3.2.3 \(\int (c+d x)^2 (a+i a \sinh (e+f x))^2 \, dx\) [103]

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

[Out]

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

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Rubi [A]
time = 0.14, antiderivative size = 174, normalized size of antiderivative = 1.00, 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 = {3398, 3377, 2718, 3392, 32, 2715, 8} \begin {gather*} \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} \end {gather*}

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 8

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

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 2715

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] && Integ
erQ[2*n]

Rule 2718

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

Rule 3377

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 3392

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 3398

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])

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*}

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Mathematica [A]
time = 0.40, size = 189, normalized size = 1.09 \begin {gather*} \frac {a^2 \left (12 c^2 f^3 x+12 c d f^3 x^2+4 d^2 f^3 x^3+16 i \left (c^2 f^2+2 c d f^2 x+d^2 \left (2+f^2 x^2\right )\right ) \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))\right )}{8 f^3} \end {gather*}

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] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 549 vs. \(2 (161 ) = 322\).
time = 0.63, size = 550, normalized size = 3.16

method	result	size
risch	\(\frac {a^{2} d^{2} x^{3}}{2}+\frac {3 a^{2} c d \,x^{2}}{2}+\frac {3 a^{2} c^{2} x}{2}+\frac {a^{2} c^{3}}{2 d}-\frac {a^{2} \left (2 d^{2} x^{2} f^{2}+4 c d \,f^{2} x +2 c^{2} f^{2}-2 d^{2} f x -2 c d f +d^{2}\right ) {\mathrm e}^{2 f x +2 e}}{16 f^{3}}+\frac {i a^{2} \left (d^{2} x^{2} f^{2}+2 c d \,f^{2} x +c^{2} f^{2}-2 d^{2} f x -2 c d f +2 d^{2}\right ) {\mathrm e}^{f x +e}}{f^{3}}+\frac {i a^{2} \left (d^{2} x^{2} f^{2}+2 c d \,f^{2} x +c^{2} f^{2}+2 d^{2} f x +2 c d f +2 d^{2}\right ) {\mathrm e}^{-f x -e}}{f^{3}}+\frac {a^{2} \left (2 d^{2} x^{2} f^{2}+4 c d \,f^{2} x +2 c^{2} f^{2}+2 d^{2} f x +2 c d f +d^{2}\right ) {\mathrm e}^{-2 f x -2 e}}{16 f^{3}}\)	\(282\)
derivativedivides	\(\frac {\frac {d^{2} a^{2} \left (f x +e \right )^{3}}{3 f^{2}}+2 i c^{2} a^{2} \cosh \left (f x +e \right )-\frac {d^{2} a^{2} \left (\frac {\left (f x +e \right )^{2} \cosh \left (f x +e \right ) \sinh \left (f x +e \right )}{2}-\frac {\left (f x +e \right )^{3}}{6}-\frac {\left (f x +e \right ) \left (\cosh ^{2}\left (f x +e \right )\right )}{2}+\frac {\cosh \left (f x +e \right ) \sinh \left (f x +e \right )}{4}+\frac {f x}{4}+\frac {e}{4}\right )}{f^{2}}-\frac {d^{2} e \,a^{2} \left (f x +e \right )^{2}}{f^{2}}+\frac {2 i d^{2} a^{2} \left (\left (f x +e \right )^{2} \cosh \left (f x +e \right )-2 \left (f x +e \right ) \sinh \left (f x +e \right )+2 \cosh \left (f x +e \right )\right )}{f^{2}}+\frac {2 d^{2} e \,a^{2} \left (\frac {\left (f x +e \right ) \cosh \left (f x +e \right ) \sinh \left (f x +e \right )}{2}-\frac {\left (f x +e \right )^{2}}{4}-\frac {\left (\cosh ^{2}\left (f x +e \right )\right )}{4}\right )}{f^{2}}+\frac {d c \,a^{2} \left (f x +e \right )^{2}}{f}-\frac {4 i d e c \,a^{2} \cosh \left (f x +e \right )}{f}-\frac {2 d c \,a^{2} \left (\frac {\left (f x +e \right ) \cosh \left (f x +e \right ) \sinh \left (f x +e \right )}{2}-\frac {\left (f x +e \right )^{2}}{4}-\frac {\left (\cosh ^{2}\left (f x +e \right )\right )}{4}\right )}{f}+\frac {d^{2} e^{2} a^{2} \left (f x +e \right )}{f^{2}}+\frac {4 i d c \,a^{2} \left (\left (f x +e \right ) \cosh \left (f x +e \right )-\sinh \left (f x +e \right )\right )}{f}-\frac {d^{2} e^{2} a^{2} \left (\frac {\cosh \left (f x +e \right ) \sinh \left (f x +e \right )}{2}-\frac {f x}{2}-\frac {e}{2}\right )}{f^{2}}-\frac {2 d e c \,a^{2} \left (f x +e \right )}{f}-\frac {4 i d^{2} e \,a^{2} \left (\left (f x +e \right ) \cosh \left (f x +e \right )-\sinh \left (f x +e \right )\right )}{f^{2}}+\frac {2 d e c \,a^{2} \left (\frac {\cosh \left (f x +e \right ) \sinh \left (f x +e \right )}{2}-\frac {f x}{2}-\frac {e}{2}\right )}{f}+a^{2} c^{2} \left (f x +e \right )+\frac {2 i d^{2} e^{2} a^{2} \cosh \left (f x +e \right )}{f^{2}}-a^{2} c^{2} \left (\frac {\cosh \left (f x +e \right ) \sinh \left (f x +e \right )}{2}-\frac {f x}{2}-\frac {e}{2}\right )}{f}\)	\(550\)
default	\(\frac {\frac {d^{2} a^{2} \left (f x +e \right )^{3}}{3 f^{2}}+2 i c^{2} a^{2} \cosh \left (f x +e \right )-\frac {d^{2} a^{2} \left (\frac {\left (f x +e \right )^{2} \cosh \left (f x +e \right ) \sinh \left (f x +e \right )}{2}-\frac {\left (f x +e \right )^{3}}{6}-\frac {\left (f x +e \right ) \left (\cosh ^{2}\left (f x +e \right )\right )}{2}+\frac {\cosh \left (f x +e \right ) \sinh \left (f x +e \right )}{4}+\frac {f x}{4}+\frac {e}{4}\right )}{f^{2}}-\frac {d^{2} e \,a^{2} \left (f x +e \right )^{2}}{f^{2}}+\frac {2 i d^{2} a^{2} \left (\left (f x +e \right )^{2} \cosh \left (f x +e \right )-2 \left (f x +e \right ) \sinh \left (f x +e \right )+2 \cosh \left (f x +e \right )\right )}{f^{2}}+\frac {2 d^{2} e \,a^{2} \left (\frac {\left (f x +e \right ) \cosh \left (f x +e \right ) \sinh \left (f x +e \right )}{2}-\frac {\left (f x +e \right )^{2}}{4}-\frac {\left (\cosh ^{2}\left (f x +e \right )\right )}{4}\right )}{f^{2}}+\frac {d c \,a^{2} \left (f x +e \right )^{2}}{f}-\frac {4 i d e c \,a^{2} \cosh \left (f x +e \right )}{f}-\frac {2 d c \,a^{2} \left (\frac {\left (f x +e \right ) \cosh \left (f x +e \right ) \sinh \left (f x +e \right )}{2}-\frac {\left (f x +e \right )^{2}}{4}-\frac {\left (\cosh ^{2}\left (f x +e \right )\right )}{4}\right )}{f}+\frac {d^{2} e^{2} a^{2} \left (f x +e \right )}{f^{2}}+\frac {4 i d c \,a^{2} \left (\left (f x +e \right ) \cosh \left (f x +e \right )-\sinh \left (f x +e \right )\right )}{f}-\frac {d^{2} e^{2} a^{2} \left (\frac {\cosh \left (f x +e \right ) \sinh \left (f x +e \right )}{2}-\frac {f x}{2}-\frac {e}{2}\right )}{f^{2}}-\frac {2 d e c \,a^{2} \left (f x +e \right )}{f}-\frac {4 i d^{2} e \,a^{2} \left (\left (f x +e \right ) \cosh \left (f x +e \right )-\sinh \left (f x +e \right )\right )}{f^{2}}+\frac {2 d e c \,a^{2} \left (\frac {\cosh \left (f x +e \right ) \sinh \left (f x +e \right )}{2}-\frac {f x}{2}-\frac {e}{2}\right )}{f}+a^{2} c^{2} \left (f x +e \right )+\frac {2 i d^{2} e^{2} a^{2} \cosh \left (f x +e \right )}{f^{2}}-a^{2} c^{2} \left (\frac {\cosh \left (f x +e \right ) \sinh \left (f x +e \right )}{2}-\frac {f x}{2}-\frac {e}{2}\right )}{f}\)	\(550\)

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,method=_RETURNVERBOSE)

[Out]

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

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Maxima [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 343 vs. \(2 (166) = 332\).
time = 0.28, size = 343, normalized size = 1.97 \begin {gather*} \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 {gather*}

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] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 356 vs. \(2 (166) = 332\).
time = 0.40, size = 356, normalized size = 2.05 \begin {gather*} \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 )} - 16 \, {\left (-i \, a^{2} d^{2} f^{2} x^{2} - i \, a^{2} c^{2} f^{2} + 2 i \, a^{2} c d f - 2 i \, a^{2} d^{2} + 2 \, {\left (-i \, a^{2} c d f^{2} + 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 )} - 16 \, {\left (-i \, a^{2} d^{2} f^{2} x^{2} - i \, a^{2} c^{2} f^{2} - 2 i \, a^{2} c d f - 2 i \, a^{2} d^{2} + 2 \, {\left (-i \, a^{2} c d f^{2} - 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 {gather*}

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 - I*a^2*c^2*f^2 + 2*I*a^2*c*d*f - 2*I*a^2*d^2 + 2*(-I*a^2*c*d*f^2 + 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 - I*
a^2*c^2*f^2 - 2*I*a^2*c*d*f - 2*I*a^2*d^2 + 2*(-I*a^2*c*d*f^2 - 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 = 0.57, size = 694, normalized size = 3.99 \begin {gather*} \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^{11} e^{e} + 64 a^{2} c d f^{11} x e^{e} + 32 a^{2} c d f^{10} e^{e} + 32 a^{2} d^{2} f^{11} x^{2} e^{e} + 32 a^{2} d^{2} f^{10} x e^{e} + 16 a^{2} d^{2} f^{9} e^{e}\right ) e^{- 2 f x} + \left (- 32 a^{2} c^{2} f^{11} e^{5 e} - 64 a^{2} c d f^{11} x e^{5 e} + 32 a^{2} c d f^{10} e^{5 e} - 32 a^{2} d^{2} f^{11} x^{2} e^{5 e} + 32 a^{2} d^{2} f^{10} x e^{5 e} - 16 a^{2} d^{2} f^{9} e^{5 e}\right ) e^{2 f x} + \left (256 i a^{2} c^{2} f^{11} e^{2 e} + 512 i a^{2} c d f^{11} x e^{2 e} + 512 i a^{2} c d f^{10} e^{2 e} + 256 i a^{2} d^{2} f^{11} x^{2} e^{2 e} + 512 i a^{2} d^{2} f^{10} x e^{2 e} + 512 i a^{2} d^{2} f^{9} e^{2 e}\right ) e^{- f x} + \left (256 i a^{2} c^{2} f^{11} e^{4 e} + 512 i a^{2} c d f^{11} x e^{4 e} - 512 i a^{2} c d f^{10} e^{4 e} + 256 i a^{2} d^{2} f^{11} x^{2} e^{4 e} - 512 i a^{2} d^{2} f^{10} x e^{4 e} + 512 i a^{2} d^{2} f^{9} e^{4 e}\right ) e^{f x}\right ) e^{- 3 e}}{256 f^{12}} & \text {for}\: f^{12} e^{3 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 {gather*}

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**11*exp(e) + 64*a**2*c*d*
f**11*x*exp(e) + 32*a**2*c*d*f**10*exp(e) + 32*a**2*d**2*f**11*x**2*exp(e) + 32*a**2*d**2*f**10*x*exp(e) + 16*
a**2*d**2*f**9*exp(e))*exp(-2*f*x) + (-32*a**2*c**2*f**11*exp(5*e) - 64*a**2*c*d*f**11*x*exp(5*e) + 32*a**2*c*
d*f**10*exp(5*e) - 32*a**2*d**2*f**11*x**2*exp(5*e) + 32*a**2*d**2*f**10*x*exp(5*e) - 16*a**2*d**2*f**9*exp(5*
e))*exp(2*f*x) + (256*I*a**2*c**2*f**11*exp(2*e) + 512*I*a**2*c*d*f**11*x*exp(2*e) + 512*I*a**2*c*d*f**10*exp(
2*e) + 256*I*a**2*d**2*f**11*x**2*exp(2*e) + 512*I*a**2*d**2*f**10*x*exp(2*e) + 512*I*a**2*d**2*f**9*exp(2*e))
*exp(-f*x) + (256*I*a**2*c**2*f**11*exp(4*e) + 512*I*a**2*c*d*f**11*x*exp(4*e) - 512*I*a**2*c*d*f**10*exp(4*e)
+ 256*I*a**2*d**2*f**11*x**2*exp(4*e) - 512*I*a**2*d**2*f**10*x*exp(4*e) + 512*I*a**2*d**2*f**9*exp(4*e))*exp
(f*x))*exp(-3*e)/(256*f**12), Ne(f**12*exp(3*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] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 333 vs. \(2 (158) = 316\).
time = 0.47, size = 333, normalized size = 1.91 \begin {gather*} \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 {gather*}

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

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Mupad [B]
time = 0.68, size = 217, normalized size = 1.25 \begin {gather*} \frac {a^2\,\left (12\,c^2\,x+12\,c\,d\,x^2+4\,d^2\,x^3\right )}{8}+\frac {\frac {a^2\,\left (-d^2\,\mathrm {sinh}\left (2\,e+2\,f\,x\right )+d^2\,\mathrm {cosh}\left (e+f\,x\right )\,32{}\mathrm {i}\right )}{8}+\frac {a^2\,f^2\,\left (-2\,c^2\,\mathrm {sinh}\left (2\,e+2\,f\,x\right )-2\,d^2\,x^2\,\mathrm {sinh}\left (2\,e+2\,f\,x\right )-4\,c\,d\,x\,\mathrm {sinh}\left (2\,e+2\,f\,x\right )+c^2\,\mathrm {cosh}\left (e+f\,x\right )\,16{}\mathrm {i}+d^2\,x^2\,\mathrm {cosh}\left (e+f\,x\right )\,16{}\mathrm {i}+c\,d\,x\,\mathrm {cosh}\left (e+f\,x\right )\,32{}\mathrm {i}\right )}{8}-\frac {a^2\,f\,\left (-2\,d^2\,x\,\mathrm {cosh}\left (2\,e+2\,f\,x\right )-2\,c\,d\,\mathrm {cosh}\left (2\,e+2\,f\,x\right )+d^2\,x\,\mathrm {sinh}\left (e+f\,x\right )\,32{}\mathrm {i}+c\,d\,\mathrm {sinh}\left (e+f\,x\right )\,32{}\mathrm {i}\right )}{8}}{f^3} \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]

(a^2*(12*c^2*x + 4*d^2*x^3 + 12*c*d*x^2))/8 + ((a^2*(d^2*cosh(e + f*x)*32i - d^2*sinh(2*e + 2*f*x)))/8 + (a^2*
f^2*(c^2*cosh(e + f*x)*16i - 2*c^2*sinh(2*e + 2*f*x) + d^2*x^2*cosh(e + f*x)*16i - 2*d^2*x^2*sinh(2*e + 2*f*x)
+ c*d*x*cosh(e + f*x)*32i - 4*c*d*x*sinh(2*e + 2*f*x)))/8 - (a^2*f*(d^2*x*sinh(e + f*x)*32i - 2*d^2*x*cosh(2*
e + 2*f*x) + c*d*sinh(e + f*x)*32i - 2*c*d*cosh(2*e + 2*f*x)))/8)/f^3

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