3.3.56 \(\int \frac {(a+a \sin (e+f x))^2 (A+B \sin (e+f x))}{(c+d \sin (e+f x))^2} \, dx\) [256]

Optimal. Leaf size=198 \[ -\frac {a^2 (2 B c-A d-2 B d) x}{d^3}-\frac {2 a^2 (c-d) \left (A d (c+2 d)-B \left (2 c^2+2 c d-d^2\right )\right ) \tan ^{-1}\left (\frac {d+c \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c^2-d^2}}\right )}{d^3 (c+d) \sqrt {c^2-d^2} f}+\frac {a^2 (A d-B (2 c+d)) \cos (e+f x)}{d^2 (c+d) f}+\frac {(B c-A d) \cos (e+f x) \left (a^2+a^2 \sin (e+f x)\right )}{d (c+d) f (c+d \sin (e+f x))} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.40, antiderivative size = 198, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3054, 3047, 3102, 2814, 2739, 632, 210} \begin {gather*} -\frac {2 a^2 (c-d) \left (A d (c+2 d)-B \left (2 c^2+2 c d-d^2\right )\right ) \text {ArcTan}\left (\frac {c \tan \left (\frac {1}{2} (e+f x)\right )+d}{\sqrt {c^2-d^2}}\right )}{d^3 f (c+d) \sqrt {c^2-d^2}}-\frac {a^2 x (-A d+2 B c-2 B d)}{d^3}+\frac {a^2 (A d-B (2 c+d)) \cos (e+f x)}{d^2 f (c+d)}+\frac {(B c-A d) \cos (e+f x) \left (a^2 \sin (e+f x)+a^2\right )}{d f (c+d) (c+d \sin (e+f x))} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((a^2*(2*B*c - A*d - 2*B*d)*x)/d^3) - (2*a^2*(c - d)*(A*d*(c + 2*d) - B*(2*c^2 + 2*c*d - d^2))*ArcTan[(d + c*
Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/(d^3*(c + d)*Sqrt[c^2 - d^2]*f) + (a^2*(A*d - B*(2*c + d))*Cos[e + f*x])/(
d^2*(c + d)*f) + ((B*c - A*d)*Cos[e + f*x]*(a^2 + a^2*Sin[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 3054

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_
.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b^2)*(B*c - A*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 1)*((c + d
*Sin[e + f*x])^(n + 1)/(d*f*(n + 1)*(b*c + a*d))), x] - Dist[b/(d*(n + 1)*(b*c + a*d)), Int[(a + b*Sin[e + f*x
])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1)*Simp[a*A*d*(m - n - 2) - B*(a*c*(m - 1) + b*d*(n + 1)) - (A*b*d*(m + n
+ 1) - B*(b*c*m - a*d*(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] && GtQ[m, 1/2] && LtQ[n, -1] && 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*} \int \frac {(a+a \sin (e+f x))^2 (A+B \sin (e+f x))}{(c+d \sin (e+f x))^2} \, dx &=\frac {(B c-A d) \cos (e+f x) \left (a^2+a^2 \sin (e+f x)\right )}{d (c+d) f (c+d \sin (e+f x))}+\frac {\int \frac {(a+a \sin (e+f x)) (-a (B (c-d)-2 A d)-a (A d-B (2 c+d)) \sin (e+f x))}{c+d \sin (e+f x)} \, dx}{d (c+d)}\\ &=\frac {(B c-A d) \cos (e+f x) \left (a^2+a^2 \sin (e+f x)\right )}{d (c+d) f (c+d \sin (e+f x))}+\frac {\int \frac {-a^2 (B (c-d)-2 A d)+\left (-a^2 (B (c-d)-2 A d)-a^2 (A d-B (2 c+d))\right ) \sin (e+f x)-a^2 (A d-B (2 c+d)) \sin ^2(e+f x)}{c+d \sin (e+f x)} \, dx}{d (c+d)}\\ &=\frac {a^2 (A d-B (2 c+d)) \cos (e+f x)}{d^2 (c+d) f}+\frac {(B c-A d) \cos (e+f x) \left (a^2+a^2 \sin (e+f x)\right )}{d (c+d) f (c+d \sin (e+f x))}+\frac {\int \frac {-a^2 d (B (c-d)-2 A d)-a^2 (c+d) (2 B (c-d)-A d) \sin (e+f x)}{c+d \sin (e+f x)} \, dx}{d^2 (c+d)}\\ &=-\frac {a^2 (2 B c-A d-2 B d) x}{d^3}+\frac {a^2 (A d-B (2 c+d)) \cos (e+f x)}{d^2 (c+d) f}+\frac {(B c-A d) \cos (e+f x) \left (a^2+a^2 \sin (e+f x)\right )}{d (c+d) f (c+d \sin (e+f x))}-\frac {\left (a^2 (c-d) \left (A d (c+2 d)-B \left (2 c^2+2 c d-d^2\right )\right )\right ) \int \frac {1}{c+d \sin (e+f x)} \, dx}{d^3 (c+d)}\\ &=-\frac {a^2 (2 B c-A d-2 B d) x}{d^3}+\frac {a^2 (A d-B (2 c+d)) \cos (e+f x)}{d^2 (c+d) f}+\frac {(B c-A d) \cos (e+f x) \left (a^2+a^2 \sin (e+f x)\right )}{d (c+d) f (c+d \sin (e+f x))}-\frac {\left (2 a^2 (c-d) \left (A d (c+2 d)-B \left (2 c^2+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^3 (c+d) f}\\ &=-\frac {a^2 (2 B c-A d-2 B d) x}{d^3}+\frac {a^2 (A d-B (2 c+d)) \cos (e+f x)}{d^2 (c+d) f}+\frac {(B c-A d) \cos (e+f x) \left (a^2+a^2 \sin (e+f x)\right )}{d (c+d) f (c+d \sin (e+f x))}+\frac {\left (4 a^2 (c-d) \left (A d (c+2 d)-B \left (2 c^2+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^3 (c+d) f}\\ &=-\frac {a^2 (2 B c-A d-2 B d) x}{d^3}-\frac {2 a^2 (c-d) \left (A d (c+2 d)-B \left (2 c^2+2 c d-d^2\right )\right ) \tan ^{-1}\left (\frac {d+c \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c^2-d^2}}\right )}{d^3 (c+d) \sqrt {c^2-d^2} f}+\frac {a^2 (A d-B (2 c+d)) \cos (e+f x)}{d^2 (c+d) f}+\frac {(B c-A d) \cos (e+f x) \left (a^2+a^2 \sin (e+f x)\right )}{d (c+d) f (c+d \sin (e+f x))}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.69, size = 192, normalized size = 0.97 \begin {gather*} \frac {a^2 (1+\sin (e+f x))^2 \left ((-2 B c+A d+2 B d) (e+f x)+\frac {2 (c-d) \left (-A d (c+2 d)+B \left (2 c^2+2 c d-d^2\right )\right ) \tan ^{-1}\left (\frac {d+c \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c^2-d^2}}\right )}{(c+d) \sqrt {c^2-d^2}}-B d \cos (e+f x)-\frac {d (-c+d) (-B c+A d) \cos (e+f x)}{(c+d) (c+d \sin (e+f x))}\right )}{d^3 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(a^2*(1 + Sin[e + f*x])^2*((-2*B*c + A*d + 2*B*d)*(e + f*x) + (2*(c - d)*(-(A*d*(c + 2*d)) + B*(2*c^2 + 2*c*d
- d^2))*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/((c + d)*Sqrt[c^2 - d^2]) - B*d*Cos[e + f*x] - (d*(-
c + d)*(-(B*c) + A*d)*Cos[e + f*x])/((c + d)*(c + d*Sin[e + f*x]))))/(d^3*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/
2])^4)

________________________________________________________________________________________

Maple [A]
time = 0.52, size = 251, normalized size = 1.27

method	result	size
derivativedivides	\(\frac {2 a^{2} \left (-\frac {\frac {-\frac {d^{2} \left (A c d -A \,d^{2}-B \,c^{2}+B c d \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{\left (c +d \right ) c}-\frac {d \left (A c d -A \,d^{2}-B \,c^{2}+B c d \right )}{c +d}}{c \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+2 d \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+c}+\frac {\left (A \,c^{2} d +A c \,d^{2}-2 A \,d^{3}-2 B \,c^{3}+3 B c \,d^{2}-B \,d^{3}\right ) \arctan \left (\frac {2 c \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+2 d}{2 \sqrt {c^{2}-d^{2}}}\right )}{\left (c +d \right ) \sqrt {c^{2}-d^{2}}}}{d^{3}}+\frac {-\frac {B d}{1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )}+\left (A d -2 B c +2 B d \right ) \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{d^{3}}\right )}{f}\)	\(251\)
default	\(\frac {2 a^{2} \left (-\frac {\frac {-\frac {d^{2} \left (A c d -A \,d^{2}-B \,c^{2}+B c d \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{\left (c +d \right ) c}-\frac {d \left (A c d -A \,d^{2}-B \,c^{2}+B c d \right )}{c +d}}{c \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+2 d \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+c}+\frac {\left (A \,c^{2} d +A c \,d^{2}-2 A \,d^{3}-2 B \,c^{3}+3 B c \,d^{2}-B \,d^{3}\right ) \arctan \left (\frac {2 c \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+2 d}{2 \sqrt {c^{2}-d^{2}}}\right )}{\left (c +d \right ) \sqrt {c^{2}-d^{2}}}}{d^{3}}+\frac {-\frac {B d}{1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )}+\left (A d -2 B c +2 B d \right ) \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{d^{3}}\right )}{f}\)	\(251\)
risch	\(\frac {a^{2} x A}{d^{2}}-\frac {2 a^{2} x B c}{d^{3}}+\frac {2 a^{2} x B}{d^{2}}-\frac {B \,a^{2} {\mathrm e}^{i \left (f x +e \right )}}{2 d^{2} f}-\frac {B \,a^{2} {\mathrm e}^{-i \left (f x +e \right )}}{2 d^{2} f}+\frac {2 i a^{2} \left (-A c d +A \,d^{2}+B \,c^{2}-B c d \right ) \left (i d +c \,{\mathrm e}^{i \left (f x +e \right )}\right )}{d^{3} \left (c +d \right ) f \left (i d -i d \,{\mathrm e}^{2 i \left (f x +e \right )}+2 c \,{\mathrm e}^{i \left (f x +e \right )}\right )}+\frac {\sqrt {-\left (c -d \right ) \left (c +d \right )}\, a^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}-\frac {-i c +\sqrt {-\left (c -d \right ) \left (c +d \right )}}{d}\right ) A c}{\left (c +d \right )^{2} f \,d^{2}}+\frac {2 \sqrt {-\left (c -d \right ) \left (c +d \right )}\, a^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}-\frac {-i c +\sqrt {-\left (c -d \right ) \left (c +d \right )}}{d}\right ) A}{\left (c +d \right )^{2} f d}-\frac {2 \sqrt {-\left (c -d \right ) \left (c +d \right )}\, a^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}-\frac {-i c +\sqrt {-\left (c -d \right ) \left (c +d \right )}}{d}\right ) B \,c^{2}}{\left (c +d \right )^{2} f \,d^{3}}-\frac {2 \sqrt {-\left (c -d \right ) \left (c +d \right )}\, a^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}-\frac {-i c +\sqrt {-\left (c -d \right ) \left (c +d \right )}}{d}\right ) B c}{\left (c +d \right )^{2} f \,d^{2}}+\frac {\sqrt {-\left (c -d \right ) \left (c +d \right )}\, a^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}-\frac {-i c +\sqrt {-\left (c -d \right ) \left (c +d \right )}}{d}\right ) B}{\left (c +d \right )^{2} f d}-\frac {\sqrt {-\left (c -d \right ) \left (c +d \right )}\, a^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i c +\sqrt {-\left (c -d \right ) \left (c +d \right )}}{d}\right ) A c}{\left (c +d \right )^{2} f \,d^{2}}-\frac {2 \sqrt {-\left (c -d \right ) \left (c +d \right )}\, a^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i c +\sqrt {-\left (c -d \right ) \left (c +d \right )}}{d}\right ) A}{\left (c +d \right )^{2} f d}+\frac {2 \sqrt {-\left (c -d \right ) \left (c +d \right )}\, a^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i c +\sqrt {-\left (c -d \right ) \left (c +d \right )}}{d}\right ) B \,c^{2}}{\left (c +d \right )^{2} f \,d^{3}}+\frac {2 \sqrt {-\left (c -d \right ) \left (c +d \right )}\, a^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i c +\sqrt {-\left (c -d \right ) \left (c +d \right )}}{d}\right ) B c}{\left (c +d \right )^{2} f \,d^{2}}-\frac {\sqrt {-\left (c -d \right ) \left (c +d \right )}\, a^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i c +\sqrt {-\left (c -d \right ) \left (c +d \right )}}{d}\right ) B}{\left (c +d \right )^{2} f d}\)	\(783\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(f*x+e))^2*(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

________________________________________________________________________________________

Fricas [A]
time = 0.43, size = 750, normalized size = 3.79 \begin {gather*} \left [-\frac {2 \, {\left (2 \, B a^{2} c^{3} - A a^{2} c^{2} d - {\left (A + 2 \, B\right )} a^{2} c d^{2}\right )} f x + {\left (2 \, B a^{2} c^{3} - {\left (A - 2 \, B\right )} a^{2} c^{2} d - {\left (2 \, A + B\right )} a^{2} c d^{2} + {\left (2 \, B a^{2} c^{2} d - {\left (A - 2 \, B\right )} a^{2} c d^{2} - {\left (2 \, A + B\right )} a^{2} d^{3}\right )} \sin \left (f x + e\right )\right )} \sqrt {-\frac {c - d}{c + d}} \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 ({\left (c^{2} + c d\right )} \cos \left (f x + e\right ) \sin \left (f x + e\right ) + {\left (c d + d^{2}\right )} \cos \left (f x + e\right )\right )} \sqrt {-\frac {c - d}{c + d}}}{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 (2 \, B a^{2} c^{2} d - A a^{2} c d^{2} + A a^{2} d^{3}\right )} \cos \left (f x + e\right ) + 2 \, {\left ({\left (2 \, B a^{2} c^{2} d - A a^{2} c d^{2} - {\left (A + 2 \, B\right )} a^{2} d^{3}\right )} f x + {\left (B a^{2} c d^{2} + B a^{2} d^{3}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{2 \, {\left ({\left (c d^{4} + d^{5}\right )} f \sin \left (f x + e\right ) + {\left (c^{2} d^{3} + c d^{4}\right )} f\right )}}, -\frac {{\left (2 \, B a^{2} c^{3} - A a^{2} c^{2} d - {\left (A + 2 \, B\right )} a^{2} c d^{2}\right )} f x + {\left (2 \, B a^{2} c^{3} - {\left (A - 2 \, B\right )} a^{2} c^{2} d - {\left (2 \, A + B\right )} a^{2} c d^{2} + {\left (2 \, B a^{2} c^{2} d - {\left (A - 2 \, B\right )} a^{2} c d^{2} - {\left (2 \, A + B\right )} a^{2} d^{3}\right )} \sin \left (f x + e\right )\right )} \sqrt {\frac {c - d}{c + d}} \arctan \left (-\frac {{\left (c \sin \left (f x + e\right ) + d\right )} \sqrt {\frac {c - d}{c + d}}}{{\left (c - d\right )} \cos \left (f x + e\right )}\right ) + {\left (2 \, B a^{2} c^{2} d - A a^{2} c d^{2} + A a^{2} d^{3}\right )} \cos \left (f x + e\right ) + {\left ({\left (2 \, B a^{2} c^{2} d - A a^{2} c d^{2} - {\left (A + 2 \, B\right )} a^{2} d^{3}\right )} f x + {\left (B a^{2} c d^{2} + B a^{2} d^{3}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{{\left (c d^{4} + d^{5}\right )} f \sin \left (f x + e\right ) + {\left (c^{2} d^{3} + c d^{4}\right )} f}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/2*(2*(2*B*a^2*c^3 - A*a^2*c^2*d - (A + 2*B)*a^2*c*d^2)*f*x + (2*B*a^2*c^3 - (A - 2*B)*a^2*c^2*d - (2*A + B
)*a^2*c*d^2 + (2*B*a^2*c^2*d - (A - 2*B)*a^2*c*d^2 - (2*A + B)*a^2*d^3)*sin(f*x + e))*sqrt(-(c - d)/(c + d))*l
og(((2*c^2 - d^2)*cos(f*x + e)^2 - 2*c*d*sin(f*x + e) - c^2 - d^2 + 2*((c^2 + c*d)*cos(f*x + e)*sin(f*x + e) +
(c*d + d^2)*cos(f*x + e))*sqrt(-(c - d)/(c + d)))/(d^2*cos(f*x + e)^2 - 2*c*d*sin(f*x + e) - c^2 - d^2)) + 2*
(2*B*a^2*c^2*d - A*a^2*c*d^2 + A*a^2*d^3)*cos(f*x + e) + 2*((2*B*a^2*c^2*d - A*a^2*c*d^2 - (A + 2*B)*a^2*d^3)*
f*x + (B*a^2*c*d^2 + B*a^2*d^3)*cos(f*x + e))*sin(f*x + e))/((c*d^4 + d^5)*f*sin(f*x + e) + (c^2*d^3 + c*d^4)*
f), -((2*B*a^2*c^3 - A*a^2*c^2*d - (A + 2*B)*a^2*c*d^2)*f*x + (2*B*a^2*c^3 - (A - 2*B)*a^2*c^2*d - (2*A + B)*a
^2*c*d^2 + (2*B*a^2*c^2*d - (A - 2*B)*a^2*c*d^2 - (2*A + B)*a^2*d^3)*sin(f*x + e))*sqrt((c - d)/(c + d))*arcta
n(-(c*sin(f*x + e) + d)*sqrt((c - d)/(c + d))/((c - d)*cos(f*x + e))) + (2*B*a^2*c^2*d - A*a^2*c*d^2 + A*a^2*d
^3)*cos(f*x + e) + ((2*B*a^2*c^2*d - A*a^2*c*d^2 - (A + 2*B)*a^2*d^3)*f*x + (B*a^2*c*d^2 + B*a^2*d^3)*cos(f*x
+ e))*sin(f*x + e))/((c*d^4 + d^5)*f*sin(f*x + e) + (c^2*d^3 + c*d^4)*f)]

________________________________________________________________________________________

Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 498 vs. \(2 (199) = 398\).
time = 0.79, size = 498, normalized size = 2.52 \begin {gather*} \frac {\frac {2 \, {\left (2 \, B a^{2} c^{3} - A a^{2} c^{2} d - A a^{2} c d^{2} - 3 \, B a^{2} c d^{2} + 2 \, A a^{2} d^{3} + B a^{2} d^{3}\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^{3} + d^{4}\right )} \sqrt {c^{2} - d^{2}}} - \frac {2 \, {\left (B a^{2} c^{2} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - A a^{2} c d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - B a^{2} c d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + A a^{2} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 2 \, B a^{2} c^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - A a^{2} c^{2} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + A a^{2} c d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 3 \, B a^{2} c^{2} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - A a^{2} c d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + B a^{2} c d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + A a^{2} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 2 \, B a^{2} c^{3} - A a^{2} c^{2} d + A a^{2} c d^{2}\right )}}{{\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 2 \, d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 2 \, 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 )} {\left (c^{2} d^{2} + c d^{3}\right )}} - \frac {{\left (2 \, B a^{2} c - A a^{2} d - 2 \, B a^{2} d\right )} {\left (f x + e\right )}}{d^{3}}}{f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Mupad [B]
time = 21.58, size = 2500, normalized size = 12.63 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

- ((2*(A*a^2*d^2 + 2*B*a^2*c^2 - A*a^2*c*d))/(d^2*(c + d)) + (2*tan(e/2 + (f*x)/2)^2*(A*a^2*d^2 + 2*B*a^2*c^2
- A*a^2*c*d))/(d^2*(c + d)) + (2*tan(e/2 + (f*x)/2)*(A*a^2*d^2 + 3*B*a^2*c^2 - A*a^2*c*d + B*a^2*c*d))/(c*d*(c
+ d)) + (2*tan(e/2 + (f*x)/2)^3*(A*a^2*d^2 + B*a^2*c^2 - A*a^2*c*d - B*a^2*c*d))/(c*d*(c + d)))/(f*(c + 2*d*t
an(e/2 + (f*x)/2) + 2*c*tan(e/2 + (f*x)/2)^2 + c*tan(e/2 + (f*x)/2)^4 + 2*d*tan(e/2 + (f*x)/2)^3)) - (atan((((
B*a^2*c*2i - a^2*d*(A + 2*B)*1i)*((32*(A^2*a^4*c^2*d^6 + 2*A^2*a^4*c^3*d^5 + A^2*a^4*c^4*d^4 + 4*B^2*a^4*c^2*d
^6 - 8*B^2*a^4*c^4*d^4 + 4*B^2*a^4*c^6*d^2 + 4*A*B*a^4*c^2*d^6 + 4*A*B*a^4*c^3*d^5 - 4*A*B*a^4*c^4*d^4 - 4*A*B
*a^4*c^5*d^3))/(2*c*d^6 + d^7 + c^2*d^5) + ((B*a^2*c*2i - a^2*d*(A + 2*B)*1i)*((((32*(c^2*d^10 + 2*c^3*d^9 + c
^4*d^8))/(2*c*d^6 + d^7 + c^2*d^5) + (32*tan(e/2 + (f*x)/2)*(3*c*d^12 + 6*c^2*d^11 + c^3*d^10 - 4*c^4*d^9 - 2*
c^5*d^8))/(2*c*d^7 + d^8 + c^2*d^6))*(B*a^2*c*2i - a^2*d*(A + 2*B)*1i))/d^3 - (32*(A*a^2*c*d^9 + 2*B*a^2*c*d^9
- A*a^2*c^3*d^7 + B*a^2*c^2*d^8 - 2*B*a^2*c^3*d^7 - B*a^2*c^4*d^6))/(2*c*d^6 + d^7 + c^2*d^5) + (32*tan(e/2 +
(f*x)/2)*(4*A*a^2*c*d^10 + 2*B*a^2*c*d^10 + 2*A*a^2*c^2*d^9 - 4*A*a^2*c^3*d^8 - 2*A*a^2*c^4*d^7 - 4*B*a^2*c^2
*d^9 - 6*B*a^2*c^3*d^8 + 4*B*a^2*c^4*d^7 + 4*B*a^2*c^5*d^6))/(2*c*d^7 + d^8 + c^2*d^6)))/d^3 + (32*tan(e/2 + (
f*x)/2)*(8*A^2*a^4*c^2*d^7 + 4*A^2*a^4*c^3*d^6 - 4*A^2*a^4*c^4*d^5 - 2*A^2*a^4*c^5*d^4 + 6*B^2*a^4*c^2*d^7 - 2
9*B^2*a^4*c^3*d^6 - 4*B^2*a^4*c^4*d^5 + 28*B^2*a^4*c^5*d^4 - 8*B^2*a^4*c^7*d^2 - 2*A^2*a^4*c*d^8 + 7*B^2*a^4*c
*d^8 + 22*A*B*a^4*c^2*d^7 - 16*A*B*a^4*c^3*d^6 - 26*A*B*a^4*c^4*d^5 + 8*A*B*a^4*c^5*d^4 + 8*A*B*a^4*c^6*d^3 +
4*A*B*a^4*c*d^8))/(2*c*d^7 + d^8 + c^2*d^6))*1i)/d^3 + ((B*a^2*c*2i - a^2*d*(A + 2*B)*1i)*((32*(A^2*a^4*c^2*d^
6 + 2*A^2*a^4*c^3*d^5 + A^2*a^4*c^4*d^4 + 4*B^2*a^4*c^2*d^6 - 8*B^2*a^4*c^4*d^4 + 4*B^2*a^4*c^6*d^2 + 4*A*B*a^
4*c^2*d^6 + 4*A*B*a^4*c^3*d^5 - 4*A*B*a^4*c^4*d^4 - 4*A*B*a^4*c^5*d^3))/(2*c*d^6 + d^7 + c^2*d^5) + ((B*a^2*c*
2i - a^2*d*(A + 2*B)*1i)*((32*(A*a^2*c*d^9 + 2*B*a^2*c*d^9 - A*a^2*c^3*d^7 + B*a^2*c^2*d^8 - 2*B*a^2*c^3*d^7 -
B*a^2*c^4*d^6))/(2*c*d^6 + d^7 + c^2*d^5) + (((32*(c^2*d^10 + 2*c^3*d^9 + c^4*d^8))/(2*c*d^6 + d^7 + c^2*d^5)
+ (32*tan(e/2 + (f*x)/2)*(3*c*d^12 + 6*c^2*d^11 + c^3*d^10 - 4*c^4*d^9 - 2*c^5*d^8))/(2*c*d^7 + d^8 + c^2*d^6
))*(B*a^2*c*2i - a^2*d*(A + 2*B)*1i))/d^3 - (32*tan(e/2 + (f*x)/2)*(4*A*a^2*c*d^10 + 2*B*a^2*c*d^10 + 2*A*a^2*
c^2*d^9 - 4*A*a^2*c^3*d^8 - 2*A*a^2*c^4*d^7 - 4*B*a^2*c^2*d^9 - 6*B*a^2*c^3*d^8 + 4*B*a^2*c^4*d^7 + 4*B*a^2*c^
5*d^6))/(2*c*d^7 + d^8 + c^2*d^6)))/d^3 + (32*tan(e/2 + (f*x)/2)*(8*A^2*a^4*c^2*d^7 + 4*A^2*a^4*c^3*d^6 - 4*A^
2*a^4*c^4*d^5 - 2*A^2*a^4*c^5*d^4 + 6*B^2*a^4*c^2*d^7 - 29*B^2*a^4*c^3*d^6 - 4*B^2*a^4*c^4*d^5 + 28*B^2*a^4*c^
5*d^4 - 8*B^2*a^4*c^7*d^2 - 2*A^2*a^4*c*d^8 + 7*B^2*a^4*c*d^8 + 22*A*B*a^4*c^2*d^7 - 16*A*B*a^4*c^3*d^6 - 26*A
*B*a^4*c^4*d^5 + 8*A*B*a^4*c^5*d^4 + 8*A*B*a^4*c^6*d^3 + 4*A*B*a^4*c*d^8))/(2*c*d^7 + d^8 + c^2*d^6))*1i)/d^3)
/((64*(4*B^3*a^6*c^6 - 2*A^3*a^6*c^2*d^4 - 2*A^3*a^6*c^3*d^3 - 10*B^3*a^6*c^2*d^4 + 14*B^3*a^6*c^3*d^3 - 2*B^3
*a^6*c^4*d^2 + 4*A^3*a^6*c*d^5 + 2*B^3*a^6*c*d^5 - 8*B^3*a^6*c^5*d + 9*A*B^2*a^6*c*d^5 - 12*A*B^2*a^6*c^5*d +
12*A^2*B*a^6*c*d^5 - 30*A*B^2*a^6*c^2*d^4 + 21*A*B^2*a^6*c^3*d^3 + 12*A*B^2*a^6*c^4*d^2 - 21*A^2*B*a^6*c^2*d^4
+ 9*A^2*B*a^6*c^4*d^2))/(2*c*d^6 + d^7 + c^2*d^5) + ((B*a^2*c*2i - a^2*d*(A + 2*B)*1i)*((32*(A^2*a^4*c^2*d^6
+ 2*A^2*a^4*c^3*d^5 + A^2*a^4*c^4*d^4 + 4*B^2*a^4*c^2*d^6 - 8*B^2*a^4*c^4*d^4 + 4*B^2*a^4*c^6*d^2 + 4*A*B*a^4*
c^2*d^6 + 4*A*B*a^4*c^3*d^5 - 4*A*B*a^4*c^4*d^4 - 4*A*B*a^4*c^5*d^3))/(2*c*d^6 + d^7 + c^2*d^5) + ((B*a^2*c*2i
- a^2*d*(A + 2*B)*1i)*((((32*(c^2*d^10 + 2*c^3*d^9 + c^4*d^8))/(2*c*d^6 + d^7 + c^2*d^5) + (32*tan(e/2 + (f*x
)/2)*(3*c*d^12 + 6*c^2*d^11 + c^3*d^10 - 4*c^4*d^9 - 2*c^5*d^8))/(2*c*d^7 + d^8 + c^2*d^6))*(B*a^2*c*2i - a^2*
d*(A + 2*B)*1i))/d^3 - (32*(A*a^2*c*d^9 + 2*B*a^2*c*d^9 - A*a^2*c^3*d^7 + B*a^2*c^2*d^8 - 2*B*a^2*c^3*d^7 - B*
a^2*c^4*d^6))/(2*c*d^6 + d^7 + c^2*d^5) + (32*tan(e/2 + (f*x)/2)*(4*A*a^2*c*d^10 + 2*B*a^2*c*d^10 + 2*A*a^2*c^
2*d^9 - 4*A*a^2*c^3*d^8 - 2*A*a^2*c^4*d^7 - 4*B*a^2*c^2*d^9 - 6*B*a^2*c^3*d^8 + 4*B*a^2*c^4*d^7 + 4*B*a^2*c^5*
d^6))/(2*c*d^7 + d^8 + c^2*d^6)))/d^3 + (32*tan(e/2 + (f*x)/2)*(8*A^2*a^4*c^2*d^7 + 4*A^2*a^4*c^3*d^6 - 4*A^2*
a^4*c^4*d^5 - 2*A^2*a^4*c^5*d^4 + 6*B^2*a^4*c^2*d^7 - 29*B^2*a^4*c^3*d^6 - 4*B^2*a^4*c^4*d^5 + 28*B^2*a^4*c^5*
d^4 - 8*B^2*a^4*c^7*d^2 - 2*A^2*a^4*c*d^8 + 7*B^2*a^4*c*d^8 + 22*A*B*a^4*c^2*d^7 - 16*A*B*a^4*c^3*d^6 - 26*A*B
*a^4*c^4*d^5 + 8*A*B*a^4*c^5*d^4 + 8*A*B*a^4*c^6*d^3 + 4*A*B*a^4*c*d^8))/(2*c*d^7 + d^8 + c^2*d^6)))/d^3 - ((B
*a^2*c*2i - a^2*d*(A + 2*B)*1i)*((32*(A^2*a^4*c^2*d^6 + 2*A^2*a^4*c^3*d^5 + A^2*a^4*c^4*d^4 + 4*B^2*a^4*c^2*d^
6 - 8*B^2*a^4*c^4*d^4 + 4*B^2*a^4*c^6*d^2 + 4*A*B*a^4*c^2*d^6 + 4*A*B*a^4*c^3*d^5 - 4*A*B*a^4*c^4*d^4 - 4*A*B*
a^4*c^5*d^3))/(2*c*d^6 + d^7 + c^2*d^5) + ((B*a^2*c*2i - a^2*d*(A + 2*B)*1i)*((32*(A*a^2*c*d^9 + 2*B*a^2*c*d^9
- A*a^2*c^3*d^7 + B*a^2*c^2*d^8 - 2*B*a^2*c^3*...

________________________________________________________________________________________