\(\int \frac {(a+a \sin (e+f x)) (A+B \sin (e+f x))}{(c+d \sin (e+f x))^2} \, dx\) [249]

Optimal result

Integrand size = 33, antiderivative size = 124 \[ \int \frac {(a+a \sin (e+f x)) (A+B \sin (e+f x))}{(c+d \sin (e+f x))^2} \, dx=\frac {a B x}{d^2}+\frac {2 a \left ((A+B) (c-d) d^2-B c \left (c^2-d^2\right )\right ) \arctan \left (\frac {d+c \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c^2-d^2}}\right )}{d^2 \left (c^2-d^2\right )^{3/2} f}+\frac {a (B c-A d) \cos (e+f x)}{d (c+d) f (c+d \sin (e+f x))} \]

[Out]

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

Rubi [A] (verified)

Time = 0.24 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {3047, 3100, 2814, 2739, 632, 210} \[ \int \frac {(a+a \sin (e+f x)) (A+B \sin (e+f x))}{(c+d \sin (e+f x))^2} \, dx=\frac {2 a \left (d^2 (A+B) (c-d)-B c \left (c^2-d^2\right )\right ) \arctan \left (\frac {c \tan \left (\frac {1}{2} (e+f x)\right )+d}{\sqrt {c^2-d^2}}\right )}{d^2 f \left (c^2-d^2\right )^{3/2}}+\frac {a (B c-A d) \cos (e+f x)}{d f (c+d) (c+d \sin (e+f x))}+\frac {a B x}{d^2} \]

[In]

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

[Out]

(a*B*x)/d^2 + (2*a*((A + B)*(c - d)*d^2 - B*c*(c^2 - d^2))*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/(
d^2*(c^2 - d^2)^(3/2)*f) + (a*(B*c - A*d)*Cos[e + f*x])/(d*(c + d)*f*(c + d*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 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 3100

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^2 - a*b*B + a^2*C))*Cos[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m
+ 1)*(a^2 - b^2))), x] + Dist[1/(b*(m + 1)*(a^2 - b^2)), Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[b*(a*A - b*B +
a*C)*(m + 1) - (A*b^2 - a*b*B + a^2*C + b*(A*b - a*B + b*C)*(m + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b,
e, f, A, B, C}, x] && LtQ[m, -1] && NeQ[a^2 - b^2, 0]

Rubi steps \begin{align*} \text {integral}& = \int \frac {a A+(a A+a B) \sin (e+f x)+a B \sin ^2(e+f x)}{(c+d \sin (e+f x))^2} \, dx \\ & = \frac {a (B c-A d) \cos (e+f x)}{d (c+d) f (c+d \sin (e+f x))}-\frac {\int \frac {-a (A+B) (c-d) d-a B \left (c^2-d^2\right ) \sin (e+f x)}{c+d \sin (e+f x)} \, dx}{d \left (c^2-d^2\right )} \\ & = \frac {a B x}{d^2}+\frac {a (B c-A d) \cos (e+f x)}{d (c+d) f (c+d \sin (e+f x))}+\frac {\left (a \left (A d^2-B \left (c^2+c d-d^2\right )\right )\right ) \int \frac {1}{c+d \sin (e+f x)} \, dx}{d^2 (c+d)} \\ & = \frac {a B x}{d^2}+\frac {a (B c-A d) \cos (e+f x)}{d (c+d) f (c+d \sin (e+f x))}+\frac {\left (2 a \left (A d^2-B \left (c^2+c d-d^2\right )\right )\right ) \text {Subst}\left (\int \frac {1}{c+2 d x+c x^2} \, dx,x,\tan \left (\frac {1}{2} (e+f x)\right )\right )}{d^2 (c+d) f} \\ & = \frac {a B x}{d^2}+\frac {a (B c-A d) \cos (e+f x)}{d (c+d) f (c+d \sin (e+f x))}-\frac {\left (4 a \left (A d^2-B \left (c^2+c d-d^2\right )\right )\right ) \text {Subst}\left (\int \frac {1}{-4 \left (c^2-d^2\right )-x^2} \, dx,x,2 d+2 c \tan \left (\frac {1}{2} (e+f x)\right )\right )}{d^2 (c+d) f} \\ & = \frac {a B x}{d^2}+\frac {2 a \left (A d^2-B \left (c^2+c d-d^2\right )\right ) \arctan \left (\frac {d+c \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c^2-d^2}}\right )}{d^2 (c+d) \sqrt {c^2-d^2} f}+\frac {a (B c-A d) \cos (e+f x)}{d (c+d) f (c+d \sin (e+f x))} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 1.90 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.75 \[ \int \frac {(a+a \sin (e+f x)) (A+B \sin (e+f x))}{(c+d \sin (e+f x))^2} \, dx=\frac {a (1+\sin (e+f x)) \left (B x+\frac {2 \left (A d^2-B \left (c^2+c d-d^2\right )\right ) \arctan \left (\frac {\sec \left (\frac {f x}{2}\right ) (\cos (e)-i \sin (e)) \left (d \cos \left (e+\frac {f x}{2}\right )+c \sin \left (\frac {f x}{2}\right )\right )}{\sqrt {c^2-d^2} \sqrt {(\cos (e)-i \sin (e))^2}}\right ) (\cos (e)-i \sin (e))}{(c+d) \sqrt {c^2-d^2} f \sqrt {(\cos (e)-i \sin (e))^2}}+\frac {(-B c+A d) \csc (e) (c \cos (e)+d \sin (f x))}{(c+d) f (c+d \sin (e+f x))}\right )}{d^2 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^2} \]

[In]

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

[Out]

(a*(1 + Sin[e + f*x])*(B*x + (2*(A*d^2 - B*(c^2 + c*d - d^2))*ArcTan[(Sec[(f*x)/2]*(Cos[e] - I*Sin[e])*(d*Cos[
e + (f*x)/2] + c*Sin[(f*x)/2]))/(Sqrt[c^2 - d^2]*Sqrt[(Cos[e] - I*Sin[e])^2])]*(Cos[e] - I*Sin[e]))/((c + d)*S
qrt[c^2 - d^2]*f*Sqrt[(Cos[e] - I*Sin[e])^2]) + ((-(B*c) + A*d)*Csc[e]*(c*Cos[e] + d*Sin[f*x]))/((c + d)*f*(c
+ d*Sin[e + f*x]))))/(d^2*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2)

Maple [A] (verified)

Time = 0.84 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.40

method	result	size
derivativedivides	\(\frac {2 a \left (\frac {B \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{d^{2}}+\frac {\frac {-\frac {d^{2} \left (d A -B c \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{\left (c +d \right ) c}-\frac {d \left (d A -B c \right )}{c +d}}{\left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right ) c +2 d \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+c}+\frac {\left (A \,d^{2}-B \,c^{2}-c d B +d^{2} B \right ) \arctan \left (\frac {2 c \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+2 d}{2 \sqrt {c^{2}-d^{2}}}\right )}{\left (c +d \right ) \sqrt {c^{2}-d^{2}}}}{d^{2}}\right )}{f}\)	\(174\)
default	\(\frac {2 a \left (\frac {B \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{d^{2}}+\frac {\frac {-\frac {d^{2} \left (d A -B c \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{\left (c +d \right ) c}-\frac {d \left (d A -B c \right )}{c +d}}{\left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right ) c +2 d \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+c}+\frac {\left (A \,d^{2}-B \,c^{2}-c d B +d^{2} B \right ) \arctan \left (\frac {2 c \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+2 d}{2 \sqrt {c^{2}-d^{2}}}\right )}{\left (c +d \right ) \sqrt {c^{2}-d^{2}}}}{d^{2}}\right )}{f}\)	\(174\)
risch	\(\frac {a B x}{d^{2}}-\frac {2 i a \left (-d A +B c \right ) \left (i d +c \,{\mathrm e}^{i \left (f x +e \right )}\right )}{d^{2} \left (c +d \right ) f \left (-i {\mathrm e}^{2 i \left (f x +e \right )} d +i d +2 c \,{\mathrm e}^{i \left (f x +e \right )}\right )}-\frac {a \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i c \sqrt {-c^{2}+d^{2}}-c^{2}+d^{2}}{\sqrt {-c^{2}+d^{2}}\, d}\right ) A}{\sqrt {-c^{2}+d^{2}}\, \left (c +d \right ) f}+\frac {a \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i c \sqrt {-c^{2}+d^{2}}-c^{2}+d^{2}}{\sqrt {-c^{2}+d^{2}}\, d}\right ) B \,c^{2}}{\sqrt {-c^{2}+d^{2}}\, \left (c +d \right ) f \,d^{2}}+\frac {a \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i c \sqrt {-c^{2}+d^{2}}-c^{2}+d^{2}}{\sqrt {-c^{2}+d^{2}}\, d}\right ) c B}{\sqrt {-c^{2}+d^{2}}\, \left (c +d \right ) f d}-\frac {a \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i c \sqrt {-c^{2}+d^{2}}-c^{2}+d^{2}}{\sqrt {-c^{2}+d^{2}}\, d}\right ) B}{\sqrt {-c^{2}+d^{2}}\, \left (c +d \right ) f}+\frac {a \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i c \sqrt {-c^{2}+d^{2}}+c^{2}-d^{2}}{\sqrt {-c^{2}+d^{2}}\, d}\right ) A}{\sqrt {-c^{2}+d^{2}}\, \left (c +d \right ) f}-\frac {a \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i c \sqrt {-c^{2}+d^{2}}+c^{2}-d^{2}}{\sqrt {-c^{2}+d^{2}}\, d}\right ) B \,c^{2}}{\sqrt {-c^{2}+d^{2}}\, \left (c +d \right ) f \,d^{2}}-\frac {a \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i c \sqrt {-c^{2}+d^{2}}+c^{2}-d^{2}}{\sqrt {-c^{2}+d^{2}}\, d}\right ) c B}{\sqrt {-c^{2}+d^{2}}\, \left (c +d \right ) f d}+\frac {a \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i c \sqrt {-c^{2}+d^{2}}+c^{2}-d^{2}}{\sqrt {-c^{2}+d^{2}}\, d}\right ) B}{\sqrt {-c^{2}+d^{2}}\, \left (c +d \right ) f}\)	\(680\)

[In]

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

[Out]

2/f*a*(B/d^2*arctan(tan(1/2*f*x+1/2*e))+1/d^2*((-d^2*(A*d-B*c)/(c+d)/c*tan(1/2*f*x+1/2*e)-d*(A*d-B*c)/(c+d))/(
tan(1/2*f*x+1/2*e)^2*c+2*d*tan(1/2*f*x+1/2*e)+c)+(A*d^2-B*c^2-B*c*d+B*d^2)/(c+d)/(c^2-d^2)^(1/2)*arctan(1/2*(2
*c*tan(1/2*f*x+1/2*e)+2*d)/(c^2-d^2)^(1/2))))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 284 vs. \(2 (119) = 238\).

Time = 0.29 (sec) , antiderivative size = 655, normalized size of antiderivative = 5.28 \[ \int \frac {(a+a \sin (e+f x)) (A+B \sin (e+f x))}{(c+d \sin (e+f x))^2} \, dx=\left [\frac {2 \, {\left (B a c^{3} d + B a c^{2} d^{2} - B a c d^{3} - B a d^{4}\right )} f x \sin \left (f x + e\right ) + 2 \, {\left (B a c^{4} + B a c^{3} d - B a c^{2} d^{2} - B a c d^{3}\right )} f x + {\left (B a c^{3} + B a c^{2} d - {\left (A + B\right )} a c d^{2} + {\left (B a c^{2} d + B a c d^{2} - {\left (A + B\right )} a d^{3}\right )} \sin \left (f x + e\right )\right )} \sqrt {-c^{2} + d^{2}} \log \left (\frac {{\left (2 \, c^{2} - d^{2}\right )} \cos \left (f x + e\right )^{2} - 2 \, c d \sin \left (f x + e\right ) - c^{2} - d^{2} + 2 \, {\left (c \cos \left (f x + e\right ) \sin \left (f x + e\right ) + d \cos \left (f x + e\right )\right )} \sqrt {-c^{2} + d^{2}}}{d^{2} \cos \left (f x + e\right )^{2} - 2 \, c d \sin \left (f x + e\right ) - c^{2} - d^{2}}\right ) + 2 \, {\left (B a c^{3} d - A a c^{2} d^{2} - B a c d^{3} + A a d^{4}\right )} \cos \left (f x + e\right )}{2 \, {\left ({\left (c^{3} d^{3} + c^{2} d^{4} - c d^{5} - d^{6}\right )} f \sin \left (f x + e\right ) + {\left (c^{4} d^{2} + c^{3} d^{3} - c^{2} d^{4} - c d^{5}\right )} f\right )}}, \frac {{\left (B a c^{3} d + B a c^{2} d^{2} - B a c d^{3} - B a d^{4}\right )} f x \sin \left (f x + e\right ) + {\left (B a c^{4} + B a c^{3} d - B a c^{2} d^{2} - B a c d^{3}\right )} f x + {\left (B a c^{3} + B a c^{2} d - {\left (A + B\right )} a c d^{2} + {\left (B a c^{2} d + B a c d^{2} - {\left (A + B\right )} a d^{3}\right )} \sin \left (f x + e\right )\right )} \sqrt {c^{2} - d^{2}} \arctan \left (-\frac {c \sin \left (f x + e\right ) + d}{\sqrt {c^{2} - d^{2}} \cos \left (f x + e\right )}\right ) + {\left (B a c^{3} d - A a c^{2} d^{2} - B a c d^{3} + A a d^{4}\right )} \cos \left (f x + e\right )}{{\left (c^{3} d^{3} + c^{2} d^{4} - c d^{5} - d^{6}\right )} f \sin \left (f x + e\right ) + {\left (c^{4} d^{2} + c^{3} d^{3} - c^{2} d^{4} - c d^{5}\right )} f}\right ] \]

[In]

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

[Out]

[1/2*(2*(B*a*c^3*d + B*a*c^2*d^2 - B*a*c*d^3 - B*a*d^4)*f*x*sin(f*x + e) + 2*(B*a*c^4 + B*a*c^3*d - B*a*c^2*d^
2 - B*a*c*d^3)*f*x + (B*a*c^3 + B*a*c^2*d - (A + B)*a*c*d^2 + (B*a*c^2*d + B*a*c*d^2 - (A + B)*a*d^3)*sin(f*x
+ e))*sqrt(-c^2 + d^2)*log(((2*c^2 - d^2)*cos(f*x + e)^2 - 2*c*d*sin(f*x + e) - c^2 - d^2 + 2*(c*cos(f*x + e)*
sin(f*x + e) + d*cos(f*x + e))*sqrt(-c^2 + d^2))/(d^2*cos(f*x + e)^2 - 2*c*d*sin(f*x + e) - c^2 - d^2)) + 2*(B
*a*c^3*d - A*a*c^2*d^2 - B*a*c*d^3 + A*a*d^4)*cos(f*x + e))/((c^3*d^3 + c^2*d^4 - c*d^5 - d^6)*f*sin(f*x + e)
+ (c^4*d^2 + c^3*d^3 - c^2*d^4 - c*d^5)*f), ((B*a*c^3*d + B*a*c^2*d^2 - B*a*c*d^3 - B*a*d^4)*f*x*sin(f*x + e)
+ (B*a*c^4 + B*a*c^3*d - B*a*c^2*d^2 - B*a*c*d^3)*f*x + (B*a*c^3 + B*a*c^2*d - (A + B)*a*c*d^2 + (B*a*c^2*d +
B*a*c*d^2 - (A + B)*a*d^3)*sin(f*x + e))*sqrt(c^2 - d^2)*arctan(-(c*sin(f*x + e) + d)/(sqrt(c^2 - d^2)*cos(f*x
+ e))) + (B*a*c^3*d - A*a*c^2*d^2 - B*a*c*d^3 + A*a*d^4)*cos(f*x + e))/((c^3*d^3 + c^2*d^4 - c*d^5 - d^6)*f*s
in(f*x + e) + (c^4*d^2 + c^3*d^3 - c^2*d^4 - c*d^5)*f)]

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F(-2)]

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

[In]

integrate((a+a*sin(f*x+e))*(A+B*sin(f*x+e))/(c+d*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*d^2-4*c^2>0)', see `assume?`
for more de

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.59 \[ \int \frac {(a+a \sin (e+f x)) (A+B \sin (e+f x))}{(c+d \sin (e+f x))^2} \, dx=\frac {\frac {{\left (f x + e\right )} B a}{d^{2}} - \frac {2 \, {\left (B a c^{2} + B a c d - A a d^{2} - B a d^{2}\right )} {\left (\pi \left \lfloor \frac {f x + e}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (c\right ) + \arctan \left (\frac {c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + d}{\sqrt {c^{2} - d^{2}}}\right )\right )}}{{\left (c d^{2} + d^{3}\right )} \sqrt {c^{2} - d^{2}}} + \frac {2 \, {\left (B a c d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - A a d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + B a c^{2} - A a c d\right )}}{{\left (c^{2} d + c d^{2}\right )} {\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 2 \, d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + c\right )}}}{f} \]

[In]

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

[Out]

((f*x + e)*B*a/d^2 - 2*(B*a*c^2 + B*a*c*d - A*a*d^2 - B*a*d^2)*(pi*floor(1/2*(f*x + e)/pi + 1/2)*sgn(c) + arct
an((c*tan(1/2*f*x + 1/2*e) + d)/sqrt(c^2 - d^2)))/((c*d^2 + d^3)*sqrt(c^2 - d^2)) + 2*(B*a*c*d*tan(1/2*f*x + 1
/2*e) - A*a*d^2*tan(1/2*f*x + 1/2*e) + B*a*c^2 - A*a*c*d)/((c^2*d + c*d^2)*(c*tan(1/2*f*x + 1/2*e)^2 + 2*d*tan
(1/2*f*x + 1/2*e) + c)))/f

Mupad [B] (verification not implemented)

Time = 20.53 (sec) , antiderivative size = 5102, normalized size of antiderivative = 41.15 \[ \int \frac {(a+a \sin (e+f x)) (A+B \sin (e+f x))}{(c+d \sin (e+f x))^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

(2*B*a*atan(((B*a*((32*(B^2*a^2*c^2*d^3 + 2*B^2*a^2*c^3*d^2 + B^2*a^2*c^4*d))/(2*c*d^3 + d^4 + c^2*d^2) + (32*
tan(e/2 + (f*x)/2)*(6*B^2*a^2*c^2*d^4 + 2*B^2*a^2*c^3*d^3 - 4*B^2*a^2*c^4*d^2 - A^2*a^2*c*d^5 + B^2*a^2*c*d^5
- 2*B^2*a^2*c^5*d + 2*A*B*a^2*c^2*d^4 + 2*A*B*a^2*c^3*d^3 - 2*A*B*a^2*c*d^5))/(2*c*d^4 + d^5 + c^2*d^3) + (B*a
*((32*tan(e/2 + (f*x)/2)*(2*A*a*c*d^7 + 2*B*a*c*d^7 + 2*A*a*c^2*d^6 - 4*B*a*c^3*d^5 - 2*B*a*c^4*d^4))/(2*c*d^4
+ d^5 + c^2*d^3) - (32*(B*a*c*d^6 - A*a*c^2*d^5 - A*a*c^3*d^4 + B*a*c^2*d^5))/(2*c*d^3 + d^4 + c^2*d^2) + (B*
a*((32*(c^2*d^7 + 2*c^3*d^6 + c^4*d^5))/(2*c*d^3 + d^4 + c^2*d^2) + (32*tan(e/2 + (f*x)/2)*(3*c*d^9 + 6*c^2*d^
8 + c^3*d^7 - 4*c^4*d^6 - 2*c^5*d^5))/(2*c*d^4 + d^5 + c^2*d^3))*1i)/d^2)*1i)/d^2))/d^2 + (B*a*((32*(B^2*a^2*c
^2*d^3 + 2*B^2*a^2*c^3*d^2 + B^2*a^2*c^4*d))/(2*c*d^3 + d^4 + c^2*d^2) + (32*tan(e/2 + (f*x)/2)*(6*B^2*a^2*c^2
*d^4 + 2*B^2*a^2*c^3*d^3 - 4*B^2*a^2*c^4*d^2 - A^2*a^2*c*d^5 + B^2*a^2*c*d^5 - 2*B^2*a^2*c^5*d + 2*A*B*a^2*c^2
*d^4 + 2*A*B*a^2*c^3*d^3 - 2*A*B*a^2*c*d^5))/(2*c*d^4 + d^5 + c^2*d^3) + (B*a*((32*(B*a*c*d^6 - A*a*c^2*d^5 -
A*a*c^3*d^4 + B*a*c^2*d^5))/(2*c*d^3 + d^4 + c^2*d^2) - (32*tan(e/2 + (f*x)/2)*(2*A*a*c*d^7 + 2*B*a*c*d^7 + 2*
A*a*c^2*d^6 - 4*B*a*c^3*d^5 - 2*B*a*c^4*d^4))/(2*c*d^4 + d^5 + c^2*d^3) + (B*a*((32*(c^2*d^7 + 2*c^3*d^6 + c^4
*d^5))/(2*c*d^3 + d^4 + c^2*d^2) + (32*tan(e/2 + (f*x)/2)*(3*c*d^9 + 6*c^2*d^8 + c^3*d^7 - 4*c^4*d^6 - 2*c^5*d
^5))/(2*c*d^4 + d^5 + c^2*d^3))*1i)/d^2)*1i)/d^2))/d^2)/((64*(B^3*a^3*c^3 + A*B^2*a^3*c^3 - B^3*a^3*c*d^2 + B^
3*a^3*c^2*d - 2*A*B^2*a^3*c*d^2 + A*B^2*a^3*c^2*d - A^2*B*a^3*c*d^2))/(2*c*d^3 + d^4 + c^2*d^2) - (64*tan(e/2
+ (f*x)/2)*(2*B^3*a^3*c*d^3 - 2*B^3*a^3*c^4 - 4*B^3*a^3*c^3*d + 2*A*B^2*a^3*c*d^3 + 2*A*B^2*a^3*c^2*d^2))/(2*c
*d^4 + d^5 + c^2*d^3) - (B*a*((32*(B^2*a^2*c^2*d^3 + 2*B^2*a^2*c^3*d^2 + B^2*a^2*c^4*d))/(2*c*d^3 + d^4 + c^2*
d^2) + (32*tan(e/2 + (f*x)/2)*(6*B^2*a^2*c^2*d^4 + 2*B^2*a^2*c^3*d^3 - 4*B^2*a^2*c^4*d^2 - A^2*a^2*c*d^5 + B^2
*a^2*c*d^5 - 2*B^2*a^2*c^5*d + 2*A*B*a^2*c^2*d^4 + 2*A*B*a^2*c^3*d^3 - 2*A*B*a^2*c*d^5))/(2*c*d^4 + d^5 + c^2*
d^3) + (B*a*((32*tan(e/2 + (f*x)/2)*(2*A*a*c*d^7 + 2*B*a*c*d^7 + 2*A*a*c^2*d^6 - 4*B*a*c^3*d^5 - 2*B*a*c^4*d^4
))/(2*c*d^4 + d^5 + c^2*d^3) - (32*(B*a*c*d^6 - A*a*c^2*d^5 - A*a*c^3*d^4 + B*a*c^2*d^5))/(2*c*d^3 + d^4 + c^2
*d^2) + (B*a*((32*(c^2*d^7 + 2*c^3*d^6 + c^4*d^5))/(2*c*d^3 + d^4 + c^2*d^2) + (32*tan(e/2 + (f*x)/2)*(3*c*d^9
+ 6*c^2*d^8 + c^3*d^7 - 4*c^4*d^6 - 2*c^5*d^5))/(2*c*d^4 + d^5 + c^2*d^3))*1i)/d^2)*1i)/d^2)*1i)/d^2 + (B*a*(
(32*(B^2*a^2*c^2*d^3 + 2*B^2*a^2*c^3*d^2 + B^2*a^2*c^4*d))/(2*c*d^3 + d^4 + c^2*d^2) + (32*tan(e/2 + (f*x)/2)*
(6*B^2*a^2*c^2*d^4 + 2*B^2*a^2*c^3*d^3 - 4*B^2*a^2*c^4*d^2 - A^2*a^2*c*d^5 + B^2*a^2*c*d^5 - 2*B^2*a^2*c^5*d +
2*A*B*a^2*c^2*d^4 + 2*A*B*a^2*c^3*d^3 - 2*A*B*a^2*c*d^5))/(2*c*d^4 + d^5 + c^2*d^3) + (B*a*((32*(B*a*c*d^6 -
A*a*c^2*d^5 - A*a*c^3*d^4 + B*a*c^2*d^5))/(2*c*d^3 + d^4 + c^2*d^2) - (32*tan(e/2 + (f*x)/2)*(2*A*a*c*d^7 + 2*
B*a*c*d^7 + 2*A*a*c^2*d^6 - 4*B*a*c^3*d^5 - 2*B*a*c^4*d^4))/(2*c*d^4 + d^5 + c^2*d^3) + (B*a*((32*(c^2*d^7 + 2
*c^3*d^6 + c^4*d^5))/(2*c*d^3 + d^4 + c^2*d^2) + (32*tan(e/2 + (f*x)/2)*(3*c*d^9 + 6*c^2*d^8 + c^3*d^7 - 4*c^4
*d^6 - 2*c^5*d^5))/(2*c*d^4 + d^5 + c^2*d^3))*1i)/d^2)*1i)/d^2)*1i)/d^2)))/(d^2*f) - ((2*(A*a*d - B*a*c))/(d*(
c + d)) + (2*a*tan(e/2 + (f*x)/2)*(A*d - B*c))/(c*(c + d)))/(f*(c + 2*d*tan(e/2 + (f*x)/2) + c*tan(e/2 + (f*x)
/2)^2)) + (a*atan(((a*(-(c + d)^3*(c - d))^(1/2)*((32*(B^2*a^2*c^2*d^3 + 2*B^2*a^2*c^3*d^2 + B^2*a^2*c^4*d))/(
2*c*d^3 + d^4 + c^2*d^2) + (32*tan(e/2 + (f*x)/2)*(6*B^2*a^2*c^2*d^4 + 2*B^2*a^2*c^3*d^3 - 4*B^2*a^2*c^4*d^2 -
A^2*a^2*c*d^5 + B^2*a^2*c*d^5 - 2*B^2*a^2*c^5*d + 2*A*B*a^2*c^2*d^4 + 2*A*B*a^2*c^3*d^3 - 2*A*B*a^2*c*d^5))/(
2*c*d^4 + d^5 + c^2*d^3) + (a*(-(c + d)^3*(c - d))^(1/2)*((32*tan(e/2 + (f*x)/2)*(2*A*a*c*d^7 + 2*B*a*c*d^7 +
2*A*a*c^2*d^6 - 4*B*a*c^3*d^5 - 2*B*a*c^4*d^4))/(2*c*d^4 + d^5 + c^2*d^3) - (32*(B*a*c*d^6 - A*a*c^2*d^5 - A*a
*c^3*d^4 + B*a*c^2*d^5))/(2*c*d^3 + d^4 + c^2*d^2) + (a*((32*(c^2*d^7 + 2*c^3*d^6 + c^4*d^5))/(2*c*d^3 + d^4 +
c^2*d^2) + (32*tan(e/2 + (f*x)/2)*(3*c*d^9 + 6*c^2*d^8 + c^3*d^7 - 4*c^4*d^6 - 2*c^5*d^5))/(2*c*d^4 + d^5 + c
^2*d^3))*(-(c + d)^3*(c - d))^(1/2)*(A*d^2 - B*c^2 + B*d^2 - B*c*d))/(2*c*d^5 + d^6 - 2*c^3*d^3 - c^4*d^2))*(A
*d^2 - B*c^2 + B*d^2 - B*c*d))/(2*c*d^5 + d^6 - 2*c^3*d^3 - c^4*d^2))*(A*d^2 - B*c^2 + B*d^2 - B*c*d)*1i)/(2*c
*d^5 + d^6 - 2*c^3*d^3 - c^4*d^2) + (a*(-(c + d)^3*(c - d))^(1/2)*((32*(B^2*a^2*c^2*d^3 + 2*B^2*a^2*c^3*d^2 +
B^2*a^2*c^4*d))/(2*c*d^3 + d^4 + c^2*d^2) + (32*tan(e/2 + (f*x)/2)*(6*B^2*a^2*c^2*d^4 + 2*B^2*a^2*c^3*d^3 - 4*
B^2*a^2*c^4*d^2 - A^2*a^2*c*d^5 + B^2*a^2*c*d^5 - 2*B^2*a^2*c^5*d + 2*A*B*a^2*c^2*d^4 + 2*A*B*a^2*c^3*d^3 - 2*
A*B*a^2*c*d^5))/(2*c*d^4 + d^5 + c^2*d^3) + (a*(-(c + d)^3*(c - d))^(1/2)*((32*(B*a*c*d^6 - A*a*c^2*d^5 - A*a*
c^3*d^4 + B*a*c^2*d^5))/(2*c*d^3 + d^4 + c^2*d^2) - (32*tan(e/2 + (f*x)/2)*(2*A*a*c*d^7 + 2*B*a*c*d^7 + 2*A*a*
c^2*d^6 - 4*B*a*c^3*d^5 - 2*B*a*c^4*d^4))/(2*c*d^4 + d^5 + c^2*d^3) + (a*((32*(c^2*d^7 + 2*c^3*d^6 + c^4*d^5))
/(2*c*d^3 + d^4 + c^2*d^2) + (32*tan(e/2 + (f*x)/2)*(3*c*d^9 + 6*c^2*d^8 + c^3*d^7 - 4*c^4*d^6 - 2*c^5*d^5))/(
2*c*d^4 + d^5 + c^2*d^3))*(-(c + d)^3*(c - d))^(1/2)*(A*d^2 - B*c^2 + B*d^2 - B*c*d))/(2*c*d^5 + d^6 - 2*c^3*d
^3 - c^4*d^2))*(A*d^2 - B*c^2 + B*d^2 - B*c*d))/(2*c*d^5 + d^6 - 2*c^3*d^3 - c^4*d^2))*(A*d^2 - B*c^2 + B*d^2
- B*c*d)*1i)/(2*c*d^5 + d^6 - 2*c^3*d^3 - c^4*d^2))/((64*(B^3*a^3*c^3 + A*B^2*a^3*c^3 - B^3*a^3*c*d^2 + B^3*a^
3*c^2*d - 2*A*B^2*a^3*c*d^2 + A*B^2*a^3*c^2*d - A^2*B*a^3*c*d^2))/(2*c*d^3 + d^4 + c^2*d^2) - (64*tan(e/2 + (f
*x)/2)*(2*B^3*a^3*c*d^3 - 2*B^3*a^3*c^4 - 4*B^3*a^3*c^3*d + 2*A*B^2*a^3*c*d^3 + 2*A*B^2*a^3*c^2*d^2))/(2*c*d^4
+ d^5 + c^2*d^3) - (a*(-(c + d)^3*(c - d))^(1/2)*((32*(B^2*a^2*c^2*d^3 + 2*B^2*a^2*c^3*d^2 + B^2*a^2*c^4*d))/
(2*c*d^3 + d^4 + c^2*d^2) + (32*tan(e/2 + (f*x)/2)*(6*B^2*a^2*c^2*d^4 + 2*B^2*a^2*c^3*d^3 - 4*B^2*a^2*c^4*d^2
- A^2*a^2*c*d^5 + B^2*a^2*c*d^5 - 2*B^2*a^2*c^5*d + 2*A*B*a^2*c^2*d^4 + 2*A*B*a^2*c^3*d^3 - 2*A*B*a^2*c*d^5))/
(2*c*d^4 + d^5 + c^2*d^3) + (a*(-(c + d)^3*(c - d))^(1/2)*((32*tan(e/2 + (f*x)/2)*(2*A*a*c*d^7 + 2*B*a*c*d^7 +
2*A*a*c^2*d^6 - 4*B*a*c^3*d^5 - 2*B*a*c^4*d^4))/(2*c*d^4 + d^5 + c^2*d^3) - (32*(B*a*c*d^6 - A*a*c^2*d^5 - A*
a*c^3*d^4 + B*a*c^2*d^5))/(2*c*d^3 + d^4 + c^2*d^2) + (a*((32*(c^2*d^7 + 2*c^3*d^6 + c^4*d^5))/(2*c*d^3 + d^4
+ c^2*d^2) + (32*tan(e/2 + (f*x)/2)*(3*c*d^9 + 6*c^2*d^8 + c^3*d^7 - 4*c^4*d^6 - 2*c^5*d^5))/(2*c*d^4 + d^5 +
c^2*d^3))*(-(c + d)^3*(c - d))^(1/2)*(A*d^2 - B*c^2 + B*d^2 - B*c*d))/(2*c*d^5 + d^6 - 2*c^3*d^3 - c^4*d^2))*(
A*d^2 - B*c^2 + B*d^2 - B*c*d))/(2*c*d^5 + d^6 - 2*c^3*d^3 - c^4*d^2))*(A*d^2 - B*c^2 + B*d^2 - B*c*d))/(2*c*d
^5 + d^6 - 2*c^3*d^3 - c^4*d^2) + (a*(-(c + d)^3*(c - d))^(1/2)*((32*(B^2*a^2*c^2*d^3 + 2*B^2*a^2*c^3*d^2 + B^
2*a^2*c^4*d))/(2*c*d^3 + d^4 + c^2*d^2) + (32*tan(e/2 + (f*x)/2)*(6*B^2*a^2*c^2*d^4 + 2*B^2*a^2*c^3*d^3 - 4*B^
2*a^2*c^4*d^2 - A^2*a^2*c*d^5 + B^2*a^2*c*d^5 - 2*B^2*a^2*c^5*d + 2*A*B*a^2*c^2*d^4 + 2*A*B*a^2*c^3*d^3 - 2*A*
B*a^2*c*d^5))/(2*c*d^4 + d^5 + c^2*d^3) + (a*(-(c + d)^3*(c - d))^(1/2)*((32*(B*a*c*d^6 - A*a*c^2*d^5 - A*a*c^
3*d^4 + B*a*c^2*d^5))/(2*c*d^3 + d^4 + c^2*d^2) - (32*tan(e/2 + (f*x)/2)*(2*A*a*c*d^7 + 2*B*a*c*d^7 + 2*A*a*c^
2*d^6 - 4*B*a*c^3*d^5 - 2*B*a*c^4*d^4))/(2*c*d^4 + d^5 + c^2*d^3) + (a*((32*(c^2*d^7 + 2*c^3*d^6 + c^4*d^5))/(
2*c*d^3 + d^4 + c^2*d^2) + (32*tan(e/2 + (f*x)/2)*(3*c*d^9 + 6*c^2*d^8 + c^3*d^7 - 4*c^4*d^6 - 2*c^5*d^5))/(2*
c*d^4 + d^5 + c^2*d^3))*(-(c + d)^3*(c - d))^(1/2)*(A*d^2 - B*c^2 + B*d^2 - B*c*d))/(2*c*d^5 + d^6 - 2*c^3*d^3
- c^4*d^2))*(A*d^2 - B*c^2 + B*d^2 - B*c*d))/(2*c*d^5 + d^6 - 2*c^3*d^3 - c^4*d^2))*(A*d^2 - B*c^2 + B*d^2 -
B*c*d))/(2*c*d^5 + d^6 - 2*c^3*d^3 - c^4*d^2)))*(-(c + d)^3*(c - d))^(1/2)*(A*d^2 - B*c^2 + B*d^2 - B*c*d)*2i)
/(f*(2*c*d^5 + d^6 - 2*c^3*d^3 - c^4*d^2))