3.8.20 \(\int \frac {1}{(a+b \sin (e+f x))^3 (c+d \sin (e+f x))} \, dx\) [720]

Optimal. Leaf size=285 \[ -\frac {b \left (6 a^3 b c d-6 a^4 d^2-a^2 b^2 \left (2 c^2-5 d^2\right )-b^4 \left (c^2+2 d^2\right )\right ) \tan ^{-1}\left (\frac {b+a \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{5/2} (b c-a d)^3 f}-\frac {2 d^3 \tan ^{-1}\left (\frac {d+c \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c^2-d^2}}\right )}{(b c-a d)^3 \sqrt {c^2-d^2} f}+\frac {b^2 \cos (e+f x)}{2 \left (a^2-b^2\right ) (b c-a d) f (a+b \sin (e+f x))^2}+\frac {b^2 \left (3 a b c-5 a^2 d+2 b^2 d\right ) \cos (e+f x)}{2 \left (a^2-b^2\right )^2 (b c-a d)^2 f (a+b \sin (e+f x))} \]

[Out]

-b*(6*a^3*b*c*d-6*a^4*d^2-a^2*b^2*(2*c^2-5*d^2)-b^4*(c^2+2*d^2))*arctan((b+a*tan(1/2*f*x+1/2*e))/(a^2-b^2)^(1/
2))/(a^2-b^2)^(5/2)/(-a*d+b*c)^3/f+1/2*b^2*cos(f*x+e)/(a^2-b^2)/(-a*d+b*c)/f/(a+b*sin(f*x+e))^2+1/2*b^2*(-5*a^
2*d+3*a*b*c+2*b^2*d)*cos(f*x+e)/(a^2-b^2)^2/(-a*d+b*c)^2/f/(a+b*sin(f*x+e))-2*d^3*arctan((d+c*tan(1/2*f*x+1/2*
e))/(c^2-d^2)^(1/2))/(-a*d+b*c)^3/f/(c^2-d^2)^(1/2)

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Rubi [A]
time = 0.71, antiderivative size = 285, 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 {b^2 \left (-5 a^2 d+3 a b c+2 b^2 d\right ) \cos (e+f x)}{2 f \left (a^2-b^2\right )^2 (b c-a d)^2 (a+b \sin (e+f x))}+\frac {b^2 \cos (e+f x)}{2 f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x))^2}-\frac {b \left (-6 a^4 d^2+6 a^3 b c d-a^2 b^2 \left (2 c^2-5 d^2\right )-b^4 \left (c^2+2 d^2\right )\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 )^{5/2} (b c-a d)^3}-\frac {2 d^3 \text {ArcTan}\left (\frac {c \tan \left (\frac {1}{2} (e+f x)\right )+d}{\sqrt {c^2-d^2}}\right )}{f \sqrt {c^2-d^2} (b c-a d)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((a + b*Sin[e + f*x])^3*(c + d*Sin[e + f*x])),x]

[Out]

-((b*(6*a^3*b*c*d - 6*a^4*d^2 - a^2*b^2*(2*c^2 - 5*d^2) - b^4*(c^2 + 2*d^2))*ArcTan[(b + a*Tan[(e + f*x)/2])/S
qrt[a^2 - b^2]])/((a^2 - b^2)^(5/2)*(b*c - a*d)^3*f)) - (2*d^3*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]
])/((b*c - a*d)^3*Sqrt[c^2 - d^2]*f) + (b^2*Cos[e + f*x])/(2*(a^2 - b^2)*(b*c - a*d)*f*(a + b*Sin[e + f*x])^2)
+ (b^2*(3*a*b*c - 5*a^2*d + 2*b^2*d)*Cos[e + f*x])/(2*(a^2 - b^2)^2*(b*c - a*d)^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 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))^3 (c+d \sin (e+f x))} \, dx &=\frac {b^2 \cos (e+f x)}{2 \left (a^2-b^2\right ) (b c-a d) f (a+b \sin (e+f x))^2}-\frac {\int \frac {-2 \left (a b c-a^2 d+b^2 d\right )+b (b c-2 a d) \sin (e+f x)+b^2 d \sin ^2(e+f x)}{(a+b \sin (e+f x))^2 (c+d \sin (e+f x))} \, dx}{2 \left (a^2-b^2\right ) (b c-a d)}\\ &=\frac {b^2 \cos (e+f x)}{2 \left (a^2-b^2\right ) (b c-a d) f (a+b \sin (e+f x))^2}+\frac {b^2 \left (3 a b c-5 a^2 d+2 b^2 d\right ) \cos (e+f x)}{2 \left (a^2-b^2\right )^2 (b c-a d)^2 f (a+b \sin (e+f x))}+\frac {\int \frac {-4 a^3 b c d+a b^3 c d+2 a^4 d^2+2 a^2 b^2 \left (c^2-2 d^2\right )+b^4 \left (c^2+2 d^2\right )+b d \left (2 a^2 b c+b^3 c-4 a^3 d+a b^2 d\right ) \sin (e+f x)}{(a+b \sin (e+f x)) (c+d \sin (e+f x))} \, dx}{2 \left (a^2-b^2\right )^2 (b c-a d)^2}\\ &=\frac {b^2 \cos (e+f x)}{2 \left (a^2-b^2\right ) (b c-a d) f (a+b \sin (e+f x))^2}+\frac {b^2 \left (3 a b c-5 a^2 d+2 b^2 d\right ) \cos (e+f x)}{2 \left (a^2-b^2\right )^2 (b c-a d)^2 f (a+b \sin (e+f x))}-\frac {d^3 \int \frac {1}{c+d \sin (e+f x)} \, dx}{(b c-a d)^3}-\frac {\left (b \left (6 a^3 b c d-6 a^4 d^2-a^2 b^2 \left (2 c^2-5 d^2\right )-b^4 \left (c^2+2 d^2\right )\right )\right ) \int \frac {1}{a+b \sin (e+f x)} \, dx}{2 \left (a^2-b^2\right )^2 (b c-a d)^3}\\ &=\frac {b^2 \cos (e+f x)}{2 \left (a^2-b^2\right ) (b c-a d) f (a+b \sin (e+f x))^2}+\frac {b^2 \left (3 a b c-5 a^2 d+2 b^2 d\right ) \cos (e+f x)}{2 \left (a^2-b^2\right )^2 (b c-a d)^2 f (a+b \sin (e+f x))}-\frac {\left (2 d^3\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 f}-\frac {\left (b \left (6 a^3 b c d-6 a^4 d^2-a^2 b^2 \left (2 c^2-5 d^2\right )-b^4 \left (c^2+2 d^2\right )\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 )^2 (b c-a d)^3 f}\\ &=\frac {b^2 \cos (e+f x)}{2 \left (a^2-b^2\right ) (b c-a d) f (a+b \sin (e+f x))^2}+\frac {b^2 \left (3 a b c-5 a^2 d+2 b^2 d\right ) \cos (e+f x)}{2 \left (a^2-b^2\right )^2 (b c-a d)^2 f (a+b \sin (e+f x))}+\frac {\left (4 d^3\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 f}+\frac {\left (2 b \left (6 a^3 b c d-6 a^4 d^2-a^2 b^2 \left (2 c^2-5 d^2\right )-b^4 \left (c^2+2 d^2\right )\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 )^2 (b c-a d)^3 f}\\ &=-\frac {b \left (6 a^3 b c d-6 a^4 d^2-a^2 b^2 \left (2 c^2-5 d^2\right )-b^4 \left (c^2+2 d^2\right )\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 )^{5/2} (b c-a d)^3 f}-\frac {2 d^3 \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 \sqrt {c^2-d^2} f}+\frac {b^2 \cos (e+f x)}{2 \left (a^2-b^2\right ) (b c-a d) f (a+b \sin (e+f x))^2}+\frac {b^2 \left (3 a b c-5 a^2 d+2 b^2 d\right ) \cos (e+f x)}{2 \left (a^2-b^2\right )^2 (b c-a d)^2 f (a+b \sin (e+f x))}\\ \end {align*}

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Mathematica [A]
time = 2.38, size = 275, normalized size = 0.96 \begin {gather*} \frac {-\frac {2 b \left (-6 a^3 b c d+6 a^4 d^2+a^2 b^2 \left (2 c^2-5 d^2\right )+b^4 \left (c^2+2 d^2\right )\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 )^{5/2} (-b c+a d)^3}+\frac {4 d^3 \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 \sqrt {c^2-d^2}}-\frac {b^2 \cos (e+f x)}{(a-b) (a+b) (-b c+a d) (a+b \sin (e+f x))^2}+\frac {b^2 \left (3 a b c-5 a^2 d+2 b^2 d\right ) \cos (e+f x)}{(a-b)^2 (a+b)^2 (b c-a d)^2 (a+b \sin (e+f x))}}{2 f} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b*Sin[e + f*x])^3*(c + d*Sin[e + f*x])),x]

[Out]

((-2*b*(-6*a^3*b*c*d + 6*a^4*d^2 + a^2*b^2*(2*c^2 - 5*d^2) + b^4*(c^2 + 2*d^2))*ArcTan[(b + a*Tan[(e + f*x)/2]
)/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(5/2)*(-(b*c) + a*d)^3) + (4*d^3*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d
^2]])/((-(b*c) + a*d)^3*Sqrt[c^2 - d^2]) - (b^2*Cos[e + f*x])/((a - b)*(a + b)*(-(b*c) + a*d)*(a + b*Sin[e + f
*x])^2) + (b^2*(3*a*b*c - 5*a^2*d + 2*b^2*d)*Cos[e + f*x])/((a - b)^2*(a + b)^2*(b*c - a*d)^2*(a + b*Sin[e + f
*x])))/(2*f)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(624\) vs. \(2(271)=542\).
time = 3.66, size = 625, normalized size = 2.19

method	result	size
derivativedivides	\(\frac {-\frac {2 b \left (\frac {\frac {b^{2} \left (7 a^{4} d^{2}-12 a^{3} b c d +5 a^{2} b^{2} c^{2}-4 a^{2} b^{2} d^{2}+6 a \,b^{3} c d -2 b^{4} c^{2}\right ) \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) a}+\frac {b \left (6 a^{6} d^{2}-10 a^{5} b c d +4 a^{4} b^{2} c^{2}+9 a^{4} b^{2} d^{2}-16 a^{3} b^{3} c d +7 a^{2} b^{4} c^{2}-6 a^{2} b^{4} d^{2}+8 a \,b^{5} c d -2 b^{6} c^{2}\right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) a^{2}}+\frac {b^{2} \left (17 a^{4} d^{2}-28 a^{3} b c d +11 a^{2} b^{2} c^{2}-8 a^{2} b^{2} d^{2}+10 a \,b^{3} c d -2 b^{4} c^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{2 a \left (a^{4}-2 a^{2} b^{2}+b^{4}\right )}+\frac {b \left (6 a^{4} d^{2}-10 a^{3} b c d +4 a^{2} b^{2} c^{2}-3 a^{2} b^{2} d^{2}+4 a \,b^{3} c d -b^{4} c^{2}\right )}{2 a^{4}-4 a^{2} b^{2}+2 b^{4}}}{\left (a \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+2 b \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+a \right )^{2}}+\frac {\left (6 a^{4} d^{2}-6 a^{3} b c d +2 a^{2} b^{2} c^{2}-5 a^{2} b^{2} d^{2}+b^{4} c^{2}+2 b^{4} d^{2}\right ) \arctan \left (\frac {2 a \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{2 \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \sqrt {a^{2}-b^{2}}}\right )}{\left (a d -b c \right )^{3}}+\frac {2 d^{3} \arctan \left (\frac {2 c \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+2 d}{2 \sqrt {c^{2}-d^{2}}}\right )}{\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \sqrt {c^{2}-d^{2}}}}{f}\)	\(625\)
default	\(\frac {-\frac {2 b \left (\frac {\frac {b^{2} \left (7 a^{4} d^{2}-12 a^{3} b c d +5 a^{2} b^{2} c^{2}-4 a^{2} b^{2} d^{2}+6 a \,b^{3} c d -2 b^{4} c^{2}\right ) \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) a}+\frac {b \left (6 a^{6} d^{2}-10 a^{5} b c d +4 a^{4} b^{2} c^{2}+9 a^{4} b^{2} d^{2}-16 a^{3} b^{3} c d +7 a^{2} b^{4} c^{2}-6 a^{2} b^{4} d^{2}+8 a \,b^{5} c d -2 b^{6} c^{2}\right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) a^{2}}+\frac {b^{2} \left (17 a^{4} d^{2}-28 a^{3} b c d +11 a^{2} b^{2} c^{2}-8 a^{2} b^{2} d^{2}+10 a \,b^{3} c d -2 b^{4} c^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{2 a \left (a^{4}-2 a^{2} b^{2}+b^{4}\right )}+\frac {b \left (6 a^{4} d^{2}-10 a^{3} b c d +4 a^{2} b^{2} c^{2}-3 a^{2} b^{2} d^{2}+4 a \,b^{3} c d -b^{4} c^{2}\right )}{2 a^{4}-4 a^{2} b^{2}+2 b^{4}}}{\left (a \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+2 b \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+a \right )^{2}}+\frac {\left (6 a^{4} d^{2}-6 a^{3} b c d +2 a^{2} b^{2} c^{2}-5 a^{2} b^{2} d^{2}+b^{4} c^{2}+2 b^{4} d^{2}\right ) \arctan \left (\frac {2 a \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{2 \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \sqrt {a^{2}-b^{2}}}\right )}{\left (a d -b c \right )^{3}}+\frac {2 d^{3} \arctan \left (\frac {2 c \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+2 d}{2 \sqrt {c^{2}-d^{2}}}\right )}{\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \sqrt {c^{2}-d^{2}}}}{f}\)	\(625\)
risch	\(\text {Expression too large to display}\)	\(1603\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b*sin(f*x+e))^3/(c+d*sin(f*x+e)),x,method=_RETURNVERBOSE)

[Out]

1/f*(-2*b/(a*d-b*c)^3*((1/2*b^2*(7*a^4*d^2-12*a^3*b*c*d+5*a^2*b^2*c^2-4*a^2*b^2*d^2+6*a*b^3*c*d-2*b^4*c^2)/(a^
4-2*a^2*b^2+b^4)/a*tan(1/2*f*x+1/2*e)^3+1/2*b*(6*a^6*d^2-10*a^5*b*c*d+4*a^4*b^2*c^2+9*a^4*b^2*d^2-16*a^3*b^3*c
*d+7*a^2*b^4*c^2-6*a^2*b^4*d^2+8*a*b^5*c*d-2*b^6*c^2)/(a^4-2*a^2*b^2+b^4)/a^2*tan(1/2*f*x+1/2*e)^2+1/2*b^2*(17
*a^4*d^2-28*a^3*b*c*d+11*a^2*b^2*c^2-8*a^2*b^2*d^2+10*a*b^3*c*d-2*b^4*c^2)/a/(a^4-2*a^2*b^2+b^4)*tan(1/2*f*x+1
/2*e)+1/2*b*(6*a^4*d^2-10*a^3*b*c*d+4*a^2*b^2*c^2-3*a^2*b^2*d^2+4*a*b^3*c*d-b^4*c^2)/(a^4-2*a^2*b^2+b^4))/(a*t
an(1/2*f*x+1/2*e)^2+2*b*tan(1/2*f*x+1/2*e)+a)^2+1/2*(6*a^4*d^2-6*a^3*b*c*d+2*a^2*b^2*c^2-5*a^2*b^2*d^2+b^4*c^2
+2*b^4*d^2)/(a^4-2*a^2*b^2+b^4)/(a^2-b^2)^(1/2)*arctan(1/2*(2*a*tan(1/2*f*x+1/2*e)+2*b)/(a^2-b^2)^(1/2)))+2*d^
3/(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c^3)/(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)))

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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))^3/(c+d*sin(f*x+e)),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))^3/(c+d*sin(f*x+e)),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))**3/(c+d*sin(f*x+e)),x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 788 vs. \(2 (277) = 554\).
time = 0.53, size = 788, normalized size = 2.76 \begin {gather*} -\frac {\frac {2 \, {\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 )} d^{3}}{{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt {c^{2} - d^{2}}} - \frac {{\left (2 \, a^{2} b^{3} c^{2} + b^{5} c^{2} - 6 \, a^{3} b^{2} c d + 6 \, a^{4} b d^{2} - 5 \, a^{2} b^{3} d^{2} + 2 \, b^{5} d^{2}\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^{4} b^{3} c^{3} - 2 \, a^{2} b^{5} c^{3} + b^{7} c^{3} - 3 \, a^{5} b^{2} c^{2} d + 6 \, a^{3} b^{4} c^{2} d - 3 \, a b^{6} c^{2} d + 3 \, a^{6} b c d^{2} - 6 \, a^{4} b^{3} c d^{2} + 3 \, a^{2} b^{5} c d^{2} - a^{7} d^{3} + 2 \, a^{5} b^{2} d^{3} - a^{3} b^{4} d^{3}\right )} \sqrt {a^{2} - b^{2}}} - \frac {5 \, a^{3} b^{4} c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 2 \, a b^{6} c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 7 \, a^{4} b^{3} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 4 \, a^{2} b^{5} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 4 \, a^{4} b^{3} c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 7 \, a^{2} b^{5} c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 2 \, b^{7} c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 6 \, a^{5} b^{2} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 9 \, a^{3} b^{4} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 6 \, a b^{6} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 11 \, a^{3} b^{4} c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 2 \, a b^{6} c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 17 \, a^{4} b^{3} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 8 \, a^{2} b^{5} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 4 \, a^{4} b^{3} c - a^{2} b^{5} c - 6 \, a^{5} b^{2} d + 3 \, a^{3} b^{4} d}{{\left (a^{6} b^{2} c^{2} - 2 \, a^{4} b^{4} c^{2} + a^{2} b^{6} c^{2} - 2 \, a^{7} b c d + 4 \, a^{5} b^{3} c d - 2 \, a^{3} b^{5} c d + a^{8} d^{2} - 2 \, a^{6} b^{2} d^{2} + a^{4} b^{4} d^{2}\right )} {\left (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 )}^{2}}}{f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*sin(f*x+e))^3/(c+d*sin(f*x+e)),x, algorithm="giac")

[Out]

-(2*(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)))*d^3/((b^3
*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*sqrt(c^2 - d^2)) - (2*a^2*b^3*c^2 + b^5*c^2 - 6*a^3*b^2*c*d +
6*a^4*b*d^2 - 5*a^2*b^3*d^2 + 2*b^5*d^2)*(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^4*b^3*c^3 - 2*a^2*b^5*c^3 + b^7*c^3 - 3*a^5*b^2*c^2*d + 6*a^3*b^4*c^2*d - 3*a*
b^6*c^2*d + 3*a^6*b*c*d^2 - 6*a^4*b^3*c*d^2 + 3*a^2*b^5*c*d^2 - a^7*d^3 + 2*a^5*b^2*d^3 - a^3*b^4*d^3)*sqrt(a^
2 - b^2)) - (5*a^3*b^4*c*tan(1/2*f*x + 1/2*e)^3 - 2*a*b^6*c*tan(1/2*f*x + 1/2*e)^3 - 7*a^4*b^3*d*tan(1/2*f*x +
1/2*e)^3 + 4*a^2*b^5*d*tan(1/2*f*x + 1/2*e)^3 + 4*a^4*b^3*c*tan(1/2*f*x + 1/2*e)^2 + 7*a^2*b^5*c*tan(1/2*f*x
+ 1/2*e)^2 - 2*b^7*c*tan(1/2*f*x + 1/2*e)^2 - 6*a^5*b^2*d*tan(1/2*f*x + 1/2*e)^2 - 9*a^3*b^4*d*tan(1/2*f*x + 1
/2*e)^2 + 6*a*b^6*d*tan(1/2*f*x + 1/2*e)^2 + 11*a^3*b^4*c*tan(1/2*f*x + 1/2*e) - 2*a*b^6*c*tan(1/2*f*x + 1/2*e
) - 17*a^4*b^3*d*tan(1/2*f*x + 1/2*e) + 8*a^2*b^5*d*tan(1/2*f*x + 1/2*e) + 4*a^4*b^3*c - a^2*b^5*c - 6*a^5*b^2
*d + 3*a^3*b^4*d)/((a^6*b^2*c^2 - 2*a^4*b^4*c^2 + a^2*b^6*c^2 - 2*a^7*b*c*d + 4*a^5*b^3*c*d - 2*a^3*b^5*c*d +
a^8*d^2 - 2*a^6*b^2*d^2 + a^4*b^4*d^2)*(a*tan(1/2*f*x + 1/2*e)^2 + 2*b*tan(1/2*f*x + 1/2*e) + a)^2))/f

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Mupad [B]
time = 30.17, size = 2500, normalized size = 8.77 \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))^3*(c + d*sin(e + f*x))),x)

[Out]

(d^3*atan(((d^3*(d^2 - c^2)^(1/2)*((8*(4*a*b^12*c^4*d^5 + 4*a*b^12*c^6*d^3 + 4*a^3*b^10*c^8*d + 4*a^4*b^9*c*d^
8 + 4*a^5*b^8*c^8*d - 16*a^6*b^7*c*d^8 + 24*a^8*b^5*c*d^8 - 16*a^10*b^3*c*d^8 - 4*a^2*b^11*c^3*d^6 - 8*a^2*b^1
1*c^5*d^4 - 2*a^2*b^11*c^7*d^2 - 4*a^3*b^10*c^2*d^7 - 16*a^3*b^10*c^4*d^5 - a^3*b^10*c^6*d^3 + 24*a^4*b^9*c^3*
d^6 - 20*a^4*b^9*c^5*d^4 - 20*a^4*b^9*c^7*d^2 + 12*a^5*b^8*c^2*d^7 + 95*a^5*b^8*c^4*d^5 + 20*a^5*b^8*c^6*d^3 -
98*a^6*b^7*c^3*d^6 + 64*a^6*b^7*c^5*d^4 - 32*a^6*b^7*c^7*d^2 + a^7*b^6*c^2*d^7 - 188*a^7*b^6*c^4*d^5 + 112*a^
7*b^6*c^6*d^3 + 164*a^8*b^5*c^3*d^6 - 216*a^8*b^5*c^5*d^4 - 28*a^9*b^4*c^2*d^7 + 240*a^9*b^4*c^4*d^5 - 140*a^1
0*b^3*c^3*d^6 + 28*a^11*b^2*c^2*d^7 + a*b^12*c^8*d + 4*a^12*b*c*d^8))/(a^14*d^6 + b^14*c^6 - 4*a^2*b^12*c^6 +
6*a^4*b^10*c^6 - 4*a^6*b^8*c^6 + a^8*b^6*c^6 + a^6*b^8*d^6 - 4*a^8*b^6*d^6 + 6*a^10*b^4*d^6 - 4*a^12*b^2*d^6 +
24*a^3*b^11*c^5*d - 6*a^5*b^9*c*d^5 - 36*a^5*b^9*c^5*d + 24*a^7*b^7*c*d^5 + 24*a^7*b^7*c^5*d - 36*a^9*b^5*c*d
^5 - 6*a^9*b^5*c^5*d + 24*a^11*b^3*c*d^5 + 15*a^2*b^12*c^4*d^2 - 20*a^3*b^11*c^3*d^3 + 15*a^4*b^10*c^2*d^4 - 6
0*a^4*b^10*c^4*d^2 + 80*a^5*b^9*c^3*d^3 - 60*a^6*b^8*c^2*d^4 + 90*a^6*b^8*c^4*d^2 - 120*a^7*b^7*c^3*d^3 + 90*a
^8*b^6*c^2*d^4 - 60*a^8*b^6*c^4*d^2 + 80*a^9*b^5*c^3*d^3 - 60*a^10*b^4*c^2*d^4 + 15*a^10*b^4*c^4*d^2 - 20*a^11
*b^3*c^3*d^3 + 15*a^12*b^2*c^2*d^4 - 6*a*b^13*c^5*d - 6*a^13*b*c*d^5) - (8*tan(e/2 + (f*x)/2)*(a*b^12*c^9 + 4*
a^13*c*d^8 + 4*a^3*b^10*c^9 + 4*a^5*b^8*c^9 - 16*a*b^12*c^3*d^6 - 4*a*b^12*c^5*d^4 + 2*a*b^12*c^7*d^2 - 2*a^2*
b^11*c^8*d - 16*a^3*b^10*c*d^8 - 20*a^4*b^9*c^8*d + 76*a^5*b^8*c*d^8 - 32*a^6*b^7*c^8*d - 162*a^7*b^6*c*d^8 +
176*a^9*b^4*c*d^8 - 96*a^11*b^2*c*d^8 - 8*a^12*b*c^2*d^7 + 32*a^2*b^11*c^2*d^7 + 8*a^2*b^11*c^4*d^5 - 4*a^2*b^
11*c^6*d^3 + 72*a^3*b^10*c^3*d^6 - 14*a^3*b^10*c^5*d^4 - 9*a^3*b^10*c^7*d^2 - 152*a^4*b^9*c^2*d^7 + 80*a^4*b^9
*c^4*d^5 + 20*a^4*b^9*c^6*d^3 - 274*a^5*b^8*c^3*d^6 + 55*a^5*b^8*c^5*d^4 + 12*a^5*b^8*c^7*d^2 + 372*a^6*b^7*c^
2*d^7 - 250*a^6*b^7*c^4*d^5 + 128*a^6*b^7*c^6*d^3 + 481*a^7*b^6*c^3*d^6 - 412*a^7*b^6*c^5*d^4 + 112*a^7*b^6*c^
7*d^2 - 472*a^8*b^5*c^2*d^7 + 612*a^8*b^5*c^4*d^5 - 216*a^8*b^5*c^6*d^3 - 564*a^9*b^4*c^3*d^6 + 240*a^9*b^4*c^
5*d^4 + 336*a^10*b^3*c^2*d^7 - 144*a^10*b^3*c^4*d^5 + 40*a^11*b^2*c^3*d^6))/(a^14*d^6 + b^14*c^6 - 4*a^2*b^12*
c^6 + 6*a^4*b^10*c^6 - 4*a^6*b^8*c^6 + a^8*b^6*c^6 + a^6*b^8*d^6 - 4*a^8*b^6*d^6 + 6*a^10*b^4*d^6 - 4*a^12*b^2
*d^6 + 24*a^3*b^11*c^5*d - 6*a^5*b^9*c*d^5 - 36*a^5*b^9*c^5*d + 24*a^7*b^7*c*d^5 + 24*a^7*b^7*c^5*d - 36*a^9*b
^5*c*d^5 - 6*a^9*b^5*c^5*d + 24*a^11*b^3*c*d^5 + 15*a^2*b^12*c^4*d^2 - 20*a^3*b^11*c^3*d^3 + 15*a^4*b^10*c^2*d
^4 - 60*a^4*b^10*c^4*d^2 + 80*a^5*b^9*c^3*d^3 - 60*a^6*b^8*c^2*d^4 + 90*a^6*b^8*c^4*d^2 - 120*a^7*b^7*c^3*d^3
+ 90*a^8*b^6*c^2*d^4 - 60*a^8*b^6*c^4*d^2 + 80*a^9*b^5*c^3*d^3 - 60*a^10*b^4*c^2*d^4 + 15*a^10*b^4*c^4*d^2 - 2
0*a^11*b^3*c^3*d^3 + 15*a^12*b^2*c^2*d^4 - 6*a*b^13*c^5*d - 6*a^13*b*c*d^5) + (d^3*(d^2 - c^2)^(1/2)*((8*(2*a^
2*b^14*c^10 - 6*a^6*b^10*c^10 + 4*a^8*b^8*c^10 + 4*a^16*c^2*d^8 + 4*a*b^15*c^7*d^3 - 10*a^3*b^13*c^9*d - 12*a^
5*b^11*c^9*d + 4*a^7*b^9*c*d^9 + 54*a^7*b^9*c^9*d - 18*a^9*b^7*c*d^9 - 32*a^9*b^7*c^9*d + 36*a^11*b^5*c*d^9 -
34*a^13*b^3*c*d^9 - 32*a^15*b*c^3*d^7 - 24*a^2*b^14*c^6*d^4 + 2*a^2*b^14*c^8*d^2 + 60*a^3*b^13*c^5*d^5 - 30*a^
3*b^13*c^7*d^3 - 80*a^4*b^12*c^4*d^6 + 138*a^4*b^12*c^6*d^4 + 2*a^4*b^12*c^8*d^2 + 60*a^5*b^11*c^3*d^7 - 310*a
^5*b^11*c^5*d^5 + 122*a^5*b^11*c^7*d^3 - 24*a^6*b^10*c^2*d^8 + 390*a^6*b^10*c^4*d^6 - 466*a^6*b^10*c^6*d^4 + 1
02*a^6*b^10*c^8*d^2 - 282*a^7*b^9*c^3*d^7 + 878*a^7*b^9*c^5*d^5 - 394*a^7*b^9*c^7*d^3 + 110*a^8*b^8*c^2*d^8 -
970*a^8*b^8*c^4*d^6 + 894*a^8*b^8*c^6*d^4 - 218*a^8*b^8*c^8*d^2 + 638*a^9*b^7*c^3*d^7 - 1290*a^9*b^7*c^5*d^5 +
522*a^9*b^7*c^7*d^3 - 232*a^10*b^6*c^2*d^8 + 1202*a^10*b^6*c^4*d^6 - 822*a^10*b^6*c^6*d^4 + 112*a^10*b^6*c^8*
d^2 - 702*a^11*b^5*c^3*d^7 + 886*a^11*b^5*c^5*d^5 - 224*a^11*b^5*c^7*d^3 + 234*a^12*b^4*c^2*d^8 - 654*a^12*b^4
*c^4*d^6 + 280*a^12*b^4*c^6*d^4 + 318*a^13*b^3*c^3*d^7 - 224*a^13*b^3*c^5*d^5 - 92*a^14*b^2*c^2*d^8 + 112*a^14
*b^2*c^4*d^6 + 12*a^15*b*c*d^9))/(a^14*d^6 + b^14*c^6 - 4*a^2*b^12*c^6 + 6*a^4*b^10*c^6 - 4*a^6*b^8*c^6 + a^8*
b^6*c^6 + a^6*b^8*d^6 - 4*a^8*b^6*d^6 + 6*a^10*b^4*d^6 - 4*a^12*b^2*d^6 + 24*a^3*b^11*c^5*d - 6*a^5*b^9*c*d^5
- 36*a^5*b^9*c^5*d + 24*a^7*b^7*c*d^5 + 24*a^7*b^7*c^5*d - 36*a^9*b^5*c*d^5 - 6*a^9*b^5*c^5*d + 24*a^11*b^3*c*
d^5 + 15*a^2*b^12*c^4*d^2 - 20*a^3*b^11*c^3*d^3 + 15*a^4*b^10*c^2*d^4 - 60*a^4*b^10*c^4*d^2 + 80*a^5*b^9*c^3*d
^3 - 60*a^6*b^8*c^2*d^4 + 90*a^6*b^8*c^4*d^2 - 120*a^7*b^7*c^3*d^3 + 90*a^8*b^6*c^2*d^4 - 60*a^8*b^6*c^4*d^2 +
80*a^9*b^5*c^3*d^3 - 60*a^10*b^4*c^2*d^4 + 15*a^10*b^4*c^4*d^2 - 20*a^11*b^3*c^3*d^3 + 15*a^12*b^2*c^2*d^4 -
6*a*b^13*c^5*d - 6*a^13*b*c*d^5) + (8*tan(e/2 + (f*x)/2)*(4*a*b^15*c^10 + 8*a^16*c*d^9 - 12*a^5*b^11*c^10 + 8*
a^7*b^9*c^10 + 4*a*b^15*c^8*d^2 - 20*a^2*b^14*c^9*d - 24*a^4*b^12*c^9*d + 108*a^6*b^10*c^9*d + 4*a^8*b^8*c*d^9
- 64*a^8*b^8*c^9*d - 8*a^10*b^6*c*d^9 + 12*a^1...

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