3.101 \(\int (c+d x)^3 (a+a \sin (e+f x))^2 \, dx\)

Optimal. Leaf size=237 \[ \frac{12 a^2 d^2 (c+d x) \cos (e+f x)}{f^3}+\frac{3 a^2 d^2 (c+d x) \sin (e+f x) \cos (e+f x)}{4 f^3}-\frac{3 a^2 c d^2 x}{4 f^2}+\frac{3 a^2 d (c+d x)^2 \sin ^2(e+f x)}{4 f^2}+\frac{6 a^2 d (c+d x)^2 \sin (e+f x)}{f^2}-\frac{2 a^2 (c+d x)^3 \cos (e+f x)}{f}-\frac{a^2 (c+d x)^3 \sin (e+f x) \cos (e+f x)}{2 f}+\frac{3 a^2 (c+d x)^4}{8 d}-\frac{3 a^2 d^3 \sin ^2(e+f x)}{8 f^4}-\frac{12 a^2 d^3 \sin (e+f x)}{f^4}-\frac{3 a^2 d^3 x^2}{8 f^2} \]

[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*a^2*d^2*(c + d*x)*Cos[e +
f*x])/f^3 - (2*a^2*(c + d*x)^3*Cos[e + f*x])/f - (12*a^2*d^3*Sin[e + f*x])/f^4 + (6*a^2*d*(c + d*x)^2*Sin[e +
f*x])/f^2 + (3*a^2*d^2*(c + d*x)*Cos[e + f*x]*Sin[e + f*x])/(4*f^3) - (a^2*(c + d*x)^3*Cos[e + f*x]*Sin[e + f
*x])/(2*f) - (3*a^2*d^3*Sin[e + f*x]^2)/(8*f^4) + (3*a^2*d*(c + d*x)^2*Sin[e + f*x]^2)/(4*f^2)

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Rubi [A]  time = 0.295401, antiderivative size = 237, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {3317, 3296, 2637, 3311, 32, 3310} \[ \frac{12 a^2 d^2 (c+d x) \cos (e+f x)}{f^3}+\frac{3 a^2 d^2 (c+d x) \sin (e+f x) \cos (e+f x)}{4 f^3}-\frac{3 a^2 c d^2 x}{4 f^2}+\frac{3 a^2 d (c+d x)^2 \sin ^2(e+f x)}{4 f^2}+\frac{6 a^2 d (c+d x)^2 \sin (e+f x)}{f^2}-\frac{2 a^2 (c+d x)^3 \cos (e+f x)}{f}-\frac{a^2 (c+d x)^3 \sin (e+f x) \cos (e+f x)}{2 f}+\frac{3 a^2 (c+d x)^4}{8 d}-\frac{3 a^2 d^3 \sin ^2(e+f x)}{8 f^4}-\frac{12 a^2 d^3 \sin (e+f x)}{f^4}-\frac{3 a^2 d^3 x^2}{8 f^2} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^3*(a + a*Sin[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*a^2*d^2*(c + d*x)*Cos[e +
f*x])/f^3 - (2*a^2*(c + d*x)^3*Cos[e + f*x])/f - (12*a^2*d^3*Sin[e + f*x])/f^4 + (6*a^2*d*(c + d*x)^2*Sin[e +
f*x])/f^2 + (3*a^2*d^2*(c + d*x)*Cos[e + f*x]*Sin[e + f*x])/(4*f^3) - (a^2*(c + d*x)^3*Cos[e + f*x]*Sin[e + f
*x])/(2*f) - (3*a^2*d^3*Sin[e + f*x]^2)/(8*f^4) + (3*a^2*d*(c + d*x)^2*Sin[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+a \sin (e+f x))^2 \, dx &=\int \left (a^2 (c+d x)^3+2 a^2 (c+d x)^3 \sin (e+f x)+a^2 (c+d x)^3 \sin ^2(e+f x)\right ) \, dx\\ &=\frac{a^2 (c+d x)^4}{4 d}+a^2 \int (c+d x)^3 \sin ^2(e+f x) \, dx+\left (2 a^2\right ) \int (c+d x)^3 \sin (e+f x) \, dx\\ &=\frac{a^2 (c+d x)^4}{4 d}-\frac{2 a^2 (c+d x)^3 \cos (e+f x)}{f}-\frac{a^2 (c+d x)^3 \cos (e+f x) \sin (e+f x)}{2 f}+\frac{3 a^2 d (c+d x)^2 \sin ^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) \sin ^2(e+f x) \, dx}{2 f^2}+\frac{\left (6 a^2 d\right ) \int (c+d x)^2 \cos (e+f x) \, dx}{f}\\ &=\frac{3 a^2 (c+d x)^4}{8 d}-\frac{2 a^2 (c+d x)^3 \cos (e+f x)}{f}+\frac{6 a^2 d (c+d x)^2 \sin (e+f x)}{f^2}+\frac{3 a^2 d^2 (c+d x) \cos (e+f x) \sin (e+f x)}{4 f^3}-\frac{a^2 (c+d x)^3 \cos (e+f x) \sin (e+f x)}{2 f}-\frac{3 a^2 d^3 \sin ^2(e+f x)}{8 f^4}+\frac{3 a^2 d (c+d x)^2 \sin ^2(e+f x)}{4 f^2}-\frac{\left (3 a^2 d^2\right ) \int (c+d x) \, dx}{4 f^2}-\frac{\left (12 a^2 d^2\right ) \int (c+d x) \sin (e+f x) \, dx}{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 a^2 d^2 (c+d x) \cos (e+f x)}{f^3}-\frac{2 a^2 (c+d x)^3 \cos (e+f x)}{f}+\frac{6 a^2 d (c+d x)^2 \sin (e+f x)}{f^2}+\frac{3 a^2 d^2 (c+d x) \cos (e+f x) \sin (e+f x)}{4 f^3}-\frac{a^2 (c+d x)^3 \cos (e+f x) \sin (e+f x)}{2 f}-\frac{3 a^2 d^3 \sin ^2(e+f x)}{8 f^4}+\frac{3 a^2 d (c+d x)^2 \sin ^2(e+f x)}{4 f^2}-\frac{\left (12 a^2 d^3\right ) \int \cos (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 a^2 d^2 (c+d x) \cos (e+f x)}{f^3}-\frac{2 a^2 (c+d x)^3 \cos (e+f x)}{f}-\frac{12 a^2 d^3 \sin (e+f x)}{f^4}+\frac{6 a^2 d (c+d x)^2 \sin (e+f x)}{f^2}+\frac{3 a^2 d^2 (c+d x) \cos (e+f x) \sin (e+f x)}{4 f^3}-\frac{a^2 (c+d x)^3 \cos (e+f x) \sin (e+f x)}{2 f}-\frac{3 a^2 d^3 \sin ^2(e+f x)}{8 f^4}+\frac{3 a^2 d (c+d x)^2 \sin ^2(e+f x)}{4 f^2}\\ \end{align*}

Mathematica [A]  time = 1.34344, size = 216, normalized size = 0.91 \[ \frac{a^2 \left (-2 f (c+d x) \left (2 c^2 f^2+4 c d f^2 x+d^2 \left (2 f^2 x^2-3\right )\right ) \sin (2 (e+f x))+96 d \left (c^2 f^2+2 c d f^2 x+d^2 \left (f^2 x^2-2\right )\right ) \sin (e+f x)-32 f (c+d x) \left (c^2 f^2+2 c d f^2 x+d^2 \left (f^2 x^2-6\right )\right ) \cos (e+f x)-3 d \left (2 c^2 f^2+4 c d f^2 x+d^2 \left (2 f^2 x^2-1\right )\right ) \cos (2 (e+f x))+6 f^4 x \left (6 c^2 d x+4 c^3+4 c d^2 x^2+d^3 x^3\right )\right )}{16 f^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^3*(a + a*Sin[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*f*(c + d*x)*(c^2*f^2 + 2*c*d*f^2*x + d^2*(-6 +
f^2*x^2))*Cos[e + f*x] - 3*d*(2*c^2*f^2 + 4*c*d*f^2*x + d^2*(-1 + 2*f^2*x^2))*Cos[2*(e + f*x)] + 96*d*(c^2*f^2
+ 2*c*d*f^2*x + d^2*(-2 + f^2*x^2))*Sin[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))*Sin[2*(e + f*x)]))/(16*f^4)

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Maple [B]  time = 0.022, size = 1135, normalized size = 4.8 \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+a*sin(f*x+e))^2,x)

[Out]

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

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Maxima [B]  time = 1.10759, size = 1308, normalized size = 5.52 \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+a*sin(f*x+e))^2,x, algorithm="maxima")

[Out]

1/16*(4*(2*f*x + 2*e - sin(2*f*x + 2*e))*a^2*c^3 + 16*(f*x + e)*a^2*c^3 + 4*(f*x + e)^4*a^2*d^3/f^3 - 16*(f*x
+ e)^3*a^2*d^3*e/f^3 + 24*(f*x + e)^2*a^2*d^3*e^2/f^3 - 4*(2*f*x + 2*e - sin(2*f*x + 2*e))*a^2*d^3*e^3/f^3 - 1
6*(f*x + e)*a^2*d^3*e^3/f^3 + 16*(f*x + e)^3*a^2*c*d^2/f^2 - 48*(f*x + e)^2*a^2*c*d^2*e/f^2 + 12*(2*f*x + 2*e
- sin(2*f*x + 2*e))*a^2*c*d^2*e^2/f^2 + 48*(f*x + e)*a^2*c*d^2*e^2/f^2 + 24*(f*x + e)^2*a^2*c^2*d/f - 12*(2*f*
x + 2*e - sin(2*f*x + 2*e))*a^2*c^2*d*e/f - 48*(f*x + e)*a^2*c^2*d*e/f - 32*a^2*c^3*cos(f*x + e) + 32*a^2*d^3*
e^3*cos(f*x + e)/f^3 - 96*a^2*c*d^2*e^2*cos(f*x + e)/f^2 + 96*a^2*c^2*d*e*cos(f*x + e)/f + 6*(2*(f*x + e)^2 -
2*(f*x + e)*sin(2*f*x + 2*e) - cos(2*f*x + 2*e))*a^2*d^3*e^2/f^3 - 96*((f*x + e)*cos(f*x + e) - sin(f*x + e))*
a^2*d^3*e^2/f^3 - 12*(2*(f*x + e)^2 - 2*(f*x + e)*sin(2*f*x + 2*e) - cos(2*f*x + 2*e))*a^2*c*d^2*e/f^2 + 192*(
(f*x + e)*cos(f*x + e) - sin(f*x + e))*a^2*c*d^2*e/f^2 + 6*(2*(f*x + e)^2 - 2*(f*x + e)*sin(2*f*x + 2*e) - cos
(2*f*x + 2*e))*a^2*c^2*d/f - 96*((f*x + e)*cos(f*x + e) - sin(f*x + e))*a^2*c^2*d/f - 2*(4*(f*x + e)^3 - 6*(f*
x + e)*cos(2*f*x + 2*e) - 3*(2*(f*x + e)^2 - 1)*sin(2*f*x + 2*e))*a^2*d^3*e/f^3 + 96*(((f*x + e)^2 - 2)*cos(f*
x + e) - 2*(f*x + e)*sin(f*x + e))*a^2*d^3*e/f^3 + 2*(4*(f*x + e)^3 - 6*(f*x + e)*cos(2*f*x + 2*e) - 3*(2*(f*x
+ e)^2 - 1)*sin(2*f*x + 2*e))*a^2*c*d^2/f^2 - 96*(((f*x + e)^2 - 2)*cos(f*x + e) - 2*(f*x + e)*sin(f*x + e))*
a^2*c*d^2/f^2 + (2*(f*x + e)^4 - 3*(2*(f*x + e)^2 - 1)*cos(2*f*x + 2*e) - 2*(2*(f*x + e)^3 - 3*f*x - 3*e)*sin(
2*f*x + 2*e))*a^2*d^3/f^3 - 32*(((f*x + e)^3 - 6*f*x - 6*e)*cos(f*x + e) - 3*((f*x + e)^2 - 2)*sin(f*x + e))*a
^2*d^3/f^3)/f

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Fricas [A]  time = 1.88544, size = 752, normalized size = 3.17 \begin{align*} \frac{3 \, a^{2} d^{3} f^{4} x^{4} + 12 \, a^{2} c d^{2} f^{4} x^{3} + 3 \,{\left (6 \, a^{2} c^{2} d f^{4} + a^{2} d^{3} f^{2}\right )} x^{2} - 3 \,{\left (2 \, a^{2} d^{3} f^{2} x^{2} + 4 \, a^{2} c d^{2} f^{2} x + 2 \, a^{2} c^{2} d f^{2} - a^{2} d^{3}\right )} \cos \left (f x + e\right )^{2} + 6 \,{\left (2 \, a^{2} c^{3} f^{4} + a^{2} c d^{2} f^{2}\right )} x - 16 \,{\left (a^{2} d^{3} f^{3} x^{3} + 3 \, a^{2} c d^{2} f^{3} x^{2} + a^{2} c^{3} f^{3} - 6 \, a^{2} c d^{2} f + 3 \,{\left (a^{2} c^{2} d f^{3} - 2 \, a^{2} d^{3} f\right )} x\right )} \cos \left (f x + e\right ) + 2 \,{\left (24 \, a^{2} d^{3} f^{2} x^{2} + 48 \, a^{2} c d^{2} f^{2} x + 24 \, a^{2} c^{2} d f^{2} - 48 \, a^{2} d^{3} -{\left (2 \, a^{2} d^{3} f^{3} x^{3} + 6 \, a^{2} c d^{2} f^{3} x^{2} + 2 \, a^{2} c^{3} f^{3} - 3 \, a^{2} c d^{2} f + 3 \,{\left (2 \, a^{2} c^{2} d f^{3} - a^{2} d^{3} f\right )} x\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{8 \, f^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(3*a^2*d^3*f^4*x^4 + 12*a^2*c*d^2*f^4*x^3 + 3*(6*a^2*c^2*d*f^4 + a^2*d^3*f^2)*x^2 - 3*(2*a^2*d^3*f^2*x^2 +
4*a^2*c*d^2*f^2*x + 2*a^2*c^2*d*f^2 - a^2*d^3)*cos(f*x + e)^2 + 6*(2*a^2*c^3*f^4 + a^2*c*d^2*f^2)*x - 16*(a^2
*d^3*f^3*x^3 + 3*a^2*c*d^2*f^3*x^2 + a^2*c^3*f^3 - 6*a^2*c*d^2*f + 3*(a^2*c^2*d*f^3 - 2*a^2*d^3*f)*x)*cos(f*x
+ e) + 2*(24*a^2*d^3*f^2*x^2 + 48*a^2*c*d^2*f^2*x + 24*a^2*c^2*d*f^2 - 48*a^2*d^3 - (2*a^2*d^3*f^3*x^3 + 6*a^2
*c*d^2*f^3*x^2 + 2*a^2*c^3*f^3 - 3*a^2*c*d^2*f + 3*(2*a^2*c^2*d*f^3 - a^2*d^3*f)*x)*cos(f*x + e))*sin(f*x + e)
)/f^4

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Sympy [A]  time = 4.78442, size = 779, normalized size = 3.29 \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+a*sin(f*x+e))**2,x)

[Out]

Piecewise((a**2*c**3*x*sin(e + f*x)**2/2 + a**2*c**3*x*cos(e + f*x)**2/2 + a**2*c**3*x - a**2*c**3*sin(e + f*x
)*cos(e + f*x)/(2*f) - 2*a**2*c**3*cos(e + f*x)/f + 3*a**2*c**2*d*x**2*sin(e + f*x)**2/4 + 3*a**2*c**2*d*x**2*
cos(e + f*x)**2/4 + 3*a**2*c**2*d*x**2/2 - 3*a**2*c**2*d*x*sin(e + f*x)*cos(e + f*x)/(2*f) - 6*a**2*c**2*d*x*c
os(e + f*x)/f + 6*a**2*c**2*d*sin(e + f*x)/f**2 - 3*a**2*c**2*d*cos(e + f*x)**2/(4*f**2) + a**2*c*d**2*x**3*si
n(e + f*x)**2/2 + a**2*c*d**2*x**3*cos(e + f*x)**2/2 + a**2*c*d**2*x**3 - 3*a**2*c*d**2*x**2*sin(e + f*x)*cos(
e + f*x)/(2*f) - 6*a**2*c*d**2*x**2*cos(e + f*x)/f + 3*a**2*c*d**2*x*sin(e + f*x)**2/(4*f**2) + 12*a**2*c*d**2
*x*sin(e + f*x)/f**2 - 3*a**2*c*d**2*x*cos(e + f*x)**2/(4*f**2) + 3*a**2*c*d**2*sin(e + f*x)*cos(e + f*x)/(4*f
**3) + 12*a**2*c*d**2*cos(e + f*x)/f**3 + a**2*d**3*x**4*sin(e + f*x)**2/8 + a**2*d**3*x**4*cos(e + f*x)**2/8
+ a**2*d**3*x**4/4 - a**2*d**3*x**3*sin(e + f*x)*cos(e + f*x)/(2*f) - 2*a**2*d**3*x**3*cos(e + f*x)/f + 3*a**2
*d**3*x**2*sin(e + f*x)**2/(8*f**2) + 6*a**2*d**3*x**2*sin(e + f*x)/f**2 - 3*a**2*d**3*x**2*cos(e + f*x)**2/(8
*f**2) + 3*a**2*d**3*x*sin(e + f*x)*cos(e + f*x)/(4*f**3) + 12*a**2*d**3*x*cos(e + f*x)/f**3 - 12*a**2*d**3*si
n(e + f*x)/f**4 + 3*a**2*d**3*cos(e + f*x)**2/(8*f**4), Ne(f, 0)), ((a*sin(e) + a)**2*(c**3*x + 3*c**2*d*x**2/
2 + c*d**2*x**3 + d**3*x**4/4), True))

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Giac [A]  time = 1.15698, size = 458, normalized size = 1.93 \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{3 \,{\left (2 \, a^{2} d^{3} f^{2} x^{2} + 4 \, a^{2} c d^{2} f^{2} x + 2 \, a^{2} c^{2} d f^{2} - a^{2} d^{3}\right )} \cos \left (2 \, f x + 2 \, e\right )}{16 \, f^{4}} - \frac{2 \,{\left (a^{2} d^{3} f^{3} x^{3} + 3 \, a^{2} c d^{2} f^{3} x^{2} + 3 \, a^{2} c^{2} d f^{3} x + a^{2} c^{3} f^{3} - 6 \, a^{2} d^{3} f x - 6 \, a^{2} c d^{2} f\right )} \cos \left (f x + e\right )}{f^{4}} - \frac{{\left (2 \, a^{2} d^{3} f^{3} x^{3} + 6 \, a^{2} c d^{2} f^{3} x^{2} + 6 \, a^{2} c^{2} d f^{3} x + 2 \, a^{2} c^{3} f^{3} - 3 \, a^{2} d^{3} f x - 3 \, a^{2} c d^{2} f\right )} \sin \left (2 \, f x + 2 \, e\right )}{8 \, f^{4}} + \frac{6 \,{\left (a^{2} d^{3} f^{2} x^{2} + 2 \, a^{2} c d^{2} f^{2} x + a^{2} c^{2} d f^{2} - 2 \, a^{2} d^{3}\right )} \sin \left (f x + e\right )}{f^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^3*(a+a*sin(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 - 3/16*(2*a^2*d^3*f^2*x^2 + 4*a^2*c*d^
2*f^2*x + 2*a^2*c^2*d*f^2 - a^2*d^3)*cos(2*f*x + 2*e)/f^4 - 2*(a^2*d^3*f^3*x^3 + 3*a^2*c*d^2*f^3*x^2 + 3*a^2*c
^2*d*f^3*x + a^2*c^3*f^3 - 6*a^2*d^3*f*x - 6*a^2*c*d^2*f)*cos(f*x + e)/f^4 - 1/8*(2*a^2*d^3*f^3*x^3 + 6*a^2*c*
d^2*f^3*x^2 + 6*a^2*c^2*d*f^3*x + 2*a^2*c^3*f^3 - 3*a^2*d^3*f*x - 3*a^2*c*d^2*f)*sin(2*f*x + 2*e)/f^4 + 6*(a^2
*d^3*f^2*x^2 + 2*a^2*c*d^2*f^2*x + a^2*c^2*d*f^2 - 2*a^2*d^3)*sin(f*x + e)/f^4