3.107 \(\int \frac {(a+i a \sinh (e+f x))^2}{(c+d x)^3} \, dx\)

Optimal. Leaf size=236 \[ \frac {i a^2 f^2 \text {Chi}\left (x f+\frac {c f}{d}\right ) \sinh \left (e-\frac {c f}{d}\right )}{d^3}-\frac {a^2 f^2 \text {Chi}\left (2 x f+\frac {2 c f}{d}\right ) \cosh \left (2 e-\frac {2 c f}{d}\right )}{d^3}-\frac {a^2 f^2 \sinh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (2 x f+\frac {2 c f}{d}\right )}{d^3}+\frac {i a^2 f^2 \cosh \left (e-\frac {c f}{d}\right ) \text {Shi}\left (x f+\frac {c f}{d}\right )}{d^3}-\frac {4 a^2 f \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \cosh ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{d^2 (c+d x)}-\frac {2 a^2 \cosh ^4\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{d (c+d x)^2} \]

[Out]

-a^2*f^2*Chi(2*c*f/d+2*f*x)*cosh(-2*e+2*c*f/d)/d^3-2*a^2*cosh(1/2*e+1/4*I*Pi+1/2*f*x)^4/d/(d*x+c)^2+I*a^2*f^2*
cosh(-e+c*f/d)*Shi(c*f/d+f*x)/d^3+a^2*f^2*Shi(2*c*f/d+2*f*x)*sinh(-2*e+2*c*f/d)/d^3-I*a^2*f^2*Chi(c*f/d+f*x)*s
inh(-e+c*f/d)/d^3-4*a^2*f*cosh(1/2*e+1/4*I*Pi+1/2*f*x)^3*sinh(1/2*e+1/4*I*Pi+1/2*f*x)/d^2/(d*x+c)

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Rubi [A]  time = 0.53, antiderivative size = 236, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {3318, 3314, 3312, 3303, 3298, 3301} \[ \frac {i a^2 f^2 \text {Chi}\left (x f+\frac {c f}{d}\right ) \sinh \left (e-\frac {c f}{d}\right )}{d^3}-\frac {a^2 f^2 \text {Chi}\left (2 x f+\frac {2 c f}{d}\right ) \cosh \left (2 e-\frac {2 c f}{d}\right )}{d^3}-\frac {a^2 f^2 \sinh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (2 x f+\frac {2 c f}{d}\right )}{d^3}+\frac {i a^2 f^2 \cosh \left (e-\frac {c f}{d}\right ) \text {Shi}\left (x f+\frac {c f}{d}\right )}{d^3}-\frac {4 a^2 f \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \cosh ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{d^2 (c+d x)}-\frac {2 a^2 \cosh ^4\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{d (c+d x)^2} \]

Antiderivative was successfully verified.

[In]

Int[(a + I*a*Sinh[e + f*x])^2/(c + d*x)^3,x]

[Out]

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

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 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 3312

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin
[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1])
)

Rule 3314

Int[((c_.) + (d_.)*(x_))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[((c + d*x)^(m + 1)*(b*Si
n[e + f*x])^n)/(d*(m + 1)), x] + (Dist[(b^2*f^2*n*(n - 1))/(d^2*(m + 1)*(m + 2)), Int[(c + d*x)^(m + 2)*(b*Sin
[e + f*x])^(n - 2), x], x] - Dist[(f^2*n^2)/(d^2*(m + 1)*(m + 2)), Int[(c + d*x)^(m + 2)*(b*Sin[e + f*x])^n, x
], x] - Simp[(b*f*n*(c + d*x)^(m + 2)*Cos[e + f*x]*(b*Sin[e + f*x])^(n - 1))/(d^2*(m + 1)*(m + 2)), x]) /; Fre
eQ[{b, c, d, e, f}, x] && GtQ[n, 1] && LtQ[m, -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])

Rubi steps

\begin {align*} \int \frac {(a+i a \sinh (e+f x))^2}{(c+d x)^3} \, 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)^3} \, dx\\ &=-\frac {2 a^2 \cosh ^4\left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{d (c+d x)^2}-\frac {4 a^2 f \cosh ^3\left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \sinh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{d^2 (c+d x)}+\frac {\left (6 a^2 f^2\right ) \int \frac {\sinh ^2\left (\frac {e}{2}-\frac {i \pi }{4}+\frac {f x}{2}\right )}{c+d x} \, dx}{d^2}+\frac {\left (8 a^2 f^2\right ) \int \frac {\sinh ^4\left (\frac {e}{2}-\frac {i \pi }{4}+\frac {f x}{2}\right )}{c+d x} \, dx}{d^2}\\ &=-\frac {2 a^2 \cosh ^4\left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{d (c+d x)^2}-\frac {4 a^2 f \cosh ^3\left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \sinh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{d^2 (c+d x)}-\frac {\left (6 a^2 f^2\right ) \int \left (\frac {1}{2 (c+d x)}+\frac {i \sinh (e+f x)}{2 (c+d x)}\right ) \, dx}{d^2}+\frac {\left (8 a^2 f^2\right ) \int \left (\frac {3}{8 (c+d x)}-\frac {\cosh (2 e+2 f x)}{8 (c+d x)}+\frac {i \sinh (e+f x)}{2 (c+d x)}\right ) \, dx}{d^2}\\ &=-\frac {2 a^2 \cosh ^4\left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{d (c+d x)^2}-\frac {4 a^2 f \cosh ^3\left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \sinh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{d^2 (c+d x)}-\frac {\left (3 i a^2 f^2\right ) \int \frac {\sinh (e+f x)}{c+d x} \, dx}{d^2}+\frac {\left (4 i a^2 f^2\right ) \int \frac {\sinh (e+f x)}{c+d x} \, dx}{d^2}-\frac {\left (a^2 f^2\right ) \int \frac {\cosh (2 e+2 f x)}{c+d x} \, dx}{d^2}\\ &=-\frac {2 a^2 \cosh ^4\left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{d (c+d x)^2}-\frac {4 a^2 f \cosh ^3\left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \sinh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{d^2 (c+d x)}-\frac {\left (a^2 f^2 \cosh \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^2}-\frac {\left (3 i a^2 f^2 \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^2}+\frac {\left (4 i a^2 f^2 \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^2}-\frac {\left (a^2 f^2 \sinh \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^2}-\frac {\left (3 i a^2 f^2 \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^2}+\frac {\left (4 i a^2 f^2 \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^2}\\ &=-\frac {2 a^2 \cosh ^4\left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{d (c+d x)^2}-\frac {a^2 f^2 \cosh \left (2 e-\frac {2 c f}{d}\right ) \text {Chi}\left (\frac {2 c f}{d}+2 f x\right )}{d^3}+\frac {i a^2 f^2 \text {Chi}\left (\frac {c f}{d}+f x\right ) \sinh \left (e-\frac {c f}{d}\right )}{d^3}-\frac {4 a^2 f \cosh ^3\left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \sinh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{d^2 (c+d x)}+\frac {i a^2 f^2 \cosh \left (e-\frac {c f}{d}\right ) \text {Shi}\left (\frac {c f}{d}+f x\right )}{d^3}-\frac {a^2 f^2 \sinh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (\frac {2 c f}{d}+2 f x\right )}{d^3}\\ \end {align*}

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Mathematica [A]  time = 2.21, size = 198, normalized size = 0.84 \[ \frac {a^2 \left (4 i f^2 \text {Chi}\left (f \left (\frac {c}{d}+x\right )\right ) \sinh \left (e-\frac {c f}{d}\right )-4 f^2 \text {Chi}\left (\frac {2 f (c+d x)}{d}\right ) \cosh \left (2 e-\frac {2 c f}{d}\right )-4 f^2 \sinh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (\frac {2 f (c+d x)}{d}\right )+4 i f^2 \cosh \left (e-\frac {c f}{d}\right ) \text {Shi}\left (f \left (\frac {c}{d}+x\right )\right )+\frac {d (-4 i f (c+d x) \cosh (e+f x)+2 c f \sinh (2 (e+f x))-4 i d \sinh (e+f x)+2 d f x \sinh (2 (e+f x))+d \cosh (2 (e+f x))-3 d)}{(c+d x)^2}\right )}{4 d^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + I*a*Sinh[e + f*x])^2/(c + d*x)^3,x]

[Out]

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

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fricas [B]  time = 0.75, size = 453, normalized size = 1.92 \[ -\frac {{\left (2 \, a^{2} d^{2} f x + 2 \, a^{2} c d f - a^{2} d^{2} - {\left (2 \, a^{2} d^{2} f x + 2 \, a^{2} c d f + a^{2} d^{2}\right )} e^{\left (4 \, f x + 4 \, e\right )} - {\left (-4 i \, a^{2} d^{2} f x - 4 i \, a^{2} c d f - 4 i \, a^{2} d^{2}\right )} e^{\left (3 \, f x + 3 \, e\right )} + {\left (6 \, a^{2} d^{2} + 4 \, {\left (a^{2} d^{2} f^{2} x^{2} + 2 \, a^{2} c d f^{2} x + a^{2} c^{2} f^{2}\right )} {\rm Ei}\left (\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) e^{\left (\frac {2 \, {\left (d e - c f\right )}}{d}\right )} - {\left (4 i \, a^{2} d^{2} f^{2} x^{2} + 8 i \, a^{2} c d f^{2} x + 4 i \, a^{2} c^{2} f^{2}\right )} {\rm Ei}\left (\frac {d f x + c f}{d}\right ) e^{\left (\frac {d e - c f}{d}\right )} - {\left (-4 i \, a^{2} d^{2} f^{2} x^{2} - 8 i \, a^{2} c d f^{2} x - 4 i \, a^{2} c^{2} f^{2}\right )} {\rm Ei}\left (-\frac {d f x + c f}{d}\right ) e^{\left (-\frac {d e - c f}{d}\right )} + 4 \, {\left (a^{2} d^{2} f^{2} x^{2} + 2 \, a^{2} c d f^{2} x + a^{2} c^{2} f^{2}\right )} {\rm Ei}\left (-\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) e^{\left (-\frac {2 \, {\left (d e - c f\right )}}{d}\right )}\right )} e^{\left (2 \, f x + 2 \, e\right )} - {\left (-4 i \, a^{2} d^{2} f x - 4 i \, a^{2} c d f + 4 i \, a^{2} d^{2}\right )} e^{\left (f x + e\right )}\right )} e^{\left (-2 \, f x - 2 \, e\right )}}{8 \, {\left (d^{5} x^{2} + 2 \, c d^{4} x + c^{2} d^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+I*a*sinh(f*x+e))^2/(d*x+c)^3,x, algorithm="fricas")

[Out]

-1/8*(2*a^2*d^2*f*x + 2*a^2*c*d*f - a^2*d^2 - (2*a^2*d^2*f*x + 2*a^2*c*d*f + a^2*d^2)*e^(4*f*x + 4*e) - (-4*I*
a^2*d^2*f*x - 4*I*a^2*c*d*f - 4*I*a^2*d^2)*e^(3*f*x + 3*e) + (6*a^2*d^2 + 4*(a^2*d^2*f^2*x^2 + 2*a^2*c*d*f^2*x
+ a^2*c^2*f^2)*Ei(2*(d*f*x + c*f)/d)*e^(2*(d*e - c*f)/d) - (4*I*a^2*d^2*f^2*x^2 + 8*I*a^2*c*d*f^2*x + 4*I*a^2
*c^2*f^2)*Ei((d*f*x + c*f)/d)*e^((d*e - c*f)/d) - (-4*I*a^2*d^2*f^2*x^2 - 8*I*a^2*c*d*f^2*x - 4*I*a^2*c^2*f^2)
*Ei(-(d*f*x + c*f)/d)*e^(-(d*e - c*f)/d) + 4*(a^2*d^2*f^2*x^2 + 2*a^2*c*d*f^2*x + a^2*c^2*f^2)*Ei(-2*(d*f*x +
c*f)/d)*e^(-2*(d*e - c*f)/d))*e^(2*f*x + 2*e) - (-4*I*a^2*d^2*f*x - 4*I*a^2*c*d*f + 4*I*a^2*d^2)*e^(f*x + e))*
e^(-2*f*x - 2*e)/(d^5*x^2 + 2*c*d^4*x + c^2*d^3)

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giac [B]  time = 0.17, size = 706, normalized size = 2.99 \[ -\frac {4 \, a^{2} d^{2} f^{2} x^{2} {\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 i \, a^{2} d^{2} f^{2} x^{2} {\rm Ei}\left (-\frac {d f x + c f}{d}\right ) e^{\left (\frac {c f}{d} - e\right )} - 4 i \, a^{2} d^{2} f^{2} x^{2} {\rm Ei}\left (\frac {d f x + c f}{d}\right ) e^{\left (-\frac {c f}{d} + e\right )} + 4 \, a^{2} d^{2} f^{2} x^{2} {\rm Ei}\left (\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) e^{\left (-\frac {2 \, c f}{d} + 2 \, e\right )} + 8 \, a^{2} c d f^{2} 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 )} + 8 i \, a^{2} c d f^{2} x {\rm Ei}\left (-\frac {d f x + c f}{d}\right ) e^{\left (\frac {c f}{d} - e\right )} - 8 i \, a^{2} c d f^{2} x {\rm Ei}\left (\frac {d f x + c f}{d}\right ) e^{\left (-\frac {c f}{d} + e\right )} + 8 \, a^{2} c d f^{2} 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} c^{2} f^{2} {\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 i \, a^{2} c^{2} f^{2} {\rm Ei}\left (-\frac {d f x + c f}{d}\right ) e^{\left (\frac {c f}{d} - e\right )} - 4 i \, a^{2} c^{2} f^{2} {\rm Ei}\left (\frac {d f x + c f}{d}\right ) e^{\left (-\frac {c f}{d} + e\right )} + 4 \, a^{2} c^{2} f^{2} {\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} d^{2} f x e^{\left (2 \, f x + 2 \, e\right )} + 4 i \, a^{2} d^{2} f x e^{\left (f x + e\right )} + 4 i \, a^{2} d^{2} f x e^{\left (-f x - e\right )} + 2 \, a^{2} d^{2} f x e^{\left (-2 \, f x - 2 \, e\right )} - 2 \, a^{2} c d f e^{\left (2 \, f x + 2 \, e\right )} + 4 i \, a^{2} c d f e^{\left (f x + e\right )} + 4 i \, a^{2} c d f e^{\left (-f x - e\right )} + 2 \, a^{2} c d f e^{\left (-2 \, f x - 2 \, e\right )} - a^{2} d^{2} e^{\left (2 \, f x + 2 \, e\right )} + 4 i \, a^{2} d^{2} e^{\left (f x + e\right )} - 4 i \, a^{2} d^{2} e^{\left (-f x - e\right )} - a^{2} d^{2} e^{\left (-2 \, f x - 2 \, e\right )} + 6 \, a^{2} d^{2}}{8 \, {\left (d^{5} x^{2} + 2 \, c d^{4} x + c^{2} d^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+I*a*sinh(f*x+e))^2/(d*x+c)^3,x, algorithm="giac")

[Out]

-1/8*(4*a^2*d^2*f^2*x^2*Ei(-2*(d*f*x + c*f)/d)*e^(2*c*f/d - 2*e) + 4*I*a^2*d^2*f^2*x^2*Ei(-(d*f*x + c*f)/d)*e^
(c*f/d - e) - 4*I*a^2*d^2*f^2*x^2*Ei((d*f*x + c*f)/d)*e^(-c*f/d + e) + 4*a^2*d^2*f^2*x^2*Ei(2*(d*f*x + c*f)/d)
*e^(-2*c*f/d + 2*e) + 8*a^2*c*d*f^2*x*Ei(-2*(d*f*x + c*f)/d)*e^(2*c*f/d - 2*e) + 8*I*a^2*c*d*f^2*x*Ei(-(d*f*x
+ c*f)/d)*e^(c*f/d - e) - 8*I*a^2*c*d*f^2*x*Ei((d*f*x + c*f)/d)*e^(-c*f/d + e) + 8*a^2*c*d*f^2*x*Ei(2*(d*f*x +
c*f)/d)*e^(-2*c*f/d + 2*e) + 4*a^2*c^2*f^2*Ei(-2*(d*f*x + c*f)/d)*e^(2*c*f/d - 2*e) + 4*I*a^2*c^2*f^2*Ei(-(d*
f*x + c*f)/d)*e^(c*f/d - e) - 4*I*a^2*c^2*f^2*Ei((d*f*x + c*f)/d)*e^(-c*f/d + e) + 4*a^2*c^2*f^2*Ei(2*(d*f*x +
c*f)/d)*e^(-2*c*f/d + 2*e) - 2*a^2*d^2*f*x*e^(2*f*x + 2*e) + 4*I*a^2*d^2*f*x*e^(f*x + e) + 4*I*a^2*d^2*f*x*e^
(-f*x - e) + 2*a^2*d^2*f*x*e^(-2*f*x - 2*e) - 2*a^2*c*d*f*e^(2*f*x + 2*e) + 4*I*a^2*c*d*f*e^(f*x + e) + 4*I*a^
2*c*d*f*e^(-f*x - e) + 2*a^2*c*d*f*e^(-2*f*x - 2*e) - a^2*d^2*e^(2*f*x + 2*e) + 4*I*a^2*d^2*e^(f*x + e) - 4*I*
a^2*d^2*e^(-f*x - e) - a^2*d^2*e^(-2*f*x - 2*e) + 6*a^2*d^2)/(d^5*x^2 + 2*c*d^4*x + c^2*d^3)

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maple [B]  time = 0.25, size = 625, normalized size = 2.65 \[ -\frac {i a^{2} f^{2} {\mathrm e}^{f x +e}}{2 d^{3} \left (\frac {c f}{d}+f x \right )^{2}}-\frac {i a^{2} f^{2} {\mathrm e}^{f x +e}}{2 d^{3} \left (\frac {c f}{d}+f x \right )}-\frac {i a^{2} f^{2} {\mathrm e}^{-\frac {c f -d e}{d}} \Ei \left (1, -f x -e -\frac {c f -d e}{d}\right )}{2 d^{3}}-\frac {3 a^{2}}{4 d \left (d x +c \right )^{2}}-\frac {f^{3} a^{2} {\mathrm e}^{-2 f x -2 e} x}{4 d \left (d^{2} f^{2} x^{2}+2 c d \,f^{2} x +c^{2} f^{2}\right )}-\frac {f^{3} a^{2} {\mathrm e}^{-2 f x -2 e} c}{4 d^{2} \left (d^{2} f^{2} x^{2}+2 c d \,f^{2} x +c^{2} f^{2}\right )}+\frac {f^{2} a^{2} {\mathrm e}^{-2 f x -2 e}}{8 d \left (d^{2} f^{2} x^{2}+2 c d \,f^{2} x +c^{2} f^{2}\right )}+\frac {f^{2} a^{2} {\mathrm e}^{\frac {2 c f -2 d e}{d}} \Ei \left (1, 2 f x +2 e +\frac {2 c f -2 d e}{d}\right )}{2 d^{3}}+\frac {f^{2} a^{2} {\mathrm e}^{2 f x +2 e}}{8 d^{3} \left (\frac {c f}{d}+f x \right )^{2}}+\frac {f^{2} a^{2} {\mathrm e}^{2 f x +2 e}}{4 d^{3} \left (\frac {c f}{d}+f x \right )}+\frac {f^{2} a^{2} {\mathrm e}^{-\frac {2 \left (c f -d e \right )}{d}} \Ei \left (1, -2 f x -2 e -\frac {2 \left (c f -d e \right )}{d}\right )}{2 d^{3}}-\frac {i a^{2} f^{3} {\mathrm e}^{-f x -e} x}{2 d \left (d^{2} f^{2} x^{2}+2 c d \,f^{2} x +c^{2} f^{2}\right )}-\frac {i a^{2} f^{3} {\mathrm e}^{-f x -e} c}{2 d^{2} \left (d^{2} f^{2} x^{2}+2 c d \,f^{2} x +c^{2} f^{2}\right )}+\frac {i a^{2} f^{2} {\mathrm e}^{-f x -e}}{2 d \left (d^{2} f^{2} x^{2}+2 c d \,f^{2} x +c^{2} f^{2}\right )}+\frac {i a^{2} f^{2} {\mathrm e}^{\frac {c f -d e}{d}} \Ei \left (1, f x +e +\frac {c f -d e}{d}\right )}{2 d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+I*a*sinh(f*x+e))^2/(d*x+c)^3,x)

[Out]

-1/2*I*a^2*f^2/d^3*exp(f*x+e)/(c*f/d+f*x)^2-1/2*I*a^2*f^2/d^3*exp(f*x+e)/(c*f/d+f*x)-1/2*I*a^2*f^2/d^3*exp(-(c
*f-d*e)/d)*Ei(1,-f*x-e-(c*f-d*e)/d)-3/4*a^2/d/(d*x+c)^2-1/4*f^3*a^2*exp(-2*f*x-2*e)/d/(d^2*f^2*x^2+2*c*d*f^2*x
+c^2*f^2)*x-1/4*f^3*a^2*exp(-2*f*x-2*e)/d^2/(d^2*f^2*x^2+2*c*d*f^2*x+c^2*f^2)*c+1/8*f^2*a^2*exp(-2*f*x-2*e)/d/
(d^2*f^2*x^2+2*c*d*f^2*x+c^2*f^2)+1/2*f^2*a^2/d^3*exp(2*(c*f-d*e)/d)*Ei(1,2*f*x+2*e+2*(c*f-d*e)/d)+1/8*f^2*a^2
/d^3*exp(2*f*x+2*e)/(c*f/d+f*x)^2+1/4*f^2*a^2/d^3*exp(2*f*x+2*e)/(c*f/d+f*x)+1/2*f^2*a^2/d^3*exp(-2*(c*f-d*e)/
d)*Ei(1,-2*f*x-2*e-2*(c*f-d*e)/d)-1/2*I*a^2*f^3*exp(-f*x-e)/d/(d^2*f^2*x^2+2*c*d*f^2*x+c^2*f^2)*x-1/2*I*a^2*f^
3*exp(-f*x-e)/d^2/(d^2*f^2*x^2+2*c*d*f^2*x+c^2*f^2)*c+1/2*I*a^2*f^2*exp(-f*x-e)/d/(d^2*f^2*x^2+2*c*d*f^2*x+c^2
*f^2)+1/2*I*a^2*f^2/d^3*exp((c*f-d*e)/d)*Ei(1,f*x+e+(c*f-d*e)/d)

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maxima [A]  time = 0.46, size = 205, normalized size = 0.87 \[ -\frac {1}{4} \, a^{2} {\left (\frac {1}{d^{3} x^{2} + 2 \, c d^{2} x + c^{2} d} - \frac {e^{\left (-2 \, e + \frac {2 \, c f}{d}\right )} E_{3}\left (\frac {2 \, {\left (d x + c\right )} f}{d}\right )}{{\left (d x + c\right )}^{2} d} - \frac {e^{\left (2 \, e - \frac {2 \, c f}{d}\right )} E_{3}\left (-\frac {2 \, {\left (d x + c\right )} f}{d}\right )}{{\left (d x + c\right )}^{2} d}\right )} + i \, a^{2} {\left (\frac {e^{\left (-e + \frac {c f}{d}\right )} E_{3}\left (\frac {{\left (d x + c\right )} f}{d}\right )}{{\left (d x + c\right )}^{2} d} - \frac {e^{\left (e - \frac {c f}{d}\right )} E_{3}\left (-\frac {{\left (d x + c\right )} f}{d}\right )}{{\left (d x + c\right )}^{2} d}\right )} - \frac {a^{2}}{2 \, {\left (d^{3} x^{2} + 2 \, c d^{2} x + c^{2} d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+I*a*sinh(f*x+e))^2/(d*x+c)^3,x, algorithm="maxima")

[Out]

-1/4*a^2*(1/(d^3*x^2 + 2*c*d^2*x + c^2*d) - e^(-2*e + 2*c*f/d)*exp_integral_e(3, 2*(d*x + c)*f/d)/((d*x + c)^2
*d) - e^(2*e - 2*c*f/d)*exp_integral_e(3, -2*(d*x + c)*f/d)/((d*x + c)^2*d)) + I*a^2*(e^(-e + c*f/d)*exp_integ
ral_e(3, (d*x + c)*f/d)/((d*x + c)^2*d) - e^(e - c*f/d)*exp_integral_e(3, -(d*x + c)*f/d)/((d*x + c)^2*d)) - 1
/2*a^2/(d^3*x^2 + 2*c*d^2*x + c^2*d)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (a+a\,\mathrm {sinh}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^2}{{\left (c+d\,x\right )}^3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + a*sinh(e + f*x)*1i)^2/(c + d*x)^3,x)

[Out]

int((a + a*sinh(e + f*x)*1i)^2/(c + d*x)^3, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ - a^{2} \left (\int \frac {\sinh ^{2}{\left (e + f x \right )}}{c^{3} + 3 c^{2} d x + 3 c d^{2} x^{2} + d^{3} x^{3}}\, dx + \int \left (- \frac {2 i \sinh {\left (e + f x \right )}}{c^{3} + 3 c^{2} d x + 3 c d^{2} x^{2} + d^{3} x^{3}}\right )\, dx + \int \left (- \frac {1}{c^{3} + 3 c^{2} d x + 3 c d^{2} x^{2} + d^{3} x^{3}}\right )\, dx\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+I*a*sinh(f*x+e))**2/(d*x+c)**3,x)

[Out]

-a**2*(Integral(sinh(e + f*x)**2/(c**3 + 3*c**2*d*x + 3*c*d**2*x**2 + d**3*x**3), x) + Integral(-2*I*sinh(e +
f*x)/(c**3 + 3*c**2*d*x + 3*c*d**2*x**2 + d**3*x**3), x) + Integral(-1/(c**3 + 3*c**2*d*x + 3*c*d**2*x**2 + d*
*3*x**3), x))

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