\(\int \frac {\sin ^2(e+f x) (A+B \sin (e+f x))}{(a+b \sin (e+f x))^2} \, dx\) [16]

Optimal result

Integrand size = 31, antiderivative size = 153 \[ \int \frac {\sin ^2(e+f x) (A+B \sin (e+f x))}{(a+b \sin (e+f x))^2} \, dx=\frac {(A b-2 a B) x}{b^3}-\frac {2 a \left (a^2 A b-2 A b^3-2 a^3 B+3 a b^2 B\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {a^2-b^2}}\right )}{b^3 \left (a^2-b^2\right )^{3/2} f}-\frac {B \cos (e+f x)}{b^2 f}+\frac {a^2 (A b-a B) \cos (e+f x)}{b^2 \left (a^2-b^2\right ) f (a+b \sin (e+f x))} \]

[Out]

(A*b-2*B*a)*x/b^3-2*a*(A*a^2*b-2*A*b^3-2*B*a^3+3*B*a*b^2)*arctan((b+a*tan(1/2*f*x+1/2*e))/(a^2-b^2)^(1/2))/b^3
/(a^2-b^2)^(3/2)/f-B*cos(f*x+e)/b^2/f+a^2*(A*b-B*a)*cos(f*x+e)/b^2/(a^2-b^2)/f/(a+b*sin(f*x+e))

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {3067, 3102, 2814, 2739, 632, 210} \[ \int \frac {\sin ^2(e+f x) (A+B \sin (e+f x))}{(a+b \sin (e+f x))^2} \, dx=\frac {a^2 (A b-a B) \cos (e+f x)}{b^2 f \left (a^2-b^2\right ) (a+b \sin (e+f x))}-\frac {2 a \left (-2 a^3 B+a^2 A b+3 a b^2 B-2 A b^3\right ) \arctan \left (\frac {a \tan \left (\frac {1}{2} (e+f x)\right )+b}{\sqrt {a^2-b^2}}\right )}{b^3 f \left (a^2-b^2\right )^{3/2}}+\frac {x (A b-2 a B)}{b^3}-\frac {B \cos (e+f x)}{b^2 f} \]

[In]

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

[Out]

((A*b - 2*a*B)*x)/b^3 - (2*a*(a^2*A*b - 2*A*b^3 - 2*a^3*B + 3*a*b^2*B)*ArcTan[(b + a*Tan[(e + f*x)/2])/Sqrt[a^
2 - b^2]])/(b^3*(a^2 - b^2)^(3/2)*f) - (B*Cos[e + f*x])/(b^2*f) + (a^2*(A*b - a*B)*Cos[e + f*x])/(b^2*(a^2 - b
^2)*f*(a + b*Sin[e + f*x]))

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 632

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 2739

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x]}, Dis
t[2*(e/d), Subst[Int[1/(a + 2*b*e*x + a*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}, x] &&
NeQ[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 3067

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

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^2 (A b-a B) \cos (e+f x)}{b^2 \left (a^2-b^2\right ) f (a+b \sin (e+f x))}+\frac {\int \frac {a b (A b-a B)+\left (a^2-b^2\right ) (A b-a B) \sin (e+f x)+b \left (a^2-b^2\right ) B \sin ^2(e+f x)}{a+b \sin (e+f x)} \, dx}{b^2 \left (a^2-b^2\right )} \\ & = -\frac {B \cos (e+f x)}{b^2 f}+\frac {a^2 (A b-a B) \cos (e+f x)}{b^2 \left (a^2-b^2\right ) f (a+b \sin (e+f x))}+\frac {\int \frac {a b^2 (A b-a B)+b \left (a^2-b^2\right ) (A b-2 a B) \sin (e+f x)}{a+b \sin (e+f x)} \, dx}{b^3 \left (a^2-b^2\right )} \\ & = \frac {(A b-2 a B) x}{b^3}-\frac {B \cos (e+f x)}{b^2 f}+\frac {a^2 (A b-a B) \cos (e+f x)}{b^2 \left (a^2-b^2\right ) f (a+b \sin (e+f x))}-\frac {\left (a \left (a^2 A b-2 A b^3-2 a^3 B+3 a b^2 B\right )\right ) \int \frac {1}{a+b \sin (e+f x)} \, dx}{b^3 \left (a^2-b^2\right )} \\ & = \frac {(A b-2 a B) x}{b^3}-\frac {B \cos (e+f x)}{b^2 f}+\frac {a^2 (A b-a B) \cos (e+f x)}{b^2 \left (a^2-b^2\right ) f (a+b \sin (e+f x))}-\frac {\left (2 a \left (a^2 A b-2 A b^3-2 a^3 B+3 a b^2 B\right )\right ) \text {Subst}\left (\int \frac {1}{a+2 b x+a x^2} \, dx,x,\tan \left (\frac {1}{2} (e+f x)\right )\right )}{b^3 \left (a^2-b^2\right ) f} \\ & = \frac {(A b-2 a B) x}{b^3}-\frac {B \cos (e+f x)}{b^2 f}+\frac {a^2 (A b-a B) \cos (e+f x)}{b^2 \left (a^2-b^2\right ) f (a+b \sin (e+f x))}+\frac {\left (4 a \left (a^2 A b-2 A b^3-2 a^3 B+3 a b^2 B\right )\right ) \text {Subst}\left (\int \frac {1}{-4 \left (a^2-b^2\right )-x^2} \, dx,x,2 b+2 a \tan \left (\frac {1}{2} (e+f x)\right )\right )}{b^3 \left (a^2-b^2\right ) f} \\ & = \frac {(A b-2 a B) x}{b^3}-\frac {2 a \left (a^2 A b-2 A b^3-2 a^3 B+3 a b^2 B\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {a^2-b^2}}\right )}{b^3 \left (a^2-b^2\right )^{3/2} f}-\frac {B \cos (e+f x)}{b^2 f}+\frac {a^2 (A b-a B) \cos (e+f x)}{b^2 \left (a^2-b^2\right ) f (a+b \sin (e+f x))} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.97 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.96 \[ \int \frac {\sin ^2(e+f x) (A+B \sin (e+f x))}{(a+b \sin (e+f x))^2} \, dx=\frac {(A b-2 a B) (e+f x)+\frac {2 a \left (-a^2 A b+2 A b^3+2 a^3 B-3 a b^2 B\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2}}-b B \cos (e+f x)+\frac {a^2 b (A b-a B) \cos (e+f x)}{(a-b) (a+b) (a+b \sin (e+f x))}}{b^3 f} \]

[In]

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

[Out]

((A*b - 2*a*B)*(e + f*x) + (2*a*(-(a^2*A*b) + 2*A*b^3 + 2*a^3*B - 3*a*b^2*B)*ArcTan[(b + a*Tan[(e + f*x)/2])/S
qrt[a^2 - b^2]])/(a^2 - b^2)^(3/2) - b*B*Cos[e + f*x] + (a^2*b*(A*b - a*B)*Cos[e + f*x])/((a - b)*(a + b)*(a +
b*Sin[e + f*x])))/(b^3*f)

Maple [A] (verified)

Time = 0.90 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.39

method	result	size
derivativedivides	\(\frac {-\frac {2 a \left (\frac {-\frac {b^{2} \left (A b -B a \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{a^{2}-b^{2}}-\frac {b a \left (A b -B a \right )}{a^{2}-b^{2}}}{\left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right ) a +2 b \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+a}+\frac {\left (A \,a^{2} b -2 A \,b^{3}-2 B \,a^{3}+3 B a \,b^{2}\right ) \arctan \left (\frac {2 a \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{\left (a^{2}-b^{2}\right )^{\frac {3}{2}}}\right )}{b^{3}}+\frac {-\frac {2 B b}{1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )}+2 \left (A b -2 B a \right ) \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{b^{3}}}{f}\)	\(212\)
default	\(\frac {-\frac {2 a \left (\frac {-\frac {b^{2} \left (A b -B a \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{a^{2}-b^{2}}-\frac {b a \left (A b -B a \right )}{a^{2}-b^{2}}}{\left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right ) a +2 b \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+a}+\frac {\left (A \,a^{2} b -2 A \,b^{3}-2 B \,a^{3}+3 B a \,b^{2}\right ) \arctan \left (\frac {2 a \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{\left (a^{2}-b^{2}\right )^{\frac {3}{2}}}\right )}{b^{3}}+\frac {-\frac {2 B b}{1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )}+2 \left (A b -2 B a \right ) \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{b^{3}}}{f}\)	\(212\)
risch	\(\frac {x A}{b^{2}}-\frac {2 x B a}{b^{3}}-\frac {B \,{\mathrm e}^{i \left (f x +e \right )}}{2 b^{2} f}-\frac {B \,{\mathrm e}^{-i \left (f x +e \right )}}{2 b^{2} f}+\frac {2 i a^{2} \left (-A b +B a \right ) \left (i b +a \,{\mathrm e}^{i \left (f x +e \right )}\right )}{b^{3} \left (a^{2}-b^{2}\right ) f \left (-i {\mathrm e}^{2 i \left (f x +e \right )} b +2 a \,{\mathrm e}^{i \left (f x +e \right )}+i b \right )}+\frac {a^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i \sqrt {-a^{2}+b^{2}}\, a -a^{2}+b^{2}}{b \sqrt {-a^{2}+b^{2}}}\right ) A}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right ) \left (a -b \right ) f \,b^{2}}-\frac {2 a \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i \sqrt {-a^{2}+b^{2}}\, a -a^{2}+b^{2}}{b \sqrt {-a^{2}+b^{2}}}\right ) A}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right ) \left (a -b \right ) f}-\frac {2 a^{4} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i \sqrt {-a^{2}+b^{2}}\, a -a^{2}+b^{2}}{b \sqrt {-a^{2}+b^{2}}}\right ) B}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right ) \left (a -b \right ) f \,b^{3}}+\frac {3 a^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i \sqrt {-a^{2}+b^{2}}\, a -a^{2}+b^{2}}{b \sqrt {-a^{2}+b^{2}}}\right ) B}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right ) \left (a -b \right ) f b}-\frac {a^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i \sqrt {-a^{2}+b^{2}}\, a +a^{2}-b^{2}}{b \sqrt {-a^{2}+b^{2}}}\right ) A}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right ) \left (a -b \right ) f \,b^{2}}+\frac {2 a \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i \sqrt {-a^{2}+b^{2}}\, a +a^{2}-b^{2}}{b \sqrt {-a^{2}+b^{2}}}\right ) A}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right ) \left (a -b \right ) f}+\frac {2 a^{4} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i \sqrt {-a^{2}+b^{2}}\, a +a^{2}-b^{2}}{b \sqrt {-a^{2}+b^{2}}}\right ) B}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right ) \left (a -b \right ) f \,b^{3}}-\frac {3 a^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i \sqrt {-a^{2}+b^{2}}\, a +a^{2}-b^{2}}{b \sqrt {-a^{2}+b^{2}}}\right ) B}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right ) \left (a -b \right ) f b}\)	\(800\)

[In]

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

[Out]

1/f*(-2*a/b^3*((-b^2*(A*b-B*a)/(a^2-b^2)*tan(1/2*f*x+1/2*e)-b*a*(A*b-B*a)/(a^2-b^2))/(tan(1/2*f*x+1/2*e)^2*a+2
*b*tan(1/2*f*x+1/2*e)+a)+(A*a^2*b-2*A*b^3-2*B*a^3+3*B*a*b^2)/(a^2-b^2)^(3/2)*arctan(1/2*(2*a*tan(1/2*f*x+1/2*e
)+2*b)/(a^2-b^2)^(1/2)))+2/b^3*(-B*b/(1+tan(1/2*f*x+1/2*e)^2)+(A*b-2*B*a)*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. 359 vs. \(2 (152) = 304\).

Time = 0.33 (sec) , antiderivative size = 804, normalized size of antiderivative = 5.25 \[ \int \frac {\sin ^2(e+f x) (A+B \sin (e+f x))}{(a+b \sin (e+f x))^2} \, dx=\left [-\frac {2 \, {\left (2 \, B a^{6} - A a^{5} b - 4 \, B a^{4} b^{2} + 2 \, A a^{3} b^{3} + 2 \, B a^{2} b^{4} - A a b^{5}\right )} f x + {\left (2 \, B a^{5} - A a^{4} b - 3 \, B a^{3} b^{2} + 2 \, A a^{2} b^{3} + {\left (2 \, B a^{4} b - A a^{3} b^{2} - 3 \, B a^{2} b^{3} + 2 \, A a b^{4}\right )} \sin \left (f x + e\right )\right )} \sqrt {-a^{2} + b^{2}} \log \left (\frac {{\left (2 \, a^{2} - b^{2}\right )} \cos \left (f x + e\right )^{2} - 2 \, a b \sin \left (f x + e\right ) - a^{2} - b^{2} + 2 \, {\left (a \cos \left (f x + e\right ) \sin \left (f x + e\right ) + b \cos \left (f x + e\right )\right )} \sqrt {-a^{2} + b^{2}}}{b^{2} \cos \left (f x + e\right )^{2} - 2 \, a b \sin \left (f x + e\right ) - a^{2} - b^{2}}\right ) + 2 \, {\left (2 \, B a^{5} b - A a^{4} b^{2} - 3 \, B a^{3} b^{3} + A a^{2} b^{4} + B a b^{5}\right )} \cos \left (f x + e\right ) + 2 \, {\left ({\left (2 \, B a^{5} b - A a^{4} b^{2} - 4 \, B a^{3} b^{3} + 2 \, A a^{2} b^{4} + 2 \, B a b^{5} - A b^{6}\right )} f x + {\left (B a^{4} b^{2} - 2 \, B a^{2} b^{4} + B b^{6}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{2 \, {\left ({\left (a^{4} b^{4} - 2 \, a^{2} b^{6} + b^{8}\right )} f \sin \left (f x + e\right ) + {\left (a^{5} b^{3} - 2 \, a^{3} b^{5} + a b^{7}\right )} f\right )}}, -\frac {{\left (2 \, B a^{6} - A a^{5} b - 4 \, B a^{4} b^{2} + 2 \, A a^{3} b^{3} + 2 \, B a^{2} b^{4} - A a b^{5}\right )} f x + {\left (2 \, B a^{5} - A a^{4} b - 3 \, B a^{3} b^{2} + 2 \, A a^{2} b^{3} + {\left (2 \, B a^{4} b - A a^{3} b^{2} - 3 \, B a^{2} b^{3} + 2 \, A a b^{4}\right )} \sin \left (f x + e\right )\right )} \sqrt {a^{2} - b^{2}} \arctan \left (-\frac {a \sin \left (f x + e\right ) + b}{\sqrt {a^{2} - b^{2}} \cos \left (f x + e\right )}\right ) + {\left (2 \, B a^{5} b - A a^{4} b^{2} - 3 \, B a^{3} b^{3} + A a^{2} b^{4} + B a b^{5}\right )} \cos \left (f x + e\right ) + {\left ({\left (2 \, B a^{5} b - A a^{4} b^{2} - 4 \, B a^{3} b^{3} + 2 \, A a^{2} b^{4} + 2 \, B a b^{5} - A b^{6}\right )} f x + {\left (B a^{4} b^{2} - 2 \, B a^{2} b^{4} + B b^{6}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{{\left (a^{4} b^{4} - 2 \, a^{2} b^{6} + b^{8}\right )} f \sin \left (f x + e\right ) + {\left (a^{5} b^{3} - 2 \, a^{3} b^{5} + a b^{7}\right )} f}\right ] \]

[In]

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

[Out]

[-1/2*(2*(2*B*a^6 - A*a^5*b - 4*B*a^4*b^2 + 2*A*a^3*b^3 + 2*B*a^2*b^4 - A*a*b^5)*f*x + (2*B*a^5 - A*a^4*b - 3*
B*a^3*b^2 + 2*A*a^2*b^3 + (2*B*a^4*b - A*a^3*b^2 - 3*B*a^2*b^3 + 2*A*a*b^4)*sin(f*x + e))*sqrt(-a^2 + b^2)*log
(((2*a^2 - b^2)*cos(f*x + e)^2 - 2*a*b*sin(f*x + e) - a^2 - b^2 + 2*(a*cos(f*x + e)*sin(f*x + e) + b*cos(f*x +
e))*sqrt(-a^2 + b^2))/(b^2*cos(f*x + e)^2 - 2*a*b*sin(f*x + e) - a^2 - b^2)) + 2*(2*B*a^5*b - A*a^4*b^2 - 3*B
*a^3*b^3 + A*a^2*b^4 + B*a*b^5)*cos(f*x + e) + 2*((2*B*a^5*b - A*a^4*b^2 - 4*B*a^3*b^3 + 2*A*a^2*b^4 + 2*B*a*b
^5 - A*b^6)*f*x + (B*a^4*b^2 - 2*B*a^2*b^4 + B*b^6)*cos(f*x + e))*sin(f*x + e))/((a^4*b^4 - 2*a^2*b^6 + b^8)*f
*sin(f*x + e) + (a^5*b^3 - 2*a^3*b^5 + a*b^7)*f), -((2*B*a^6 - A*a^5*b - 4*B*a^4*b^2 + 2*A*a^3*b^3 + 2*B*a^2*b
^4 - A*a*b^5)*f*x + (2*B*a^5 - A*a^4*b - 3*B*a^3*b^2 + 2*A*a^2*b^3 + (2*B*a^4*b - A*a^3*b^2 - 3*B*a^2*b^3 + 2*
A*a*b^4)*sin(f*x + e))*sqrt(a^2 - b^2)*arctan(-(a*sin(f*x + e) + b)/(sqrt(a^2 - b^2)*cos(f*x + e))) + (2*B*a^5
*b - A*a^4*b^2 - 3*B*a^3*b^3 + A*a^2*b^4 + B*a*b^5)*cos(f*x + e) + ((2*B*a^5*b - A*a^4*b^2 - 4*B*a^3*b^3 + 2*A
*a^2*b^4 + 2*B*a*b^5 - A*b^6)*f*x + (B*a^4*b^2 - 2*B*a^2*b^4 + B*b^6)*cos(f*x + e))*sin(f*x + e))/((a^4*b^4 -
2*a^2*b^6 + b^8)*f*sin(f*x + e) + (a^5*b^3 - 2*a^3*b^5 + a*b^7)*f)]

Sympy [F(-1)]

Timed out. \[ \int \frac {\sin ^2(e+f x) (A+B \sin (e+f x))}{(a+b \sin (e+f x))^2} \, dx=\text {Timed out} \]

[In]

integrate(sin(f*x+e)**2*(A+B*sin(f*x+e))/(a+b*sin(f*x+e))**2,x)

[Out]

Timed out

Maxima [F(-2)]

Exception generated. \[ \int \frac {\sin ^2(e+f x) (A+B \sin (e+f x))}{(a+b \sin (e+f x))^2} \, dx=\text {Exception raised: ValueError} \]

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*b^2-4*a^2>0)', see `assume?`
for more de

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 356 vs. \(2 (152) = 304\).

Time = 0.66 (sec) , antiderivative size = 356, normalized size of antiderivative = 2.33 \[ \int \frac {\sin ^2(e+f x) (A+B \sin (e+f x))}{(a+b \sin (e+f x))^2} \, dx=\frac {\frac {2 \, {\left (2 \, B a^{4} - A a^{3} b - 3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} {\left (\pi \left \lfloor \frac {f x + e}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (a\right ) + \arctan \left (\frac {a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + b}{\sqrt {a^{2} - b^{2}}}\right )\right )}}{{\left (a^{2} b^{3} - b^{5}\right )} \sqrt {a^{2} - b^{2}}} - \frac {2 \, {\left (B a^{2} b \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - A a b^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 2 \, B a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - A a^{2} b \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - B a b^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 3 \, B a^{2} b \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - A a b^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 2 \, B b^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 2 \, B a^{3} - A a^{2} b - B a b^{2}\right )}}{{\left (a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 2 \, b \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 2 \, a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 2 \, b \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + a\right )} {\left (a^{2} b^{2} - b^{4}\right )}} - \frac {{\left (2 \, B a - A b\right )} {\left (f x + e\right )}}{b^{3}}}{f} \]

[In]

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

[Out]

(2*(2*B*a^4 - A*a^3*b - 3*B*a^2*b^2 + 2*A*a*b^3)*(pi*floor(1/2*(f*x + e)/pi + 1/2)*sgn(a) + arctan((a*tan(1/2*
f*x + 1/2*e) + b)/sqrt(a^2 - b^2)))/((a^2*b^3 - b^5)*sqrt(a^2 - b^2)) - 2*(B*a^2*b*tan(1/2*f*x + 1/2*e)^3 - A*
a*b^2*tan(1/2*f*x + 1/2*e)^3 + 2*B*a^3*tan(1/2*f*x + 1/2*e)^2 - A*a^2*b*tan(1/2*f*x + 1/2*e)^2 - B*a*b^2*tan(1
/2*f*x + 1/2*e)^2 + 3*B*a^2*b*tan(1/2*f*x + 1/2*e) - A*a*b^2*tan(1/2*f*x + 1/2*e) - 2*B*b^3*tan(1/2*f*x + 1/2*
e) + 2*B*a^3 - A*a^2*b - B*a*b^2)/((a*tan(1/2*f*x + 1/2*e)^4 + 2*b*tan(1/2*f*x + 1/2*e)^3 + 2*a*tan(1/2*f*x +
1/2*e)^2 + 2*b*tan(1/2*f*x + 1/2*e) + a)*(a^2*b^2 - b^4)) - (2*B*a - A*b)*(f*x + e)/b^3)/f

Mupad [B] (verification not implemented)

Time = 17.91 (sec) , antiderivative size = 3718, normalized size of antiderivative = 24.30 \[ \int \frac {\sin ^2(e+f x) (A+B \sin (e+f x))}{(a+b \sin (e+f x))^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

((2*(A*a^2*b - 2*B*a^3 + B*a*b^2))/(b^2*(a^2 - b^2)) - (2*tan(e/2 + (f*x)/2)^3*(B*a^2 - A*a*b))/(b*(a^2 - b^2)
) + (2*tan(e/2 + (f*x)/2)*(2*B*b^2 - 3*B*a^2 + A*a*b))/(b*(a^2 - b^2)) + (2*tan(e/2 + (f*x)/2)^2*(A*a^2*b - 2*
B*a^3 + B*a*b^2))/(b^2*(a^2 - b^2)))/(f*(a + 2*b*tan(e/2 + (f*x)/2) + 2*a*tan(e/2 + (f*x)/2)^2 + a*tan(e/2 + (
f*x)/2)^4 + 2*b*tan(e/2 + (f*x)/2)^3)) + (log(tan(e/2 + (f*x)/2) + 1i)*(A*b - 2*B*a)*1i)/(b^3*f) - (log(tan(e/
2 + (f*x)/2) - 1i)*(A*b*1i - B*a*2i))/(b^3*f) - (a*atan(((a*(-(a + b)^3*(a - b)^3)^(1/2)*((32*(A^2*a^2*b^8 - 2
*A^2*a^4*b^6 + A^2*a^6*b^4 + 4*B^2*a^4*b^6 - 8*B^2*a^6*b^4 + 4*B^2*a^8*b^2 - 4*A*B*a^3*b^7 + 8*A*B*a^5*b^5 - 4
*A*B*a^7*b^3))/(b^9 - 2*a^2*b^7 + a^4*b^5) + (32*tan(e/2 + (f*x)/2)*(2*A^2*a*b^10 - 9*A^2*a^3*b^8 + 8*A^2*a^5*
b^6 - 2*A^2*a^7*b^4 + 8*B^2*a^3*b^8 - 29*B^2*a^5*b^6 + 28*B^2*a^7*b^4 - 8*B^2*a^9*b^2 - 8*A*B*a^2*b^9 + 32*A*B
*a^4*b^7 - 30*A*B*a^6*b^5 + 8*A*B*a^8*b^3))/(b^10 - 2*a^2*b^8 + a^4*b^6) + (a*(-(a + b)^3*(a - b)^3)^(1/2)*((3
2*tan(e/2 + (f*x)/2)*(4*A*a^2*b^11 - 6*A*a^4*b^9 + 2*A*a^6*b^7 - 6*B*a^3*b^10 + 10*B*a^5*b^8 - 4*B*a^7*b^6))/(
b^10 - 2*a^2*b^8 + a^4*b^6) - (32*(A*a^3*b^9 + 2*B*a^2*b^10 - 3*B*a^4*b^8 + B*a^6*b^6 - A*a*b^11))/(b^9 - 2*a^
2*b^7 + a^4*b^5) + (a*((32*(a^2*b^12 - 2*a^4*b^10 + a^6*b^8))/(b^9 - 2*a^2*b^7 + a^4*b^5) + (32*tan(e/2 + (f*x
)/2)*(3*a*b^14 - 8*a^3*b^12 + 7*a^5*b^10 - 2*a^7*b^8))/(b^10 - 2*a^2*b^8 + a^4*b^6))*(-(a + b)^3*(a - b)^3)^(1
/2)*(2*A*b^3 + 2*B*a^3 - A*a^2*b - 3*B*a*b^2))/(b^9 - 3*a^2*b^7 + 3*a^4*b^5 - a^6*b^3))*(2*A*b^3 + 2*B*a^3 - A
*a^2*b - 3*B*a*b^2))/(b^9 - 3*a^2*b^7 + 3*a^4*b^5 - a^6*b^3))*(2*A*b^3 + 2*B*a^3 - A*a^2*b - 3*B*a*b^2)*1i)/(b
^9 - 3*a^2*b^7 + 3*a^4*b^5 - a^6*b^3) + (a*(-(a + b)^3*(a - b)^3)^(1/2)*((32*(A^2*a^2*b^8 - 2*A^2*a^4*b^6 + A^
2*a^6*b^4 + 4*B^2*a^4*b^6 - 8*B^2*a^6*b^4 + 4*B^2*a^8*b^2 - 4*A*B*a^3*b^7 + 8*A*B*a^5*b^5 - 4*A*B*a^7*b^3))/(b
^9 - 2*a^2*b^7 + a^4*b^5) + (32*tan(e/2 + (f*x)/2)*(2*A^2*a*b^10 - 9*A^2*a^3*b^8 + 8*A^2*a^5*b^6 - 2*A^2*a^7*b
^4 + 8*B^2*a^3*b^8 - 29*B^2*a^5*b^6 + 28*B^2*a^7*b^4 - 8*B^2*a^9*b^2 - 8*A*B*a^2*b^9 + 32*A*B*a^4*b^7 - 30*A*B
*a^6*b^5 + 8*A*B*a^8*b^3))/(b^10 - 2*a^2*b^8 + a^4*b^6) + (a*(-(a + b)^3*(a - b)^3)^(1/2)*((32*(A*a^3*b^9 + 2*
B*a^2*b^10 - 3*B*a^4*b^8 + B*a^6*b^6 - A*a*b^11))/(b^9 - 2*a^2*b^7 + a^4*b^5) - (32*tan(e/2 + (f*x)/2)*(4*A*a^
2*b^11 - 6*A*a^4*b^9 + 2*A*a^6*b^7 - 6*B*a^3*b^10 + 10*B*a^5*b^8 - 4*B*a^7*b^6))/(b^10 - 2*a^2*b^8 + a^4*b^6)
+ (a*((32*(a^2*b^12 - 2*a^4*b^10 + a^6*b^8))/(b^9 - 2*a^2*b^7 + a^4*b^5) + (32*tan(e/2 + (f*x)/2)*(3*a*b^14 -
8*a^3*b^12 + 7*a^5*b^10 - 2*a^7*b^8))/(b^10 - 2*a^2*b^8 + a^4*b^6))*(-(a + b)^3*(a - b)^3)^(1/2)*(2*A*b^3 + 2*
B*a^3 - A*a^2*b - 3*B*a*b^2))/(b^9 - 3*a^2*b^7 + 3*a^4*b^5 - a^6*b^3))*(2*A*b^3 + 2*B*a^3 - A*a^2*b - 3*B*a*b^
2))/(b^9 - 3*a^2*b^7 + 3*a^4*b^5 - a^6*b^3))*(2*A*b^3 + 2*B*a^3 - A*a^2*b - 3*B*a*b^2)*1i)/(b^9 - 3*a^2*b^7 +
3*a^4*b^5 - a^6*b^3))/((64*(4*B^3*a^8 + 2*A^3*a^3*b^5 - A^3*a^5*b^3 - 6*B^3*a^6*b^2 - 8*A*B^2*a^7*b + 13*A*B^2
*a^5*b^3 - 9*A^2*B*a^4*b^4 + 5*A^2*B*a^6*b^2))/(b^9 - 2*a^2*b^7 + a^4*b^5) + (64*tan(e/2 + (f*x)/2)*(16*B^3*a^
9 - 4*A^3*a^2*b^7 + 6*A^3*a^4*b^5 - 2*A^3*a^6*b^3 + 24*B^3*a^5*b^4 - 40*B^3*a^7*b^2 - 24*A*B^2*a^8*b - 40*A*B^
2*a^4*b^5 + 64*A*B^2*a^6*b^3 + 22*A^2*B*a^3*b^6 - 34*A^2*B*a^5*b^4 + 12*A^2*B*a^7*b^2))/(b^10 - 2*a^2*b^8 + a^
4*b^6) - (a*(-(a + b)^3*(a - b)^3)^(1/2)*((32*(A^2*a^2*b^8 - 2*A^2*a^4*b^6 + A^2*a^6*b^4 + 4*B^2*a^4*b^6 - 8*B
^2*a^6*b^4 + 4*B^2*a^8*b^2 - 4*A*B*a^3*b^7 + 8*A*B*a^5*b^5 - 4*A*B*a^7*b^3))/(b^9 - 2*a^2*b^7 + a^4*b^5) + (32
*tan(e/2 + (f*x)/2)*(2*A^2*a*b^10 - 9*A^2*a^3*b^8 + 8*A^2*a^5*b^6 - 2*A^2*a^7*b^4 + 8*B^2*a^3*b^8 - 29*B^2*a^5
*b^6 + 28*B^2*a^7*b^4 - 8*B^2*a^9*b^2 - 8*A*B*a^2*b^9 + 32*A*B*a^4*b^7 - 30*A*B*a^6*b^5 + 8*A*B*a^8*b^3))/(b^1
0 - 2*a^2*b^8 + a^4*b^6) + (a*(-(a + b)^3*(a - b)^3)^(1/2)*((32*tan(e/2 + (f*x)/2)*(4*A*a^2*b^11 - 6*A*a^4*b^9
+ 2*A*a^6*b^7 - 6*B*a^3*b^10 + 10*B*a^5*b^8 - 4*B*a^7*b^6))/(b^10 - 2*a^2*b^8 + a^4*b^6) - (32*(A*a^3*b^9 + 2
*B*a^2*b^10 - 3*B*a^4*b^8 + B*a^6*b^6 - A*a*b^11))/(b^9 - 2*a^2*b^7 + a^4*b^5) + (a*((32*(a^2*b^12 - 2*a^4*b^1
0 + a^6*b^8))/(b^9 - 2*a^2*b^7 + a^4*b^5) + (32*tan(e/2 + (f*x)/2)*(3*a*b^14 - 8*a^3*b^12 + 7*a^5*b^10 - 2*a^7
*b^8))/(b^10 - 2*a^2*b^8 + a^4*b^6))*(-(a + b)^3*(a - b)^3)^(1/2)*(2*A*b^3 + 2*B*a^3 - A*a^2*b - 3*B*a*b^2))/(
b^9 - 3*a^2*b^7 + 3*a^4*b^5 - a^6*b^3))*(2*A*b^3 + 2*B*a^3 - A*a^2*b - 3*B*a*b^2))/(b^9 - 3*a^2*b^7 + 3*a^4*b^
5 - a^6*b^3))*(2*A*b^3 + 2*B*a^3 - A*a^2*b - 3*B*a*b^2))/(b^9 - 3*a^2*b^7 + 3*a^4*b^5 - a^6*b^3) + (a*(-(a + b
)^3*(a - b)^3)^(1/2)*((32*(A^2*a^2*b^8 - 2*A^2*a^4*b^6 + A^2*a^6*b^4 + 4*B^2*a^4*b^6 - 8*B^2*a^6*b^4 + 4*B^2*a
^8*b^2 - 4*A*B*a^3*b^7 + 8*A*B*a^5*b^5 - 4*A*B*a^7*b^3))/(b^9 - 2*a^2*b^7 + a^4*b^5) + (32*tan(e/2 + (f*x)/2)*
(2*A^2*a*b^10 - 9*A^2*a^3*b^8 + 8*A^2*a^5*b^6 - 2*A^2*a^7*b^4 + 8*B^2*a^3*b^8 - 29*B^2*a^5*b^6 + 28*B^2*a^7*b^
4 - 8*B^2*a^9*b^2 - 8*A*B*a^2*b^9 + 32*A*B*a^4*b^7 - 30*A*B*a^6*b^5 + 8*A*B*a^8*b^3))/(b^10 - 2*a^2*b^8 + a^4*
b^6) + (a*(-(a + b)^3*(a - b)^3)^(1/2)*((32*(A*a^3*b^9 + 2*B*a^2*b^10 - 3*B*a^4*b^8 + B*a^6*b^6 - A*a*b^11))/(
b^9 - 2*a^2*b^7 + a^4*b^5) - (32*tan(e/2 + (f*x)/2)*(4*A*a^2*b^11 - 6*A*a^4*b^9 + 2*A*a^6*b^7 - 6*B*a^3*b^10 +
10*B*a^5*b^8 - 4*B*a^7*b^6))/(b^10 - 2*a^2*b^8 + a^4*b^6) + (a*((32*(a^2*b^12 - 2*a^4*b^10 + a^6*b^8))/(b^9 -
2*a^2*b^7 + a^4*b^5) + (32*tan(e/2 + (f*x)/2)*(3*a*b^14 - 8*a^3*b^12 + 7*a^5*b^10 - 2*a^7*b^8))/(b^10 - 2*a^2
*b^8 + a^4*b^6))*(-(a + b)^3*(a - b)^3)^(1/2)*(2*A*b^3 + 2*B*a^3 - A*a^2*b - 3*B*a*b^2))/(b^9 - 3*a^2*b^7 + 3*
a^4*b^5 - a^6*b^3))*(2*A*b^3 + 2*B*a^3 - A*a^2*b - 3*B*a*b^2))/(b^9 - 3*a^2*b^7 + 3*a^4*b^5 - a^6*b^3))*(2*A*b
^3 + 2*B*a^3 - A*a^2*b - 3*B*a*b^2))/(b^9 - 3*a^2*b^7 + 3*a^4*b^5 - a^6*b^3)))*(-(a + b)^3*(a - b)^3)^(1/2)*(2
*A*b^3 + 2*B*a^3 - A*a^2*b - 3*B*a*b^2)*2i)/(f*(b^9 - 3*a^2*b^7 + 3*a^4*b^5 - a^6*b^3))