\(\int (c+d x)^3 (a+a \sin (e+f x))^2 \, dx\) [101]

Optimal result

Integrand size = 20, antiderivative size = 237 \[ \int (c+d x)^3 (a+a \sin (e+f x))^2 \, dx=-\frac {3 a^2 c d^2 x}{4 f^2}-\frac {3 a^2 d^3 x^2}{8 f^2}+\frac {3 a^2 (c+d x)^4}{8 d}+\frac {12 a^2 d^2 (c+d x) \cos (e+f x)}{f^3}-\frac {2 a^2 (c+d x)^3 \cos (e+f x)}{f}-\frac {12 a^2 d^3 \sin (e+f x)}{f^4}+\frac {6 a^2 d (c+d x)^2 \sin (e+f x)}{f^2}+\frac {3 a^2 d^2 (c+d x) \cos (e+f x) \sin (e+f x)}{4 f^3}-\frac {a^2 (c+d x)^3 \cos (e+f x) \sin (e+f x)}{2 f}-\frac {3 a^2 d^3 \sin ^2(e+f x)}{8 f^4}+\frac {3 a^2 d (c+d x)^2 \sin ^2(e+f x)}{4 f^2} \]

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

Rubi [A] (verified)

Time = 0.17 (sec) , antiderivative size = 237, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3398, 3377, 2717, 3392, 32, 3391} \[ \int (c+d x)^3 (a+a \sin (e+f x))^2 \, dx=\frac {12 a^2 d^2 (c+d x) \cos (e+f x)}{f^3}+\frac {3 a^2 d^2 (c+d x) \sin (e+f x) \cos (e+f x)}{4 f^3}-\frac {3 a^2 c d^2 x}{4 f^2}+\frac {3 a^2 d (c+d x)^2 \sin ^2(e+f x)}{4 f^2}+\frac {6 a^2 d (c+d x)^2 \sin (e+f x)}{f^2}-\frac {2 a^2 (c+d x)^3 \cos (e+f x)}{f}-\frac {a^2 (c+d x)^3 \sin (e+f x) \cos (e+f x)}{2 f}+\frac {3 a^2 (c+d x)^4}{8 d}-\frac {3 a^2 d^3 \sin ^2(e+f x)}{8 f^4}-\frac {12 a^2 d^3 \sin (e+f x)}{f^4}-\frac {3 a^2 d^3 x^2}{8 f^2} \]

[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 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*} \text {integral}& = \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] (verified)

Time = 0.85 (sec) , antiderivative size = 216, normalized size of antiderivative = 0.91 \[ \int (c+d x)^3 (a+a \sin (e+f x))^2 \, dx=\frac {a^2 \left (6 f^4 x \left (4 c^3+6 c^2 d x+4 c d^2 x^2+d^3 x^3\right )-32 f (c+d x) \left (c^2 f^2+2 c d f^2 x+d^2 \left (-6+f^2 x^2\right )\right ) \cos (e+f x)-3 d \left (2 c^2 f^2+4 c d f^2 x+d^2 \left (-1+2 f^2 x^2\right )\right ) \cos (2 (e+f x))+96 d \left (c^2 f^2+2 c d f^2 x+d^2 \left (-2+f^2 x^2\right )\right ) \sin (e+f x)-2 f (c+d x) \left (2 c^2 f^2+4 c d f^2 x+d^2 \left (-3+2 f^2 x^2\right )\right ) \sin (2 (e+f x))\right )}{16 f^4} \]

[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)

Maple [A] (verified)

Time = 0.43 (sec) , antiderivative size = 195, normalized size of antiderivative = 0.82

method	result	size
parallelrisch	\(-\frac {\left (\left (d x +c \right ) f \left (\left (d x +c \right )^{2} f^{2}-\frac {3 d^{2}}{2}\right ) \sin \left (2 f x +2 e \right )+\frac {3 \left (\left (d x +c \right )^{2} f^{2}-\frac {d^{2}}{2}\right ) d \cos \left (2 f x +2 e \right )}{2}+8 \left (\left (d x +c \right )^{2} f^{2}-6 d^{2}\right ) \left (d x +c \right ) f \cos \left (f x +e \right )-24 d \left (\left (d x +c \right )^{2} f^{2}-2 d^{2}\right ) \sin \left (f x +e \right )+\left (-6 x^{3} c \,d^{2}-9 x^{2} c^{2} d -\frac {3}{2} d^{3} x^{4}-6 x \,c^{3}\right ) f^{4}+8 c^{3} f^{3}-\frac {3 c^{2} d \,f^{2}}{2}-48 c \,d^{2} f +\frac {3 d^{3}}{4}\right ) a^{2}}{4 f^{4}}\)	\(195\)
risch	\(\frac {3 a^{2} d^{3} x^{4}}{8}+\frac {3 a^{2} c \,d^{2} x^{3}}{2}+\frac {9 a^{2} d \,c^{2} x^{2}}{4}+\frac {3 a^{2} c^{3} x}{2}+\frac {3 a^{2} c^{4}}{8 d}-\frac {2 a^{2} \left (d^{3} f^{2} x^{3}+3 c \,d^{2} f^{2} x^{2}+3 c^{2} d \,f^{2} x +c^{3} f^{2}-6 d^{3} x -6 c \,d^{2}\right ) \cos \left (f x +e \right )}{f^{3}}+\frac {6 a^{2} d \left (d^{2} x^{2} f^{2}+2 c d \,f^{2} x +c^{2} f^{2}-2 d^{2}\right ) \sin \left (f x +e \right )}{f^{4}}-\frac {3 a^{2} d \left (2 d^{2} x^{2} f^{2}+4 c d \,f^{2} x +2 c^{2} f^{2}-d^{2}\right ) \cos \left (2 f x +2 e \right )}{16 f^{4}}-\frac {a^{2} \left (2 d^{3} f^{2} x^{3}+6 c \,d^{2} f^{2} x^{2}+6 c^{2} d \,f^{2} x +2 c^{3} f^{2}-3 d^{3} x -3 c \,d^{2}\right ) \sin \left (2 f x +2 e \right )}{8 f^{3}}\)	\(291\)
norman	\(\frac {\frac {a^{2} d^{3} x^{3} \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}-\frac {4 a^{2} c^{3} f^{2}-24 a^{2} c \,d^{2}}{f^{3}}+\frac {3 a^{2} d^{3} x^{4}}{8}-\frac {\left (8 a^{2} c^{3} f^{3}-6 a^{2} c^{2} d \,f^{2}-48 a^{2} c \,d^{2} f +3 a^{2} d^{3}\right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 f^{4}}+\frac {3 a^{2} d^{3} x^{4} \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{4}+\frac {3 a^{2} d^{3} x^{4} \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{8}-\frac {a^{2} \left (2 c^{3} f^{3}-24 c^{2} d \,f^{2}-3 c \,d^{2} f +48 d^{3}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{2 f^{4}}+\frac {3 a^{2} \left (2 c^{3} f^{3}-8 c^{2} d \,f^{2}-c \,d^{2} f +16 d^{3}\right ) x}{4 f^{3}}+\frac {a^{2} \left (2 c^{3} f^{3}+24 c^{2} d \,f^{2}-3 c \,d^{2} f -48 d^{3}\right ) \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 f^{4}}+3 a^{2} c \,d^{2} x^{3} \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\frac {3 a^{2} d \left (6 c^{2} f^{2}-16 c d f -d^{2}\right ) x^{2}}{8 f^{2}}-\frac {a^{2} d^{3} x^{3} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{f}+\frac {3 a^{2} \left (2 c^{3} f^{3}+8 c^{2} d \,f^{2}-c \,d^{2} f -16 d^{3}\right ) x \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{4 f^{3}}+\frac {d^{2} a^{2} \left (3 c f -4 d \right ) x^{3}}{2 f}+\frac {3 a^{2} c \left (2 c^{2} f^{2}+3 d^{2}\right ) x \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 f^{2}}+\frac {9 a^{2} d \left (2 c^{2} f^{2}+d^{2}\right ) x^{2} \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{4 f^{2}}-\frac {3 a^{2} d \left (2 c^{2} f^{2}-16 c d f -d^{2}\right ) x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{2 f^{3}}+\frac {3 a^{2} d \left (2 c^{2} f^{2}+16 c d f -d^{2}\right ) x \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 f^{3}}+\frac {3 a^{2} d \left (6 c^{2} f^{2}+16 c d f -d^{2}\right ) x^{2} \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{8 f^{2}}-\frac {3 d^{2} a^{2} \left (c f -4 d \right ) x^{2} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{f^{2}}+\frac {3 d^{2} a^{2} \left (c f +4 d \right ) x^{2} \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f^{2}}+\frac {d^{2} a^{2} \left (3 c f +4 d \right ) x^{3} \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 f}}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{2}}\)	\(750\)
parts	\(\text {Expression too large to display}\)	\(917\)
derivativedivides	\(\text {Expression too large to display}\)	\(1135\)
default	\(\text {Expression too large to display}\)	\(1135\)

[In]

int((d*x+c)^3*(a+a*sin(f*x+e))^2,x,method=_RETURNVERBOSE)

[Out]

-1/4*((d*x+c)*f*((d*x+c)^2*f^2-3/2*d^2)*sin(2*f*x+2*e)+3/2*((d*x+c)^2*f^2-1/2*d^2)*d*cos(2*f*x+2*e)+8*((d*x+c)
^2*f^2-6*d^2)*(d*x+c)*f*cos(f*x+e)-24*d*((d*x+c)^2*f^2-2*d^2)*sin(f*x+e)+(-6*x^3*c*d^2-9*x^2*c^2*d-3/2*d^3*x^4
-6*x*c^3)*f^4+8*c^3*f^3-3/2*c^2*d*f^2-48*c*d^2*f+3/4*d^3)*a^2/f^4

Fricas [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 368, normalized size of antiderivative = 1.55 \[ \int (c+d x)^3 (a+a \sin (e+f x))^2 \, dx=\frac {3 \, a^{2} d^{3} f^{4} x^{4} + 12 \, a^{2} c d^{2} f^{4} x^{3} + 3 \, {\left (6 \, a^{2} c^{2} d f^{4} + a^{2} d^{3} f^{2}\right )} x^{2} - 3 \, {\left (2 \, a^{2} d^{3} f^{2} x^{2} + 4 \, a^{2} c d^{2} f^{2} x + 2 \, a^{2} c^{2} d f^{2} - a^{2} d^{3}\right )} \cos \left (f x + e\right )^{2} + 6 \, {\left (2 \, a^{2} c^{3} f^{4} + a^{2} c d^{2} f^{2}\right )} x - 16 \, {\left (a^{2} d^{3} f^{3} x^{3} + 3 \, a^{2} c d^{2} f^{3} x^{2} + a^{2} c^{3} f^{3} - 6 \, a^{2} c d^{2} f + 3 \, {\left (a^{2} c^{2} d f^{3} - 2 \, a^{2} d^{3} f\right )} x\right )} \cos \left (f x + e\right ) + 2 \, {\left (24 \, a^{2} d^{3} f^{2} x^{2} + 48 \, a^{2} c d^{2} f^{2} x + 24 \, a^{2} c^{2} d f^{2} - 48 \, a^{2} d^{3} - {\left (2 \, a^{2} d^{3} f^{3} x^{3} + 6 \, a^{2} c d^{2} f^{3} x^{2} + 2 \, a^{2} c^{3} f^{3} - 3 \, a^{2} c d^{2} f + 3 \, {\left (2 \, a^{2} c^{2} d f^{3} - a^{2} d^{3} f\right )} x\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{8 \, f^{4}} \]

[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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 779 vs. \(2 (243) = 486\).

Time = 0.43 (sec) , antiderivative size = 779, normalized size of antiderivative = 3.29 \[ \int (c+d x)^3 (a+a \sin (e+f x))^2 \, dx=\begin {cases} \frac {a^{2} c^{3} x \sin ^{2}{\left (e + f x \right )}}{2} + \frac {a^{2} c^{3} x \cos ^{2}{\left (e + f x \right )}}{2} + a^{2} c^{3} x - \frac {a^{2} c^{3} \sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{2 f} - \frac {2 a^{2} c^{3} \cos {\left (e + f x \right )}}{f} + \frac {3 a^{2} c^{2} d x^{2} \sin ^{2}{\left (e + f x \right )}}{4} + \frac {3 a^{2} c^{2} d x^{2} \cos ^{2}{\left (e + f x \right )}}{4} + \frac {3 a^{2} c^{2} d x^{2}}{2} - \frac {3 a^{2} c^{2} d x \sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{2 f} - \frac {6 a^{2} c^{2} d x \cos {\left (e + f x \right )}}{f} + \frac {3 a^{2} c^{2} d \sin ^{2}{\left (e + f x \right )}}{4 f^{2}} + \frac {6 a^{2} c^{2} d \sin {\left (e + f x \right )}}{f^{2}} + \frac {a^{2} c d^{2} x^{3} \sin ^{2}{\left (e + f x \right )}}{2} + \frac {a^{2} c d^{2} x^{3} \cos ^{2}{\left (e + f x \right )}}{2} + a^{2} c d^{2} x^{3} - \frac {3 a^{2} c d^{2} x^{2} \sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{2 f} - \frac {6 a^{2} c d^{2} x^{2} \cos {\left (e + f x \right )}}{f} + \frac {3 a^{2} c d^{2} x \sin ^{2}{\left (e + f x \right )}}{4 f^{2}} + \frac {12 a^{2} c d^{2} x \sin {\left (e + f x \right )}}{f^{2}} - \frac {3 a^{2} c d^{2} x \cos ^{2}{\left (e + f x \right )}}{4 f^{2}} + \frac {3 a^{2} c d^{2} \sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{4 f^{3}} + \frac {12 a^{2} c d^{2} \cos {\left (e + f x \right )}}{f^{3}} + \frac {a^{2} d^{3} x^{4} \sin ^{2}{\left (e + f x \right )}}{8} + \frac {a^{2} d^{3} x^{4} \cos ^{2}{\left (e + f x \right )}}{8} + \frac {a^{2} d^{3} x^{4}}{4} - \frac {a^{2} d^{3} x^{3} \sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{2 f} - \frac {2 a^{2} d^{3} x^{3} \cos {\left (e + f x \right )}}{f} + \frac {3 a^{2} d^{3} x^{2} \sin ^{2}{\left (e + f x \right )}}{8 f^{2}} + \frac {6 a^{2} d^{3} x^{2} \sin {\left (e + f x \right )}}{f^{2}} - \frac {3 a^{2} d^{3} x^{2} \cos ^{2}{\left (e + f x \right )}}{8 f^{2}} + \frac {3 a^{2} d^{3} x \sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{4 f^{3}} + \frac {12 a^{2} d^{3} x \cos {\left (e + f x \right )}}{f^{3}} - \frac {3 a^{2} d^{3} \sin ^{2}{\left (e + f x \right )}}{8 f^{4}} - \frac {12 a^{2} d^{3} \sin {\left (e + f x \right )}}{f^{4}} & \text {for}\: f \neq 0 \\\left (a \sin {\left (e \right )} + a\right )^{2} \left (c^{3} x + \frac {3 c^{2} d x^{2}}{2} + c d^{2} x^{3} + \frac {d^{3} x^{4}}{4}\right ) & \text {otherwise} \end {cases} \]

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 969 vs. \(2 (223) = 446\).

Time = 0.24 (sec) , antiderivative size = 969, normalized size of antiderivative = 4.09 \[ \int (c+d x)^3 (a+a \sin (e+f x))^2 \, dx=\text {Too large to display} \]

[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

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 335, normalized size of antiderivative = 1.41 \[ \int (c+d x)^3 (a+a \sin (e+f x))^2 \, dx=\frac {3}{8} \, a^{2} d^{3} x^{4} + \frac {3}{2} \, a^{2} c d^{2} x^{3} + \frac {9}{4} \, a^{2} c^{2} d x^{2} + \frac {3}{2} \, a^{2} c^{3} x - \frac {3 \, {\left (2 \, a^{2} d^{3} f^{2} x^{2} + 4 \, a^{2} c d^{2} f^{2} x + 2 \, a^{2} c^{2} d f^{2} - a^{2} d^{3}\right )} \cos \left (2 \, f x + 2 \, e\right )}{16 \, f^{4}} - \frac {2 \, {\left (a^{2} d^{3} f^{3} x^{3} + 3 \, a^{2} c d^{2} f^{3} x^{2} + 3 \, a^{2} c^{2} d f^{3} x + a^{2} c^{3} f^{3} - 6 \, a^{2} d^{3} f x - 6 \, a^{2} c d^{2} f\right )} \cos \left (f x + e\right )}{f^{4}} - \frac {{\left (2 \, a^{2} d^{3} f^{3} x^{3} + 6 \, a^{2} c d^{2} f^{3} x^{2} + 6 \, a^{2} c^{2} d f^{3} x + 2 \, a^{2} c^{3} f^{3} - 3 \, a^{2} d^{3} f x - 3 \, a^{2} c d^{2} f\right )} \sin \left (2 \, f x + 2 \, e\right )}{8 \, f^{4}} + \frac {6 \, {\left (a^{2} d^{3} f^{2} x^{2} + 2 \, a^{2} c d^{2} f^{2} x + a^{2} c^{2} d f^{2} - 2 \, a^{2} d^{3}\right )} \sin \left (f x + e\right )}{f^{4}} \]

[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

Mupad [B] (verification not implemented)

Time = 1.11 (sec) , antiderivative size = 452, normalized size of antiderivative = 1.91 \[ \int (c+d x)^3 (a+a \sin (e+f x))^2 \, dx=-\frac {96\,a^2\,d^3\,\sin \left (e+f\,x\right )-\frac {3\,a^2\,d^3\,\cos \left (2\,e+2\,f\,x\right )}{2}+16\,a^2\,c^3\,f^3\,\cos \left (e+f\,x\right )-12\,a^2\,c^3\,f^4\,x+2\,a^2\,c^3\,f^3\,\sin \left (2\,e+2\,f\,x\right )-3\,a^2\,d^3\,f^4\,x^4-96\,a^2\,c\,d^2\,f\,\cos \left (e+f\,x\right )-96\,a^2\,d^3\,f\,x\,\cos \left (e+f\,x\right )+3\,a^2\,d^3\,f^2\,x^2\,\cos \left (2\,e+2\,f\,x\right )+2\,a^2\,d^3\,f^3\,x^3\,\sin \left (2\,e+2\,f\,x\right )-3\,a^2\,c\,d^2\,f\,\sin \left (2\,e+2\,f\,x\right )-48\,a^2\,c^2\,d\,f^2\,\sin \left (e+f\,x\right )-3\,a^2\,d^3\,f\,x\,\sin \left (2\,e+2\,f\,x\right )+3\,a^2\,c^2\,d\,f^2\,\cos \left (2\,e+2\,f\,x\right )-18\,a^2\,c^2\,d\,f^4\,x^2-12\,a^2\,c\,d^2\,f^4\,x^3+16\,a^2\,d^3\,f^3\,x^3\,\cos \left (e+f\,x\right )-48\,a^2\,d^3\,f^2\,x^2\,\sin \left (e+f\,x\right )+6\,a^2\,c\,d^2\,f^2\,x\,\cos \left (2\,e+2\,f\,x\right )+48\,a^2\,c\,d^2\,f^3\,x^2\,\cos \left (e+f\,x\right )+6\,a^2\,c^2\,d\,f^3\,x\,\sin \left (2\,e+2\,f\,x\right )+6\,a^2\,c\,d^2\,f^3\,x^2\,\sin \left (2\,e+2\,f\,x\right )+48\,a^2\,c^2\,d\,f^3\,x\,\cos \left (e+f\,x\right )-96\,a^2\,c\,d^2\,f^2\,x\,\sin \left (e+f\,x\right )}{8\,f^4} \]

[In]

int((a + a*sin(e + f*x))^2*(c + d*x)^3,x)

[Out]

-(96*a^2*d^3*sin(e + f*x) - (3*a^2*d^3*cos(2*e + 2*f*x))/2 + 16*a^2*c^3*f^3*cos(e + f*x) - 12*a^2*c^3*f^4*x +
2*a^2*c^3*f^3*sin(2*e + 2*f*x) - 3*a^2*d^3*f^4*x^4 - 96*a^2*c*d^2*f*cos(e + f*x) - 96*a^2*d^3*f*x*cos(e + f*x)
+ 3*a^2*d^3*f^2*x^2*cos(2*e + 2*f*x) + 2*a^2*d^3*f^3*x^3*sin(2*e + 2*f*x) - 3*a^2*c*d^2*f*sin(2*e + 2*f*x) -
48*a^2*c^2*d*f^2*sin(e + f*x) - 3*a^2*d^3*f*x*sin(2*e + 2*f*x) + 3*a^2*c^2*d*f^2*cos(2*e + 2*f*x) - 18*a^2*c^2
*d*f^4*x^2 - 12*a^2*c*d^2*f^4*x^3 + 16*a^2*d^3*f^3*x^3*cos(e + f*x) - 48*a^2*d^3*f^2*x^2*sin(e + f*x) + 6*a^2*
c*d^2*f^2*x*cos(2*e + 2*f*x) + 48*a^2*c*d^2*f^3*x^2*cos(e + f*x) + 6*a^2*c^2*d*f^3*x*sin(2*e + 2*f*x) + 6*a^2*
c*d^2*f^3*x^2*sin(2*e + 2*f*x) + 48*a^2*c^2*d*f^3*x*cos(e + f*x) - 96*a^2*c*d^2*f^2*x*sin(e + f*x))/(8*f^4)