3.2.2 \(\int (c+d x)^3 (a+i a \sinh (e+f x))^2 \, dx\) [102]

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

[Out]

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

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Rubi [A]
time = 0.20, antiderivative size = 245, normalized size of antiderivative = 1.00, 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 = {3398, 3377, 2717, 3392, 32, 3391} \begin {gather*} \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} \end {gather*}

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 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 2717

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[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 3391

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]

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)^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*}

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Mathematica [A]
time = 0.86, size = 220, normalized size = 0.90 \begin {gather*} \frac {a^2 \left (6 f^4 x \left (4 c^3+6 c^2 d x+4 c d^2 x^2+d^3 x^3\right )+32 i f (c+d x) \left (c^2 f^2+2 c d f^2 x+d^2 \left (6+f^2 x^2\right )\right ) \cosh (e+f x)+3 d \left (2 c^2 f^2+4 c d f^2 x+d^2 \left (1+2 f^2 x^2\right )\right ) \cosh (2 (e+f x))-96 i d \left (c^2 f^2+2 c d f^2 x+d^2 \left (2+f^2 x^2\right )\right ) \sinh (e+f x)-2 f (c+d x) \left (2 c^2 f^2+4 c d f^2 x+d^2 \left (3+2 f^2 x^2\right )\right ) \sinh (2 (e+f x))\right )}{16 f^4} \end {gather*}

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] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1081 vs. \(2 (227 ) = 454\).
time = 0.64, size = 1082, normalized size = 4.42

method	result	size
risch	\(\frac {3 a^{2} d^{3} x^{4}}{8}+\frac {3 a^{2} c \,d^{2} x^{3}}{2}+\frac {9 a^{2} c^{2} d \,x^{2}}{4}+\frac {3 c^{3} a^{2} x}{2}+\frac {3 a^{2} c^{4}}{8 d}-\frac {a^{2} \left (4 d^{3} x^{3} f^{3}+12 c \,d^{2} f^{3} x^{2}+12 c^{2} d \,f^{3} x -6 d^{3} f^{2} x^{2}+4 c^{3} f^{3}-12 c \,d^{2} f^{2} x -6 c^{2} d \,f^{2}+6 d^{3} f x +6 c \,d^{2} f -3 d^{3}\right ) {\mathrm e}^{2 f x +2 e}}{32 f^{4}}+\frac {i a^{2} \left (d^{3} x^{3} f^{3}+3 c \,d^{2} f^{3} x^{2}+3 c^{2} d \,f^{3} x -3 d^{3} f^{2} x^{2}+c^{3} f^{3}-6 c \,d^{2} f^{2} x -3 c^{2} d \,f^{2}+6 d^{3} f x +6 c \,d^{2} f -6 d^{3}\right ) {\mathrm e}^{f x +e}}{f^{4}}+\frac {i a^{2} \left (d^{3} x^{3} f^{3}+3 c \,d^{2} f^{3} x^{2}+3 c^{2} d \,f^{3} x +3 d^{3} f^{2} x^{2}+c^{3} f^{3}+6 c \,d^{2} f^{2} x +3 c^{2} d \,f^{2}+6 d^{3} f x +6 c \,d^{2} f +6 d^{3}\right ) {\mathrm e}^{-f x -e}}{f^{4}}+\frac {a^{2} \left (4 d^{3} x^{3} f^{3}+12 c \,d^{2} f^{3} x^{2}+12 c^{2} d \,f^{3} x +6 d^{3} f^{2} x^{2}+4 c^{3} f^{3}+12 c \,d^{2} f^{2} x +6 c^{2} d \,f^{2}+6 d^{3} f x +6 c \,d^{2} f +3 d^{3}\right ) {\mathrm e}^{-2 f x -2 e}}{32 f^{4}}\)	\(484\)
derivativedivides	\(\text {Expression too large to display}\)	\(1082\)
default	\(\text {Expression too large to display}\)	\(1082\)

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,method=_RETURNVERBOSE)

[Out]

1/f*(2*I*d^3/f^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))+d^2/f
^2*c*a^2*(f*x+e)^3-3*d/f*e*c^2*a^2*(f*x+e)-3*d^2/f^2*e^2*c*a^2*(1/2*cosh(f*x+e)*sinh(f*x+e)-1/2*f*x-1/2*e)+3*d
/f*e*c^2*a^2*(1/2*cosh(f*x+e)*sinh(f*x+e)-1/2*f*x-1/2*e)+6*d^2/f^2*e*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)-6*I*d^3/f^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
))+6*I*d^2/f^2*c*a^2*((f*x+e)^2*cosh(f*x+e)-2*(f*x+e)*sinh(f*x+e)+2*cosh(f*x+e))+6*I*d^3/f^3*e^2*a^2*((f*x+e)*
cosh(f*x+e)-sinh(f*x+e))+6*I*d/f*c^2*a^2*((f*x+e)*cosh(f*x+e)-sinh(f*x+e))-3*d^2/f^2*e*c*a^2*(f*x+e)^2-2*I*d^3
/f^3*e^3*a^2*cosh(f*x+e)+3*d^2/f^2*e^2*c*a^2*(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)+1/4*d^3/f^3*a^2*(f*x+e)^4-d^3/f^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)+2*I*c^3*a^2*cosh(f*x+
e)-3*d^2/f^2*c*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*d^3/f^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)-d^3/f^3*e*a^2*(f*x+e)^3-d^3/f^3*e^3*a^2*(f*x+e)+3/2
*d^3/f^3*e^2*a^2*(f*x+e)^2+3/2*d/f*c^2*a^2*(f*x+e)^2+d^3/f^3*e^3*a^2*(1/2*cosh(f*x+e)*sinh(f*x+e)-1/2*f*x-1/2*
e)-3*d^3/f^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)-3*d/f*c^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)-6*I*d/f*e*c^2*a^2*cosh(f*x+e)+6*I*d^2/f^2*e^2
*c*a^2*cosh(f*x+e)-12*I*d^2/f^2*e*c*a^2*((f*x+e)*cosh(f*x+e)-sinh(f*x+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. 552 vs. \(2 (233) = 466\).
time = 0.30, size = 552, normalized size = 2.25 \begin {gather*} \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 {gather*}

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] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 601 vs. \(2 (233) = 466\).
time = 0.49, size = 601, normalized size = 2.45 \begin {gather*} \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 )} - 32 \, {\left (-i \, a^{2} d^{3} f^{3} x^{3} - i \, a^{2} c^{3} f^{3} + 3 i \, a^{2} c^{2} d f^{2} - 6 i \, a^{2} c d^{2} f + 6 i \, a^{2} d^{3} + 3 \, {\left (-i \, a^{2} c d^{2} f^{3} + i \, a^{2} d^{3} f^{2}\right )} x^{2} + 3 \, {\left (-i \, a^{2} c^{2} d f^{3} + 2 i \, a^{2} c d^{2} f^{2} - 2 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 )} - 32 \, {\left (-i \, a^{2} d^{3} f^{3} x^{3} - i \, a^{2} c^{3} f^{3} - 3 i \, a^{2} c^{2} d f^{2} - 6 i \, a^{2} c d^{2} f - 6 i \, a^{2} d^{3} + 3 \, {\left (-i \, a^{2} c d^{2} f^{3} - i \, a^{2} d^{3} f^{2}\right )} x^{2} + 3 \, {\left (-i \, a^{2} c^{2} d f^{3} - 2 i \, a^{2} c d^{2} f^{2} - 2 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 {gather*}

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 - I*a^2*c^3*f^3 + 3*I*a^2*c^2*d*f^2 - 6*I
*a^2*c*d^2*f + 6*I*a^2*d^3 + 3*(-I*a^2*c*d^2*f^3 + I*a^2*d^3*f^2)*x^2 + 3*(-I*a^2*c^2*d*f^3 + 2*I*a^2*c*d^2*f^
2 - 2*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 - I*a^2*c^3*f^3 - 3*I*a^2*c^2*d*f^2 - 6*I*a^2*c*d^2*f -
6*I*a^2*d^3 + 3*(-I*a^2*c*d^2*f^3 - I*a^2*d^3*f^2)*x^2 + 3*(-I*a^2*c^2*d*f^3 - 2*I*a^2*c*d^2*f^2 - 2*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 = 0.77, size = 1134, normalized size = 4.63 \begin {gather*} \frac {3 a^{2} c^{3} x}{2} + \frac {9 a^{2} c^{2} d x^{2}}{4} + \frac {3 a^{2} c d^{2} x^{3}}{2} + \frac {3 a^{2} d^{3} x^{4}}{8} + \begin {cases} \frac {\left (\left (128 a^{2} c^{3} f^{15} e^{e} + 384 a^{2} c^{2} d f^{15} x e^{e} + 192 a^{2} c^{2} d f^{14} e^{e} + 384 a^{2} c d^{2} f^{15} x^{2} e^{e} + 384 a^{2} c d^{2} f^{14} x e^{e} + 192 a^{2} c d^{2} f^{13} e^{e} + 128 a^{2} d^{3} f^{15} x^{3} e^{e} + 192 a^{2} d^{3} f^{14} x^{2} e^{e} + 192 a^{2} d^{3} f^{13} x e^{e} + 96 a^{2} d^{3} f^{12} e^{e}\right ) e^{- 2 f x} + \left (- 128 a^{2} c^{3} f^{15} e^{5 e} - 384 a^{2} c^{2} d f^{15} x e^{5 e} + 192 a^{2} c^{2} d f^{14} e^{5 e} - 384 a^{2} c d^{2} f^{15} x^{2} e^{5 e} + 384 a^{2} c d^{2} f^{14} x e^{5 e} - 192 a^{2} c d^{2} f^{13} e^{5 e} - 128 a^{2} d^{3} f^{15} x^{3} e^{5 e} + 192 a^{2} d^{3} f^{14} x^{2} e^{5 e} - 192 a^{2} d^{3} f^{13} x e^{5 e} + 96 a^{2} d^{3} f^{12} e^{5 e}\right ) e^{2 f x} + \left (1024 i a^{2} c^{3} f^{15} e^{2 e} + 3072 i a^{2} c^{2} d f^{15} x e^{2 e} + 3072 i a^{2} c^{2} d f^{14} e^{2 e} + 3072 i a^{2} c d^{2} f^{15} x^{2} e^{2 e} + 6144 i a^{2} c d^{2} f^{14} x e^{2 e} + 6144 i a^{2} c d^{2} f^{13} e^{2 e} + 1024 i a^{2} d^{3} f^{15} x^{3} e^{2 e} + 3072 i a^{2} d^{3} f^{14} x^{2} e^{2 e} + 6144 i a^{2} d^{3} f^{13} x e^{2 e} + 6144 i a^{2} d^{3} f^{12} e^{2 e}\right ) e^{- f x} + \left (1024 i a^{2} c^{3} f^{15} e^{4 e} + 3072 i a^{2} c^{2} d f^{15} x e^{4 e} - 3072 i a^{2} c^{2} d f^{14} e^{4 e} + 3072 i a^{2} c d^{2} f^{15} x^{2} e^{4 e} - 6144 i a^{2} c d^{2} f^{14} x e^{4 e} + 6144 i a^{2} c d^{2} f^{13} e^{4 e} + 1024 i a^{2} d^{3} f^{15} x^{3} e^{4 e} - 3072 i a^{2} d^{3} f^{14} x^{2} e^{4 e} + 6144 i a^{2} d^{3} f^{13} x e^{4 e} - 6144 i a^{2} d^{3} f^{12} e^{4 e}\right ) e^{f x}\right ) e^{- 3 e}}{1024 f^{16}} & \text {for}\: f^{16} e^{3 e} \neq 0 \\\frac {x^{4} \left (- a^{2} d^{3} e^{4 e} + 4 i a^{2} d^{3} e^{3 e} - 4 i a^{2} d^{3} e^{e} - a^{2} d^{3}\right ) e^{- 2 e}}{16} + \frac {x^{3} \left (- a^{2} c d^{2} e^{4 e} + 4 i a^{2} c d^{2} e^{3 e} - 4 i a^{2} c d^{2} e^{e} - a^{2} c d^{2}\right ) e^{- 2 e}}{4} + \frac {x^{2} \left (- 3 a^{2} c^{2} d e^{4 e} + 12 i a^{2} c^{2} d e^{3 e} - 12 i a^{2} c^{2} d e^{e} - 3 a^{2} c^{2} d\right ) e^{- 2 e}}{8} + \frac {x \left (- a^{2} c^{3} e^{4 e} + 4 i a^{2} c^{3} e^{3 e} - 4 i a^{2} c^{3} e^{e} - a^{2} c^{3}\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)**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**15*exp(e) + 384*a**2*c**2*d*f**15*x*exp(e) + 192*a**2*c**2*d*f**14*exp(e) + 384*a**2*c*d**2*f**15*x**2*ex
p(e) + 384*a**2*c*d**2*f**14*x*exp(e) + 192*a**2*c*d**2*f**13*exp(e) + 128*a**2*d**3*f**15*x**3*exp(e) + 192*a
**2*d**3*f**14*x**2*exp(e) + 192*a**2*d**3*f**13*x*exp(e) + 96*a**2*d**3*f**12*exp(e))*exp(-2*f*x) + (-128*a**
2*c**3*f**15*exp(5*e) - 384*a**2*c**2*d*f**15*x*exp(5*e) + 192*a**2*c**2*d*f**14*exp(5*e) - 384*a**2*c*d**2*f*
*15*x**2*exp(5*e) + 384*a**2*c*d**2*f**14*x*exp(5*e) - 192*a**2*c*d**2*f**13*exp(5*e) - 128*a**2*d**3*f**15*x*
*3*exp(5*e) + 192*a**2*d**3*f**14*x**2*exp(5*e) - 192*a**2*d**3*f**13*x*exp(5*e) + 96*a**2*d**3*f**12*exp(5*e)
)*exp(2*f*x) + (1024*I*a**2*c**3*f**15*exp(2*e) + 3072*I*a**2*c**2*d*f**15*x*exp(2*e) + 3072*I*a**2*c**2*d*f**
14*exp(2*e) + 3072*I*a**2*c*d**2*f**15*x**2*exp(2*e) + 6144*I*a**2*c*d**2*f**14*x*exp(2*e) + 6144*I*a**2*c*d**
2*f**13*exp(2*e) + 1024*I*a**2*d**3*f**15*x**3*exp(2*e) + 3072*I*a**2*d**3*f**14*x**2*exp(2*e) + 6144*I*a**2*d
**3*f**13*x*exp(2*e) + 6144*I*a**2*d**3*f**12*exp(2*e))*exp(-f*x) + (1024*I*a**2*c**3*f**15*exp(4*e) + 3072*I*
a**2*c**2*d*f**15*x*exp(4*e) - 3072*I*a**2*c**2*d*f**14*exp(4*e) + 3072*I*a**2*c*d**2*f**15*x**2*exp(4*e) - 61
44*I*a**2*c*d**2*f**14*x*exp(4*e) + 6144*I*a**2*c*d**2*f**13*exp(4*e) + 1024*I*a**2*d**3*f**15*x**3*exp(4*e) -
3072*I*a**2*d**3*f**14*x**2*exp(4*e) + 6144*I*a**2*d**3*f**13*x*exp(4*e) - 6144*I*a**2*d**3*f**12*exp(4*e))*e
xp(f*x))*exp(-3*e)/(1024*f**16), Ne(f**16*exp(3*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) - 1
2*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] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 580 vs. \(2 (223) = 446\).
time = 0.44, size = 580, normalized size = 2.37 \begin {gather*} \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 {gather*}

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

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Mupad [B]
time = 1.05, size = 393, normalized size = 1.60 \begin {gather*} \frac {a^2\,\left (3\,d^3\,\mathrm {cosh}\left (2\,e+2\,f\,x\right )+24\,c^3\,f^4\,x-4\,c^3\,f^3\,\mathrm {sinh}\left (2\,e+2\,f\,x\right )+6\,d^3\,f^4\,x^4+6\,c^2\,d\,f^2\,\mathrm {cosh}\left (2\,e+2\,f\,x\right )+36\,c^2\,d\,f^4\,x^2+24\,c\,d^2\,f^4\,x^3+6\,d^3\,f^2\,x^2\,\mathrm {cosh}\left (2\,e+2\,f\,x\right )-4\,d^3\,f^3\,x^3\,\mathrm {sinh}\left (2\,e+2\,f\,x\right )-6\,c\,d^2\,f\,\mathrm {sinh}\left (2\,e+2\,f\,x\right )-6\,d^3\,f\,x\,\mathrm {sinh}\left (2\,e+2\,f\,x\right )+12\,c\,d^2\,f^2\,x\,\mathrm {cosh}\left (2\,e+2\,f\,x\right )-12\,c^2\,d\,f^3\,x\,\mathrm {sinh}\left (2\,e+2\,f\,x\right )-12\,c\,d^2\,f^3\,x^2\,\mathrm {sinh}\left (2\,e+2\,f\,x\right )-d^3\,\mathrm {sinh}\left (e+f\,x\right )\,192{}\mathrm {i}+c^3\,f^3\,\mathrm {cosh}\left (e+f\,x\right )\,32{}\mathrm {i}+d^3\,f^3\,x^3\,\mathrm {cosh}\left (e+f\,x\right )\,32{}\mathrm {i}-d^3\,f^2\,x^2\,\mathrm {sinh}\left (e+f\,x\right )\,96{}\mathrm {i}+c\,d^2\,f\,\mathrm {cosh}\left (e+f\,x\right )\,192{}\mathrm {i}+d^3\,f\,x\,\mathrm {cosh}\left (e+f\,x\right )\,192{}\mathrm {i}-c^2\,d\,f^2\,\mathrm {sinh}\left (e+f\,x\right )\,96{}\mathrm {i}+c^2\,d\,f^3\,x\,\mathrm {cosh}\left (e+f\,x\right )\,96{}\mathrm {i}-c\,d^2\,f^2\,x\,\mathrm {sinh}\left (e+f\,x\right )\,192{}\mathrm {i}+c\,d^2\,f^3\,x^2\,\mathrm {cosh}\left (e+f\,x\right )\,96{}\mathrm {i}\right )}{16\,f^4} \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)^3,x)

[Out]

(a^2*(3*d^3*cosh(2*e + 2*f*x) - d^3*sinh(e + f*x)*192i + c^3*f^3*cosh(e + f*x)*32i + 24*c^3*f^4*x - 4*c^3*f^3*
sinh(2*e + 2*f*x) + 6*d^3*f^4*x^4 + 6*c^2*d*f^2*cosh(2*e + 2*f*x) + 36*c^2*d*f^4*x^2 + 24*c*d^2*f^4*x^3 + d^3*
f^3*x^3*cosh(e + f*x)*32i - d^3*f^2*x^2*sinh(e + f*x)*96i + c*d^2*f*cosh(e + f*x)*192i + d^3*f*x*cosh(e + f*x)
*192i + 6*d^3*f^2*x^2*cosh(2*e + 2*f*x) - 4*d^3*f^3*x^3*sinh(2*e + 2*f*x) - 6*c*d^2*f*sinh(2*e + 2*f*x) - c^2*
d*f^2*sinh(e + f*x)*96i - 6*d^3*f*x*sinh(2*e + 2*f*x) + c^2*d*f^3*x*cosh(e + f*x)*96i - c*d^2*f^2*x*sinh(e + f
*x)*192i + 12*c*d^2*f^2*x*cosh(2*e + 2*f*x) + c*d^2*f^3*x^2*cosh(e + f*x)*96i - 12*c^2*d*f^3*x*sinh(2*e + 2*f*
x) - 12*c*d^2*f^3*x^2*sinh(2*e + 2*f*x)))/(16*f^4)

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