\(\int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))^2}{(a+a \sin (e+f x))^3} \, dx\) [280]

Optimal result

Integrand size = 35, antiderivative size = 164 \[ \int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))^2}{(a+a \sin (e+f x))^3} \, dx=\frac {B d^2 x}{a^3}-\frac {(c-d) (B (3 c-7 d)+2 A (c+d)) \cos (e+f x)}{15 a f (a+a \sin (e+f x))^2}-\frac {\left (B \left (3 c^2+14 c d-29 d^2\right )+2 A \left (c^2+3 c d+2 d^2\right )\right ) \cos (e+f x)}{15 f \left (a^3+a^3 \sin (e+f x)\right )}-\frac {(A-B) \cos (e+f x) (c+d \sin (e+f x))^2}{5 f (a+a \sin (e+f x))^3} \]

[Out]

B*d^2*x/a^3-1/15*(c-d)*(B*(3*c-7*d)+2*A*(c+d))*cos(f*x+e)/a/f/(a+a*sin(f*x+e))^2-1/15*(B*(3*c^2+14*c*d-29*d^2)
+2*A*(c^2+3*c*d+2*d^2))*cos(f*x+e)/f/(a^3+a^3*sin(f*x+e))-1/5*(A-B)*cos(f*x+e)*(c+d*sin(f*x+e))^2/f/(a+a*sin(f
*x+e))^3

Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3056, 3047, 3098, 2814, 2727} \[ \int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))^2}{(a+a \sin (e+f x))^3} \, dx=-\frac {\left (2 A \left (c^2+3 c d+2 d^2\right )+B \left (3 c^2+14 c d-29 d^2\right )\right ) \cos (e+f x)}{15 f \left (a^3 \sin (e+f x)+a^3\right )}+\frac {B d^2 x}{a^3}-\frac {(A-B) \cos (e+f x) (c+d \sin (e+f x))^2}{5 f (a \sin (e+f x)+a)^3}-\frac {(c-d) (2 A (c+d)+B (3 c-7 d)) \cos (e+f x)}{15 a f (a \sin (e+f x)+a)^2} \]

[In]

Int[((A + B*Sin[e + f*x])*(c + d*Sin[e + f*x])^2)/(a + a*Sin[e + f*x])^3,x]

[Out]

(B*d^2*x)/a^3 - ((c - d)*(B*(3*c - 7*d) + 2*A*(c + d))*Cos[e + f*x])/(15*a*f*(a + a*Sin[e + f*x])^2) - ((B*(3*
c^2 + 14*c*d - 29*d^2) + 2*A*(c^2 + 3*c*d + 2*d^2))*Cos[e + f*x])/(15*f*(a^3 + a^3*Sin[e + f*x])) - ((A - B)*C
os[e + f*x]*(c + d*Sin[e + f*x])^2)/(5*f*(a + a*Sin[e + f*x])^3)

Rule 2727

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> Simp[-Cos[c + d*x]/(d*(b + a*Sin[c + d*x])), x]
/; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]

Rule 2814

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

Rule 3047

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

Rule 3056

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

Rule 3098

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_
.)*(x_)]^2), x_Symbol] :> Simp[(A*b - a*B + b*C)*Cos[e + f*x]*((a + b*Sin[e + f*x])^m/(a*f*(2*m + 1))), x] + D
ist[1/(a^2*(2*m + 1)), Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[a*A*(m + 1) + m*(b*B - a*C) + b*C*(2*m + 1)*Sin[e
+ f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C}, x] && LtQ[m, -1] && EqQ[a^2 - b^2, 0]

Rubi steps \begin{align*} \text {integral}& = -\frac {(A-B) \cos (e+f x) (c+d \sin (e+f x))^2}{5 f (a+a \sin (e+f x))^3}+\frac {\int \frac {(c+d \sin (e+f x)) (a (B (3 c-2 d)+2 A (c+d))+5 a B d \sin (e+f x))}{(a+a \sin (e+f x))^2} \, dx}{5 a^2} \\ & = -\frac {(A-B) \cos (e+f x) (c+d \sin (e+f x))^2}{5 f (a+a \sin (e+f x))^3}+\frac {\int \frac {a c (B (3 c-2 d)+2 A (c+d))+(5 a B c d+a d (B (3 c-2 d)+2 A (c+d))) \sin (e+f x)+5 a B d^2 \sin ^2(e+f x)}{(a+a \sin (e+f x))^2} \, dx}{5 a^2} \\ & = -\frac {(c-d) (B (3 c-7 d)+2 A (c+d)) \cos (e+f x)}{15 a f (a+a \sin (e+f x))^2}-\frac {(A-B) \cos (e+f x) (c+d \sin (e+f x))^2}{5 f (a+a \sin (e+f x))^3}-\frac {\int \frac {-a^2 \left (B \left (3 c^2+14 c d-14 d^2\right )+2 A \left (c^2+3 c d+2 d^2\right )\right )-15 a^2 B d^2 \sin (e+f x)}{a+a \sin (e+f x)} \, dx}{15 a^4} \\ & = \frac {B d^2 x}{a^3}-\frac {(c-d) (B (3 c-7 d)+2 A (c+d)) \cos (e+f x)}{15 a f (a+a \sin (e+f x))^2}-\frac {(A-B) \cos (e+f x) (c+d \sin (e+f x))^2}{5 f (a+a \sin (e+f x))^3}+\frac {\left (B \left (3 c^2+14 c d-29 d^2\right )+2 A \left (c^2+3 c d+2 d^2\right )\right ) \int \frac {1}{a+a \sin (e+f x)} \, dx}{15 a^2} \\ & = \frac {B d^2 x}{a^3}-\frac {(c-d) (B (3 c-7 d)+2 A (c+d)) \cos (e+f x)}{15 a f (a+a \sin (e+f x))^2}-\frac {\left (B \left (3 c^2+14 c d-29 d^2\right )+2 A \left (c^2+3 c d+2 d^2\right )\right ) \cos (e+f x)}{15 f \left (a^3+a^3 \sin (e+f x)\right )}-\frac {(A-B) \cos (e+f x) (c+d \sin (e+f x))^2}{5 f (a+a \sin (e+f x))^3} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(514\) vs. \(2(164)=328\).

Time = 0.81 (sec) , antiderivative size = 514, normalized size of antiderivative = 3.13 \[ \int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))^2}{(a+a \sin (e+f x))^3} \, dx=\frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (30 \left (2 A d (c+d)+B \left (c^2+4 c d+d^2 (-9+5 e+5 f x)\right )\right ) \cos \left (\frac {1}{2} (e+f x)\right )-5 \left (4 A \left (c^2+3 c d+2 d^2\right )+B \left (6 c^2+16 c d+d^2 (-46+15 e+15 f x)\right )\right ) \cos \left (\frac {3}{2} (e+f x)\right )-15 B d^2 e \cos \left (\frac {5}{2} (e+f x)\right )-15 B d^2 f x \cos \left (\frac {5}{2} (e+f x)\right )+40 A c^2 \sin \left (\frac {1}{2} (e+f x)\right )+30 B c^2 \sin \left (\frac {1}{2} (e+f x)\right )+60 A c d \sin \left (\frac {1}{2} (e+f x)\right )+160 B c d \sin \left (\frac {1}{2} (e+f x)\right )+80 A d^2 \sin \left (\frac {1}{2} (e+f x)\right )-370 B d^2 \sin \left (\frac {1}{2} (e+f x)\right )+150 B d^2 e \sin \left (\frac {1}{2} (e+f x)\right )+150 B d^2 f x \sin \left (\frac {1}{2} (e+f x)\right )+60 B c d \sin \left (\frac {3}{2} (e+f x)\right )+30 A d^2 \sin \left (\frac {3}{2} (e+f x)\right )-90 B d^2 \sin \left (\frac {3}{2} (e+f x)\right )+75 B d^2 e \sin \left (\frac {3}{2} (e+f x)\right )+75 B d^2 f x \sin \left (\frac {3}{2} (e+f x)\right )-4 A c^2 \sin \left (\frac {5}{2} (e+f x)\right )-6 B c^2 \sin \left (\frac {5}{2} (e+f x)\right )-12 A c d \sin \left (\frac {5}{2} (e+f x)\right )-28 B c d \sin \left (\frac {5}{2} (e+f x)\right )-14 A d^2 \sin \left (\frac {5}{2} (e+f x)\right )+64 B d^2 \sin \left (\frac {5}{2} (e+f x)\right )-15 B d^2 e \sin \left (\frac {5}{2} (e+f x)\right )-15 B d^2 f x \sin \left (\frac {5}{2} (e+f x)\right )\right )}{60 a^3 f (1+\sin (e+f x))^3} \]

[In]

Integrate[((A + B*Sin[e + f*x])*(c + d*Sin[e + f*x])^2)/(a + a*Sin[e + f*x])^3,x]

[Out]

((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(30*(2*A*d*(c + d) + B*(c^2 + 4*c*d + d^2*(-9 + 5*e + 5*f*x)))*Cos[(e +
f*x)/2] - 5*(4*A*(c^2 + 3*c*d + 2*d^2) + B*(6*c^2 + 16*c*d + d^2*(-46 + 15*e + 15*f*x)))*Cos[(3*(e + f*x))/2]
- 15*B*d^2*e*Cos[(5*(e + f*x))/2] - 15*B*d^2*f*x*Cos[(5*(e + f*x))/2] + 40*A*c^2*Sin[(e + f*x)/2] + 30*B*c^2*
Sin[(e + f*x)/2] + 60*A*c*d*Sin[(e + f*x)/2] + 160*B*c*d*Sin[(e + f*x)/2] + 80*A*d^2*Sin[(e + f*x)/2] - 370*B*
d^2*Sin[(e + f*x)/2] + 150*B*d^2*e*Sin[(e + f*x)/2] + 150*B*d^2*f*x*Sin[(e + f*x)/2] + 60*B*c*d*Sin[(3*(e + f*
x))/2] + 30*A*d^2*Sin[(3*(e + f*x))/2] - 90*B*d^2*Sin[(3*(e + f*x))/2] + 75*B*d^2*e*Sin[(3*(e + f*x))/2] + 75*
B*d^2*f*x*Sin[(3*(e + f*x))/2] - 4*A*c^2*Sin[(5*(e + f*x))/2] - 6*B*c^2*Sin[(5*(e + f*x))/2] - 12*A*c*d*Sin[(5
*(e + f*x))/2] - 28*B*c*d*Sin[(5*(e + f*x))/2] - 14*A*d^2*Sin[(5*(e + f*x))/2] + 64*B*d^2*Sin[(5*(e + f*x))/2]
- 15*B*d^2*e*Sin[(5*(e + f*x))/2] - 15*B*d^2*f*x*Sin[(5*(e + f*x))/2]))/(60*a^3*f*(1 + Sin[e + f*x])^3)

Maple [A] (verified)

Time = 0.73 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.46

method	result	size
parallelrisch	\(\frac {15 B x \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right ) d^{2} f +\left (\left (75 f x +30\right ) B \,d^{2}-30 A \,c^{2}\right ) \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (\left (150 f x +150\right ) B \,d^{2}-60 A c d -60 c^{2} \left (A +\frac {B}{2}\right )\right ) \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (\left (150 f x B -40 A +290 B \right ) d^{2}-60 \left (A +\frac {4 B}{3}\right ) c d -80 c^{2} \left (A +\frac {3 B}{8}\right )\right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (\left (75 f x B -20 A +190 B \right ) d^{2}-60 c \left (A +\frac {2 B}{3}\right ) d -40 c^{2} \left (A +\frac {3 B}{4}\right )\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+\left (15 f x B -4 A +44 B \right ) d^{2}-12 c \left (A +\frac {2 B}{3}\right ) d -14 c^{2} \left (A +\frac {3 B}{7}\right )}{15 f \,a^{3} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{5}}\)	\(240\)
derivativedivides	\(\frac {-\frac {2 \left (A \,c^{2}-d^{2} B \right )}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1}-\frac {-4 A \,c^{2}+4 A c d +2 B \,c^{2}-2 d^{2} B}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}-\frac {-8 A \,c^{2}+16 A c d -8 A \,d^{2}+8 B \,c^{2}-16 c d B +8 d^{2} B}{2 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{4}}-\frac {2 \left (4 A \,c^{2}-8 A c d +4 A \,d^{2}-4 B \,c^{2}+8 c d B -4 d^{2} B \right )}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{5}}-\frac {2 \left (8 A \,c^{2}-12 A c d +4 A \,d^{2}-6 B \,c^{2}+8 c d B -2 d^{2} B \right )}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}+2 d^{2} B \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a^{3} f}\)	\(241\)
default	\(\frac {-\frac {2 \left (A \,c^{2}-d^{2} B \right )}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1}-\frac {-4 A \,c^{2}+4 A c d +2 B \,c^{2}-2 d^{2} B}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}-\frac {-8 A \,c^{2}+16 A c d -8 A \,d^{2}+8 B \,c^{2}-16 c d B +8 d^{2} B}{2 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{4}}-\frac {2 \left (4 A \,c^{2}-8 A c d +4 A \,d^{2}-4 B \,c^{2}+8 c d B -4 d^{2} B \right )}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{5}}-\frac {2 \left (8 A \,c^{2}-12 A c d +4 A \,d^{2}-6 B \,c^{2}+8 c d B -2 d^{2} B \right )}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}+2 d^{2} B \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a^{3} f}\)	\(241\)
risch	\(\frac {B \,d^{2} x}{a^{3}}-\frac {2 \left (14 c d B +2 A \,c^{2}+3 B \,c^{2}-32 d^{2} B +30 i A \,d^{2} {\mathrm e}^{3 i \left (f x +e \right )}+15 i B \,c^{2} {\mathrm e}^{3 i \left (f x +e \right )}+30 B c d \,{\mathrm e}^{4 i \left (f x +e \right )}-135 i B \,d^{2} {\mathrm e}^{3 i \left (f x +e \right )}+30 i A c d \,{\mathrm e}^{3 i \left (f x +e \right )}+60 i B c d \,{\mathrm e}^{3 i \left (f x +e \right )}+15 A \,d^{2} {\mathrm e}^{4 i \left (f x +e \right )}-45 B \,d^{2} {\mathrm e}^{4 i \left (f x +e \right )}-20 A \,c^{2} {\mathrm e}^{2 i \left (f x +e \right )}-10 i A \,c^{2} {\mathrm e}^{i \left (f x +e \right )}-20 i A \,d^{2} {\mathrm e}^{i \left (f x +e \right )}-15 i B \,c^{2} {\mathrm e}^{i \left (f x +e \right )}-80 B c d \,{\mathrm e}^{2 i \left (f x +e \right )}+115 i B \,d^{2} {\mathrm e}^{i \left (f x +e \right )}-30 A c d \,{\mathrm e}^{2 i \left (f x +e \right )}-40 A \,d^{2} {\mathrm e}^{2 i \left (f x +e \right )}+185 B \,d^{2} {\mathrm e}^{2 i \left (f x +e \right )}-15 B \,c^{2} {\mathrm e}^{2 i \left (f x +e \right )}-30 i A c d \,{\mathrm e}^{i \left (f x +e \right )}-40 i B c d \,{\mathrm e}^{i \left (f x +e \right )}+6 A c d +7 A \,d^{2}\right )}{15 f \,a^{3} \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )^{5}}\)	\(372\)
norman	\(\frac {\frac {x \,d^{2} B}{a}+\frac {x \,d^{2} B \left (\tan ^{11}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a}-\frac {14 A \,c^{2}+12 A c d +4 A \,d^{2}+6 B \,c^{2}+8 c d B -44 d^{2} B}{15 f a}-\frac {\left (2 A \,c^{2}-2 d^{2} B \right ) \left (\tan ^{10}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f a}-\frac {\left (4 A \,c^{2}+4 A c d +2 B \,c^{2}-10 d^{2} B \right ) \left (\tan ^{9}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f a}-\frac {\left (8 A \,c^{2}+12 A c d +4 A \,d^{2}+6 B \,c^{2}+8 c d B -38 d^{2} B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{3 f a}-\frac {2 \left (10 A \,c^{2}+12 A c d +2 A \,d^{2}+6 B \,c^{2}+4 c d B -34 d^{2} B \right ) \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f a}-\frac {\left (12 A \,c^{2}+16 A c d +4 A \,d^{2}+8 B \,c^{2}+8 c d B -48 d^{2} B \right ) \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f a}-\frac {2 \left (22 A \,c^{2}+24 A c d +2 A \,d^{2}+12 B \,c^{2}+4 c d B -64 d^{2} B \right ) \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 f a}-\frac {\left (34 A \,c^{2}+12 A c d +8 A \,d^{2}+6 B \,c^{2}+16 c d B -76 d^{2} B \right ) \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 f a}-\frac {2 \left (52 A \,c^{2}+36 A c d +22 A \,d^{2}+18 B \,c^{2}+44 c d B -172 d^{2} B \right ) \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{5 f a}-\frac {\left (122 A \,c^{2}+96 A c d +52 A \,d^{2}+48 B \,c^{2}+104 c d B -422 d^{2} B \right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{15 f a}-\frac {2 \left (172 A \,c^{2}+96 A c d +62 A \,d^{2}+48 B \,c^{2}+124 c d B -502 d^{2} B \right ) \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{15 f a}+\frac {5 x \,d^{2} B \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{a}+\frac {13 x \,d^{2} B \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a}+\frac {25 x \,d^{2} B \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a}+\frac {38 x \,d^{2} B \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a}+\frac {46 x \,d^{2} B \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a}+\frac {46 x \,d^{2} B \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a}+\frac {38 x \,d^{2} B \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a}+\frac {25 x \,d^{2} B \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a}+\frac {13 x \,d^{2} B \left (\tan ^{9}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a}+\frac {5 x \,d^{2} B \left (\tan ^{10}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a}}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{3} a^{2} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{5}}\)	\(819\)

[In]

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

[Out]

1/15*(15*B*x*tan(1/2*f*x+1/2*e)^5*d^2*f+((75*f*x+30)*B*d^2-30*A*c^2)*tan(1/2*f*x+1/2*e)^4+((150*f*x+150)*B*d^2
-60*A*c*d-60*c^2*(A+1/2*B))*tan(1/2*f*x+1/2*e)^3+((150*B*f*x-40*A+290*B)*d^2-60*(A+4/3*B)*c*d-80*c^2*(A+3/8*B)
)*tan(1/2*f*x+1/2*e)^2+((75*B*f*x-20*A+190*B)*d^2-60*c*(A+2/3*B)*d-40*c^2*(A+3/4*B))*tan(1/2*f*x+1/2*e)+(15*B*
f*x-4*A+44*B)*d^2-12*c*(A+2/3*B)*d-14*c^2*(A+3/7*B))/f/a^3/(tan(1/2*f*x+1/2*e)+1)^5

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 432 vs. \(2 (158) = 316\).

Time = 0.26 (sec) , antiderivative size = 432, normalized size of antiderivative = 2.63 \[ \int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))^2}{(a+a \sin (e+f x))^3} \, dx=-\frac {60 \, B d^{2} f x - {\left (15 \, B d^{2} f x - {\left (2 \, A + 3 \, B\right )} c^{2} - 2 \, {\left (3 \, A + 7 \, B\right )} c d - {\left (7 \, A - 32 \, B\right )} d^{2}\right )} \cos \left (f x + e\right )^{3} - 3 \, {\left (A - B\right )} c^{2} + 6 \, {\left (A - B\right )} c d - 3 \, {\left (A - B\right )} d^{2} - {\left (45 \, B d^{2} f x + 2 \, {\left (2 \, A + 3 \, B\right )} c^{2} + 2 \, {\left (6 \, A - B\right )} c d - {\left (A + 19 \, B\right )} d^{2}\right )} \cos \left (f x + e\right )^{2} + 3 \, {\left (10 \, B d^{2} f x - {\left (3 \, A + 2 \, B\right )} c^{2} - 2 \, {\left (2 \, A + 3 \, B\right )} c d - 3 \, {\left (A - 6 \, B\right )} d^{2}\right )} \cos \left (f x + e\right ) + {\left (60 \, B d^{2} f x + 3 \, {\left (A - B\right )} c^{2} - 6 \, {\left (A - B\right )} c d + 3 \, {\left (A - B\right )} d^{2} - {\left (15 \, B d^{2} f x + {\left (2 \, A + 3 \, B\right )} c^{2} + 2 \, {\left (3 \, A + 7 \, B\right )} c d + {\left (7 \, A - 32 \, B\right )} d^{2}\right )} \cos \left (f x + e\right )^{2} + 3 \, {\left (10 \, B d^{2} f x - {\left (2 \, A + 3 \, B\right )} c^{2} - 2 \, {\left (3 \, A + 2 \, B\right )} c d - {\left (2 \, A - 17 \, B\right )} d^{2}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{15 \, {\left (a^{3} f \cos \left (f x + e\right )^{3} + 3 \, a^{3} f \cos \left (f x + e\right )^{2} - 2 \, a^{3} f \cos \left (f x + e\right ) - 4 \, a^{3} f + {\left (a^{3} f \cos \left (f x + e\right )^{2} - 2 \, a^{3} f \cos \left (f x + e\right ) - 4 \, a^{3} f\right )} \sin \left (f x + e\right )\right )}} \]

[In]

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

[Out]

-1/15*(60*B*d^2*f*x - (15*B*d^2*f*x - (2*A + 3*B)*c^2 - 2*(3*A + 7*B)*c*d - (7*A - 32*B)*d^2)*cos(f*x + e)^3 -
3*(A - B)*c^2 + 6*(A - B)*c*d - 3*(A - B)*d^2 - (45*B*d^2*f*x + 2*(2*A + 3*B)*c^2 + 2*(6*A - B)*c*d - (A + 19
*B)*d^2)*cos(f*x + e)^2 + 3*(10*B*d^2*f*x - (3*A + 2*B)*c^2 - 2*(2*A + 3*B)*c*d - 3*(A - 6*B)*d^2)*cos(f*x + e
) + (60*B*d^2*f*x + 3*(A - B)*c^2 - 6*(A - B)*c*d + 3*(A - B)*d^2 - (15*B*d^2*f*x + (2*A + 3*B)*c^2 + 2*(3*A +
7*B)*c*d + (7*A - 32*B)*d^2)*cos(f*x + e)^2 + 3*(10*B*d^2*f*x - (2*A + 3*B)*c^2 - 2*(3*A + 2*B)*c*d - (2*A -
17*B)*d^2)*cos(f*x + e))*sin(f*x + e))/(a^3*f*cos(f*x + e)^3 + 3*a^3*f*cos(f*x + e)^2 - 2*a^3*f*cos(f*x + e) -
4*a^3*f + (a^3*f*cos(f*x + e)^2 - 2*a^3*f*cos(f*x + e) - 4*a^3*f)*sin(f*x + e))

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3468 vs. \(2 (151) = 302\).

Time = 7.85 (sec) , antiderivative size = 3468, normalized size of antiderivative = 21.15 \[ \int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))^2}{(a+a \sin (e+f x))^3} \, dx=\text {Too large to display} \]

[In]

integrate((A+B*sin(f*x+e))*(c+d*sin(f*x+e))**2/(a+a*sin(f*x+e))**3,x)

[Out]

Piecewise((-30*A*c**2*tan(e/2 + f*x/2)**4/(15*a**3*f*tan(e/2 + f*x/2)**5 + 75*a**3*f*tan(e/2 + f*x/2)**4 + 150
*a**3*f*tan(e/2 + f*x/2)**3 + 150*a**3*f*tan(e/2 + f*x/2)**2 + 75*a**3*f*tan(e/2 + f*x/2) + 15*a**3*f) - 60*A*
c**2*tan(e/2 + f*x/2)**3/(15*a**3*f*tan(e/2 + f*x/2)**5 + 75*a**3*f*tan(e/2 + f*x/2)**4 + 150*a**3*f*tan(e/2 +
f*x/2)**3 + 150*a**3*f*tan(e/2 + f*x/2)**2 + 75*a**3*f*tan(e/2 + f*x/2) + 15*a**3*f) - 80*A*c**2*tan(e/2 + f*
x/2)**2/(15*a**3*f*tan(e/2 + f*x/2)**5 + 75*a**3*f*tan(e/2 + f*x/2)**4 + 150*a**3*f*tan(e/2 + f*x/2)**3 + 150*
a**3*f*tan(e/2 + f*x/2)**2 + 75*a**3*f*tan(e/2 + f*x/2) + 15*a**3*f) - 40*A*c**2*tan(e/2 + f*x/2)/(15*a**3*f*t
an(e/2 + f*x/2)**5 + 75*a**3*f*tan(e/2 + f*x/2)**4 + 150*a**3*f*tan(e/2 + f*x/2)**3 + 150*a**3*f*tan(e/2 + f*x
/2)**2 + 75*a**3*f*tan(e/2 + f*x/2) + 15*a**3*f) - 14*A*c**2/(15*a**3*f*tan(e/2 + f*x/2)**5 + 75*a**3*f*tan(e/
2 + f*x/2)**4 + 150*a**3*f*tan(e/2 + f*x/2)**3 + 150*a**3*f*tan(e/2 + f*x/2)**2 + 75*a**3*f*tan(e/2 + f*x/2) +
15*a**3*f) - 60*A*c*d*tan(e/2 + f*x/2)**3/(15*a**3*f*tan(e/2 + f*x/2)**5 + 75*a**3*f*tan(e/2 + f*x/2)**4 + 15
0*a**3*f*tan(e/2 + f*x/2)**3 + 150*a**3*f*tan(e/2 + f*x/2)**2 + 75*a**3*f*tan(e/2 + f*x/2) + 15*a**3*f) - 60*A
*c*d*tan(e/2 + f*x/2)**2/(15*a**3*f*tan(e/2 + f*x/2)**5 + 75*a**3*f*tan(e/2 + f*x/2)**4 + 150*a**3*f*tan(e/2 +
f*x/2)**3 + 150*a**3*f*tan(e/2 + f*x/2)**2 + 75*a**3*f*tan(e/2 + f*x/2) + 15*a**3*f) - 60*A*c*d*tan(e/2 + f*x
/2)/(15*a**3*f*tan(e/2 + f*x/2)**5 + 75*a**3*f*tan(e/2 + f*x/2)**4 + 150*a**3*f*tan(e/2 + f*x/2)**3 + 150*a**3
*f*tan(e/2 + f*x/2)**2 + 75*a**3*f*tan(e/2 + f*x/2) + 15*a**3*f) - 12*A*c*d/(15*a**3*f*tan(e/2 + f*x/2)**5 + 7
5*a**3*f*tan(e/2 + f*x/2)**4 + 150*a**3*f*tan(e/2 + f*x/2)**3 + 150*a**3*f*tan(e/2 + f*x/2)**2 + 75*a**3*f*tan
(e/2 + f*x/2) + 15*a**3*f) - 40*A*d**2*tan(e/2 + f*x/2)**2/(15*a**3*f*tan(e/2 + f*x/2)**5 + 75*a**3*f*tan(e/2
+ f*x/2)**4 + 150*a**3*f*tan(e/2 + f*x/2)**3 + 150*a**3*f*tan(e/2 + f*x/2)**2 + 75*a**3*f*tan(e/2 + f*x/2) + 1
5*a**3*f) - 20*A*d**2*tan(e/2 + f*x/2)/(15*a**3*f*tan(e/2 + f*x/2)**5 + 75*a**3*f*tan(e/2 + f*x/2)**4 + 150*a*
*3*f*tan(e/2 + f*x/2)**3 + 150*a**3*f*tan(e/2 + f*x/2)**2 + 75*a**3*f*tan(e/2 + f*x/2) + 15*a**3*f) - 4*A*d**2
/(15*a**3*f*tan(e/2 + f*x/2)**5 + 75*a**3*f*tan(e/2 + f*x/2)**4 + 150*a**3*f*tan(e/2 + f*x/2)**3 + 150*a**3*f*
tan(e/2 + f*x/2)**2 + 75*a**3*f*tan(e/2 + f*x/2) + 15*a**3*f) - 30*B*c**2*tan(e/2 + f*x/2)**3/(15*a**3*f*tan(e
/2 + f*x/2)**5 + 75*a**3*f*tan(e/2 + f*x/2)**4 + 150*a**3*f*tan(e/2 + f*x/2)**3 + 150*a**3*f*tan(e/2 + f*x/2)*
*2 + 75*a**3*f*tan(e/2 + f*x/2) + 15*a**3*f) - 30*B*c**2*tan(e/2 + f*x/2)**2/(15*a**3*f*tan(e/2 + f*x/2)**5 +
75*a**3*f*tan(e/2 + f*x/2)**4 + 150*a**3*f*tan(e/2 + f*x/2)**3 + 150*a**3*f*tan(e/2 + f*x/2)**2 + 75*a**3*f*ta
n(e/2 + f*x/2) + 15*a**3*f) - 30*B*c**2*tan(e/2 + f*x/2)/(15*a**3*f*tan(e/2 + f*x/2)**5 + 75*a**3*f*tan(e/2 +
f*x/2)**4 + 150*a**3*f*tan(e/2 + f*x/2)**3 + 150*a**3*f*tan(e/2 + f*x/2)**2 + 75*a**3*f*tan(e/2 + f*x/2) + 15*
a**3*f) - 6*B*c**2/(15*a**3*f*tan(e/2 + f*x/2)**5 + 75*a**3*f*tan(e/2 + f*x/2)**4 + 150*a**3*f*tan(e/2 + f*x/2
)**3 + 150*a**3*f*tan(e/2 + f*x/2)**2 + 75*a**3*f*tan(e/2 + f*x/2) + 15*a**3*f) - 80*B*c*d*tan(e/2 + f*x/2)**2
/(15*a**3*f*tan(e/2 + f*x/2)**5 + 75*a**3*f*tan(e/2 + f*x/2)**4 + 150*a**3*f*tan(e/2 + f*x/2)**3 + 150*a**3*f*
tan(e/2 + f*x/2)**2 + 75*a**3*f*tan(e/2 + f*x/2) + 15*a**3*f) - 40*B*c*d*tan(e/2 + f*x/2)/(15*a**3*f*tan(e/2 +
f*x/2)**5 + 75*a**3*f*tan(e/2 + f*x/2)**4 + 150*a**3*f*tan(e/2 + f*x/2)**3 + 150*a**3*f*tan(e/2 + f*x/2)**2 +
75*a**3*f*tan(e/2 + f*x/2) + 15*a**3*f) - 8*B*c*d/(15*a**3*f*tan(e/2 + f*x/2)**5 + 75*a**3*f*tan(e/2 + f*x/2)
**4 + 150*a**3*f*tan(e/2 + f*x/2)**3 + 150*a**3*f*tan(e/2 + f*x/2)**2 + 75*a**3*f*tan(e/2 + f*x/2) + 15*a**3*f
) + 15*B*d**2*f*x*tan(e/2 + f*x/2)**5/(15*a**3*f*tan(e/2 + f*x/2)**5 + 75*a**3*f*tan(e/2 + f*x/2)**4 + 150*a**
3*f*tan(e/2 + f*x/2)**3 + 150*a**3*f*tan(e/2 + f*x/2)**2 + 75*a**3*f*tan(e/2 + f*x/2) + 15*a**3*f) + 75*B*d**2
*f*x*tan(e/2 + f*x/2)**4/(15*a**3*f*tan(e/2 + f*x/2)**5 + 75*a**3*f*tan(e/2 + f*x/2)**4 + 150*a**3*f*tan(e/2 +
f*x/2)**3 + 150*a**3*f*tan(e/2 + f*x/2)**2 + 75*a**3*f*tan(e/2 + f*x/2) + 15*a**3*f) + 150*B*d**2*f*x*tan(e/2
+ f*x/2)**3/(15*a**3*f*tan(e/2 + f*x/2)**5 + 75*a**3*f*tan(e/2 + f*x/2)**4 + 150*a**3*f*tan(e/2 + f*x/2)**3 +
150*a**3*f*tan(e/2 + f*x/2)**2 + 75*a**3*f*tan(e/2 + f*x/2) + 15*a**3*f) + 150*B*d**2*f*x*tan(e/2 + f*x/2)**2
/(15*a**3*f*tan(e/2 + f*x/2)**5 + 75*a**3*f*tan(e/2 + f*x/2)**4 + 150*a**3*f*tan(e/2 + f*x/2)**3 + 150*a**3*f*
tan(e/2 + f*x/2)**2 + 75*a**3*f*tan(e/2 + f*x/2) + 15*a**3*f) + 75*B*d**2*f*x*tan(e/2 + f*x/2)/(15*a**3*f*tan(
e/2 + f*x/2)**5 + 75*a**3*f*tan(e/2 + f*x/2)**4 + 150*a**3*f*tan(e/2 + f*x/2)**3 + 150*a**3*f*tan(e/2 + f*x/2)
**2 + 75*a**3*f*tan(e/2 + f*x/2) + 15*a**3*f) + 15*B*d**2*f*x/(15*a**3*f*tan(e/2 + f*x/2)**5 + 75*a**3*f*tan(e
/2 + f*x/2)**4 + 150*a**3*f*tan(e/2 + f*x/2)**3 + 150*a**3*f*tan(e/2 + f*x/2)**2 + 75*a**3*f*tan(e/2 + f*x/2)
+ 15*a**3*f) + 30*B*d**2*tan(e/2 + f*x/2)**4/(15*a**3*f*tan(e/2 + f*x/2)**5 + 75*a**3*f*tan(e/2 + f*x/2)**4 +
150*a**3*f*tan(e/2 + f*x/2)**3 + 150*a**3*f*tan(e/2 + f*x/2)**2 + 75*a**3*f*tan(e/2 + f*x/2) + 15*a**3*f) + 15
0*B*d**2*tan(e/2 + f*x/2)**3/(15*a**3*f*tan(e/2 + f*x/2)**5 + 75*a**3*f*tan(e/2 + f*x/2)**4 + 150*a**3*f*tan(e
/2 + f*x/2)**3 + 150*a**3*f*tan(e/2 + f*x/2)**2 + 75*a**3*f*tan(e/2 + f*x/2) + 15*a**3*f) + 290*B*d**2*tan(e/2
+ f*x/2)**2/(15*a**3*f*tan(e/2 + f*x/2)**5 + 75*a**3*f*tan(e/2 + f*x/2)**4 + 150*a**3*f*tan(e/2 + f*x/2)**3 +
150*a**3*f*tan(e/2 + f*x/2)**2 + 75*a**3*f*tan(e/2 + f*x/2) + 15*a**3*f) + 190*B*d**2*tan(e/2 + f*x/2)/(15*a*
*3*f*tan(e/2 + f*x/2)**5 + 75*a**3*f*tan(e/2 + f*x/2)**4 + 150*a**3*f*tan(e/2 + f*x/2)**3 + 150*a**3*f*tan(e/2
+ f*x/2)**2 + 75*a**3*f*tan(e/2 + f*x/2) + 15*a**3*f) + 44*B*d**2/(15*a**3*f*tan(e/2 + f*x/2)**5 + 75*a**3*f*
tan(e/2 + f*x/2)**4 + 150*a**3*f*tan(e/2 + f*x/2)**3 + 150*a**3*f*tan(e/2 + f*x/2)**2 + 75*a**3*f*tan(e/2 + f*
x/2) + 15*a**3*f), Ne(f, 0)), (x*(A + B*sin(e))*(c + d*sin(e))**2/(a*sin(e) + a)**3, True))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1132 vs. \(2 (158) = 316\).

Time = 0.32 (sec) , antiderivative size = 1132, normalized size of antiderivative = 6.90 \[ \int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))^2}{(a+a \sin (e+f x))^3} \, dx=\text {Too large to display} \]

[In]

integrate((A+B*sin(f*x+e))*(c+d*sin(f*x+e))^2/(a+a*sin(f*x+e))^3,x, algorithm="maxima")

[Out]

2/15*(B*d^2*((95*sin(f*x + e)/(cos(f*x + e) + 1) + 145*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 75*sin(f*x + e)^3
/(cos(f*x + e) + 1)^3 + 15*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 22)/(a^3 + 5*a^3*sin(f*x + e)/(cos(f*x + e) +
1) + 10*a^3*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 10*a^3*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 5*a^3*sin(f*x
+ e)^4/(cos(f*x + e) + 1)^4 + a^3*sin(f*x + e)^5/(cos(f*x + e) + 1)^5) + 15*arctan(sin(f*x + e)/(cos(f*x + e)
+ 1))/a^3) - A*c^2*(20*sin(f*x + e)/(cos(f*x + e) + 1) + 40*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 30*sin(f*x +
e)^3/(cos(f*x + e) + 1)^3 + 15*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 7)/(a^3 + 5*a^3*sin(f*x + e)/(cos(f*x +
e) + 1) + 10*a^3*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 10*a^3*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 5*a^3*sin(
f*x + e)^4/(cos(f*x + e) + 1)^4 + a^3*sin(f*x + e)^5/(cos(f*x + e) + 1)^5) - 4*B*c*d*(5*sin(f*x + e)/(cos(f*x
+ e) + 1) + 10*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 1)/(a^3 + 5*a^3*sin(f*x + e)/(cos(f*x + e) + 1) + 10*a^3*
sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 10*a^3*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 5*a^3*sin(f*x + e)^4/(cos(f
*x + e) + 1)^4 + a^3*sin(f*x + e)^5/(cos(f*x + e) + 1)^5) - 2*A*d^2*(5*sin(f*x + e)/(cos(f*x + e) + 1) + 10*si
n(f*x + e)^2/(cos(f*x + e) + 1)^2 + 1)/(a^3 + 5*a^3*sin(f*x + e)/(cos(f*x + e) + 1) + 10*a^3*sin(f*x + e)^2/(c
os(f*x + e) + 1)^2 + 10*a^3*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 5*a^3*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 +
a^3*sin(f*x + e)^5/(cos(f*x + e) + 1)^5) - 3*B*c^2*(5*sin(f*x + e)/(cos(f*x + e) + 1) + 5*sin(f*x + e)^2/(cos(
f*x + e) + 1)^2 + 5*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 1)/(a^3 + 5*a^3*sin(f*x + e)/(cos(f*x + e) + 1) + 10
*a^3*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 10*a^3*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 5*a^3*sin(f*x + e)^4/(
cos(f*x + e) + 1)^4 + a^3*sin(f*x + e)^5/(cos(f*x + e) + 1)^5) - 6*A*c*d*(5*sin(f*x + e)/(cos(f*x + e) + 1) +
5*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 5*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 1)/(a^3 + 5*a^3*sin(f*x + e)/(
cos(f*x + e) + 1) + 10*a^3*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 10*a^3*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 +
5*a^3*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + a^3*sin(f*x + e)^5/(cos(f*x + e) + 1)^5))/f

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 362 vs. \(2 (158) = 316\).

Time = 0.31 (sec) , antiderivative size = 362, normalized size of antiderivative = 2.21 \[ \int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))^2}{(a+a \sin (e+f x))^3} \, dx=\frac {\frac {15 \, {\left (f x + e\right )} B d^{2}}{a^{3}} - \frac {2 \, {\left (15 \, A c^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 15 \, B d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 30 \, A c^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 15 \, B c^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 30 \, A c d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 75 \, B d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 40 \, A c^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 15 \, B c^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 30 \, A c d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 40 \, B c d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 20 \, A d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 145 \, B d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 20 \, A c^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 15 \, B c^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 30 \, A c d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 20 \, B c d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 10 \, A d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 95 \, B d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 7 \, A c^{2} + 3 \, B c^{2} + 6 \, A c d + 4 \, B c d + 2 \, A d^{2} - 22 \, B d^{2}\right )}}{a^{3} {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{5}}}{15 \, f} \]

[In]

integrate((A+B*sin(f*x+e))*(c+d*sin(f*x+e))^2/(a+a*sin(f*x+e))^3,x, algorithm="giac")

[Out]

1/15*(15*(f*x + e)*B*d^2/a^3 - 2*(15*A*c^2*tan(1/2*f*x + 1/2*e)^4 - 15*B*d^2*tan(1/2*f*x + 1/2*e)^4 + 30*A*c^2
*tan(1/2*f*x + 1/2*e)^3 + 15*B*c^2*tan(1/2*f*x + 1/2*e)^3 + 30*A*c*d*tan(1/2*f*x + 1/2*e)^3 - 75*B*d^2*tan(1/2
*f*x + 1/2*e)^3 + 40*A*c^2*tan(1/2*f*x + 1/2*e)^2 + 15*B*c^2*tan(1/2*f*x + 1/2*e)^2 + 30*A*c*d*tan(1/2*f*x + 1
/2*e)^2 + 40*B*c*d*tan(1/2*f*x + 1/2*e)^2 + 20*A*d^2*tan(1/2*f*x + 1/2*e)^2 - 145*B*d^2*tan(1/2*f*x + 1/2*e)^2
+ 20*A*c^2*tan(1/2*f*x + 1/2*e) + 15*B*c^2*tan(1/2*f*x + 1/2*e) + 30*A*c*d*tan(1/2*f*x + 1/2*e) + 20*B*c*d*ta
n(1/2*f*x + 1/2*e) + 10*A*d^2*tan(1/2*f*x + 1/2*e) - 95*B*d^2*tan(1/2*f*x + 1/2*e) + 7*A*c^2 + 3*B*c^2 + 6*A*c
*d + 4*B*c*d + 2*A*d^2 - 22*B*d^2)/(a^3*(tan(1/2*f*x + 1/2*e) + 1)^5))/f

Mupad [B] (verification not implemented)

Time = 16.84 (sec) , antiderivative size = 286, normalized size of antiderivative = 1.74 \[ \int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))^2}{(a+a \sin (e+f x))^3} \, dx=\frac {B\,d^2\,x}{a^3}-\frac {{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (\frac {16\,A\,c^2}{3}+\frac {8\,A\,d^2}{3}+2\,B\,c^2-\frac {58\,B\,d^2}{3}+4\,A\,c\,d+\frac {16\,B\,c\,d}{3}\right )+\frac {14\,A\,c^2}{15}+\frac {4\,A\,d^2}{15}+\frac {2\,B\,c^2}{5}-\frac {44\,B\,d^2}{15}+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left (4\,A\,c^2+2\,B\,c^2-10\,B\,d^2+4\,A\,c\,d\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\left (2\,A\,c^2-2\,B\,d^2\right )+\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (\frac {8\,A\,c^2}{3}+\frac {4\,A\,d^2}{3}+2\,B\,c^2-\frac {38\,B\,d^2}{3}+4\,A\,c\,d+\frac {8\,B\,c\,d}{3}\right )+\frac {4\,A\,c\,d}{5}+\frac {8\,B\,c\,d}{15}}{f\,\left (a^3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5+5\,a^3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4+10\,a^3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3+10\,a^3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+5\,a^3\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )+a^3\right )} \]

[In]

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

[Out]

(B*d^2*x)/a^3 - (tan(e/2 + (f*x)/2)^2*((16*A*c^2)/3 + (8*A*d^2)/3 + 2*B*c^2 - (58*B*d^2)/3 + 4*A*c*d + (16*B*c
*d)/3) + (14*A*c^2)/15 + (4*A*d^2)/15 + (2*B*c^2)/5 - (44*B*d^2)/15 + tan(e/2 + (f*x)/2)^3*(4*A*c^2 + 2*B*c^2
- 10*B*d^2 + 4*A*c*d) + tan(e/2 + (f*x)/2)^4*(2*A*c^2 - 2*B*d^2) + tan(e/2 + (f*x)/2)*((8*A*c^2)/3 + (4*A*d^2)
/3 + 2*B*c^2 - (38*B*d^2)/3 + 4*A*c*d + (8*B*c*d)/3) + (4*A*c*d)/5 + (8*B*c*d)/15)/(f*(10*a^3*tan(e/2 + (f*x)/
2)^2 + 10*a^3*tan(e/2 + (f*x)/2)^3 + 5*a^3*tan(e/2 + (f*x)/2)^4 + a^3*tan(e/2 + (f*x)/2)^5 + a^3 + 5*a^3*tan(e
/2 + (f*x)/2)))