\(\int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))^2}{(a+a \sin (e+f x))^2} \, dx\) [273]
Optimal result
Integrand size = 35, antiderivative size = 132 \[
\int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))^2}{(a+a \sin (e+f x))^2} \, dx=\frac {d (2 B (c-d)+A d) x}{a^2}+\frac {(A-4 B) d^2 \cos (e+f x)}{3 a^2 f}-\frac {(c-d) (2 B (c-3 d)+A (c+3 d)) \cos (e+f x)}{3 a^2 f (1+\sin (e+f x))}-\frac {(A-B) \cos (e+f x) (c+d \sin (e+f x))^2}{3 f (a+a \sin (e+f x))^2}
\]
[Out]
d*(2*B*(c-d)+A*d)*x/a^2+1/3*(A-4*B)*d^2*cos(f*x+e)/a^2/f-1/3*(c-d)*(2*B*(c-3*d)+A*(c+3*d))*cos(f*x+e)/a^2/f/(1
+sin(f*x+e))-1/3*(A-B)*cos(f*x+e)*(c+d*sin(f*x+e))^2/f/(a+a*sin(f*x+e))^2
Rubi [A] (verified)
Time = 0.35 (sec) , antiderivative size = 132, 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, 3102, 2814, 2727}
\[
\int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))^2}{(a+a \sin (e+f x))^2} \, dx=-\frac {(c-d) (A (c+3 d)+2 B (c-3 d)) \cos (e+f x)}{3 a^2 f (\sin (e+f x)+1)}+\frac {d x (A d+2 B (c-d))}{a^2}+\frac {d^2 (A-4 B) \cos (e+f x)}{3 a^2 f}-\frac {(A-B) \cos (e+f x) (c+d \sin (e+f x))^2}{3 f (a \sin (e+f x)+a)^2}
\]
[In]
Int[((A + B*Sin[e + f*x])*(c + d*Sin[e + f*x])^2)/(a + a*Sin[e + f*x])^2,x]
[Out]
(d*(2*B*(c - d) + A*d)*x)/a^2 + ((A - 4*B)*d^2*Cos[e + f*x])/(3*a^2*f) - ((c - d)*(2*B*(c - 3*d) + A*(c + 3*d)
)*Cos[e + f*x])/(3*a^2*f*(1 + Sin[e + f*x])) - ((A - B)*Cos[e + f*x]*(c + d*Sin[e + f*x])^2)/(3*f*(a + a*Sin[e
+ f*x])^2)
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 3102
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (
f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Cos[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 2))), x] + Dist[1/(
b*(m + 2)), Int[(a + b*Sin[e + f*x])^m*Simp[A*b*(m + 2) + b*C*(m + 1) + (b*B*(m + 2) - a*C)*Sin[e + f*x], x],
x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && !LtQ[m, -1]
Rubi steps \begin{align*}
\text {integral}& = -\frac {(A-B) \cos (e+f x) (c+d \sin (e+f x))^2}{3 f (a+a \sin (e+f x))^2}+\frac {\int \frac {(c+d \sin (e+f x)) (a (2 B (c-d)+A (c+2 d))-a (A-4 B) d \sin (e+f x))}{a+a \sin (e+f x)} \, dx}{3 a^2} \\ & = -\frac {(A-B) \cos (e+f x) (c+d \sin (e+f x))^2}{3 f (a+a \sin (e+f x))^2}+\frac {\int \frac {a c (2 B (c-d)+A (c+2 d))+(-a (A-4 B) c d+a d (2 B (c-d)+A (c+2 d))) \sin (e+f x)-a (A-4 B) d^2 \sin ^2(e+f x)}{a+a \sin (e+f x)} \, dx}{3 a^2} \\ & = \frac {(A-4 B) d^2 \cos (e+f x)}{3 a^2 f}-\frac {(A-B) \cos (e+f x) (c+d \sin (e+f x))^2}{3 f (a+a \sin (e+f x))^2}+\frac {\int \frac {a^2 c (2 B (c-d)+A (c+2 d))+3 a^2 d (2 B (c-d)+A d) \sin (e+f x)}{a+a \sin (e+f x)} \, dx}{3 a^3} \\ & = \frac {d (2 B (c-d)+A d) x}{a^2}+\frac {(A-4 B) d^2 \cos (e+f x)}{3 a^2 f}-\frac {(A-B) \cos (e+f x) (c+d \sin (e+f x))^2}{3 f (a+a \sin (e+f x))^2}+\frac {((c-d) (2 B (c-3 d)+A (c+3 d))) \int \frac {1}{a+a \sin (e+f x)} \, dx}{3 a} \\ & = \frac {d (2 B (c-d)+A d) x}{a^2}+\frac {(A-4 B) d^2 \cos (e+f x)}{3 a^2 f}-\frac {(c-d) (2 B (c-3 d)+A (c+3 d)) \cos (e+f x)}{3 f \left (a^2+a^2 \sin (e+f x)\right )}-\frac {(A-B) \cos (e+f x) (c+d \sin (e+f x))^2}{3 f (a+a \sin (e+f x))^2} \\
\end{align*}
Mathematica [B] (verified)
Leaf count is larger than twice the leaf count of optimal. \(338\) vs. \(2(132)=264\).
Time = 1.33 (sec) , antiderivative size = 338, normalized size of antiderivative = 2.56
\[
\int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))^2}{(a+a \sin (e+f x))^2} \, dx=\frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (6 \left (A d (4 c+d (-4+3 e+3 f x))+B \left (2 c^2+d^2 (5-6 e-6 f x)+2 c d (-4+3 e+3 f x)\right )\right ) \cos \left (\frac {1}{2} (e+f x)\right )-\left (B \left (8 c^2+d^2 (41-12 e-12 f x)+4 c d (-10+3 e+3 f x)\right )+2 A \left (2 c^2+8 c d+d^2 (-10+3 e+3 f x)\right )\right ) \cos \left (\frac {3}{2} (e+f x)\right )+3 B d^2 \cos \left (\frac {5}{2} (e+f x)\right )+6 \left (2 A c^2+2 B c^2+4 A c d-12 B c d-6 A d^2+9 B d^2+8 B c d e+4 A d^2 e-8 B d^2 e+8 B c d f x+4 A d^2 f x-8 B d^2 f x-2 d (-2 B c (e+f x)-A d (e+f x)+2 B d (1+e+f x)) \cos (e+f x)-B d^2 \cos (2 (e+f x))\right ) \sin \left (\frac {1}{2} (e+f x)\right )\right )}{12 a^2 f (1+\sin (e+f x))^2}
\]
[In]
Integrate[((A + B*Sin[e + f*x])*(c + d*Sin[e + f*x])^2)/(a + a*Sin[e + f*x])^2,x]
[Out]
((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(6*(A*d*(4*c + d*(-4 + 3*e + 3*f*x)) + B*(2*c^2 + d^2*(5 - 6*e - 6*f*x)
+ 2*c*d*(-4 + 3*e + 3*f*x)))*Cos[(e + f*x)/2] - (B*(8*c^2 + d^2*(41 - 12*e - 12*f*x) + 4*c*d*(-10 + 3*e + 3*f
*x)) + 2*A*(2*c^2 + 8*c*d + d^2*(-10 + 3*e + 3*f*x)))*Cos[(3*(e + f*x))/2] + 3*B*d^2*Cos[(5*(e + f*x))/2] + 6*
(2*A*c^2 + 2*B*c^2 + 4*A*c*d - 12*B*c*d - 6*A*d^2 + 9*B*d^2 + 8*B*c*d*e + 4*A*d^2*e - 8*B*d^2*e + 8*B*c*d*f*x
+ 4*A*d^2*f*x - 8*B*d^2*f*x - 2*d*(-2*B*c*(e + f*x) - A*d*(e + f*x) + 2*B*d*(1 + e + f*x))*Cos[e + f*x] - B*d^
2*Cos[2*(e + f*x)])*Sin[(e + f*x)/2]))/(12*a^2*f*(1 + Sin[e + f*x])^2)
Maple [A] (verified)
Time = 0.79 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.46
| | |
| method | result | size |
| | |
| derivativedivides |
\(\frac {-\frac {2 \left (A \,c^{2}-A \,d^{2}-2 c d B +2 d^{2} B \right )}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1}-\frac {-2 A \,c^{2}+4 A c d -2 A \,d^{2}+2 B \,c^{2}-4 c d B +2 d^{2} B}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}-\frac {2 \left (2 A \,c^{2}-4 A c d +2 A \,d^{2}-2 B \,c^{2}+4 c d B -2 d^{2} B \right )}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}+2 d \left (-\frac {d B}{1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )}+\left (d A +2 B c -2 d B \right ) \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )\right )}{a^{2} f}\) |
\(193\) |
| default |
\(\frac {-\frac {2 \left (A \,c^{2}-A \,d^{2}-2 c d B +2 d^{2} B \right )}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1}-\frac {-2 A \,c^{2}+4 A c d -2 A \,d^{2}+2 B \,c^{2}-4 c d B +2 d^{2} B}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}-\frac {2 \left (2 A \,c^{2}-4 A c d +2 A \,d^{2}-2 B \,c^{2}+4 c d B -2 d^{2} B \right )}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}+2 d \left (-\frac {d B}{1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )}+\left (d A +2 B c -2 d B \right ) \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )\right )}{a^{2} f}\) |
\(193\) |
| risch |
\(\frac {x \,d^{2} A}{a^{2}}+\frac {2 x d B c}{a^{2}}-\frac {2 x \,d^{2} B}{a^{2}}-\frac {B \,d^{2} {\mathrm e}^{i \left (f x +e \right )}}{2 a^{2} f}-\frac {B \,d^{2} {\mathrm e}^{-i \left (f x +e \right )}}{2 a^{2} f}-\frac {2 \left (10 c d B +3 i A \,c^{2} {\mathrm e}^{i \left (f x +e \right )}-6 A \,d^{2} {\mathrm e}^{2 i \left (f x +e \right )}+9 B \,d^{2} {\mathrm e}^{2 i \left (f x +e \right )}-9 i A \,d^{2} {\mathrm e}^{i \left (f x +e \right )}-A \,c^{2}-2 B \,c^{2}-8 d^{2} B +3 i B \,c^{2} {\mathrm e}^{i \left (f x +e \right )}-12 B c d \,{\mathrm e}^{2 i \left (f x +e \right )}+6 i A c d \,{\mathrm e}^{i \left (f x +e \right )}+15 i B \,d^{2} {\mathrm e}^{i \left (f x +e \right )}+6 A c d \,{\mathrm e}^{2 i \left (f x +e \right )}-18 i B c d \,{\mathrm e}^{i \left (f x +e \right )}+3 B \,c^{2} {\mathrm e}^{2 i \left (f x +e \right )}-4 A c d +5 A \,d^{2}\right )}{3 f \,a^{2} \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )^{3}}\) |
\(296\) |
| parallelrisch |
\(\frac {\left (\left (-18 f x A +36 f x B -30 A +78 B \right ) d^{2}+12 \left (\left (-3 f x -5\right ) B +A \right ) c d +18 \left (A +\frac {B}{3}\right ) c^{2}\right ) \cos \left (\frac {f x}{2}+\frac {e}{2}\right )+\left (\left (6 f x A -12 f x B -2 A +5 B \right ) d^{2}+4 \left (\left (3 f x -1\right ) B +A \right ) c d -2 c^{2} \left (A -B \right )\right ) \cos \left (\frac {3 f x}{2}+\frac {3 e}{2}\right )+\left (\left (-18 f x A +36 f x B -18 A +42 B \right ) d^{2}+12 c \left (\left (-3 f x -3\right ) B +A \right ) d +6 c^{2} \left (A +B \right )\right ) \sin \left (\frac {f x}{2}+\frac {e}{2}\right )+\left (\left (-6 f x A +12 f x B -18 A +45 B \right ) d^{2}+12 c \left (\left (-f x -3\right ) B +A \right ) d +6 c^{2} \left (A +B \right )\right ) \sin \left (\frac {3 f x}{2}+\frac {3 e}{2}\right )-3 B \,d^{2} \left (\cos \left (\frac {5 f x}{2}+\frac {5 e}{2}\right )-\sin \left (\frac {5 f x}{2}+\frac {5 e}{2}\right )\right )}{6 f \,a^{2} \left (-3 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )+\cos \left (\frac {3 f x}{2}+\frac {3 e}{2}\right )-\sin \left (\frac {3 f x}{2}+\frac {3 e}{2}\right )-3 \sin \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}\) |
\(302\) |
| norman |
\(\frac {\frac {d \left (d A +2 B c -2 d B \right ) x}{a}+\frac {d \left (d A +2 B c -2 d B \right ) x \left (\tan ^{9}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a}-\frac {4 A \,c^{2}+4 A c d -8 A \,d^{2}+2 B \,c^{2}-16 c d B +20 d^{2} B}{3 a f}-\frac {\left (2 A \,c^{2}-2 A \,d^{2}-4 c d B +4 d^{2} B \right ) \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a f}-\frac {2 \left (A \,c^{2}+2 A c d -3 A \,d^{2}+B \,c^{2}-6 c d B +6 d^{2} B \right ) \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a f}-\frac {\left (2 A \,c^{2}+4 A c d -6 A \,d^{2}+2 B \,c^{2}-12 c d B +16 d^{2} B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{a f}-\frac {2 \left (3 A \,c^{2}+2 A c d -5 A \,d^{2}+B \,c^{2}-10 c d B +14 d^{2} B \right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a f}-\frac {2 \left (3 A \,c^{2}+6 A c d -9 A \,d^{2}+3 B \,c^{2}-18 c d B +20 d^{2} B \right ) \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a f}-\frac {2 \left (3 A \,c^{2}+6 A c d -9 A \,d^{2}+3 B \,c^{2}-18 c d B +22 d^{2} B \right ) \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a f}-\frac {2 \left (5 A \,c^{2}+2 A c d -7 A \,d^{2}+B \,c^{2}-14 c d B +20 d^{2} B \right ) \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a f}-\frac {2 \left (11 A \,c^{2}+2 A c d -13 A \,d^{2}+B \,c^{2}-26 c d B +34 d^{2} B \right ) \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 a f}+\frac {3 d \left (d A +2 B c -2 d B \right ) x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{a}+\frac {6 d \left (d A +2 B c -2 d B \right ) x \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a}+\frac {10 d \left (d A +2 B c -2 d B \right ) x \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a}+\frac {12 d \left (d A +2 B c -2 d B \right ) x \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a}+\frac {12 d \left (d A +2 B c -2 d B \right ) x \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a}+\frac {10 d \left (d A +2 B c -2 d B \right ) x \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a}+\frac {6 d \left (d A +2 B c -2 d B \right ) x \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a}+\frac {3 d \left (d A +2 B c -2 d B \right ) x \left (\tan ^{8}\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 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}\) |
\(776\) |
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[In]
int((A+B*sin(f*x+e))*(c+d*sin(f*x+e))^2/(a+a*sin(f*x+e))^2,x,method=_RETURNVERBOSE)
[Out]
2/f/a^2*(-(A*c^2-A*d^2-2*B*c*d+2*B*d^2)/(tan(1/2*f*x+1/2*e)+1)-1/2*(-2*A*c^2+4*A*c*d-2*A*d^2+2*B*c^2-4*B*c*d+2
*B*d^2)/(tan(1/2*f*x+1/2*e)+1)^2-1/3*(2*A*c^2-4*A*c*d+2*A*d^2-2*B*c^2+4*B*c*d-2*B*d^2)/(tan(1/2*f*x+1/2*e)+1)^
3+d*(-d*B/(1+tan(1/2*f*x+1/2*e)^2)+(A*d+2*B*c-2*B*d)*arctan(tan(1/2*f*x+1/2*e))))
Fricas [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 375 vs. \(2 (126) = 252\).
Time = 0.27 (sec) , antiderivative size = 375, normalized size of antiderivative = 2.84
\[
\int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))^2}{(a+a \sin (e+f x))^2} \, dx=-\frac {3 \, B d^{2} \cos \left (f x + e\right )^{3} - {\left (A - B\right )} c^{2} + 2 \, {\left (A - B\right )} c d - {\left (A - B\right )} d^{2} + 6 \, {\left (2 \, B c d + {\left (A - 2 \, B\right )} d^{2}\right )} f x - {\left ({\left (A + 2 \, B\right )} c^{2} + 2 \, {\left (2 \, A - 5 \, B\right )} c d - {\left (5 \, A - 11 \, B\right )} d^{2} + 3 \, {\left (2 \, B c d + {\left (A - 2 \, B\right )} d^{2}\right )} f x\right )} \cos \left (f x + e\right )^{2} - {\left ({\left (2 \, A + B\right )} c^{2} + 2 \, {\left (A - 4 \, B\right )} c d - {\left (4 \, A - 13 \, B\right )} d^{2} - 3 \, {\left (2 \, B c d + {\left (A - 2 \, B\right )} d^{2}\right )} f x\right )} \cos \left (f x + e\right ) - {\left (3 \, B d^{2} \cos \left (f x + e\right )^{2} - {\left (A - B\right )} c^{2} + 2 \, {\left (A - B\right )} c d - {\left (A - B\right )} d^{2} - 6 \, {\left (2 \, B c d + {\left (A - 2 \, B\right )} d^{2}\right )} f x + {\left ({\left (A + 2 \, B\right )} c^{2} + 2 \, {\left (2 \, A - 5 \, B\right )} c d - {\left (5 \, A - 14 \, B\right )} d^{2} - 3 \, {\left (2 \, B c d + {\left (A - 2 \, B\right )} d^{2}\right )} f x\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{3 \, {\left (a^{2} f \cos \left (f x + e\right )^{2} - a^{2} f \cos \left (f x + e\right ) - 2 \, a^{2} f - {\left (a^{2} f \cos \left (f x + e\right ) + 2 \, a^{2} f\right )} \sin \left (f x + e\right )\right )}}
\]
[In]
integrate((A+B*sin(f*x+e))*(c+d*sin(f*x+e))^2/(a+a*sin(f*x+e))^2,x, algorithm="fricas")
[Out]
-1/3*(3*B*d^2*cos(f*x + e)^3 - (A - B)*c^2 + 2*(A - B)*c*d - (A - B)*d^2 + 6*(2*B*c*d + (A - 2*B)*d^2)*f*x - (
(A + 2*B)*c^2 + 2*(2*A - 5*B)*c*d - (5*A - 11*B)*d^2 + 3*(2*B*c*d + (A - 2*B)*d^2)*f*x)*cos(f*x + e)^2 - ((2*A
+ B)*c^2 + 2*(A - 4*B)*c*d - (4*A - 13*B)*d^2 - 3*(2*B*c*d + (A - 2*B)*d^2)*f*x)*cos(f*x + e) - (3*B*d^2*cos(
f*x + e)^2 - (A - B)*c^2 + 2*(A - B)*c*d - (A - B)*d^2 - 6*(2*B*c*d + (A - 2*B)*d^2)*f*x + ((A + 2*B)*c^2 + 2*
(2*A - 5*B)*c*d - (5*A - 14*B)*d^2 - 3*(2*B*c*d + (A - 2*B)*d^2)*f*x)*cos(f*x + e))*sin(f*x + e))/(a^2*f*cos(f
*x + e)^2 - a^2*f*cos(f*x + e) - 2*a^2*f - (a^2*f*cos(f*x + e) + 2*a^2*f)*sin(f*x + e))
Sympy [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 5358 vs. \(2 (121) = 242\).
Time = 4.01 (sec) , antiderivative size = 5358, normalized size of antiderivative = 40.59
\[
\int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))^2}{(a+a \sin (e+f x))^2} \, dx=\text {Too large to display}
\]
[In]
integrate((A+B*sin(f*x+e))*(c+d*sin(f*x+e))**2/(a+a*sin(f*x+e))**2,x)
[Out]
Piecewise((-6*A*c**2*tan(e/2 + f*x/2)**4/(3*a**2*f*tan(e/2 + f*x/2)**5 + 9*a**2*f*tan(e/2 + f*x/2)**4 + 12*a**
2*f*tan(e/2 + f*x/2)**3 + 12*a**2*f*tan(e/2 + f*x/2)**2 + 9*a**2*f*tan(e/2 + f*x/2) + 3*a**2*f) - 6*A*c**2*tan
(e/2 + f*x/2)**3/(3*a**2*f*tan(e/2 + f*x/2)**5 + 9*a**2*f*tan(e/2 + f*x/2)**4 + 12*a**2*f*tan(e/2 + f*x/2)**3
+ 12*a**2*f*tan(e/2 + f*x/2)**2 + 9*a**2*f*tan(e/2 + f*x/2) + 3*a**2*f) - 10*A*c**2*tan(e/2 + f*x/2)**2/(3*a**
2*f*tan(e/2 + f*x/2)**5 + 9*a**2*f*tan(e/2 + f*x/2)**4 + 12*a**2*f*tan(e/2 + f*x/2)**3 + 12*a**2*f*tan(e/2 + f
*x/2)**2 + 9*a**2*f*tan(e/2 + f*x/2) + 3*a**2*f) - 6*A*c**2*tan(e/2 + f*x/2)/(3*a**2*f*tan(e/2 + f*x/2)**5 + 9
*a**2*f*tan(e/2 + f*x/2)**4 + 12*a**2*f*tan(e/2 + f*x/2)**3 + 12*a**2*f*tan(e/2 + f*x/2)**2 + 9*a**2*f*tan(e/2
+ f*x/2) + 3*a**2*f) - 4*A*c**2/(3*a**2*f*tan(e/2 + f*x/2)**5 + 9*a**2*f*tan(e/2 + f*x/2)**4 + 12*a**2*f*tan(
e/2 + f*x/2)**3 + 12*a**2*f*tan(e/2 + f*x/2)**2 + 9*a**2*f*tan(e/2 + f*x/2) + 3*a**2*f) - 12*A*c*d*tan(e/2 + f
*x/2)**3/(3*a**2*f*tan(e/2 + f*x/2)**5 + 9*a**2*f*tan(e/2 + f*x/2)**4 + 12*a**2*f*tan(e/2 + f*x/2)**3 + 12*a**
2*f*tan(e/2 + f*x/2)**2 + 9*a**2*f*tan(e/2 + f*x/2) + 3*a**2*f) - 4*A*c*d*tan(e/2 + f*x/2)**2/(3*a**2*f*tan(e/
2 + f*x/2)**5 + 9*a**2*f*tan(e/2 + f*x/2)**4 + 12*a**2*f*tan(e/2 + f*x/2)**3 + 12*a**2*f*tan(e/2 + f*x/2)**2 +
9*a**2*f*tan(e/2 + f*x/2) + 3*a**2*f) - 12*A*c*d*tan(e/2 + f*x/2)/(3*a**2*f*tan(e/2 + f*x/2)**5 + 9*a**2*f*ta
n(e/2 + f*x/2)**4 + 12*a**2*f*tan(e/2 + f*x/2)**3 + 12*a**2*f*tan(e/2 + f*x/2)**2 + 9*a**2*f*tan(e/2 + f*x/2)
+ 3*a**2*f) - 4*A*c*d/(3*a**2*f*tan(e/2 + f*x/2)**5 + 9*a**2*f*tan(e/2 + f*x/2)**4 + 12*a**2*f*tan(e/2 + f*x/2
)**3 + 12*a**2*f*tan(e/2 + f*x/2)**2 + 9*a**2*f*tan(e/2 + f*x/2) + 3*a**2*f) + 3*A*d**2*f*x*tan(e/2 + f*x/2)**
5/(3*a**2*f*tan(e/2 + f*x/2)**5 + 9*a**2*f*tan(e/2 + f*x/2)**4 + 12*a**2*f*tan(e/2 + f*x/2)**3 + 12*a**2*f*tan
(e/2 + f*x/2)**2 + 9*a**2*f*tan(e/2 + f*x/2) + 3*a**2*f) + 9*A*d**2*f*x*tan(e/2 + f*x/2)**4/(3*a**2*f*tan(e/2
+ f*x/2)**5 + 9*a**2*f*tan(e/2 + f*x/2)**4 + 12*a**2*f*tan(e/2 + f*x/2)**3 + 12*a**2*f*tan(e/2 + f*x/2)**2 + 9
*a**2*f*tan(e/2 + f*x/2) + 3*a**2*f) + 12*A*d**2*f*x*tan(e/2 + f*x/2)**3/(3*a**2*f*tan(e/2 + f*x/2)**5 + 9*a**
2*f*tan(e/2 + f*x/2)**4 + 12*a**2*f*tan(e/2 + f*x/2)**3 + 12*a**2*f*tan(e/2 + f*x/2)**2 + 9*a**2*f*tan(e/2 + f
*x/2) + 3*a**2*f) + 12*A*d**2*f*x*tan(e/2 + f*x/2)**2/(3*a**2*f*tan(e/2 + f*x/2)**5 + 9*a**2*f*tan(e/2 + f*x/2
)**4 + 12*a**2*f*tan(e/2 + f*x/2)**3 + 12*a**2*f*tan(e/2 + f*x/2)**2 + 9*a**2*f*tan(e/2 + f*x/2) + 3*a**2*f) +
9*A*d**2*f*x*tan(e/2 + f*x/2)/(3*a**2*f*tan(e/2 + f*x/2)**5 + 9*a**2*f*tan(e/2 + f*x/2)**4 + 12*a**2*f*tan(e/
2 + f*x/2)**3 + 12*a**2*f*tan(e/2 + f*x/2)**2 + 9*a**2*f*tan(e/2 + f*x/2) + 3*a**2*f) + 3*A*d**2*f*x/(3*a**2*f
*tan(e/2 + f*x/2)**5 + 9*a**2*f*tan(e/2 + f*x/2)**4 + 12*a**2*f*tan(e/2 + f*x/2)**3 + 12*a**2*f*tan(e/2 + f*x/
2)**2 + 9*a**2*f*tan(e/2 + f*x/2) + 3*a**2*f) + 6*A*d**2*tan(e/2 + f*x/2)**4/(3*a**2*f*tan(e/2 + f*x/2)**5 + 9
*a**2*f*tan(e/2 + f*x/2)**4 + 12*a**2*f*tan(e/2 + f*x/2)**3 + 12*a**2*f*tan(e/2 + f*x/2)**2 + 9*a**2*f*tan(e/2
+ f*x/2) + 3*a**2*f) + 18*A*d**2*tan(e/2 + f*x/2)**3/(3*a**2*f*tan(e/2 + f*x/2)**5 + 9*a**2*f*tan(e/2 + f*x/2
)**4 + 12*a**2*f*tan(e/2 + f*x/2)**3 + 12*a**2*f*tan(e/2 + f*x/2)**2 + 9*a**2*f*tan(e/2 + f*x/2) + 3*a**2*f) +
14*A*d**2*tan(e/2 + f*x/2)**2/(3*a**2*f*tan(e/2 + f*x/2)**5 + 9*a**2*f*tan(e/2 + f*x/2)**4 + 12*a**2*f*tan(e/
2 + f*x/2)**3 + 12*a**2*f*tan(e/2 + f*x/2)**2 + 9*a**2*f*tan(e/2 + f*x/2) + 3*a**2*f) + 18*A*d**2*tan(e/2 + f*
x/2)/(3*a**2*f*tan(e/2 + f*x/2)**5 + 9*a**2*f*tan(e/2 + f*x/2)**4 + 12*a**2*f*tan(e/2 + f*x/2)**3 + 12*a**2*f*
tan(e/2 + f*x/2)**2 + 9*a**2*f*tan(e/2 + f*x/2) + 3*a**2*f) + 8*A*d**2/(3*a**2*f*tan(e/2 + f*x/2)**5 + 9*a**2*
f*tan(e/2 + f*x/2)**4 + 12*a**2*f*tan(e/2 + f*x/2)**3 + 12*a**2*f*tan(e/2 + f*x/2)**2 + 9*a**2*f*tan(e/2 + f*x
/2) + 3*a**2*f) - 6*B*c**2*tan(e/2 + f*x/2)**3/(3*a**2*f*tan(e/2 + f*x/2)**5 + 9*a**2*f*tan(e/2 + f*x/2)**4 +
12*a**2*f*tan(e/2 + f*x/2)**3 + 12*a**2*f*tan(e/2 + f*x/2)**2 + 9*a**2*f*tan(e/2 + f*x/2) + 3*a**2*f) - 2*B*c*
*2*tan(e/2 + f*x/2)**2/(3*a**2*f*tan(e/2 + f*x/2)**5 + 9*a**2*f*tan(e/2 + f*x/2)**4 + 12*a**2*f*tan(e/2 + f*x/
2)**3 + 12*a**2*f*tan(e/2 + f*x/2)**2 + 9*a**2*f*tan(e/2 + f*x/2) + 3*a**2*f) - 6*B*c**2*tan(e/2 + f*x/2)/(3*a
**2*f*tan(e/2 + f*x/2)**5 + 9*a**2*f*tan(e/2 + f*x/2)**4 + 12*a**2*f*tan(e/2 + f*x/2)**3 + 12*a**2*f*tan(e/2 +
f*x/2)**2 + 9*a**2*f*tan(e/2 + f*x/2) + 3*a**2*f) - 2*B*c**2/(3*a**2*f*tan(e/2 + f*x/2)**5 + 9*a**2*f*tan(e/2
+ f*x/2)**4 + 12*a**2*f*tan(e/2 + f*x/2)**3 + 12*a**2*f*tan(e/2 + f*x/2)**2 + 9*a**2*f*tan(e/2 + f*x/2) + 3*a
**2*f) + 6*B*c*d*f*x*tan(e/2 + f*x/2)**5/(3*a**2*f*tan(e/2 + f*x/2)**5 + 9*a**2*f*tan(e/2 + f*x/2)**4 + 12*a**
2*f*tan(e/2 + f*x/2)**3 + 12*a**2*f*tan(e/2 + f*x/2)**2 + 9*a**2*f*tan(e/2 + f*x/2) + 3*a**2*f) + 18*B*c*d*f*x
*tan(e/2 + f*x/2)**4/(3*a**2*f*tan(e/2 + f*x/2)**5 + 9*a**2*f*tan(e/2 + f*x/2)**4 + 12*a**2*f*tan(e/2 + f*x/2)
**3 + 12*a**2*f*tan(e/2 + f*x/2)**2 + 9*a**2*f*tan(e/2 + f*x/2) + 3*a**2*f) + 24*B*c*d*f*x*tan(e/2 + f*x/2)**3
/(3*a**2*f*tan(e/2 + f*x/2)**5 + 9*a**2*f*tan(e/2 + f*x/2)**4 + 12*a**2*f*tan(e/2 + f*x/2)**3 + 12*a**2*f*tan(
e/2 + f*x/2)**2 + 9*a**2*f*tan(e/2 + f*x/2) + 3*a**2*f) + 24*B*c*d*f*x*tan(e/2 + f*x/2)**2/(3*a**2*f*tan(e/2 +
f*x/2)**5 + 9*a**2*f*tan(e/2 + f*x/2)**4 + 12*a**2*f*tan(e/2 + f*x/2)**3 + 12*a**2*f*tan(e/2 + f*x/2)**2 + 9*
a**2*f*tan(e/2 + f*x/2) + 3*a**2*f) + 18*B*c*d*f*x*tan(e/2 + f*x/2)/(3*a**2*f*tan(e/2 + f*x/2)**5 + 9*a**2*f*t
an(e/2 + f*x/2)**4 + 12*a**2*f*tan(e/2 + f*x/2)**3 + 12*a**2*f*tan(e/2 + f*x/2)**2 + 9*a**2*f*tan(e/2 + f*x/2)
+ 3*a**2*f) + 6*B*c*d*f*x/(3*a**2*f*tan(e/2 + f*x/2)**5 + 9*a**2*f*tan(e/2 + f*x/2)**4 + 12*a**2*f*tan(e/2 +
f*x/2)**3 + 12*a**2*f*tan(e/2 + f*x/2)**2 + 9*a**2*f*tan(e/2 + f*x/2) + 3*a**2*f) + 12*B*c*d*tan(e/2 + f*x/2)*
*4/(3*a**2*f*tan(e/2 + f*x/2)**5 + 9*a**2*f*tan(e/2 + f*x/2)**4 + 12*a**2*f*tan(e/2 + f*x/2)**3 + 12*a**2*f*ta
n(e/2 + f*x/2)**2 + 9*a**2*f*tan(e/2 + f*x/2) + 3*a**2*f) + 36*B*c*d*tan(e/2 + f*x/2)**3/(3*a**2*f*tan(e/2 + f
*x/2)**5 + 9*a**2*f*tan(e/2 + f*x/2)**4 + 12*a**2*f*tan(e/2 + f*x/2)**3 + 12*a**2*f*tan(e/2 + f*x/2)**2 + 9*a*
*2*f*tan(e/2 + f*x/2) + 3*a**2*f) + 28*B*c*d*tan(e/2 + f*x/2)**2/(3*a**2*f*tan(e/2 + f*x/2)**5 + 9*a**2*f*tan(
e/2 + f*x/2)**4 + 12*a**2*f*tan(e/2 + f*x/2)**3 + 12*a**2*f*tan(e/2 + f*x/2)**2 + 9*a**2*f*tan(e/2 + f*x/2) +
3*a**2*f) + 36*B*c*d*tan(e/2 + f*x/2)/(3*a**2*f*tan(e/2 + f*x/2)**5 + 9*a**2*f*tan(e/2 + f*x/2)**4 + 12*a**2*f
*tan(e/2 + f*x/2)**3 + 12*a**2*f*tan(e/2 + f*x/2)**2 + 9*a**2*f*tan(e/2 + f*x/2) + 3*a**2*f) + 16*B*c*d/(3*a**
2*f*tan(e/2 + f*x/2)**5 + 9*a**2*f*tan(e/2 + f*x/2)**4 + 12*a**2*f*tan(e/2 + f*x/2)**3 + 12*a**2*f*tan(e/2 + f
*x/2)**2 + 9*a**2*f*tan(e/2 + f*x/2) + 3*a**2*f) - 6*B*d**2*f*x*tan(e/2 + f*x/2)**5/(3*a**2*f*tan(e/2 + f*x/2)
**5 + 9*a**2*f*tan(e/2 + f*x/2)**4 + 12*a**2*f*tan(e/2 + f*x/2)**3 + 12*a**2*f*tan(e/2 + f*x/2)**2 + 9*a**2*f*
tan(e/2 + f*x/2) + 3*a**2*f) - 18*B*d**2*f*x*tan(e/2 + f*x/2)**4/(3*a**2*f*tan(e/2 + f*x/2)**5 + 9*a**2*f*tan(
e/2 + f*x/2)**4 + 12*a**2*f*tan(e/2 + f*x/2)**3 + 12*a**2*f*tan(e/2 + f*x/2)**2 + 9*a**2*f*tan(e/2 + f*x/2) +
3*a**2*f) - 24*B*d**2*f*x*tan(e/2 + f*x/2)**3/(3*a**2*f*tan(e/2 + f*x/2)**5 + 9*a**2*f*tan(e/2 + f*x/2)**4 + 1
2*a**2*f*tan(e/2 + f*x/2)**3 + 12*a**2*f*tan(e/2 + f*x/2)**2 + 9*a**2*f*tan(e/2 + f*x/2) + 3*a**2*f) - 24*B*d*
*2*f*x*tan(e/2 + f*x/2)**2/(3*a**2*f*tan(e/2 + f*x/2)**5 + 9*a**2*f*tan(e/2 + f*x/2)**4 + 12*a**2*f*tan(e/2 +
f*x/2)**3 + 12*a**2*f*tan(e/2 + f*x/2)**2 + 9*a**2*f*tan(e/2 + f*x/2) + 3*a**2*f) - 18*B*d**2*f*x*tan(e/2 + f*
x/2)/(3*a**2*f*tan(e/2 + f*x/2)**5 + 9*a**2*f*tan(e/2 + f*x/2)**4 + 12*a**2*f*tan(e/2 + f*x/2)**3 + 12*a**2*f*
tan(e/2 + f*x/2)**2 + 9*a**2*f*tan(e/2 + f*x/2) + 3*a**2*f) - 6*B*d**2*f*x/(3*a**2*f*tan(e/2 + f*x/2)**5 + 9*a
**2*f*tan(e/2 + f*x/2)**4 + 12*a**2*f*tan(e/2 + f*x/2)**3 + 12*a**2*f*tan(e/2 + f*x/2)**2 + 9*a**2*f*tan(e/2 +
f*x/2) + 3*a**2*f) - 12*B*d**2*tan(e/2 + f*x/2)**4/(3*a**2*f*tan(e/2 + f*x/2)**5 + 9*a**2*f*tan(e/2 + f*x/2)*
*4 + 12*a**2*f*tan(e/2 + f*x/2)**3 + 12*a**2*f*tan(e/2 + f*x/2)**2 + 9*a**2*f*tan(e/2 + f*x/2) + 3*a**2*f) - 3
6*B*d**2*tan(e/2 + f*x/2)**3/(3*a**2*f*tan(e/2 + f*x/2)**5 + 9*a**2*f*tan(e/2 + f*x/2)**4 + 12*a**2*f*tan(e/2
+ f*x/2)**3 + 12*a**2*f*tan(e/2 + f*x/2)**2 + 9*a**2*f*tan(e/2 + f*x/2) + 3*a**2*f) - 44*B*d**2*tan(e/2 + f*x/
2)**2/(3*a**2*f*tan(e/2 + f*x/2)**5 + 9*a**2*f*tan(e/2 + f*x/2)**4 + 12*a**2*f*tan(e/2 + f*x/2)**3 + 12*a**2*f
*tan(e/2 + f*x/2)**2 + 9*a**2*f*tan(e/2 + f*x/2) + 3*a**2*f) - 48*B*d**2*tan(e/2 + f*x/2)/(3*a**2*f*tan(e/2 +
f*x/2)**5 + 9*a**2*f*tan(e/2 + f*x/2)**4 + 12*a**2*f*tan(e/2 + f*x/2)**3 + 12*a**2*f*tan(e/2 + f*x/2)**2 + 9*a
**2*f*tan(e/2 + f*x/2) + 3*a**2*f) - 20*B*d**2/(3*a**2*f*tan(e/2 + f*x/2)**5 + 9*a**2*f*tan(e/2 + f*x/2)**4 +
12*a**2*f*tan(e/2 + f*x/2)**3 + 12*a**2*f*tan(e/2 + f*x/2)**2 + 9*a**2*f*tan(e/2 + f*x/2) + 3*a**2*f), Ne(f, 0
)), (x*(A + B*sin(e))*(c + d*sin(e))**2/(a*sin(e) + a)**2, True))
Maxima [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 831 vs. \(2 (126) = 252\).
Time = 0.32 (sec) , antiderivative size = 831, normalized size of antiderivative = 6.30
\[
\int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))^2}{(a+a \sin (e+f x))^2} \, dx=\text {Too large to display}
\]
[In]
integrate((A+B*sin(f*x+e))*(c+d*sin(f*x+e))^2/(a+a*sin(f*x+e))^2,x, algorithm="maxima")
[Out]
-2/3*(2*B*d^2*((12*sin(f*x + e)/(cos(f*x + e) + 1) + 11*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 9*sin(f*x + e)^3
/(cos(f*x + e) + 1)^3 + 3*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 5)/(a^2 + 3*a^2*sin(f*x + e)/(cos(f*x + e) + 1
) + 4*a^2*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 4*a^2*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 3*a^2*sin(f*x + e)
^4/(cos(f*x + e) + 1)^4 + a^2*sin(f*x + e)^5/(cos(f*x + e) + 1)^5) + 3*arctan(sin(f*x + e)/(cos(f*x + e) + 1))
/a^2) - 2*B*c*d*((9*sin(f*x + e)/(cos(f*x + e) + 1) + 3*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 4)/(a^2 + 3*a^2*
sin(f*x + e)/(cos(f*x + e) + 1) + 3*a^2*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + a^2*sin(f*x + e)^3/(cos(f*x + e)
+ 1)^3) + 3*arctan(sin(f*x + e)/(cos(f*x + e) + 1))/a^2) - A*d^2*((9*sin(f*x + e)/(cos(f*x + e) + 1) + 3*sin(
f*x + e)^2/(cos(f*x + e) + 1)^2 + 4)/(a^2 + 3*a^2*sin(f*x + e)/(cos(f*x + e) + 1) + 3*a^2*sin(f*x + e)^2/(cos(
f*x + e) + 1)^2 + a^2*sin(f*x + e)^3/(cos(f*x + e) + 1)^3) + 3*arctan(sin(f*x + e)/(cos(f*x + e) + 1))/a^2) +
A*c^2*(3*sin(f*x + e)/(cos(f*x + e) + 1) + 3*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 2)/(a^2 + 3*a^2*sin(f*x + e
)/(cos(f*x + e) + 1) + 3*a^2*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + a^2*sin(f*x + e)^3/(cos(f*x + e) + 1)^3) +
B*c^2*(3*sin(f*x + e)/(cos(f*x + e) + 1) + 1)/(a^2 + 3*a^2*sin(f*x + e)/(cos(f*x + e) + 1) + 3*a^2*sin(f*x + e
)^2/(cos(f*x + e) + 1)^2 + a^2*sin(f*x + e)^3/(cos(f*x + e) + 1)^3) + 2*A*c*d*(3*sin(f*x + e)/(cos(f*x + e) +
1) + 1)/(a^2 + 3*a^2*sin(f*x + e)/(cos(f*x + e) + 1) + 3*a^2*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + a^2*sin(f*x
+ e)^3/(cos(f*x + e) + 1)^3))/f
Giac [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 264 vs. \(2 (126) = 252\).
Time = 0.31 (sec) , antiderivative size = 264, normalized size of antiderivative = 2.00
\[
\int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))^2}{(a+a \sin (e+f x))^2} \, dx=\frac {\frac {3 \, {\left (2 \, B c d + A d^{2} - 2 \, B d^{2}\right )} {\left (f x + e\right )}}{a^{2}} - \frac {6 \, B d^{2}}{{\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 1\right )} a^{2}} - \frac {2 \, {\left (3 \, A c^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 6 \, B c d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 3 \, A d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 6 \, B d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 3 \, A c^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 3 \, B c^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 6 \, A c d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 18 \, B c d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 9 \, A d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 15 \, B d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 2 \, A c^{2} + B c^{2} + 2 \, A c d - 8 \, B c d - 4 \, A d^{2} + 7 \, B d^{2}\right )}}{a^{2} {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{3}}}{3 \, f}
\]
[In]
integrate((A+B*sin(f*x+e))*(c+d*sin(f*x+e))^2/(a+a*sin(f*x+e))^2,x, algorithm="giac")
[Out]
1/3*(3*(2*B*c*d + A*d^2 - 2*B*d^2)*(f*x + e)/a^2 - 6*B*d^2/((tan(1/2*f*x + 1/2*e)^2 + 1)*a^2) - 2*(3*A*c^2*tan
(1/2*f*x + 1/2*e)^2 - 6*B*c*d*tan(1/2*f*x + 1/2*e)^2 - 3*A*d^2*tan(1/2*f*x + 1/2*e)^2 + 6*B*d^2*tan(1/2*f*x +
1/2*e)^2 + 3*A*c^2*tan(1/2*f*x + 1/2*e) + 3*B*c^2*tan(1/2*f*x + 1/2*e) + 6*A*c*d*tan(1/2*f*x + 1/2*e) - 18*B*c
*d*tan(1/2*f*x + 1/2*e) - 9*A*d^2*tan(1/2*f*x + 1/2*e) + 15*B*d^2*tan(1/2*f*x + 1/2*e) + 2*A*c^2 + B*c^2 + 2*A
*c*d - 8*B*c*d - 4*A*d^2 + 7*B*d^2)/(a^2*(tan(1/2*f*x + 1/2*e) + 1)^3))/f
Mupad [B] (verification not implemented)
Time = 16.49 (sec) , antiderivative size = 365, normalized size of antiderivative = 2.77
\[
\int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))^2}{(a+a \sin (e+f x))^2} \, dx=\frac {2\,d\,\mathrm {atan}\left (\frac {2\,d\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (A\,d+2\,B\,c-2\,B\,d\right )}{2\,A\,d^2-4\,B\,d^2+4\,B\,c\,d}\right )\,\left (A\,d+2\,B\,c-2\,B\,d\right )}{a^2\,f}-\frac {{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left (2\,A\,c^2-6\,A\,d^2+2\,B\,c^2+12\,B\,d^2+4\,A\,c\,d-12\,B\,c\,d\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (\frac {10\,A\,c^2}{3}-\frac {14\,A\,d^2}{3}+\frac {2\,B\,c^2}{3}+\frac {44\,B\,d^2}{3}+\frac {4\,A\,c\,d}{3}-\frac {28\,B\,c\,d}{3}\right )+\frac {4\,A\,c^2}{3}-\frac {8\,A\,d^2}{3}+\frac {2\,B\,c^2}{3}+\frac {20\,B\,d^2}{3}+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\left (2\,A\,c^2-2\,A\,d^2+4\,B\,d^2-4\,B\,c\,d\right )+\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (2\,A\,c^2-6\,A\,d^2+2\,B\,c^2+16\,B\,d^2+4\,A\,c\,d-12\,B\,c\,d\right )+\frac {4\,A\,c\,d}{3}-\frac {16\,B\,c\,d}{3}}{f\,\left (a^2\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5+3\,a^2\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4+4\,a^2\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3+4\,a^2\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+3\,a^2\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )+a^2\right )}
\]
[In]
int(((A + B*sin(e + f*x))*(c + d*sin(e + f*x))^2)/(a + a*sin(e + f*x))^2,x)
[Out]
(2*d*atan((2*d*tan(e/2 + (f*x)/2)*(A*d + 2*B*c - 2*B*d))/(2*A*d^2 - 4*B*d^2 + 4*B*c*d))*(A*d + 2*B*c - 2*B*d))
/(a^2*f) - (tan(e/2 + (f*x)/2)^3*(2*A*c^2 - 6*A*d^2 + 2*B*c^2 + 12*B*d^2 + 4*A*c*d - 12*B*c*d) + tan(e/2 + (f*
x)/2)^2*((10*A*c^2)/3 - (14*A*d^2)/3 + (2*B*c^2)/3 + (44*B*d^2)/3 + (4*A*c*d)/3 - (28*B*c*d)/3) + (4*A*c^2)/3
- (8*A*d^2)/3 + (2*B*c^2)/3 + (20*B*d^2)/3 + tan(e/2 + (f*x)/2)^4*(2*A*c^2 - 2*A*d^2 + 4*B*d^2 - 4*B*c*d) + ta
n(e/2 + (f*x)/2)*(2*A*c^2 - 6*A*d^2 + 2*B*c^2 + 16*B*d^2 + 4*A*c*d - 12*B*c*d) + (4*A*c*d)/3 - (16*B*c*d)/3)/(
f*(4*a^2*tan(e/2 + (f*x)/2)^2 + 4*a^2*tan(e/2 + (f*x)/2)^3 + 3*a^2*tan(e/2 + (f*x)/2)^4 + a^2*tan(e/2 + (f*x)/
2)^5 + a^2 + 3*a^2*tan(e/2 + (f*x)/2)))