3.103 \(\int (c+d x)^2 (a+i a \sinh (e+f x))^2 \, dx\)

Optimal. Leaf size=174 \[ \frac{a^2 d (c+d x) \sinh ^2(e+f x)}{2 f^2}-\frac{4 i a^2 d (c+d x) \sinh (e+f x)}{f^2}+\frac{2 i a^2 (c+d x)^2 \cosh (e+f x)}{f}-\frac{a^2 (c+d x)^2 \sinh (e+f x) \cosh (e+f x)}{2 f}+\frac{a^2 (c+d x)^3}{2 d}+\frac{4 i a^2 d^2 \cosh (e+f x)}{f^3}-\frac{a^2 d^2 \sinh (e+f x) \cosh (e+f x)}{4 f^3}+\frac{a^2 d^2 x}{4 f^2} \]

[Out]

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

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Rubi [A]  time = 0.197184, antiderivative size = 174, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.304, Rules used = {3317, 3296, 2638, 3311, 32, 2635, 8} \[ \frac{a^2 d (c+d x) \sinh ^2(e+f x)}{2 f^2}-\frac{4 i a^2 d (c+d x) \sinh (e+f x)}{f^2}+\frac{2 i a^2 (c+d x)^2 \cosh (e+f x)}{f}-\frac{a^2 (c+d x)^2 \sinh (e+f x) \cosh (e+f x)}{2 f}+\frac{a^2 (c+d x)^3}{2 d}+\frac{4 i a^2 d^2 \cosh (e+f x)}{f^3}-\frac{a^2 d^2 \sinh (e+f x) \cosh (e+f x)}{4 f^3}+\frac{a^2 d^2 x}{4 f^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^2*d^2*x)/(4*f^2) + (a^2*(c + d*x)^3)/(2*d) + ((4*I)*a^2*d^2*Cosh[e + f*x])/f^3 + ((2*I)*a^2*(c + d*x)^2*Cos
h[e + f*x])/f - ((4*I)*a^2*d*(c + d*x)*Sinh[e + f*x])/f^2 - (a^2*d^2*Cosh[e + f*x]*Sinh[e + f*x])/(4*f^3) - (a
^2*(c + d*x)^2*Cosh[e + f*x]*Sinh[e + f*x])/(2*f) + (a^2*d*(c + d*x)*Sinh[e + f*x]^2)/(2*f^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 3296

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> -Simp[((c + d*x)^m*Cos[e + f*x])/f, x] +
Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 2638

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 3311

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

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rule 2635

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Sin[c + d*x])^(n - 1))/(d*n),
x] + Dist[(b^2*(n - 1))/n, Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integer
Q[2*n]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

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

Mathematica [A]  time = 0.713293, size = 189, normalized size = 1.09 \[ \frac{a^2 \left (16 i \left (c^2 f^2+2 c d f^2 x+d^2 \left (f^2 x^2+2\right )\right ) \cosh (e+f x)-2 c^2 f^2 \sinh (2 (e+f x))+12 c^2 f^3 x-4 c d f^2 x \sinh (2 (e+f x))-32 i c d f \sinh (e+f x)+2 d f (c+d x) \cosh (2 (e+f x))+12 c d f^3 x^2-2 d^2 f^2 x^2 \sinh (2 (e+f x))-32 i d^2 f x \sinh (e+f x)-d^2 \sinh (2 (e+f x))+4 d^2 f^3 x^3\right )}{8 f^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^2*(12*c^2*f^3*x + 12*c*d*f^3*x^2 + 4*d^2*f^3*x^3 + (16*I)*(c^2*f^2 + 2*c*d*f^2*x + d^2*(2 + f^2*x^2))*Cosh[
e + f*x] + 2*d*f*(c + d*x)*Cosh[2*(e + f*x)] - (32*I)*c*d*f*Sinh[e + f*x] - (32*I)*d^2*f*x*Sinh[e + f*x] - d^2
*Sinh[2*(e + f*x)] - 2*c^2*f^2*Sinh[2*(e + f*x)] - 4*c*d*f^2*x*Sinh[2*(e + f*x)] - 2*d^2*f^2*x^2*Sinh[2*(e + f
*x)]))/(8*f^3)

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Maple [B]  time = 0.014, size = 550, normalized size = 3.2 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/f*(1/3/f^2*d^2*a^2*(f*x+e)^3-4*I/f^2*d^2*e*a^2*((f*x+e)*cosh(f*x+e)-sinh(f*x+e))-1/f^2*d^2*a^2*(1/2*(f*x+e)^
2*cosh(f*x+e)*sinh(f*x+e)-1/6*(f*x+e)^3-1/2*(f*x+e)*cosh(f*x+e)^2+1/4*cosh(f*x+e)*sinh(f*x+e)+1/4*f*x+1/4*e)-1
/f^2*d^2*e*a^2*(f*x+e)^2-4*I/f*c*d*e*a^2*cosh(f*x+e)+2/f^2*d^2*e*a^2*(1/2*(f*x+e)*cosh(f*x+e)*sinh(f*x+e)-1/4*
(f*x+e)^2-1/4*cosh(f*x+e)^2)+1/f^2*d^2*e^2*a^2*(f*x+e)+4*I/f*c*d*a^2*((f*x+e)*cosh(f*x+e)-sinh(f*x+e))-1/f^2*d
^2*e^2*a^2*(1/2*cosh(f*x+e)*sinh(f*x+e)-1/2*f*x-1/2*e)+1/f*c*d*a^2*(f*x+e)^2+2*I/f^2*d^2*a^2*((f*x+e)^2*cosh(f
*x+e)-2*(f*x+e)*sinh(f*x+e)+2*cosh(f*x+e))-2/f*c*d*a^2*(1/2*(f*x+e)*cosh(f*x+e)*sinh(f*x+e)-1/4*(f*x+e)^2-1/4*
cosh(f*x+e)^2)-2/f*c*d*e*a^2*(f*x+e)+2*I*c^2*a^2*cosh(f*x+e)+2/f*c*d*e*a^2*(1/2*cosh(f*x+e)*sinh(f*x+e)-1/2*f*
x-1/2*e)+c^2*a^2*(f*x+e)+2*I/f^2*d^2*e^2*a^2*cosh(f*x+e)-c^2*a^2*(1/2*cosh(f*x+e)*sinh(f*x+e)-1/2*f*x-1/2*e))

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Maxima [B]  time = 1.26859, size = 440, normalized size = 2.53 \begin{align*} \frac{1}{3} \, a^{2} d^{2} x^{3} + a^{2} c d x^{2} + \frac{1}{8} \,{\left (4 \, x^{2} - \frac{{\left (2 \, f x e^{\left (2 \, e\right )} - e^{\left (2 \, e\right )}\right )} e^{\left (2 \, f x\right )}}{f^{2}} + \frac{{\left (2 \, f x + 1\right )} e^{\left (-2 \, f x - 2 \, e\right )}}{f^{2}}\right )} a^{2} c d + \frac{1}{48} \,{\left (8 \, x^{3} - \frac{3 \,{\left (2 \, f^{2} x^{2} e^{\left (2 \, e\right )} - 2 \, f x e^{\left (2 \, e\right )} + e^{\left (2 \, e\right )}\right )} e^{\left (2 \, f x\right )}}{f^{3}} + \frac{3 \,{\left (2 \, f^{2} x^{2} + 2 \, f x + 1\right )} e^{\left (-2 \, f x - 2 \, e\right )}}{f^{3}}\right )} a^{2} d^{2} + \frac{1}{8} \, a^{2} c^{2}{\left (4 \, x - \frac{e^{\left (2 \, f x + 2 \, e\right )}}{f} + \frac{e^{\left (-2 \, f x - 2 \, e\right )}}{f}\right )} + a^{2} c^{2} x + 2 i \, a^{2} c d{\left (\frac{{\left (f x e^{e} - e^{e}\right )} e^{\left (f x\right )}}{f^{2}} + \frac{{\left (f x + 1\right )} e^{\left (-f x - e\right )}}{f^{2}}\right )} + i \, a^{2} d^{2}{\left (\frac{{\left (f^{2} x^{2} e^{e} - 2 \, f x e^{e} + 2 \, e^{e}\right )} e^{\left (f x\right )}}{f^{3}} + \frac{{\left (f^{2} x^{2} + 2 \, f x + 2\right )} e^{\left (-f x - e\right )}}{f^{3}}\right )} + \frac{2 i \, a^{2} c^{2} \cosh \left (f x + e\right )}{f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*a^2*d^2*x^3 + a^2*c*d*x^2 + 1/8*(4*x^2 - (2*f*x*e^(2*e) - e^(2*e))*e^(2*f*x)/f^2 + (2*f*x + 1)*e^(-2*f*x -
2*e)/f^2)*a^2*c*d + 1/48*(8*x^3 - 3*(2*f^2*x^2*e^(2*e) - 2*f*x*e^(2*e) + e^(2*e))*e^(2*f*x)/f^3 + 3*(2*f^2*x^
2 + 2*f*x + 1)*e^(-2*f*x - 2*e)/f^3)*a^2*d^2 + 1/8*a^2*c^2*(4*x - e^(2*f*x + 2*e)/f + e^(-2*f*x - 2*e)/f) + a^
2*c^2*x + 2*I*a^2*c*d*((f*x*e^e - e^e)*e^(f*x)/f^2 + (f*x + 1)*e^(-f*x - e)/f^2) + I*a^2*d^2*((f^2*x^2*e^e - 2
*f*x*e^e + 2*e^e)*e^(f*x)/f^3 + (f^2*x^2 + 2*f*x + 2)*e^(-f*x - e)/f^3) + 2*I*a^2*c^2*cosh(f*x + e)/f

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Fricas [B]  time = 3.04527, size = 768, normalized size = 4.41 \begin{align*} \frac{{\left (2 \, a^{2} d^{2} f^{2} x^{2} + 2 \, a^{2} c^{2} f^{2} + 2 \, a^{2} c d f + a^{2} d^{2} + 2 \,{\left (2 \, a^{2} c d f^{2} + a^{2} d^{2} f\right )} x -{\left (2 \, a^{2} d^{2} f^{2} x^{2} + 2 \, a^{2} c^{2} f^{2} - 2 \, a^{2} c d f + a^{2} d^{2} + 2 \,{\left (2 \, a^{2} c d f^{2} - a^{2} d^{2} f\right )} x\right )} e^{\left (4 \, f x + 4 \, e\right )} +{\left (16 i \, a^{2} d^{2} f^{2} x^{2} + 16 i \, a^{2} c^{2} f^{2} - 32 i \, a^{2} c d f + 32 i \, a^{2} d^{2} +{\left (32 i \, a^{2} c d f^{2} - 32 i \, a^{2} d^{2} f\right )} x\right )} e^{\left (3 \, f x + 3 \, e\right )} + 8 \,{\left (a^{2} d^{2} f^{3} x^{3} + 3 \, a^{2} c d f^{3} x^{2} + 3 \, a^{2} c^{2} f^{3} x\right )} e^{\left (2 \, f x + 2 \, e\right )} +{\left (16 i \, a^{2} d^{2} f^{2} x^{2} + 16 i \, a^{2} c^{2} f^{2} + 32 i \, a^{2} c d f + 32 i \, a^{2} d^{2} +{\left (32 i \, a^{2} c d f^{2} + 32 i \, a^{2} d^{2} f\right )} x\right )} e^{\left (f x + e\right )}\right )} e^{\left (-2 \, f x - 2 \, e\right )}}{16 \, f^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*(2*a^2*d^2*f^2*x^2 + 2*a^2*c^2*f^2 + 2*a^2*c*d*f + a^2*d^2 + 2*(2*a^2*c*d*f^2 + a^2*d^2*f)*x - (2*a^2*d^2
*f^2*x^2 + 2*a^2*c^2*f^2 - 2*a^2*c*d*f + a^2*d^2 + 2*(2*a^2*c*d*f^2 - a^2*d^2*f)*x)*e^(4*f*x + 4*e) + (16*I*a^
2*d^2*f^2*x^2 + 16*I*a^2*c^2*f^2 - 32*I*a^2*c*d*f + 32*I*a^2*d^2 + (32*I*a^2*c*d*f^2 - 32*I*a^2*d^2*f)*x)*e^(3
*f*x + 3*e) + 8*(a^2*d^2*f^3*x^3 + 3*a^2*c*d*f^3*x^2 + 3*a^2*c^2*f^3*x)*e^(2*f*x + 2*e) + (16*I*a^2*d^2*f^2*x^
2 + 16*I*a^2*c^2*f^2 + 32*I*a^2*c*d*f + 32*I*a^2*d^2 + (32*I*a^2*c*d*f^2 + 32*I*a^2*d^2*f)*x)*e^(f*x + e))*e^(
-2*f*x - 2*e)/f^3

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Sympy [A]  time = 3.20624, size = 706, normalized size = 4.06 \begin{align*} \frac{3 a^{2} c^{2} x}{2} + \frac{3 a^{2} c d x^{2}}{2} + \frac{a^{2} d^{2} x^{3}}{2} + \begin{cases} \frac{\left (\left (32 a^{2} c^{2} f^{17} e^{3 e} + 64 a^{2} c d f^{17} x e^{3 e} + 32 a^{2} c d f^{16} e^{3 e} + 32 a^{2} d^{2} f^{17} x^{2} e^{3 e} + 32 a^{2} d^{2} f^{16} x e^{3 e} + 16 a^{2} d^{2} f^{15} e^{3 e}\right ) e^{- 2 f x} + \left (- 32 a^{2} c^{2} f^{17} e^{7 e} - 64 a^{2} c d f^{17} x e^{7 e} + 32 a^{2} c d f^{16} e^{7 e} - 32 a^{2} d^{2} f^{17} x^{2} e^{7 e} + 32 a^{2} d^{2} f^{16} x e^{7 e} - 16 a^{2} d^{2} f^{15} e^{7 e}\right ) e^{2 f x} + \left (256 i a^{2} c^{2} f^{17} e^{4 e} + 512 i a^{2} c d f^{17} x e^{4 e} + 512 i a^{2} c d f^{16} e^{4 e} + 256 i a^{2} d^{2} f^{17} x^{2} e^{4 e} + 512 i a^{2} d^{2} f^{16} x e^{4 e} + 512 i a^{2} d^{2} f^{15} e^{4 e}\right ) e^{- f x} + \left (256 i a^{2} c^{2} f^{17} e^{6 e} + 512 i a^{2} c d f^{17} x e^{6 e} - 512 i a^{2} c d f^{16} e^{6 e} + 256 i a^{2} d^{2} f^{17} x^{2} e^{6 e} - 512 i a^{2} d^{2} f^{16} x e^{6 e} + 512 i a^{2} d^{2} f^{15} e^{6 e}\right ) e^{f x}\right ) e^{- 5 e}}{256 f^{18}} & \text{for}\: 256 f^{18} e^{5 e} \neq 0 \\\frac{x^{3} \left (- a^{2} d^{2} e^{4 e} + 4 i a^{2} d^{2} e^{3 e} - 4 i a^{2} d^{2} e^{e} - a^{2} d^{2}\right ) e^{- 2 e}}{12} + \frac{x^{2} \left (- a^{2} c d e^{4 e} + 4 i a^{2} c d e^{3 e} - 4 i a^{2} c d e^{e} - a^{2} c d\right ) e^{- 2 e}}{4} + \frac{x \left (- a^{2} c^{2} e^{4 e} + 4 i a^{2} c^{2} e^{3 e} - 4 i a^{2} c^{2} e^{e} - a^{2} c^{2}\right ) e^{- 2 e}}{4} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*a**2*c**2*x/2 + 3*a**2*c*d*x**2/2 + a**2*d**2*x**3/2 + Piecewise((((32*a**2*c**2*f**17*exp(3*e) + 64*a**2*c*
d*f**17*x*exp(3*e) + 32*a**2*c*d*f**16*exp(3*e) + 32*a**2*d**2*f**17*x**2*exp(3*e) + 32*a**2*d**2*f**16*x*exp(
3*e) + 16*a**2*d**2*f**15*exp(3*e))*exp(-2*f*x) + (-32*a**2*c**2*f**17*exp(7*e) - 64*a**2*c*d*f**17*x*exp(7*e)
+ 32*a**2*c*d*f**16*exp(7*e) - 32*a**2*d**2*f**17*x**2*exp(7*e) + 32*a**2*d**2*f**16*x*exp(7*e) - 16*a**2*d**
2*f**15*exp(7*e))*exp(2*f*x) + (256*I*a**2*c**2*f**17*exp(4*e) + 512*I*a**2*c*d*f**17*x*exp(4*e) + 512*I*a**2*
c*d*f**16*exp(4*e) + 256*I*a**2*d**2*f**17*x**2*exp(4*e) + 512*I*a**2*d**2*f**16*x*exp(4*e) + 512*I*a**2*d**2*
f**15*exp(4*e))*exp(-f*x) + (256*I*a**2*c**2*f**17*exp(6*e) + 512*I*a**2*c*d*f**17*x*exp(6*e) - 512*I*a**2*c*d
*f**16*exp(6*e) + 256*I*a**2*d**2*f**17*x**2*exp(6*e) - 512*I*a**2*d**2*f**16*x*exp(6*e) + 512*I*a**2*d**2*f**
15*exp(6*e))*exp(f*x))*exp(-5*e)/(256*f**18), Ne(256*f**18*exp(5*e), 0)), (x**3*(-a**2*d**2*exp(4*e) + 4*I*a**
2*d**2*exp(3*e) - 4*I*a**2*d**2*exp(e) - a**2*d**2)*exp(-2*e)/12 + x**2*(-a**2*c*d*exp(4*e) + 4*I*a**2*c*d*exp
(3*e) - 4*I*a**2*c*d*exp(e) - a**2*c*d)*exp(-2*e)/4 + x*(-a**2*c**2*exp(4*e) + 4*I*a**2*c**2*exp(3*e) - 4*I*a*
*2*c**2*exp(e) - a**2*c**2)*exp(-2*e)/4, True))

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Giac [B]  time = 1.26013, size = 455, normalized size = 2.61 \begin{align*} \frac{1}{2} \, a^{2} d^{2} x^{3} + \frac{3}{2} \, a^{2} c d x^{2} + \frac{3}{2} \, a^{2} c^{2} x - \frac{{\left (2 \, a^{2} d^{2} f^{2} x^{2} + 4 \, a^{2} c d f^{2} x + 2 \, a^{2} c^{2} f^{2} - 2 \, a^{2} d^{2} f x - 2 \, a^{2} c d f + a^{2} d^{2}\right )} e^{\left (2 \, f x + 2 \, e\right )}}{16 \, f^{3}} + \frac{{\left (i \, a^{2} d^{2} f^{2} x^{2} + 2 i \, a^{2} c d f^{2} x + i \, a^{2} c^{2} f^{2} - 2 i \, a^{2} d^{2} f x - 2 i \, a^{2} c d f + 2 i \, a^{2} d^{2}\right )} e^{\left (f x + e\right )}}{f^{3}} - \frac{{\left (-i \, a^{2} d^{2} f^{2} x^{2} - 2 i \, a^{2} c d f^{2} x - i \, a^{2} c^{2} f^{2} - 2 i \, a^{2} d^{2} f x - 2 i \, a^{2} c d f - 2 i \, a^{2} d^{2}\right )} e^{\left (-f x - e\right )}}{f^{3}} + \frac{{\left (2 \, a^{2} d^{2} f^{2} x^{2} + 4 \, a^{2} c d f^{2} x + 2 \, a^{2} c^{2} f^{2} + 2 \, a^{2} d^{2} f x + 2 \, a^{2} c d f + a^{2} d^{2}\right )} e^{\left (-2 \, f x - 2 \, e\right )}}{16 \, f^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*a^2*d^2*x^3 + 3/2*a^2*c*d*x^2 + 3/2*a^2*c^2*x - 1/16*(2*a^2*d^2*f^2*x^2 + 4*a^2*c*d*f^2*x + 2*a^2*c^2*f^2
- 2*a^2*d^2*f*x - 2*a^2*c*d*f + a^2*d^2)*e^(2*f*x + 2*e)/f^3 + (I*a^2*d^2*f^2*x^2 + 2*I*a^2*c*d*f^2*x + I*a^2*
c^2*f^2 - 2*I*a^2*d^2*f*x - 2*I*a^2*c*d*f + 2*I*a^2*d^2)*e^(f*x + e)/f^3 - (-I*a^2*d^2*f^2*x^2 - 2*I*a^2*c*d*f
^2*x - I*a^2*c^2*f^2 - 2*I*a^2*d^2*f*x - 2*I*a^2*c*d*f - 2*I*a^2*d^2)*e^(-f*x - e)/f^3 + 1/16*(2*a^2*d^2*f^2*x
^2 + 4*a^2*c*d*f^2*x + 2*a^2*c^2*f^2 + 2*a^2*d^2*f*x + 2*a^2*c*d*f + a^2*d^2)*e^(-2*f*x - 2*e)/f^3