, {\left (d x + c\right )} f}{d}\right )}{{\left (d x + c\right )} d} + \frac {e^{\left (2 \, e - \frac {2 \, c f}{d}\right )} E_{2}\left (-\frac {2 \, {\left (d x + c\right )} f}{d}\right )}{{\left (d x + c\right )} d} + \frac {2}{d^{2} x + c d}\right )} - a^{2} {\left (\frac {e^{\left (-e + \frac {c f}{d}\right )} E_{2}\left (\frac {{\left (d x + c\right )} f}{d}\right )}{{\left (d x + c\right )} d} + \frac {e^{\left (e - \frac {c f}{d}\right )} E_{2}\left (-\frac {{\left (d x + c\right )} f}{d}\right )}{{\left (d x + c\right )} d}\right )} - \frac {a^{2}}{d^{2} x + c d} \]

[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)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1134 vs. \(2 (156) = 312\).

Time = 0.35 (sec) , antiderivative size = 1134, normalized size of antiderivative = 7.22 \[ \int \frac {(a+a \cosh (e+f x))^2}{(c+d x)^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/4*(2*(d*x + c)*a^2*(d*e/(d*x + c) - c*f/(d*x + c) + f)*f^2*Ei(2*((d*x + c)*(d*e/(d*x + c) - c*f/(d*x + c) +
f) - d*e + c*f)/d)*e^(2*(d*e - c*f)/d) - 2*a^2*d*e*f^2*Ei(2*((d*x + c)*(d*e/(d*x + c) - c*f/(d*x + c) + f) - d
*e + c*f)/d)*e^(2*(d*e - c*f)/d) + 2*a^2*c*f^3*Ei(2*((d*x + c)*(d*e/(d*x + c) - c*f/(d*x + c) + f) - d*e + c*f
)/d)*e^(2*(d*e - c*f)/d) + 4*(d*x + c)*a^2*(d*e/(d*x + c) - c*f/(d*x + c) + f)*f^2*Ei(((d*x + c)*(d*e/(d*x + c
) - c*f/(d*x + c) + f) - d*e + c*f)/d)*e^((d*e - c*f)/d) - 4*a^2*d*e*f^2*Ei(((d*x + c)*(d*e/(d*x + c) - c*f/(d
*x + c) + f) - d*e + c*f)/d)*e^((d*e - c*f)/d) + 4*a^2*c*f^3*Ei(((d*x + c)*(d*e/(d*x + c) - c*f/(d*x + c) + f)
- d*e + c*f)/d)*e^((d*e - c*f)/d) - 4*(d*x + c)*a^2*(d*e/(d*x + c) - c*f/(d*x + c) + f)*f^2*Ei(-((d*x + c)*(d
*e/(d*x + c) - c*f/(d*x + c) + f) - d*e + c*f)/d)*e^(-(d*e - c*f)/d) + 4*a^2*d*e*f^2*Ei(-((d*x + c)*(d*e/(d*x
+ c) - c*f/(d*x + c) + f) - d*e + c*f)/d)*e^(-(d*e - c*f)/d) - 4*a^2*c*f^3*Ei(-((d*x + c)*(d*e/(d*x + c) - c*f
/(d*x + c) + f) - d*e + c*f)/d)*e^(-(d*e - c*f)/d) - 2*(d*x + c)*a^2*(d*e/(d*x + c) - c*f/(d*x + c) + f)*f^2*E
i(-2*((d*x + c)*(d*e/(d*x + c) - c*f/(d*x + c) + f) - d*e + c*f)/d)*e^(-2*(d*e - c*f)/d) + 2*a^2*d*e*f^2*Ei(-2
*((d*x + c)*(d*e/(d*x + c) - c*f/(d*x + c) + f) - d*e + c*f)/d)*e^(-2*(d*e - c*f)/d) - 2*a^2*c*f^3*Ei(-2*((d*x
+ c)*(d*e/(d*x + c) - c*f/(d*x + c) + f) - d*e + c*f)/d)*e^(-2*(d*e - c*f)/d) - a^2*d*f^2*e^(2*(d*x + c)*(d*e
/(d*x + c) - c*f/(d*x + c) + f)/d) - 4*a^2*d*f^2*e^((d*x + c)*(d*e/(d*x + c) - c*f/(d*x + c) + f)/d) - 4*a^2*d
*f^2*e^(-(d*x + c)*(d*e/(d*x + c) - c*f/(d*x + c) + f)/d) - a^2*d*f^2*e^(-2*(d*x + c)*(d*e/(d*x + c) - c*f/(d*
x + c) + f)/d) - 6*a^2*d*f^2)*d^2/(((d*x + c)*d^4*(d*e/(d*x + c) - c*f/(d*x + c) + f) - d^5*e + c*d^4*f)*f)

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+a \cosh (e+f x))^2}{(c+d x)^2} \, dx=\int \frac {{\left (a+a\,\mathrm {cosh}\left (e+f\,x\right )\right )}^2}{{\left (c+d\,x\right )}^2} \,d x \]

[In]

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

[Out]

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