3.102 \(\int (c+d x)^3 (a+i a \sinh (e+f x))^2 \, dx\)

Optimal. Leaf size=245 \[ \frac{12 i a^2 d^2 (c+d x) \cosh (e+f x)}{f^3}-\frac{3 a^2 d^2 (c+d x) \sinh (e+f x) \cosh (e+f x)}{4 f^3}+\frac{3 a^2 c d^2 x}{4 f^2}+\frac{3 a^2 d (c+d x)^2 \sinh ^2(e+f x)}{4 f^2}-\frac{6 i a^2 d (c+d x)^2 \sinh (e+f x)}{f^2}+\frac{2 i a^2 (c+d x)^3 \cosh (e+f x)}{f}-\frac{a^2 (c+d x)^3 \sinh (e+f x) \cosh (e+f x)}{2 f}+\frac{3 a^2 (c+d x)^4}{8 d}+\frac{3 a^2 d^3 \sinh ^2(e+f x)}{8 f^4}-\frac{12 i a^2 d^3 \sinh (e+f x)}{f^4}+\frac{3 a^2 d^3 x^2}{8 f^2} \]

[Out]

(3*a^2*c*d^2*x)/(4*f^2) + (3*a^2*d^3*x^2)/(8*f^2) + (3*a^2*(c + d*x)^4)/(8*d) + ((12*I)*a^2*d^2*(c + d*x)*Cosh
[e + f*x])/f^3 + ((2*I)*a^2*(c + d*x)^3*Cosh[e + f*x])/f - ((12*I)*a^2*d^3*Sinh[e + f*x])/f^4 - ((6*I)*a^2*d*(
c + d*x)^2*Sinh[e + f*x])/f^2 - (3*a^2*d^2*(c + d*x)*Cosh[e + f*x]*Sinh[e + f*x])/(4*f^3) - (a^2*(c + d*x)^3*C
osh[e + f*x]*Sinh[e + f*x])/(2*f) + (3*a^2*d^3*Sinh[e + f*x]^2)/(8*f^4) + (3*a^2*d*(c + d*x)^2*Sinh[e + f*x]^2
)/(4*f^2)

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Rubi [A]  time = 0.287126, antiderivative size = 245, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261, Rules used = {3317, 3296, 2637, 3311, 32, 3310} \[ \frac{12 i a^2 d^2 (c+d x) \cosh (e+f x)}{f^3}-\frac{3 a^2 d^2 (c+d x) \sinh (e+f x) \cosh (e+f x)}{4 f^3}+\frac{3 a^2 c d^2 x}{4 f^2}+\frac{3 a^2 d (c+d x)^2 \sinh ^2(e+f x)}{4 f^2}-\frac{6 i a^2 d (c+d x)^2 \sinh (e+f x)}{f^2}+\frac{2 i a^2 (c+d x)^3 \cosh (e+f x)}{f}-\frac{a^2 (c+d x)^3 \sinh (e+f x) \cosh (e+f x)}{2 f}+\frac{3 a^2 (c+d x)^4}{8 d}+\frac{3 a^2 d^3 \sinh ^2(e+f x)}{8 f^4}-\frac{12 i a^2 d^3 \sinh (e+f x)}{f^4}+\frac{3 a^2 d^3 x^2}{8 f^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*a^2*c*d^2*x)/(4*f^2) + (3*a^2*d^3*x^2)/(8*f^2) + (3*a^2*(c + d*x)^4)/(8*d) + ((12*I)*a^2*d^2*(c + d*x)*Cosh
[e + f*x])/f^3 + ((2*I)*a^2*(c + d*x)^3*Cosh[e + f*x])/f - ((12*I)*a^2*d^3*Sinh[e + f*x])/f^4 - ((6*I)*a^2*d*(
c + d*x)^2*Sinh[e + f*x])/f^2 - (3*a^2*d^2*(c + d*x)*Cosh[e + f*x]*Sinh[e + f*x])/(4*f^3) - (a^2*(c + d*x)^3*C
osh[e + f*x]*Sinh[e + f*x])/(2*f) + (3*a^2*d^3*Sinh[e + f*x]^2)/(8*f^4) + (3*a^2*d*(c + d*x)^2*Sinh[e + f*x]^2
)/(4*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 2637

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[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 3310

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

Rubi steps

\begin{align*} \int (c+d x)^3 (a+i a \sinh (e+f x))^2 \, dx &=\int \left (a^2 (c+d x)^3+2 i a^2 (c+d x)^3 \sinh (e+f x)-a^2 (c+d x)^3 \sinh ^2(e+f x)\right ) \, dx\\ &=\frac{a^2 (c+d x)^4}{4 d}+\left (2 i a^2\right ) \int (c+d x)^3 \sinh (e+f x) \, dx-a^2 \int (c+d x)^3 \sinh ^2(e+f x) \, dx\\ &=\frac{a^2 (c+d x)^4}{4 d}+\frac{2 i a^2 (c+d x)^3 \cosh (e+f x)}{f}-\frac{a^2 (c+d x)^3 \cosh (e+f x) \sinh (e+f x)}{2 f}+\frac{3 a^2 d (c+d x)^2 \sinh ^2(e+f x)}{4 f^2}+\frac{1}{2} a^2 \int (c+d x)^3 \, dx-\frac{\left (3 a^2 d^2\right ) \int (c+d x) \sinh ^2(e+f x) \, dx}{2 f^2}-\frac{\left (6 i a^2 d\right ) \int (c+d x)^2 \cosh (e+f x) \, dx}{f}\\ &=\frac{3 a^2 (c+d x)^4}{8 d}+\frac{2 i a^2 (c+d x)^3 \cosh (e+f x)}{f}-\frac{6 i a^2 d (c+d x)^2 \sinh (e+f x)}{f^2}-\frac{3 a^2 d^2 (c+d x) \cosh (e+f x) \sinh (e+f x)}{4 f^3}-\frac{a^2 (c+d x)^3 \cosh (e+f x) \sinh (e+f x)}{2 f}+\frac{3 a^2 d^3 \sinh ^2(e+f x)}{8 f^4}+\frac{3 a^2 d (c+d x)^2 \sinh ^2(e+f x)}{4 f^2}+\frac{\left (12 i a^2 d^2\right ) \int (c+d x) \sinh (e+f x) \, dx}{f^2}+\frac{\left (3 a^2 d^2\right ) \int (c+d x) \, dx}{4 f^2}\\ &=\frac{3 a^2 c d^2 x}{4 f^2}+\frac{3 a^2 d^3 x^2}{8 f^2}+\frac{3 a^2 (c+d x)^4}{8 d}+\frac{12 i a^2 d^2 (c+d x) \cosh (e+f x)}{f^3}+\frac{2 i a^2 (c+d x)^3 \cosh (e+f x)}{f}-\frac{6 i a^2 d (c+d x)^2 \sinh (e+f x)}{f^2}-\frac{3 a^2 d^2 (c+d x) \cosh (e+f x) \sinh (e+f x)}{4 f^3}-\frac{a^2 (c+d x)^3 \cosh (e+f x) \sinh (e+f x)}{2 f}+\frac{3 a^2 d^3 \sinh ^2(e+f x)}{8 f^4}+\frac{3 a^2 d (c+d x)^2 \sinh ^2(e+f x)}{4 f^2}-\frac{\left (12 i a^2 d^3\right ) \int \cosh (e+f x) \, dx}{f^3}\\ &=\frac{3 a^2 c d^2 x}{4 f^2}+\frac{3 a^2 d^3 x^2}{8 f^2}+\frac{3 a^2 (c+d x)^4}{8 d}+\frac{12 i a^2 d^2 (c+d x) \cosh (e+f x)}{f^3}+\frac{2 i a^2 (c+d x)^3 \cosh (e+f x)}{f}-\frac{12 i a^2 d^3 \sinh (e+f x)}{f^4}-\frac{6 i a^2 d (c+d x)^2 \sinh (e+f x)}{f^2}-\frac{3 a^2 d^2 (c+d x) \cosh (e+f x) \sinh (e+f x)}{4 f^3}-\frac{a^2 (c+d x)^3 \cosh (e+f x) \sinh (e+f x)}{2 f}+\frac{3 a^2 d^3 \sinh ^2(e+f x)}{8 f^4}+\frac{3 a^2 d (c+d x)^2 \sinh ^2(e+f x)}{4 f^2}\\ \end{align*}

Mathematica [A]  time = 1.4905, size = 220, normalized size = 0.9 \[ \frac{a^2 \left (-2 f (c+d x) \left (2 c^2 f^2+4 c d f^2 x+d^2 \left (2 f^2 x^2+3\right )\right ) \sinh (2 (e+f x))-96 i d \left (c^2 f^2+2 c d f^2 x+d^2 \left (f^2 x^2+2\right )\right ) \sinh (e+f x)+32 i f (c+d x) \left (c^2 f^2+2 c d f^2 x+d^2 \left (f^2 x^2+6\right )\right ) \cosh (e+f x)+3 d \left (2 c^2 f^2+4 c d f^2 x+d^2 \left (2 f^2 x^2+1\right )\right ) \cosh (2 (e+f x))+6 f^4 x \left (6 c^2 d x+4 c^3+4 c d^2 x^2+d^3 x^3\right )\right )}{16 f^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^2*(6*f^4*x*(4*c^3 + 6*c^2*d*x + 4*c*d^2*x^2 + d^3*x^3) + (32*I)*f*(c + d*x)*(c^2*f^2 + 2*c*d*f^2*x + d^2*(6
+ f^2*x^2))*Cosh[e + f*x] + 3*d*(2*c^2*f^2 + 4*c*d*f^2*x + d^2*(1 + 2*f^2*x^2))*Cosh[2*(e + f*x)] - (96*I)*d*
(c^2*f^2 + 2*c*d*f^2*x + d^2*(2 + f^2*x^2))*Sinh[e + f*x] - 2*f*(c + d*x)*(2*c^2*f^2 + 4*c*d*f^2*x + d^2*(3 +
2*f^2*x^2))*Sinh[2*(e + f*x)]))/(16*f^4)

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Maple [B]  time = 0.018, size = 1082, normalized size = 4.4 \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)^3*(a+I*a*sinh(f*x+e))^2,x)

[Out]

1/f*(3/2/f^3*d^3*e^2*a^2*(f*x+e)^2-1/f^3*d^3*e*a^2*(f*x+e)^3+1/f^3*d^3*e^3*a^2*(1/2*cosh(f*x+e)*sinh(f*x+e)-1/
2*f*x-1/2*e)+3/2/f*c^2*d*a^2*(f*x+e)^2+1/f^2*c*d^2*a^2*(f*x+e)^3-3/f^2*c*d^2*a^2*(1/2*(f*x+e)^2*cosh(f*x+e)*si
nh(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)-3/f*c^2*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*I/f^3*d^3*a^2*((f*x+e)^3*cosh(f*x+e)-3*(f
*x+e)^2*sinh(f*x+e)+6*(f*x+e)*cosh(f*x+e)-6*sinh(f*x+e))+3/f^3*d^3*e*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)-3/f^3*d^3*e^2*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^3*d^3*e^3*a^2*(f*x+e)-1/f^3*d^3*a^2*(1/2*(f
*x+e)^3*cosh(f*x+e)*sinh(f*x+e)-1/8*(f*x+e)^4-3/4*(f*x+e)^2*cosh(f*x+e)^2+3/4*(f*x+e)*cosh(f*x+e)*sinh(f*x+e)+
3/8*(f*x+e)^2-3/8*cosh(f*x+e)^2)+1/4/f^3*d^3*a^2*(f*x+e)^4+2*I*c^3*a^2*cosh(f*x+e)+6*I/f^2*c*d^2*e^2*a^2*cosh(
f*x+e)-12*I/f^2*c*d^2*e*a^2*((f*x+e)*cosh(f*x+e)-sinh(f*x+e))-6*I/f*c^2*d*e*a^2*cosh(f*x+e)+6*I/f*c^2*d*a^2*((
f*x+e)*cosh(f*x+e)-sinh(f*x+e))-2*I/f^3*d^3*e^3*a^2*cosh(f*x+e)+6*I/f^3*d^3*e^2*a^2*((f*x+e)*cosh(f*x+e)-sinh(
f*x+e))+3/f^2*c*d^2*e^2*a^2*(f*x+e)-3/f*c^2*d*e*a^2*(f*x+e)+3/f*c^2*d*e*a^2*(1/2*cosh(f*x+e)*sinh(f*x+e)-1/2*f
*x-1/2*e)+6/f^2*c*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)-6*I/f^3*d^3*
e*a^2*((f*x+e)^2*cosh(f*x+e)-2*(f*x+e)*sinh(f*x+e)+2*cosh(f*x+e))-3/f^2*c*d^2*e^2*a^2*(1/2*cosh(f*x+e)*sinh(f*
x+e)-1/2*f*x-1/2*e)-3/f^2*c*d^2*e*a^2*(f*x+e)^2+6*I/f^2*c*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))-c^3*a^2*(1/2*cosh(f*x+e)*sinh(f*x+e)-1/2*f*x-1/2*e)+c^3*a^2*(f*x+e))

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Maxima [B]  time = 1.15161, size = 709, normalized size = 2.89 \begin{align*} \frac{1}{4} \, a^{2} d^{3} x^{4} + a^{2} c d^{2} x^{3} + \frac{3}{2} \, a^{2} c^{2} d x^{2} + \frac{3}{16} \,{\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^{2} d + \frac{1}{16} \,{\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} c d^{2} + \frac{1}{32} \,{\left (4 \, x^{4} - \frac{{\left (4 \, f^{3} x^{3} e^{\left (2 \, e\right )} - 6 \, f^{2} x^{2} e^{\left (2 \, e\right )} + 6 \, f x e^{\left (2 \, e\right )} - 3 \, e^{\left (2 \, e\right )}\right )} e^{\left (2 \, f x\right )}}{f^{4}} + \frac{{\left (4 \, f^{3} x^{3} + 6 \, f^{2} x^{2} + 6 \, f x + 3\right )} e^{\left (-2 \, f x - 2 \, e\right )}}{f^{4}}\right )} a^{2} d^{3} + \frac{1}{8} \, a^{2} c^{3}{\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^{3} x + 3 i \, a^{2} c^{2} 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 )} + 3 i \, a^{2} c 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 )} + i \, a^{2} d^{3}{\left (\frac{{\left (f^{3} x^{3} e^{e} - 3 \, f^{2} x^{2} e^{e} + 6 \, f x e^{e} - 6 \, e^{e}\right )} e^{\left (f x\right )}}{f^{4}} + \frac{{\left (f^{3} x^{3} + 3 \, f^{2} x^{2} + 6 \, f x + 6\right )} e^{\left (-f x - e\right )}}{f^{4}}\right )} + \frac{2 i \, a^{2} c^{3} \cosh \left (f x + e\right )}{f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*a^2*d^3*x^4 + a^2*c*d^2*x^3 + 3/2*a^2*c^2*d*x^2 + 3/16*(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^2*d + 1/16*(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*c*d^2 + 1/32*(4*x^4 - (4*f^3*x^3*e^(2*e) - 6
*f^2*x^2*e^(2*e) + 6*f*x*e^(2*e) - 3*e^(2*e))*e^(2*f*x)/f^4 + (4*f^3*x^3 + 6*f^2*x^2 + 6*f*x + 3)*e^(-2*f*x -
2*e)/f^4)*a^2*d^3 + 1/8*a^2*c^3*(4*x - e^(2*f*x + 2*e)/f + e^(-2*f*x - 2*e)/f) + a^2*c^3*x + 3*I*a^2*c^2*d*((f
*x*e^e - e^e)*e^(f*x)/f^2 + (f*x + 1)*e^(-f*x - e)/f^2) + 3*I*a^2*c*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) + I*a^2*d^3*((f^3*x^3*e^e - 3*f^2*x^2*e^e + 6*f*x*e^e - 6*e
^e)*e^(f*x)/f^4 + (f^3*x^3 + 3*f^2*x^2 + 6*f*x + 6)*e^(-f*x - e)/f^4) + 2*I*a^2*c^3*cosh(f*x + e)/f

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Fricas [B]  time = 3.27347, size = 1283, normalized size = 5.24 \begin{align*} \frac{{\left (4 \, a^{2} d^{3} f^{3} x^{3} + 4 \, a^{2} c^{3} f^{3} + 6 \, a^{2} c^{2} d f^{2} + 6 \, a^{2} c d^{2} f + 3 \, a^{2} d^{3} + 6 \,{\left (2 \, a^{2} c d^{2} f^{3} + a^{2} d^{3} f^{2}\right )} x^{2} + 6 \,{\left (2 \, a^{2} c^{2} d f^{3} + 2 \, a^{2} c d^{2} f^{2} + a^{2} d^{3} f\right )} x -{\left (4 \, a^{2} d^{3} f^{3} x^{3} + 4 \, a^{2} c^{3} f^{3} - 6 \, a^{2} c^{2} d f^{2} + 6 \, a^{2} c d^{2} f - 3 \, a^{2} d^{3} + 6 \,{\left (2 \, a^{2} c d^{2} f^{3} - a^{2} d^{3} f^{2}\right )} x^{2} + 6 \,{\left (2 \, a^{2} c^{2} d f^{3} - 2 \, a^{2} c d^{2} f^{2} + a^{2} d^{3} f\right )} x\right )} e^{\left (4 \, f x + 4 \, e\right )} +{\left (32 i \, a^{2} d^{3} f^{3} x^{3} + 32 i \, a^{2} c^{3} f^{3} - 96 i \, a^{2} c^{2} d f^{2} + 192 i \, a^{2} c d^{2} f - 192 i \, a^{2} d^{3} +{\left (96 i \, a^{2} c d^{2} f^{3} - 96 i \, a^{2} d^{3} f^{2}\right )} x^{2} +{\left (96 i \, a^{2} c^{2} d f^{3} - 192 i \, a^{2} c d^{2} f^{2} + 192 i \, a^{2} d^{3} f\right )} x\right )} e^{\left (3 \, f x + 3 \, e\right )} + 12 \,{\left (a^{2} d^{3} f^{4} x^{4} + 4 \, a^{2} c d^{2} f^{4} x^{3} + 6 \, a^{2} c^{2} d f^{4} x^{2} + 4 \, a^{2} c^{3} f^{4} x\right )} e^{\left (2 \, f x + 2 \, e\right )} +{\left (32 i \, a^{2} d^{3} f^{3} x^{3} + 32 i \, a^{2} c^{3} f^{3} + 96 i \, a^{2} c^{2} d f^{2} + 192 i \, a^{2} c d^{2} f + 192 i \, a^{2} d^{3} +{\left (96 i \, a^{2} c d^{2} f^{3} + 96 i \, a^{2} d^{3} f^{2}\right )} x^{2} +{\left (96 i \, a^{2} c^{2} d f^{3} + 192 i \, a^{2} c d^{2} f^{2} + 192 i \, a^{2} d^{3} f\right )} x\right )} e^{\left (f x + e\right )}\right )} e^{\left (-2 \, f x - 2 \, e\right )}}{32 \, f^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/32*(4*a^2*d^3*f^3*x^3 + 4*a^2*c^3*f^3 + 6*a^2*c^2*d*f^2 + 6*a^2*c*d^2*f + 3*a^2*d^3 + 6*(2*a^2*c*d^2*f^3 + a
^2*d^3*f^2)*x^2 + 6*(2*a^2*c^2*d*f^3 + 2*a^2*c*d^2*f^2 + a^2*d^3*f)*x - (4*a^2*d^3*f^3*x^3 + 4*a^2*c^3*f^3 - 6
*a^2*c^2*d*f^2 + 6*a^2*c*d^2*f - 3*a^2*d^3 + 6*(2*a^2*c*d^2*f^3 - a^2*d^3*f^2)*x^2 + 6*(2*a^2*c^2*d*f^3 - 2*a^
2*c*d^2*f^2 + a^2*d^3*f)*x)*e^(4*f*x + 4*e) + (32*I*a^2*d^3*f^3*x^3 + 32*I*a^2*c^3*f^3 - 96*I*a^2*c^2*d*f^2 +
192*I*a^2*c*d^2*f - 192*I*a^2*d^3 + (96*I*a^2*c*d^2*f^3 - 96*I*a^2*d^3*f^2)*x^2 + (96*I*a^2*c^2*d*f^3 - 192*I*
a^2*c*d^2*f^2 + 192*I*a^2*d^3*f)*x)*e^(3*f*x + 3*e) + 12*(a^2*d^3*f^4*x^4 + 4*a^2*c*d^2*f^4*x^3 + 6*a^2*c^2*d*
f^4*x^2 + 4*a^2*c^3*f^4*x)*e^(2*f*x + 2*e) + (32*I*a^2*d^3*f^3*x^3 + 32*I*a^2*c^3*f^3 + 96*I*a^2*c^2*d*f^2 + 1
92*I*a^2*c*d^2*f + 192*I*a^2*d^3 + (96*I*a^2*c*d^2*f^3 + 96*I*a^2*d^3*f^2)*x^2 + (96*I*a^2*c^2*d*f^3 + 192*I*a
^2*c*d^2*f^2 + 192*I*a^2*d^3*f)*x)*e^(f*x + e))*e^(-2*f*x - 2*e)/f^4

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Sympy [A]  time = 4.62814, size = 1153, normalized size = 4.71 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*a**2*c**3*x/2 + 9*a**2*c**2*d*x**2/4 + 3*a**2*c*d**2*x**3/2 + 3*a**2*d**3*x**4/8 + Piecewise((((128*a**2*c**
3*f**27*exp(4*e) + 384*a**2*c**2*d*f**27*x*exp(4*e) + 192*a**2*c**2*d*f**26*exp(4*e) + 384*a**2*c*d**2*f**27*x
**2*exp(4*e) + 384*a**2*c*d**2*f**26*x*exp(4*e) + 192*a**2*c*d**2*f**25*exp(4*e) + 128*a**2*d**3*f**27*x**3*ex
p(4*e) + 192*a**2*d**3*f**26*x**2*exp(4*e) + 192*a**2*d**3*f**25*x*exp(4*e) + 96*a**2*d**3*f**24*exp(4*e))*exp
(-2*f*x) + (-128*a**2*c**3*f**27*exp(8*e) - 384*a**2*c**2*d*f**27*x*exp(8*e) + 192*a**2*c**2*d*f**26*exp(8*e)
- 384*a**2*c*d**2*f**27*x**2*exp(8*e) + 384*a**2*c*d**2*f**26*x*exp(8*e) - 192*a**2*c*d**2*f**25*exp(8*e) - 12
8*a**2*d**3*f**27*x**3*exp(8*e) + 192*a**2*d**3*f**26*x**2*exp(8*e) - 192*a**2*d**3*f**25*x*exp(8*e) + 96*a**2
*d**3*f**24*exp(8*e))*exp(2*f*x) + (1024*I*a**2*c**3*f**27*exp(5*e) + 3072*I*a**2*c**2*d*f**27*x*exp(5*e) + 30
72*I*a**2*c**2*d*f**26*exp(5*e) + 3072*I*a**2*c*d**2*f**27*x**2*exp(5*e) + 6144*I*a**2*c*d**2*f**26*x*exp(5*e)
+ 6144*I*a**2*c*d**2*f**25*exp(5*e) + 1024*I*a**2*d**3*f**27*x**3*exp(5*e) + 3072*I*a**2*d**3*f**26*x**2*exp(
5*e) + 6144*I*a**2*d**3*f**25*x*exp(5*e) + 6144*I*a**2*d**3*f**24*exp(5*e))*exp(-f*x) + (1024*I*a**2*c**3*f**2
7*exp(7*e) + 3072*I*a**2*c**2*d*f**27*x*exp(7*e) - 3072*I*a**2*c**2*d*f**26*exp(7*e) + 3072*I*a**2*c*d**2*f**2
7*x**2*exp(7*e) - 6144*I*a**2*c*d**2*f**26*x*exp(7*e) + 6144*I*a**2*c*d**2*f**25*exp(7*e) + 1024*I*a**2*d**3*f
**27*x**3*exp(7*e) - 3072*I*a**2*d**3*f**26*x**2*exp(7*e) + 6144*I*a**2*d**3*f**25*x*exp(7*e) - 6144*I*a**2*d*
*3*f**24*exp(7*e))*exp(f*x))*exp(-6*e)/(1024*f**28), Ne(1024*f**28*exp(6*e), 0)), (x**4*(-a**2*d**3*exp(4*e) +
4*I*a**2*d**3*exp(3*e) - 4*I*a**2*d**3*exp(e) - a**2*d**3)*exp(-2*e)/16 + x**3*(-a**2*c*d**2*exp(4*e) + 4*I*a
**2*c*d**2*exp(3*e) - 4*I*a**2*c*d**2*exp(e) - a**2*c*d**2)*exp(-2*e)/4 + x**2*(-3*a**2*c**2*d*exp(4*e) + 12*I
*a**2*c**2*d*exp(3*e) - 12*I*a**2*c**2*d*exp(e) - 3*a**2*c**2*d)*exp(-2*e)/8 + x*(-a**2*c**3*exp(4*e) + 4*I*a*
*2*c**3*exp(3*e) - 4*I*a**2*c**3*exp(e) - a**2*c**3)*exp(-2*e)/4, True))

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Giac [B]  time = 1.34564, size = 788, normalized size = 3.22 \begin{align*} \frac{3}{8} \, a^{2} d^{3} x^{4} + \frac{3}{2} \, a^{2} c d^{2} x^{3} + \frac{9}{4} \, a^{2} c^{2} d x^{2} + \frac{3}{2} \, a^{2} c^{3} x - \frac{{\left (4 \, a^{2} d^{3} f^{3} x^{3} + 12 \, a^{2} c d^{2} f^{3} x^{2} + 12 \, a^{2} c^{2} d f^{3} x - 6 \, a^{2} d^{3} f^{2} x^{2} + 4 \, a^{2} c^{3} f^{3} - 12 \, a^{2} c d^{2} f^{2} x - 6 \, a^{2} c^{2} d f^{2} + 6 \, a^{2} d^{3} f x + 6 \, a^{2} c d^{2} f - 3 \, a^{2} d^{3}\right )} e^{\left (2 \, f x + 2 \, e\right )}}{32 \, f^{4}} + \frac{{\left (i \, a^{2} d^{3} f^{3} x^{3} + 3 i \, a^{2} c d^{2} f^{3} x^{2} + 3 i \, a^{2} c^{2} d f^{3} x - 3 i \, a^{2} d^{3} f^{2} x^{2} + i \, a^{2} c^{3} f^{3} - 6 i \, a^{2} c d^{2} f^{2} x - 3 i \, a^{2} c^{2} d f^{2} + 6 i \, a^{2} d^{3} f x + 6 i \, a^{2} c d^{2} f - 6 i \, a^{2} d^{3}\right )} e^{\left (f x + e\right )}}{f^{4}} + \frac{{\left (i \, a^{2} d^{3} f^{3} x^{3} + 3 i \, a^{2} c d^{2} f^{3} x^{2} + 3 i \, a^{2} c^{2} d f^{3} x + 3 i \, a^{2} d^{3} f^{2} x^{2} + i \, a^{2} c^{3} f^{3} + 6 i \, a^{2} c d^{2} f^{2} x + 3 i \, a^{2} c^{2} d f^{2} + 6 i \, a^{2} d^{3} f x + 6 i \, a^{2} c d^{2} f + 6 i \, a^{2} d^{3}\right )} e^{\left (-f x - e\right )}}{f^{4}} + \frac{{\left (4 \, a^{2} d^{3} f^{3} x^{3} + 12 \, a^{2} c d^{2} f^{3} x^{2} + 12 \, a^{2} c^{2} d f^{3} x + 6 \, a^{2} d^{3} f^{2} x^{2} + 4 \, a^{2} c^{3} f^{3} + 12 \, a^{2} c d^{2} f^{2} x + 6 \, a^{2} c^{2} d f^{2} + 6 \, a^{2} d^{3} f x + 6 \, a^{2} c d^{2} f + 3 \, a^{2} d^{3}\right )} e^{\left (-2 \, f x - 2 \, e\right )}}{32 \, f^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/8*a^2*d^3*x^4 + 3/2*a^2*c*d^2*x^3 + 9/4*a^2*c^2*d*x^2 + 3/2*a^2*c^3*x - 1/32*(4*a^2*d^3*f^3*x^3 + 12*a^2*c*d
^2*f^3*x^2 + 12*a^2*c^2*d*f^3*x - 6*a^2*d^3*f^2*x^2 + 4*a^2*c^3*f^3 - 12*a^2*c*d^2*f^2*x - 6*a^2*c^2*d*f^2 + 6
*a^2*d^3*f*x + 6*a^2*c*d^2*f - 3*a^2*d^3)*e^(2*f*x + 2*e)/f^4 + (I*a^2*d^3*f^3*x^3 + 3*I*a^2*c*d^2*f^3*x^2 + 3
*I*a^2*c^2*d*f^3*x - 3*I*a^2*d^3*f^2*x^2 + I*a^2*c^3*f^3 - 6*I*a^2*c*d^2*f^2*x - 3*I*a^2*c^2*d*f^2 + 6*I*a^2*d
^3*f*x + 6*I*a^2*c*d^2*f - 6*I*a^2*d^3)*e^(f*x + e)/f^4 + (I*a^2*d^3*f^3*x^3 + 3*I*a^2*c*d^2*f^3*x^2 + 3*I*a^2
*c^2*d*f^3*x + 3*I*a^2*d^3*f^2*x^2 + I*a^2*c^3*f^3 + 6*I*a^2*c*d^2*f^2*x + 3*I*a^2*c^2*d*f^2 + 6*I*a^2*d^3*f*x
+ 6*I*a^2*c*d^2*f + 6*I*a^2*d^3)*e^(-f*x - e)/f^4 + 1/32*(4*a^2*d^3*f^3*x^3 + 12*a^2*c*d^2*f^3*x^2 + 12*a^2*c
^2*d*f^3*x + 6*a^2*d^3*f^2*x^2 + 4*a^2*c^3*f^3 + 12*a^2*c*d^2*f^2*x + 6*a^2*c^2*d*f^2 + 6*a^2*d^3*f*x + 6*a^2*
c*d^2*f + 3*a^2*d^3)*e^(-2*f*x - 2*e)/f^4