3.8.12 \(\int \frac {1}{(a+b \sin (e+f x))^2 (c+d \sin (e+f x))^2} \, dx\) [712]

Optimal. Leaf size=290 \[ \frac {2 b^2 \left (a b c-3 a^2 d+2 b^2 d\right ) \tan ^{-1}\left (\frac {b+a \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} (b c-a d)^3 f}+\frac {2 d^2 \left (3 b c^2-a c d-2 b d^2\right ) \tan ^{-1}\left (\frac {d+c \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c^2-d^2}}\right )}{(b c-a d)^3 \left (c^2-d^2\right )^{3/2} f}+\frac {d \left (a^2 d^2+b^2 \left (c^2-2 d^2\right )\right ) \cos (e+f x)}{\left (a^2-b^2\right ) (b c-a d)^2 \left (c^2-d^2\right ) f (c+d \sin (e+f x))}+\frac {b^2 \cos (e+f x)}{\left (a^2-b^2\right ) (b c-a d) f (a+b \sin (e+f x)) (c+d \sin (e+f x))} \]

[Out]

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

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Rubi [A]
time = 0.80, antiderivative size = 290, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {2881, 3134, 3080, 2739, 632, 210} \begin {gather*} \frac {2 b^2 \left (-3 a^2 d+a b c+2 b^2 d\right ) \text {ArcTan}\left (\frac {a \tan \left (\frac {1}{2} (e+f x)\right )+b}{\sqrt {a^2-b^2}}\right )}{f \left (a^2-b^2\right )^{3/2} (b c-a d)^3}+\frac {d \left (a^2 d^2+b^2 \left (c^2-2 d^2\right )\right ) \cos (e+f x)}{f \left (a^2-b^2\right ) \left (c^2-d^2\right ) (b c-a d)^2 (c+d \sin (e+f x))}+\frac {b^2 \cos (e+f x)}{f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x)) (c+d \sin (e+f x))}+\frac {2 d^2 \left (-a c d+3 b c^2-2 b d^2\right ) \text {ArcTan}\left (\frac {c \tan \left (\frac {1}{2} (e+f x)\right )+d}{\sqrt {c^2-d^2}}\right )}{f \left (c^2-d^2\right )^{3/2} (b c-a d)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((a + b*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^2),x]

[Out]

(2*b^2*(a*b*c - 3*a^2*d + 2*b^2*d)*ArcTan[(b + a*Tan[(e + f*x)/2])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(3/2)*(b*c -
a*d)^3*f) + (2*d^2*(3*b*c^2 - a*c*d - 2*b*d^2)*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/((b*c - a*d)
^3*(c^2 - d^2)^(3/2)*f) + (d*(a^2*d^2 + b^2*(c^2 - 2*d^2))*Cos[e + f*x])/((a^2 - b^2)*(b*c - a*d)^2*(c^2 - d^2
)*f*(c + d*Sin[e + f*x])) + (b^2*Cos[e + f*x])/((a^2 - b^2)*(b*c - a*d)*f*(a + b*Sin[e + f*x])*(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 2881

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si
mp[(-b^2)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)*((c + d*Sin[e + f*x])^(n + 1)/(f*(m + 1)*(b*c - a*d)*(a^2
- b^2))), x] + Dist[1/((m + 1)*(b*c - a*d)*(a^2 - b^2)), Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])
^n*Simp[a*(b*c - a*d)*(m + 1) + b^2*d*(m + n + 2) - (b^2*c + b*(b*c - a*d)*(m + 1))*Sin[e + f*x] - b^2*d*(m +
n + 3)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0]
&& NeQ[c^2 - d^2, 0] && LtQ[m, -1] && IntegersQ[2*m, 2*n] && ((EqQ[a, 0] && IntegerQ[m] &&  !IntegerQ[n]) ||
!(IntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n] &&  !IntegerQ[m]) || EqQ[a, 0])))

Rule 3080

Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.)
+ (f_.)*(x_)])), x_Symbol] :> Dist[(A*b - a*B)/(b*c - a*d), Int[1/(a + b*Sin[e + f*x]), x], x] + Dist[(B*c - A
*d)/(b*c - a*d), Int[1/(c + d*Sin[e + f*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]

Rule 3134

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*s
in[(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)*((c + d*Sin[e + f*x])^(n + 1)/(f*(m + 1)*(b*c - a*d)*(a^2 - b^2))), x] + D
ist[1/((m + 1)*(b*c - a*d)*(a^2 - b^2)), Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[(m + 1)*
(b*c - a*d)*(a*A - b*B + a*C) + d*(A*b^2 - a*b*B + a^2*C)*(m + n + 2) - (c*(A*b^2 - a*b*B + a^2*C) + (m + 1)*(
b*c - a*d)*(A*b - a*B + b*C))*Sin[e + f*x] - d*(A*b^2 - a*b*B + a^2*C)*(m + n + 3)*Sin[e + f*x]^2, x], x], x]
/; FreeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] &&
LtQ[m, -1] && ((EqQ[a, 0] && IntegerQ[m] &&  !IntegerQ[n]) ||  !(IntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n]
&&  !IntegerQ[m]) || EqQ[a, 0])))

Rubi steps

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

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Mathematica [A]
time = 3.04, size = 227, normalized size = 0.78 \begin {gather*} \frac {\frac {2 b^2 \left (a b c-3 a^2 d+2 b^2 d\right ) \tan ^{-1}\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}}-\frac {2 d^2 \left (-3 b c^2+a c d+2 b d^2\right ) \tan ^{-1}\left (\frac {d+c \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c^2-d^2}}\right )}{\left (c^2-d^2\right )^{3/2}}+\frac {b^3 (b c-a d) \cos (e+f x)}{(a-b) (a+b) (a+b \sin (e+f x))}+\frac {d^3 (b c-a d) \cos (e+f x)}{(c-d) (c+d) (c+d \sin (e+f x))}}{(b c-a d)^3 f} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^2),x]

[Out]

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

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Maple [A]
time = 3.31, size = 331, normalized size = 1.14

method	result	size
derivativedivides	\(\frac {\frac {2 d^{2} \left (\frac {\frac {d^{2} \left (a d -b c \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{c \left (c^{2}-d^{2}\right )}+\frac {d \left (a d -b c \right )}{c^{2}-d^{2}}}{c \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+2 d \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+c}+\frac {\left (a c d -3 b \,c^{2}+2 b \,d^{2}\right ) \arctan \left (\frac {2 c \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+2 d}{2 \sqrt {c^{2}-d^{2}}}\right )}{\left (c^{2}-d^{2}\right )^{\frac {3}{2}}}\right )}{\left (a d -b c \right )^{3}}+\frac {2 b^{2} \left (\frac {\frac {b^{2} \left (a d -b c \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{a \left (a^{2}-b^{2}\right )}+\frac {b \left (a d -b c \right )}{a^{2}-b^{2}}}{a \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+2 b \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+a}+\frac {\left (3 a^{2} d -a b c -2 b^{2} d \right ) \arctan \left (\frac {2 a \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{\left (a^{2}-b^{2}\right )^{\frac {3}{2}}}\right )}{\left (a d -b c \right )^{3}}}{f}\)	\(331\)
default	\(\frac {\frac {2 d^{2} \left (\frac {\frac {d^{2} \left (a d -b c \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{c \left (c^{2}-d^{2}\right )}+\frac {d \left (a d -b c \right )}{c^{2}-d^{2}}}{c \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+2 d \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+c}+\frac {\left (a c d -3 b \,c^{2}+2 b \,d^{2}\right ) \arctan \left (\frac {2 c \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+2 d}{2 \sqrt {c^{2}-d^{2}}}\right )}{\left (c^{2}-d^{2}\right )^{\frac {3}{2}}}\right )}{\left (a d -b c \right )^{3}}+\frac {2 b^{2} \left (\frac {\frac {b^{2} \left (a d -b c \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{a \left (a^{2}-b^{2}\right )}+\frac {b \left (a d -b c \right )}{a^{2}-b^{2}}}{a \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+2 b \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+a}+\frac {\left (3 a^{2} d -a b c -2 b^{2} d \right ) \arctan \left (\frac {2 a \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{\left (a^{2}-b^{2}\right )^{\frac {3}{2}}}\right )}{\left (a d -b c \right )^{3}}}{f}\)	\(331\)
risch	\(\text {Expression too large to display}\)	\(1547\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b*sin(f*x+e))^2/(c+d*sin(f*x+e))^2,x,method=_RETURNVERBOSE)

[Out]

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

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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(1/(a+b*sin(f*x+e))^2/(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

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Fricas [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(1/(a+b*sin(f*x+e))^2/(c+d*sin(f*x+e))^2,x, algorithm="fricas")

[Out]

Timed out

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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(1/(a+b*sin(f*x+e))**2/(c+d*sin(f*x+e))**2,x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1005 vs. \(2 (287) = 574\).
time = 0.70, size = 1005, normalized size = 3.47 \begin {gather*} \frac {2 \, {\left (\frac {{\left (a b^{3} c - 3 \, a^{2} b^{2} d + 2 \, b^{4} d\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} c^{3} - b^{5} c^{3} - 3 \, a^{3} b^{2} c^{2} d + 3 \, a b^{4} c^{2} d + 3 \, a^{4} b c d^{2} - 3 \, a^{2} b^{3} c d^{2} - a^{5} d^{3} + a^{3} b^{2} d^{3}\right )} \sqrt {a^{2} - b^{2}}} + \frac {{\left (3 \, b c^{2} d^{2} - a c d^{3} - 2 \, b d^{4}\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 (b^{3} c^{5} - 3 \, a b^{2} c^{4} d + 3 \, a^{2} b c^{3} d^{2} - b^{3} c^{3} d^{2} - a^{3} c^{2} d^{3} + 3 \, a b^{2} c^{2} d^{3} - 3 \, a^{2} b c d^{4} + a^{3} d^{5}\right )} \sqrt {c^{2} - d^{2}}} + \frac {b^{4} c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - b^{4} c^{2} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + a^{4} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - a^{2} b^{2} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + a b^{3} c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 2 \, b^{4} c^{3} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - a b^{3} c^{2} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + a^{4} c d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - a^{2} b^{2} c d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 2 \, b^{4} c d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 2 \, a^{3} b d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 2 \, a b^{3} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + b^{4} c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 2 \, a b^{3} c^{3} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - b^{4} c^{2} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 2 \, a^{3} b c d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 4 \, a b^{3} c d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + a^{4} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - a^{2} b^{2} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + a b^{3} c^{4} - a b^{3} c^{2} d^{2} + a^{4} c d^{3} - a^{2} b^{2} c d^{3}}{{\left (a^{3} b^{2} c^{5} - a b^{4} c^{5} - 2 \, a^{4} b c^{4} d + 2 \, a^{2} b^{3} c^{4} d + a^{5} c^{3} d^{2} - 2 \, a^{3} b^{2} c^{3} d^{2} + a b^{4} c^{3} d^{2} + 2 \, a^{4} b c^{2} d^{3} - 2 \, a^{2} b^{3} c^{2} d^{3} - a^{5} c d^{4} + a^{3} b^{2} c d^{4}\right )} {\left (a c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 2 \, b c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 2 \, a d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 2 \, a c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 4 \, b d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 2 \, b c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 2 \, a d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + a c\right )}}\right )}}{f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*sin(f*x+e))^2/(c+d*sin(f*x+e))^2,x, algorithm="giac")

[Out]

2*((a*b^3*c - 3*a^2*b^2*d + 2*b^4*d)*(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*c^3 - b^5*c^3 - 3*a^3*b^2*c^2*d + 3*a*b^4*c^2*d + 3*a^4*b*c*d^2 - 3*a^2*b^3*
c*d^2 - a^5*d^3 + a^3*b^2*d^3)*sqrt(a^2 - b^2)) + (3*b*c^2*d^2 - a*c*d^3 - 2*b*d^4)*(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)))/((b^3*c^5 - 3*a*b^2*c^4*d + 3*a^2*b*c^3
*d^2 - b^3*c^3*d^2 - a^3*c^2*d^3 + 3*a*b^2*c^2*d^3 - 3*a^2*b*c*d^4 + a^3*d^5)*sqrt(c^2 - d^2)) + (b^4*c^4*tan(
1/2*f*x + 1/2*e)^3 - b^4*c^2*d^2*tan(1/2*f*x + 1/2*e)^3 + a^4*d^4*tan(1/2*f*x + 1/2*e)^3 - a^2*b^2*d^4*tan(1/2
*f*x + 1/2*e)^3 + a*b^3*c^4*tan(1/2*f*x + 1/2*e)^2 + 2*b^4*c^3*d*tan(1/2*f*x + 1/2*e)^2 - a*b^3*c^2*d^2*tan(1/
2*f*x + 1/2*e)^2 + a^4*c*d^3*tan(1/2*f*x + 1/2*e)^2 - a^2*b^2*c*d^3*tan(1/2*f*x + 1/2*e)^2 - 2*b^4*c*d^3*tan(1
/2*f*x + 1/2*e)^2 + 2*a^3*b*d^4*tan(1/2*f*x + 1/2*e)^2 - 2*a*b^3*d^4*tan(1/2*f*x + 1/2*e)^2 + b^4*c^4*tan(1/2*
f*x + 1/2*e) + 2*a*b^3*c^3*d*tan(1/2*f*x + 1/2*e) - b^4*c^2*d^2*tan(1/2*f*x + 1/2*e) + 2*a^3*b*c*d^3*tan(1/2*f
*x + 1/2*e) - 4*a*b^3*c*d^3*tan(1/2*f*x + 1/2*e) + a^4*d^4*tan(1/2*f*x + 1/2*e) - a^2*b^2*d^4*tan(1/2*f*x + 1/
2*e) + a*b^3*c^4 - a*b^3*c^2*d^2 + a^4*c*d^3 - a^2*b^2*c*d^3)/((a^3*b^2*c^5 - a*b^4*c^5 - 2*a^4*b*c^4*d + 2*a^
2*b^3*c^4*d + a^5*c^3*d^2 - 2*a^3*b^2*c^3*d^2 + a*b^4*c^3*d^2 + 2*a^4*b*c^2*d^3 - 2*a^2*b^3*c^2*d^3 - a^5*c*d^
4 + a^3*b^2*c*d^4)*(a*c*tan(1/2*f*x + 1/2*e)^4 + 2*b*c*tan(1/2*f*x + 1/2*e)^3 + 2*a*d*tan(1/2*f*x + 1/2*e)^3 +
2*a*c*tan(1/2*f*x + 1/2*e)^2 + 4*b*d*tan(1/2*f*x + 1/2*e)^2 + 2*b*c*tan(1/2*f*x + 1/2*e) + 2*a*d*tan(1/2*f*x
+ 1/2*e) + a*c)))/f

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Mupad [B]
time = 30.89, size = 2500, normalized size = 8.62 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((a + b*sin(e + f*x))^2*(c + d*sin(e + f*x))^2),x)

[Out]

((2*(a^3*d^3 + b^3*c^3 - a*b^2*d^3 - b^3*c*d^2))/((a^2*d^2 + b^2*c^2 - 2*a*b*c*d)*(a^2*c^2 - a^2*d^2 - b^2*c^2
+ b^2*d^2)) + (2*tan(e/2 + (f*x)/2)^3*(a^4*d^4 + b^4*c^4 - a^2*b^2*d^4 - b^4*c^2*d^2))/(a*c*(a^2*d^2 + b^2*c^
2 - 2*a*b*c*d)*(a^2*c^2 - a^2*d^2 - b^2*c^2 + b^2*d^2)) + (2*tan(e/2 + (f*x)/2)*(a^4*d^4 + b^4*c^4 - a^2*b^2*d
^4 - b^4*c^2*d^2 - 4*a*b^3*c*d^3 + 2*a*b^3*c^3*d + 2*a^3*b*c*d^3))/(a*c*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)*(a^2*c
^2 - a^2*d^2 - b^2*c^2 + b^2*d^2)) + (2*tan(e/2 + (f*x)/2)^2*(a*c + 2*b*d)*(a^3*d^3 + b^3*c^3 - a*b^2*d^3 - b^
3*c*d^2))/(a*c*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)*(a^2*c^2 - a^2*d^2 - b^2*c^2 + b^2*d^2)))/(f*(a*c + tan(e/2 + (
f*x)/2)^3*(2*a*d + 2*b*c) + tan(e/2 + (f*x)/2)^2*(2*a*c + 4*b*d) + tan(e/2 + (f*x)/2)*(2*a*d + 2*b*c) + a*c*ta
n(e/2 + (f*x)/2)^4)) - (b^2*atan(((b^2*(-(a + b)^3*(a - b)^3)^(1/2)*((32*(4*a*b^10*c^4*d^7 - 8*a*b^10*c^6*d^5
+ 4*a*b^10*c^8*d^3 + a^3*b^8*c^10*d + 4*a^4*b^7*c*d^10 - 8*a^6*b^5*c*d^10 + 4*a^8*b^3*c*d^10 + a^10*b*c^3*d^8
- 4*a^2*b^9*c^3*d^8 + 8*a^2*b^9*c^5*d^6 - 7*a^2*b^9*c^7*d^4 + 4*a^2*b^9*c^9*d^2 - 4*a^3*b^8*c^2*d^9 + 21*a^3*b
^8*c^6*d^5 - 22*a^3*b^8*c^8*d^3 - 18*a^4*b^7*c^5*d^6 + 26*a^4*b^7*c^7*d^4 - 8*a^4*b^7*c^9*d^2 + 8*a^5*b^6*c^2*
d^9 - 18*a^5*b^6*c^4*d^7 - 8*a^5*b^6*c^6*d^5 + 22*a^5*b^6*c^8*d^3 + 21*a^6*b^5*c^3*d^8 - 8*a^6*b^5*c^5*d^6 - 1
5*a^6*b^5*c^7*d^4 - 7*a^7*b^4*c^2*d^9 + 26*a^7*b^4*c^4*d^7 - 15*a^7*b^4*c^6*d^5 - 22*a^8*b^3*c^3*d^8 + 22*a^8*
b^3*c^5*d^6 + 4*a^9*b^2*c^2*d^9 - 8*a^9*b^2*c^4*d^7))/(a^10*d^10 + b^10*c^10 - 2*a^2*b^8*c^10 + a^4*b^6*c^10 +
a^6*b^4*d^10 - 2*a^8*b^2*d^10 - 2*a^10*c^2*d^8 + a^10*c^4*d^6 + b^10*c^6*d^4 - 2*b^10*c^8*d^2 - 6*a*b^9*c^5*d
^5 + 12*a*b^9*c^7*d^3 + 12*a^3*b^7*c^9*d - 6*a^5*b^5*c*d^9 - 6*a^5*b^5*c^9*d + 12*a^7*b^3*c*d^9 + 12*a^9*b*c^3
*d^7 - 6*a^9*b*c^5*d^5 + 15*a^2*b^8*c^4*d^6 - 32*a^2*b^8*c^6*d^4 + 19*a^2*b^8*c^8*d^2 - 20*a^3*b^7*c^3*d^7 + 5
2*a^3*b^7*c^5*d^5 - 44*a^3*b^7*c^7*d^3 + 15*a^4*b^6*c^2*d^8 - 60*a^4*b^6*c^4*d^6 + 76*a^4*b^6*c^6*d^4 - 32*a^4
*b^6*c^8*d^2 + 52*a^5*b^5*c^3*d^7 - 92*a^5*b^5*c^5*d^5 + 52*a^5*b^5*c^7*d^3 - 32*a^6*b^4*c^2*d^8 + 76*a^6*b^4*
c^4*d^6 - 60*a^6*b^4*c^6*d^4 + 15*a^6*b^4*c^8*d^2 - 44*a^7*b^3*c^3*d^7 + 52*a^7*b^3*c^5*d^5 - 20*a^7*b^3*c^7*d
^3 + 19*a^8*b^2*c^2*d^8 - 32*a^8*b^2*c^4*d^6 + 15*a^8*b^2*c^6*d^4 - 6*a*b^9*c^9*d - 6*a^9*b*c*d^9) - (32*tan(e
/2 + (f*x)/2)*(a^3*b^8*c^11 + a^11*c^3*d^8 - 16*a*b^10*c^3*d^8 + 44*a*b^10*c^5*d^6 - 34*a*b^10*c^7*d^4 + 4*a*b
^10*c^9*d^2 + 4*a^2*b^9*c^10*d - 16*a^3*b^8*c*d^10 - 8*a^4*b^7*c^10*d + 44*a^5*b^6*c*d^10 - 34*a^7*b^4*c*d^10
+ 4*a^9*b^2*c*d^10 + 4*a^10*b*c^2*d^9 - 8*a^10*b*c^4*d^7 + 32*a^2*b^9*c^2*d^9 - 104*a^2*b^9*c^4*d^7 + 100*a^2*
b^9*c^6*d^5 - 24*a^2*b^9*c^8*d^3 + 120*a^3*b^8*c^3*d^8 - 222*a^3*b^8*c^5*d^6 + 134*a^3*b^8*c^7*d^4 - 24*a^3*b^
8*c^9*d^2 - 104*a^4*b^7*c^2*d^9 + 312*a^4*b^7*c^4*d^7 - 272*a^4*b^7*c^6*d^5 + 60*a^4*b^7*c^8*d^3 - 222*a^5*b^6
*c^3*d^8 + 316*a^5*b^6*c^5*d^6 - 136*a^5*b^6*c^7*d^4 + 22*a^5*b^6*c^9*d^2 + 100*a^6*b^5*c^2*d^9 - 272*a^6*b^5*
c^4*d^7 + 192*a^6*b^5*c^6*d^5 - 24*a^6*b^5*c^8*d^3 + 134*a^7*b^4*c^3*d^8 - 136*a^7*b^4*c^5*d^6 + 18*a^7*b^4*c^
7*d^4 - 24*a^8*b^3*c^2*d^9 + 60*a^8*b^3*c^4*d^7 - 24*a^8*b^3*c^6*d^5 - 24*a^9*b^2*c^3*d^8 + 22*a^9*b^2*c^5*d^6
))/(a^10*d^10 + b^10*c^10 - 2*a^2*b^8*c^10 + a^4*b^6*c^10 + a^6*b^4*d^10 - 2*a^8*b^2*d^10 - 2*a^10*c^2*d^8 + a
^10*c^4*d^6 + b^10*c^6*d^4 - 2*b^10*c^8*d^2 - 6*a*b^9*c^5*d^5 + 12*a*b^9*c^7*d^3 + 12*a^3*b^7*c^9*d - 6*a^5*b^
5*c*d^9 - 6*a^5*b^5*c^9*d + 12*a^7*b^3*c*d^9 + 12*a^9*b*c^3*d^7 - 6*a^9*b*c^5*d^5 + 15*a^2*b^8*c^4*d^6 - 32*a^
2*b^8*c^6*d^4 + 19*a^2*b^8*c^8*d^2 - 20*a^3*b^7*c^3*d^7 + 52*a^3*b^7*c^5*d^5 - 44*a^3*b^7*c^7*d^3 + 15*a^4*b^6
*c^2*d^8 - 60*a^4*b^6*c^4*d^6 + 76*a^4*b^6*c^6*d^4 - 32*a^4*b^6*c^8*d^2 + 52*a^5*b^5*c^3*d^7 - 92*a^5*b^5*c^5*
d^5 + 52*a^5*b^5*c^7*d^3 - 32*a^6*b^4*c^2*d^8 + 76*a^6*b^4*c^4*d^6 - 60*a^6*b^4*c^6*d^4 + 15*a^6*b^4*c^8*d^2 -
44*a^7*b^3*c^3*d^7 + 52*a^7*b^3*c^5*d^5 - 20*a^7*b^3*c^7*d^3 + 19*a^8*b^2*c^2*d^8 - 32*a^8*b^2*c^4*d^6 + 15*a
^8*b^2*c^6*d^4 - 6*a*b^9*c^9*d - 6*a^9*b*c*d^9) + (b^2*(-(a + b)^3*(a - b)^3)^(1/2)*((32*tan(e/2 + (f*x)/2)*(2
*a^4*b^9*c^13 - 2*a^2*b^11*c^13 - 2*a^13*c^2*d^11 + 2*a^13*c^4*d^9 - 2*a*b^12*c^8*d^5 + 6*a*b^12*c^10*d^3 + 20
*a^3*b^10*c^12*d - 16*a^5*b^8*c^12*d - 2*a^8*b^5*c*d^12 + 6*a^10*b^3*c*d^12 + 20*a^12*b*c^3*d^10 - 16*a^12*b*c
^5*d^8 + 10*a^2*b^11*c^7*d^6 - 34*a^2*b^11*c^9*d^4 + 26*a^2*b^11*c^11*d^2 - 18*a^3*b^10*c^6*d^7 + 80*a^3*b^10*
c^8*d^5 - 82*a^3*b^10*c^10*d^3 + 10*a^4*b^9*c^5*d^8 - 96*a^4*b^9*c^7*d^6 + 160*a^4*b^9*c^9*d^4 - 76*a^4*b^9*c^
11*d^2 + 10*a^5*b^8*c^4*d^9 + 44*a^5*b^8*c^6*d^7 - 188*a^5*b^8*c^8*d^5 + 150*a^5*b^8*c^10*d^3 - 18*a^6*b^7*c^3
*d^10 + 44*a^6*b^7*c^5*d^8 + 88*a^6*b^7*c^7*d^6 - 164*a^6*b^7*c^9*d^4 + 50*a^6*b^7*c^11*d^2 + 10*a^7*b^6*c^2*d
^11 - 96*a^7*b^6*c^4*d^9 + 88*a^7*b^6*c^6*d^7 + 72*a^7*b^6*c^8*d^5 - 74*a^7*b^6*c^10*d^3 + 80*a^8*b^5*c^3*d^10
- 188*a^8*b^5*c^5*d^8 + 72*a^8*b^5*c^7*d^6 + 38*a^8*b^5*c^9*d^4 - 34*a^9*b^4*c^2*d^11 + 160*a^9*b^4*c^4*d^9 -
164*a^9*b^4*c^6*d^7 + 38*a^9*b^4*c^8*d^5 - 82*...

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