3.2.6 \(\int \frac {(a+i a \sinh (e+f x))^2}{(c+d x)^2} \, dx\) [106]

Optimal. Leaf size=170 \[ -\frac {4 a^2 \cosh ^4\left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{d (c+d x)}+\frac {2 i a^2 f \cosh \left (e-\frac {c f}{d}\right ) \text {Chi}\left (\frac {c f}{d}+f x\right )}{d^2}-\frac {a^2 f \text {Chi}\left (\frac {2 c f}{d}+2 f x\right ) \sinh \left (2 e-\frac {2 c f}{d}\right )}{d^2}+\frac {2 i a^2 f \sinh \left (e-\frac {c f}{d}\right ) \text {Shi}\left (\frac {c f}{d}+f x\right )}{d^2}-\frac {a^2 f \cosh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (\frac {2 c f}{d}+2 f x\right )}{d^2} \]

[Out]

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

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Rubi [A]
time = 0.24, antiderivative size = 170, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {3399, 3394, 3384, 3379, 3382} \begin {gather*} -\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 i a^2 f \text {Chi}\left (x f+\frac {c f}{d}\right ) \cosh \left (e-\frac {c f}{d}\right )}{d^2}+\frac {2 i a^2 f \sinh \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}+\frac {i \pi }{4}\right )}{d (c+d x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-4*a^2*Cosh[e/2 + (I/4)*Pi + (f*x)/2]^4)/(d*(c + d*x)) + ((2*I)*a^2*f*Cosh[e - (c*f)/d]*CoshIntegral[(c*f)/d
+ f*x])/d^2 - (a^2*f*CoshIntegral[(2*c*f)/d + 2*f*x]*Sinh[2*e - (2*c*f)/d])/d^2 + ((2*I)*a^2*f*Sinh[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*} \int \frac {(a+i a \sinh (e+f x))^2}{(c+d x)^2} \, dx &=\left (4 a^2\right ) \int \frac {\sin ^4\left (\frac {1}{2} \left (i e+\frac {\pi }{2}\right )+\frac {i f x}{2}\right )}{(c+d x)^2} \, dx\\ &=-\frac {4 a^2 \cosh ^4\left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{d (c+d x)}+\frac {\left (8 i a^2 f\right ) \int \left (\frac {\cosh (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 {i \pi }{4}+\frac {f x}{2}\right )}{d (c+d x)}+\frac {\left (2 i a^2 f\right ) \int \frac {\cosh (e+f x)}{c+d x} \, dx}{d}-\frac {\left (a^2 f\right ) \int \frac {\sinh (2 e+2 f x)}{c+d x} \, dx}{d}\\ &=-\frac {4 a^2 \cosh ^4\left (\frac {e}{2}+\frac {i \pi }{4}+\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 i a^2 f \cosh \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 {\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 i a^2 f \sinh \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 {4 a^2 \cosh ^4\left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{d (c+d x)}+\frac {2 i a^2 f \cosh \left (e-\frac {c f}{d}\right ) \text {Chi}\left (\frac {c f}{d}+f x\right )}{d^2}-\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 i a^2 f \sinh \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*}

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Mathematica [A]
time = 0.38, size = 214, normalized size = 1.26 \begin {gather*} \frac {a^2 \left (-3 d+d \cosh (2 (e+f x))+4 i f (c+d x) \cosh \left (e-\frac {c f}{d}\right ) \text {Chi}\left (f \left (\frac {c}{d}+x\right )\right )-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 i d \sinh (e+f x)+4 i c f \sinh \left (e-\frac {c f}{d}\right ) \text {Shi}\left (f \left (\frac {c}{d}+x\right )\right )+4 i d f x \sinh \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)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(a^2*(-3*d + d*Cosh[2*(e + f*x)] + (4*I)*f*(c + d*x)*Cosh[e - (c*f)/d]*CoshIntegral[f*(c/d + x)] - 2*f*(c + d*
x)*CoshIntegral[(2*f*(c + d*x))/d]*Sinh[2*e - (2*c*f)/d] - (4*I)*d*Sinh[e + f*x] + (4*I)*c*f*Sinh[e - (c*f)/d]
*SinhIntegral[f*(c/d + x)] + (4*I)*d*f*x*Sinh[e - (c*f)/d]*SinhIntegral[f*(c/d + x)] - 2*c*f*Cosh[2*e - (2*c*f
)/d]*SinhIntegral[(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 = 3.39, size = 313, normalized size = 1.84

method	result	size
risch	\(-\frac {i a^{2} f \,{\mathrm e}^{f x +e}}{d^{2} \left (\frac {c f}{d}+f x \right )}-\frac {i a^{2} f \,{\mathrm e}^{-\frac {c f -d e}{d}} \expIntegral \left (1, -f x -e -\frac {c f -d e}{d}\right )}{d^{2}}-\frac {3 a^{2}}{2 d \left (d x +c \right )}+\frac {f \,a^{2} {\mathrm e}^{-2 f x -2 e}}{4 d \left (d x f +c f \right )}-\frac {f \,a^{2} {\mathrm e}^{\frac {2 c f -2 d e}{d}} \expIntegral \left (1, 2 f x +2 e +\frac {2 c f -2 d e}{d}\right )}{2 d^{2}}+\frac {f \,a^{2} {\mathrm e}^{2 f x +2 e}}{4 d^{2} \left (\frac {c f}{d}+f x \right )}+\frac {f \,a^{2} {\mathrm e}^{-\frac {2 \left (c f -d e \right )}{d}} \expIntegral \left (1, -2 f x -2 e -\frac {2 \left (c f -d e \right )}{d}\right )}{2 d^{2}}+\frac {i a^{2} f \,{\mathrm e}^{-f x -e}}{d \left (d x f +c f \right )}-\frac {i a^{2} f \,{\mathrm e}^{\frac {c f -d e}{d}} \expIntegral \left (1, f x +e +\frac {c f -d e}{d}\right )}{d^{2}}\)	\(313\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.32, size = 187, normalized size = 1.10 \begin {gather*} \frac {1}{4} \, a^{2} {\left (\frac {e^{\left (\frac {2 \, c f}{d} - 2 \, e\right )} E_{2}\left (\frac {2 \, {\left (d x + c\right )} f}{d}\right )}{{\left (d x + c\right )} d} + \frac {e^{\left (-\frac {2 \, c f}{d} + 2 \, e\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 )} + i \, a^{2} {\left (\frac {e^{\left (\frac {c f}{d} - e\right )} E_{2}\left (\frac {{\left (d x + c\right )} f}{d}\right )}{{\left (d x + c\right )} d} - \frac {e^{\left (-\frac {c f}{d} + e\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 {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*a^2*(e^(2*c*f/d - 2*e)*exp_integral_e(2, 2*(d*x + c)*f/d)/((d*x + c)*d) + e^(-2*c*f/d + 2*e)*exp_integral_
e(2, -2*(d*x + c)*f/d)/((d*x + c)*d) - 2/(d^2*x + c*d)) + I*a^2*(e^(c*f/d - e)*exp_integral_e(2, (d*x + c)*f/d
)/((d*x + c)*d) - e^(-c*f/d + e)*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 = 0.40, size = 285, normalized size = 1.68 \begin {gather*} \frac {{\left (a^{2} d e^{\left (4 \, f x + 4 \, e\right )} - 4 i \, a^{2} d e^{\left (3 \, f x + 3 \, e\right )} + 4 i \, a^{2} d e^{\left (f x + e\right )} + a^{2} d - 2 \, {\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 ) e^{\left (2 \, f x - \frac {2 \, {\left (c f - d e\right )}}{d} + 2 \, e\right )} - 2 \, {\left (3 \, a^{2} d - {\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 ) e^{\left (\frac {2 \, {\left (c f - d e\right )}}{d}\right )} + 2 \, {\left (-i \, a^{2} d f x - i \, a^{2} c f\right )} {\rm Ei}\left (-\frac {d f x + c f}{d}\right ) e^{\left (\frac {c f - d e}{d}\right )} + 2 \, {\left (-i \, a^{2} d f x - i \, a^{2} c f\right )} {\rm Ei}\left (\frac {d f x + c f}{d}\right ) e^{\left (-\frac {c f - d e}{d}\right )}\right )} e^{\left (2 \, f x + 2 \, e\right )}\right )} e^{\left (-2 \, f x - 2 \, e\right )}}{4 \, {\left (d^{3} x + c d^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+I*a*sinh(f*x+e))^2/(d*x+c)^2,x, algorithm="fricas")

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - a^{2} \left (\int \frac {\sinh ^{2}{\left (e + f x \right )}}{c^{2} + 2 c d x + d^{2} x^{2}}\, dx + \int \left (- \frac {2 i \sinh {\left (e + f x \right )}}{c^{2} + 2 c d x + d^{2} x^{2}}\right )\, dx + \int \left (- \frac {1}{c^{2} + 2 c d x + d^{2} x^{2}}\right )\, dx\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a**2*(Integral(sinh(e + f*x)**2/(c**2 + 2*c*d*x + d**2*x**2), x) + Integral(-2*I*sinh(e + f*x)/(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] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1134 vs. \(2 (159) = 318\).
time = 0.51, size = 1134, normalized size = 6.67 \begin {gather*} -\frac {{\left (2 \, {\left (d x + c\right )} a^{2} {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )} f^{2} {\rm Ei}\left (\frac {2 \, {\left ({\left (d x + c\right )} {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )} - d e + c f\right )}}{d}\right ) e^{\left (\frac {2 \, {\left (d e - c f\right )}}{d}\right )} - 2 \, a^{2} d e f^{2} {\rm Ei}\left (\frac {2 \, {\left ({\left (d x + c\right )} {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )} - d e + c f\right )}}{d}\right ) e^{\left (\frac {2 \, {\left (d e - c f\right )}}{d}\right )} + 2 \, a^{2} c f^{3} {\rm Ei}\left (\frac {2 \, {\left ({\left (d x + c\right )} {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )} - d e + c f\right )}}{d}\right ) e^{\left (\frac {2 \, {\left (d e - c f\right )}}{d}\right )} - 4 i \, {\left (d x + c\right )} a^{2} {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )} f^{2} {\rm Ei}\left (\frac {{\left (d x + c\right )} {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )} - d e + c f}{d}\right ) e^{\left (\frac {d e - c f}{d}\right )} + 4 i \, a^{2} d e f^{2} {\rm Ei}\left (\frac {{\left (d x + c\right )} {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )} - d e + c f}{d}\right ) e^{\left (\frac {d e - c f}{d}\right )} - 4 i \, a^{2} c f^{3} {\rm Ei}\left (\frac {{\left (d x + c\right )} {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )} - d e + c f}{d}\right ) e^{\left (\frac {d e - c f}{d}\right )} - 4 i \, {\left (d x + c\right )} a^{2} {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )} f^{2} {\rm Ei}\left (-\frac {{\left (d x + c\right )} {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )} - d e + c f}{d}\right ) e^{\left (-\frac {d e - c f}{d}\right )} + 4 i \, a^{2} d e f^{2} {\rm Ei}\left (-\frac {{\left (d x + c\right )} {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )} - d e + c f}{d}\right ) e^{\left (-\frac {d e - c f}{d}\right )} - 4 i \, a^{2} c f^{3} {\rm Ei}\left (-\frac {{\left (d x + c\right )} {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )} - d e + c f}{d}\right ) e^{\left (-\frac {d e - c f}{d}\right )} - 2 \, {\left (d x + c\right )} a^{2} {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )} f^{2} {\rm Ei}\left (-\frac {2 \, {\left ({\left (d x + c\right )} {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )} - d e + c f\right )}}{d}\right ) e^{\left (-\frac {2 \, {\left (d e - c f\right )}}{d}\right )} + 2 \, a^{2} d e f^{2} {\rm Ei}\left (-\frac {2 \, {\left ({\left (d x + c\right )} {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )} - d e + c f\right )}}{d}\right ) e^{\left (-\frac {2 \, {\left (d e - c f\right )}}{d}\right )} - 2 \, a^{2} c f^{3} {\rm Ei}\left (-\frac {2 \, {\left ({\left (d x + c\right )} {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )} - d e + c f\right )}}{d}\right ) e^{\left (-\frac {2 \, {\left (d e - c f\right )}}{d}\right )} - a^{2} d f^{2} e^{\left (\frac {2 \, {\left (d x + c\right )} {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )}}{d}\right )} + 4 i \, a^{2} d f^{2} e^{\left (\frac {{\left (d x + c\right )} {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )}}{d}\right )} - 4 i \, a^{2} d f^{2} e^{\left (-\frac {{\left (d x + c\right )} {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )}}{d}\right )} - a^{2} d f^{2} e^{\left (-\frac {2 \, {\left (d x + c\right )} {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )}}{d}\right )} + 6 \, a^{2} d f^{2}\right )} d^{2}}{4 \, {\left ({\left (d x + c\right )} d^{4} {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )} - d^{5} e + c d^{4} f\right )} f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+I*a*sinh(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*I*(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*I*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*I*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*I*(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*I*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*I*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*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) - a^2*d*f^2*e^(2*(
d*x + c)*(d*e/(d*x + c) - c*f/(d*x + c) + f)/d) + 4*I*a^2*d*f^2*e^((d*x + c)*(d*e/(d*x + c) - c*f/(d*x + c) +
f)/d) - 4*I*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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (a+a\,\mathrm {sinh}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^2}{{\left (c+d\,x\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + a*sinh(e + f*x)*1i)^2/(c + d*x)^2,x)

[Out]

int((a + a*sinh(e + f*x)*1i)^2/(c + d*x)^2, x)

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