3.434 \(\int (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^4 \, dx\)

Optimal. Leaf size=318 \[ \frac{a^2 \left (-311 c^3 d^2-448 c^2 d^3-48 c^4 d+4 c^5-288 c d^4-64 d^5\right ) \cos (e+f x)}{60 d f}+\frac{a^2 \left (4 c^2-48 c d-55 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{120 d f}+\frac{a^2 \left (-48 c^2 d+4 c^3-123 c d^2-64 d^3\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{120 d f}+\frac{a^2 \left (-438 c^2 d^2-96 c^3 d+8 c^4-464 c d^3-165 d^4\right ) \sin (e+f x) \cos (e+f x)}{240 f}+\frac{1}{16} a^2 x \left (84 c^2 d^2+64 c^3 d+24 c^4+48 c d^3+11 d^4\right )-\frac{a^2 \cos (e+f x) (c+d \sin (e+f x))^5}{6 d f}+\frac{a^2 (c-12 d) \cos (e+f x) (c+d \sin (e+f x))^4}{30 d f} \]

[Out]

(a^2*(24*c^4 + 64*c^3*d + 84*c^2*d^2 + 48*c*d^3 + 11*d^4)*x)/16 + (a^2*(4*c^5 - 48*c^4*d - 311*c^3*d^2 - 448*c
^2*d^3 - 288*c*d^4 - 64*d^5)*Cos[e + f*x])/(60*d*f) + (a^2*(8*c^4 - 96*c^3*d - 438*c^2*d^2 - 464*c*d^3 - 165*d
^4)*Cos[e + f*x]*Sin[e + f*x])/(240*f) + (a^2*(4*c^3 - 48*c^2*d - 123*c*d^2 - 64*d^3)*Cos[e + f*x]*(c + d*Sin[
e + f*x])^2)/(120*d*f) + (a^2*(4*c^2 - 48*c*d - 55*d^2)*Cos[e + f*x]*(c + d*Sin[e + f*x])^3)/(120*d*f) + (a^2*
(c - 12*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^4)/(30*d*f) - (a^2*Cos[e + f*x]*(c + d*Sin[e + f*x])^5)/(6*d*f)

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Rubi [A]  time = 0.462589, antiderivative size = 318, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {2763, 2753, 2734} \[ \frac{a^2 \left (-311 c^3 d^2-448 c^2 d^3-48 c^4 d+4 c^5-288 c d^4-64 d^5\right ) \cos (e+f x)}{60 d f}+\frac{a^2 \left (4 c^2-48 c d-55 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{120 d f}+\frac{a^2 \left (-48 c^2 d+4 c^3-123 c d^2-64 d^3\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{120 d f}+\frac{a^2 \left (-438 c^2 d^2-96 c^3 d+8 c^4-464 c d^3-165 d^4\right ) \sin (e+f x) \cos (e+f x)}{240 f}+\frac{1}{16} a^2 x \left (84 c^2 d^2+64 c^3 d+24 c^4+48 c d^3+11 d^4\right )-\frac{a^2 \cos (e+f x) (c+d \sin (e+f x))^5}{6 d f}+\frac{a^2 (c-12 d) \cos (e+f x) (c+d \sin (e+f x))^4}{30 d f} \]

Antiderivative was successfully verified.

[In]

Int[(a + a*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^4,x]

[Out]

(a^2*(24*c^4 + 64*c^3*d + 84*c^2*d^2 + 48*c*d^3 + 11*d^4)*x)/16 + (a^2*(4*c^5 - 48*c^4*d - 311*c^3*d^2 - 448*c
^2*d^3 - 288*c*d^4 - 64*d^5)*Cos[e + f*x])/(60*d*f) + (a^2*(8*c^4 - 96*c^3*d - 438*c^2*d^2 - 464*c*d^3 - 165*d
^4)*Cos[e + f*x]*Sin[e + f*x])/(240*f) + (a^2*(4*c^3 - 48*c^2*d - 123*c*d^2 - 64*d^3)*Cos[e + f*x]*(c + d*Sin[
e + f*x])^2)/(120*d*f) + (a^2*(4*c^2 - 48*c*d - 55*d^2)*Cos[e + f*x]*(c + d*Sin[e + f*x])^3)/(120*d*f) + (a^2*
(c - 12*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^4)/(30*d*f) - (a^2*Cos[e + f*x]*(c + d*Sin[e + f*x])^5)/(6*d*f)

Rule 2763

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> -Si
mp[(b^2*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 2)*(c + d*Sin[e + f*x])^(n + 1))/(d*f*(m + n)), x] + Dist[1/(d*
(m + n)), Int[(a + b*Sin[e + f*x])^(m - 2)*(c + d*Sin[e + f*x])^n*Simp[a*b*c*(m - 2) + b^2*d*(n + 1) + a^2*d*(
m + n) - b*(b*c*(m - 1) - a*d*(3*m + 2*n - 2))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] &&
NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 1] &&  !LtQ[n, -1] && (IntegersQ[2*m, 2*
n] || IntegerQ[m + 1/2] || (IntegerQ[m] && EqQ[c, 0]))

Rule 2753

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> -Simp[(d
*Cos[e + f*x]*(a + b*Sin[e + f*x])^m)/(f*(m + 1)), x] + Dist[1/(m + 1), Int[(a + b*Sin[e + f*x])^(m - 1)*Simp[
b*d*m + a*c*(m + 1) + (a*d*m + b*c*(m + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*
c - a*d, 0] && NeQ[a^2 - b^2, 0] && GtQ[m, 0] && IntegerQ[2*m]

Rule 2734

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[((2*a*c
+ b*d)*x)/2, x] + (-Simp[((b*c + a*d)*Cos[e + f*x])/f, x] - Simp[(b*d*Cos[e + f*x]*Sin[e + f*x])/(2*f), x]) /;
FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]

Rubi steps

\begin{align*} \int (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^4 \, dx &=-\frac{a^2 \cos (e+f x) (c+d \sin (e+f x))^5}{6 d f}+\frac{\int \left (11 a^2 d-a^2 (c-12 d) \sin (e+f x)\right ) (c+d \sin (e+f x))^4 \, dx}{6 d}\\ &=\frac{a^2 (c-12 d) \cos (e+f x) (c+d \sin (e+f x))^4}{30 d f}-\frac{a^2 \cos (e+f x) (c+d \sin (e+f x))^5}{6 d f}+\frac{\int (c+d \sin (e+f x))^3 \left (3 a^2 d (17 c+16 d)-a^2 \left (4 c^2-48 c d-55 d^2\right ) \sin (e+f x)\right ) \, dx}{30 d}\\ &=\frac{a^2 \left (4 c^2-48 c d-55 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{120 d f}+\frac{a^2 (c-12 d) \cos (e+f x) (c+d \sin (e+f x))^4}{30 d f}-\frac{a^2 \cos (e+f x) (c+d \sin (e+f x))^5}{6 d f}+\frac{\int (c+d \sin (e+f x))^2 \left (3 a^2 d \left (64 c^2+112 c d+55 d^2\right )-3 a^2 \left (4 c^3-48 c^2 d-123 c d^2-64 d^3\right ) \sin (e+f x)\right ) \, dx}{120 d}\\ &=\frac{a^2 \left (4 c^3-48 c^2 d-123 c d^2-64 d^3\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{120 d f}+\frac{a^2 \left (4 c^2-48 c d-55 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{120 d f}+\frac{a^2 (c-12 d) \cos (e+f x) (c+d \sin (e+f x))^4}{30 d f}-\frac{a^2 \cos (e+f x) (c+d \sin (e+f x))^5}{6 d f}+\frac{\int (c+d \sin (e+f x)) \left (3 a^2 d \left (184 c^3+432 c^2 d+411 c d^2+128 d^3\right )-3 a^2 \left (8 c^4-96 c^3 d-438 c^2 d^2-464 c d^3-165 d^4\right ) \sin (e+f x)\right ) \, dx}{360 d}\\ &=\frac{1}{16} a^2 \left (24 c^4+64 c^3 d+84 c^2 d^2+48 c d^3+11 d^4\right ) x+\frac{a^2 \left (4 c^5-48 c^4 d-311 c^3 d^2-448 c^2 d^3-288 c d^4-64 d^5\right ) \cos (e+f x)}{60 d f}+\frac{a^2 \left (8 c^4-96 c^3 d-438 c^2 d^2-464 c d^3-165 d^4\right ) \cos (e+f x) \sin (e+f x)}{240 f}+\frac{a^2 \left (4 c^3-48 c^2 d-123 c d^2-64 d^3\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{120 d f}+\frac{a^2 \left (4 c^2-48 c d-55 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{120 d f}+\frac{a^2 (c-12 d) \cos (e+f x) (c+d \sin (e+f x))^4}{30 d f}-\frac{a^2 \cos (e+f x) (c+d \sin (e+f x))^5}{6 d f}\\ \end{align*}

Mathematica [A]  time = 1.38604, size = 262, normalized size = 0.82 \[ -\frac{a^2 \cos (e+f x) \left (30 \left (84 c^2 d^2+64 c^3 d+24 c^4+48 c d^3+11 d^4\right ) \sin ^{-1}\left (\frac{\sqrt{1-\sin (e+f x)}}{\sqrt{2}}\right )+\sqrt{\cos ^2(e+f x)} \left (10 d^2 \left (36 c^2+48 c d+11 d^2\right ) \sin ^3(e+f x)+64 d \left (15 c^2 d+5 c^3+9 c d^2+2 d^3\right ) \sin ^2(e+f x)+15 \left (84 c^2 d^2+64 c^3 d+8 c^4+48 c d^3+11 d^4\right ) \sin (e+f x)+32 \left (60 c^2 d^2+50 c^3 d+15 c^4+36 c d^3+8 d^4\right )+96 d^3 (2 c+d) \sin ^4(e+f x)+40 d^4 \sin ^5(e+f x)\right )\right )}{240 f \sqrt{\cos ^2(e+f x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + a*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^4,x]

[Out]

-(a^2*Cos[e + f*x]*(30*(24*c^4 + 64*c^3*d + 84*c^2*d^2 + 48*c*d^3 + 11*d^4)*ArcSin[Sqrt[1 - Sin[e + f*x]]/Sqrt
[2]] + Sqrt[Cos[e + f*x]^2]*(32*(15*c^4 + 50*c^3*d + 60*c^2*d^2 + 36*c*d^3 + 8*d^4) + 15*(8*c^4 + 64*c^3*d + 8
4*c^2*d^2 + 48*c*d^3 + 11*d^4)*Sin[e + f*x] + 64*d*(5*c^3 + 15*c^2*d + 9*c*d^2 + 2*d^3)*Sin[e + f*x]^2 + 10*d^
2*(36*c^2 + 48*c*d + 11*d^2)*Sin[e + f*x]^3 + 96*d^3*(2*c + d)*Sin[e + f*x]^4 + 40*d^4*Sin[e + f*x]^5)))/(240*
f*Sqrt[Cos[e + f*x]^2])

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Maple [A]  time = 0.056, size = 462, normalized size = 1.5 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+a*sin(f*x+e))^2*(c+d*sin(f*x+e))^4,x)

[Out]

1/f*(a^2*c^4*(-1/2*sin(f*x+e)*cos(f*x+e)+1/2*f*x+1/2*e)-4/3*a^2*c^3*d*(2+sin(f*x+e)^2)*cos(f*x+e)+6*a^2*c^2*d^
2*(-1/4*(sin(f*x+e)^3+3/2*sin(f*x+e))*cos(f*x+e)+3/8*f*x+3/8*e)-4/5*a^2*c*d^3*(8/3+sin(f*x+e)^4+4/3*sin(f*x+e)
^2)*cos(f*x+e)+a^2*d^4*(-1/6*(sin(f*x+e)^5+5/4*sin(f*x+e)^3+15/8*sin(f*x+e))*cos(f*x+e)+5/16*f*x+5/16*e)-2*a^2
*c^4*cos(f*x+e)+8*a^2*c^3*d*(-1/2*sin(f*x+e)*cos(f*x+e)+1/2*f*x+1/2*e)-4*a^2*c^2*d^2*(2+sin(f*x+e)^2)*cos(f*x+
e)+8*a^2*c*d^3*(-1/4*(sin(f*x+e)^3+3/2*sin(f*x+e))*cos(f*x+e)+3/8*f*x+3/8*e)-2/5*a^2*d^4*(8/3+sin(f*x+e)^4+4/3
*sin(f*x+e)^2)*cos(f*x+e)+a^2*c^4*(f*x+e)-4*a^2*c^3*d*cos(f*x+e)+6*a^2*c^2*d^2*(-1/2*sin(f*x+e)*cos(f*x+e)+1/2
*f*x+1/2*e)-4/3*a^2*c*d^3*(2+sin(f*x+e)^2)*cos(f*x+e)+a^2*d^4*(-1/4*(sin(f*x+e)^3+3/2*sin(f*x+e))*cos(f*x+e)+3
/8*f*x+3/8*e))

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Maxima [A]  time = 1.22161, size = 609, normalized size = 1.92 \begin{align*} \frac{240 \,{\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} a^{2} c^{4} + 960 \,{\left (f x + e\right )} a^{2} c^{4} + 1280 \,{\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} a^{2} c^{3} d + 1920 \,{\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} a^{2} c^{3} d + 3840 \,{\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} a^{2} c^{2} d^{2} + 180 \,{\left (12 \, f x + 12 \, e + \sin \left (4 \, f x + 4 \, e\right ) - 8 \, \sin \left (2 \, f x + 2 \, e\right )\right )} a^{2} c^{2} d^{2} + 1440 \,{\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} a^{2} c^{2} d^{2} - 256 \,{\left (3 \, \cos \left (f x + e\right )^{5} - 10 \, \cos \left (f x + e\right )^{3} + 15 \, \cos \left (f x + e\right )\right )} a^{2} c d^{3} + 1280 \,{\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} a^{2} c d^{3} + 240 \,{\left (12 \, f x + 12 \, e + \sin \left (4 \, f x + 4 \, e\right ) - 8 \, \sin \left (2 \, f x + 2 \, e\right )\right )} a^{2} c d^{3} - 128 \,{\left (3 \, \cos \left (f x + e\right )^{5} - 10 \, \cos \left (f x + e\right )^{3} + 15 \, \cos \left (f x + e\right )\right )} a^{2} d^{4} + 5 \,{\left (4 \, \sin \left (2 \, f x + 2 \, e\right )^{3} + 60 \, f x + 60 \, e + 9 \, \sin \left (4 \, f x + 4 \, e\right ) - 48 \, \sin \left (2 \, f x + 2 \, e\right )\right )} a^{2} d^{4} + 30 \,{\left (12 \, f x + 12 \, e + \sin \left (4 \, f x + 4 \, e\right ) - 8 \, \sin \left (2 \, f x + 2 \, e\right )\right )} a^{2} d^{4} - 1920 \, a^{2} c^{4} \cos \left (f x + e\right ) - 3840 \, a^{2} c^{3} d \cos \left (f x + e\right )}{960 \, f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(f*x+e))^2*(c+d*sin(f*x+e))^4,x, algorithm="maxima")

[Out]

1/960*(240*(2*f*x + 2*e - sin(2*f*x + 2*e))*a^2*c^4 + 960*(f*x + e)*a^2*c^4 + 1280*(cos(f*x + e)^3 - 3*cos(f*x
+ e))*a^2*c^3*d + 1920*(2*f*x + 2*e - sin(2*f*x + 2*e))*a^2*c^3*d + 3840*(cos(f*x + e)^3 - 3*cos(f*x + e))*a^
2*c^2*d^2 + 180*(12*f*x + 12*e + sin(4*f*x + 4*e) - 8*sin(2*f*x + 2*e))*a^2*c^2*d^2 + 1440*(2*f*x + 2*e - sin(
2*f*x + 2*e))*a^2*c^2*d^2 - 256*(3*cos(f*x + e)^5 - 10*cos(f*x + e)^3 + 15*cos(f*x + e))*a^2*c*d^3 + 1280*(cos
(f*x + e)^3 - 3*cos(f*x + e))*a^2*c*d^3 + 240*(12*f*x + 12*e + sin(4*f*x + 4*e) - 8*sin(2*f*x + 2*e))*a^2*c*d^
3 - 128*(3*cos(f*x + e)^5 - 10*cos(f*x + e)^3 + 15*cos(f*x + e))*a^2*d^4 + 5*(4*sin(2*f*x + 2*e)^3 + 60*f*x +
60*e + 9*sin(4*f*x + 4*e) - 48*sin(2*f*x + 2*e))*a^2*d^4 + 30*(12*f*x + 12*e + sin(4*f*x + 4*e) - 8*sin(2*f*x
+ 2*e))*a^2*d^4 - 1920*a^2*c^4*cos(f*x + e) - 3840*a^2*c^3*d*cos(f*x + e))/f

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Fricas [A]  time = 1.89266, size = 662, normalized size = 2.08 \begin{align*} -\frac{96 \,{\left (2 \, a^{2} c d^{3} + a^{2} d^{4}\right )} \cos \left (f x + e\right )^{5} - 320 \,{\left (a^{2} c^{3} d + 3 \, a^{2} c^{2} d^{2} + 3 \, a^{2} c d^{3} + a^{2} d^{4}\right )} \cos \left (f x + e\right )^{3} - 15 \,{\left (24 \, a^{2} c^{4} + 64 \, a^{2} c^{3} d + 84 \, a^{2} c^{2} d^{2} + 48 \, a^{2} c d^{3} + 11 \, a^{2} d^{4}\right )} f x + 480 \,{\left (a^{2} c^{4} + 4 \, a^{2} c^{3} d + 6 \, a^{2} c^{2} d^{2} + 4 \, a^{2} c d^{3} + a^{2} d^{4}\right )} \cos \left (f x + e\right ) + 5 \,{\left (8 \, a^{2} d^{4} \cos \left (f x + e\right )^{5} - 2 \,{\left (36 \, a^{2} c^{2} d^{2} + 48 \, a^{2} c d^{3} + 19 \, a^{2} d^{4}\right )} \cos \left (f x + e\right )^{3} + 3 \,{\left (8 \, a^{2} c^{4} + 64 \, a^{2} c^{3} d + 108 \, a^{2} c^{2} d^{2} + 80 \, a^{2} c d^{3} + 21 \, a^{2} d^{4}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{240 \, f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(f*x+e))^2*(c+d*sin(f*x+e))^4,x, algorithm="fricas")

[Out]

-1/240*(96*(2*a^2*c*d^3 + a^2*d^4)*cos(f*x + e)^5 - 320*(a^2*c^3*d + 3*a^2*c^2*d^2 + 3*a^2*c*d^3 + a^2*d^4)*co
s(f*x + e)^3 - 15*(24*a^2*c^4 + 64*a^2*c^3*d + 84*a^2*c^2*d^2 + 48*a^2*c*d^3 + 11*a^2*d^4)*f*x + 480*(a^2*c^4
+ 4*a^2*c^3*d + 6*a^2*c^2*d^2 + 4*a^2*c*d^3 + a^2*d^4)*cos(f*x + e) + 5*(8*a^2*d^4*cos(f*x + e)^5 - 2*(36*a^2*
c^2*d^2 + 48*a^2*c*d^3 + 19*a^2*d^4)*cos(f*x + e)^3 + 3*(8*a^2*c^4 + 64*a^2*c^3*d + 108*a^2*c^2*d^2 + 80*a^2*c
*d^3 + 21*a^2*d^4)*cos(f*x + e))*sin(f*x + e))/f

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Sympy [A]  time = 8.36999, size = 1136, normalized size = 3.57 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(f*x+e))**2*(c+d*sin(f*x+e))**4,x)

[Out]

Piecewise((a**2*c**4*x*sin(e + f*x)**2/2 + a**2*c**4*x*cos(e + f*x)**2/2 + a**2*c**4*x - a**2*c**4*sin(e + f*x
)*cos(e + f*x)/(2*f) - 2*a**2*c**4*cos(e + f*x)/f + 4*a**2*c**3*d*x*sin(e + f*x)**2 + 4*a**2*c**3*d*x*cos(e +
f*x)**2 - 4*a**2*c**3*d*sin(e + f*x)**2*cos(e + f*x)/f - 4*a**2*c**3*d*sin(e + f*x)*cos(e + f*x)/f - 8*a**2*c*
*3*d*cos(e + f*x)**3/(3*f) - 4*a**2*c**3*d*cos(e + f*x)/f + 9*a**2*c**2*d**2*x*sin(e + f*x)**4/4 + 9*a**2*c**2
*d**2*x*sin(e + f*x)**2*cos(e + f*x)**2/2 + 3*a**2*c**2*d**2*x*sin(e + f*x)**2 + 9*a**2*c**2*d**2*x*cos(e + f*
x)**4/4 + 3*a**2*c**2*d**2*x*cos(e + f*x)**2 - 15*a**2*c**2*d**2*sin(e + f*x)**3*cos(e + f*x)/(4*f) - 12*a**2*
c**2*d**2*sin(e + f*x)**2*cos(e + f*x)/f - 9*a**2*c**2*d**2*sin(e + f*x)*cos(e + f*x)**3/(4*f) - 3*a**2*c**2*d
**2*sin(e + f*x)*cos(e + f*x)/f - 8*a**2*c**2*d**2*cos(e + f*x)**3/f + 3*a**2*c*d**3*x*sin(e + f*x)**4 + 6*a**
2*c*d**3*x*sin(e + f*x)**2*cos(e + f*x)**2 + 3*a**2*c*d**3*x*cos(e + f*x)**4 - 4*a**2*c*d**3*sin(e + f*x)**4*c
os(e + f*x)/f - 5*a**2*c*d**3*sin(e + f*x)**3*cos(e + f*x)/f - 16*a**2*c*d**3*sin(e + f*x)**2*cos(e + f*x)**3/
(3*f) - 4*a**2*c*d**3*sin(e + f*x)**2*cos(e + f*x)/f - 3*a**2*c*d**3*sin(e + f*x)*cos(e + f*x)**3/f - 32*a**2*
c*d**3*cos(e + f*x)**5/(15*f) - 8*a**2*c*d**3*cos(e + f*x)**3/(3*f) + 5*a**2*d**4*x*sin(e + f*x)**6/16 + 15*a*
*2*d**4*x*sin(e + f*x)**4*cos(e + f*x)**2/16 + 3*a**2*d**4*x*sin(e + f*x)**4/8 + 15*a**2*d**4*x*sin(e + f*x)**
2*cos(e + f*x)**4/16 + 3*a**2*d**4*x*sin(e + f*x)**2*cos(e + f*x)**2/4 + 5*a**2*d**4*x*cos(e + f*x)**6/16 + 3*
a**2*d**4*x*cos(e + f*x)**4/8 - 11*a**2*d**4*sin(e + f*x)**5*cos(e + f*x)/(16*f) - 2*a**2*d**4*sin(e + f*x)**4
*cos(e + f*x)/f - 5*a**2*d**4*sin(e + f*x)**3*cos(e + f*x)**3/(6*f) - 5*a**2*d**4*sin(e + f*x)**3*cos(e + f*x)
/(8*f) - 8*a**2*d**4*sin(e + f*x)**2*cos(e + f*x)**3/(3*f) - 5*a**2*d**4*sin(e + f*x)*cos(e + f*x)**5/(16*f) -
3*a**2*d**4*sin(e + f*x)*cos(e + f*x)**3/(8*f) - 16*a**2*d**4*cos(e + f*x)**5/(15*f), Ne(f, 0)), (x*(c + d*si
n(e))**4*(a*sin(e) + a)**2, True))

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Giac [A]  time = 1.39704, size = 618, normalized size = 1.94 \begin{align*} \frac{a^{2} c d^{3} \cos \left (3 \, f x + 3 \, e\right )}{3 \, f} - \frac{a^{2} d^{4} \sin \left (6 \, f x + 6 \, e\right )}{192 \, f} + \frac{a^{2} d^{4} \sin \left (4 \, f x + 4 \, e\right )}{32 \, f} + \frac{1}{16} \,{\left (8 \, a^{2} c^{4} + 64 \, a^{2} c^{3} d + 36 \, a^{2} c^{2} d^{2} + 48 \, a^{2} c d^{3} + 5 \, a^{2} d^{4}\right )} x + \frac{1}{8} \,{\left (8 \, a^{2} c^{4} + 24 \, a^{2} c^{2} d^{2} + 3 \, a^{2} d^{4}\right )} x - \frac{{\left (2 \, a^{2} c d^{3} + a^{2} d^{4}\right )} \cos \left (5 \, f x + 5 \, e\right )}{40 \, f} + \frac{{\left (8 \, a^{2} c^{3} d + 24 \, a^{2} c^{2} d^{2} + 10 \, a^{2} c d^{3} + 5 \, a^{2} d^{4}\right )} \cos \left (3 \, f x + 3 \, e\right )}{24 \, f} - \frac{{\left (8 \, a^{2} c^{4} + 12 \, a^{2} c^{3} d + 36 \, a^{2} c^{2} d^{2} + 10 \, a^{2} c d^{3} + 5 \, a^{2} d^{4}\right )} \cos \left (f x + e\right )}{4 \, f} - \frac{{\left (4 \, a^{2} c^{3} d + 3 \, a^{2} c d^{3}\right )} \cos \left (f x + e\right )}{f} + \frac{{\left (12 \, a^{2} c^{2} d^{2} + 16 \, a^{2} c d^{3} + 3 \, a^{2} d^{4}\right )} \sin \left (4 \, f x + 4 \, e\right )}{64 \, f} - \frac{{\left (16 \, a^{2} c^{4} + 128 \, a^{2} c^{3} d + 96 \, a^{2} c^{2} d^{2} + 128 \, a^{2} c d^{3} + 15 \, a^{2} d^{4}\right )} \sin \left (2 \, f x + 2 \, e\right )}{64 \, f} - \frac{{\left (6 \, a^{2} c^{2} d^{2} + a^{2} d^{4}\right )} \sin \left (2 \, f x + 2 \, e\right )}{4 \, f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(f*x+e))^2*(c+d*sin(f*x+e))^4,x, algorithm="giac")

[Out]

1/3*a^2*c*d^3*cos(3*f*x + 3*e)/f - 1/192*a^2*d^4*sin(6*f*x + 6*e)/f + 1/32*a^2*d^4*sin(4*f*x + 4*e)/f + 1/16*(
8*a^2*c^4 + 64*a^2*c^3*d + 36*a^2*c^2*d^2 + 48*a^2*c*d^3 + 5*a^2*d^4)*x + 1/8*(8*a^2*c^4 + 24*a^2*c^2*d^2 + 3*
a^2*d^4)*x - 1/40*(2*a^2*c*d^3 + a^2*d^4)*cos(5*f*x + 5*e)/f + 1/24*(8*a^2*c^3*d + 24*a^2*c^2*d^2 + 10*a^2*c*d
^3 + 5*a^2*d^4)*cos(3*f*x + 3*e)/f - 1/4*(8*a^2*c^4 + 12*a^2*c^3*d + 36*a^2*c^2*d^2 + 10*a^2*c*d^3 + 5*a^2*d^4
)*cos(f*x + e)/f - (4*a^2*c^3*d + 3*a^2*c*d^3)*cos(f*x + e)/f + 1/64*(12*a^2*c^2*d^2 + 16*a^2*c*d^3 + 3*a^2*d^
4)*sin(4*f*x + 4*e)/f - 1/64*(16*a^2*c^4 + 128*a^2*c^3*d + 96*a^2*c^2*d^2 + 128*a^2*c*d^3 + 15*a^2*d^4)*sin(2*
f*x + 2*e)/f - 1/4*(6*a^2*c^2*d^2 + a^2*d^4)*sin(2*f*x + 2*e)/f