3.8.5 \(\int \frac {1}{(a+b \sin (e+f x)) (c+d \sin (e+f x))^3} \, dx\) [705]

Optimal. Leaf size=284 \[ \frac {2 b^3 \tan ^{-1}\left (\frac {b+a \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2} (b c-a d)^3 f}+\frac {d \left (6 a b c^3 d-a^2 d^2 \left (2 c^2+d^2\right )-b^2 \left (6 c^4-5 c^2 d^2+2 d^4\right )\right ) \tan ^{-1}\left (\frac {d+c \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c^2-d^2}}\right )}{(b c-a d)^3 \left (c^2-d^2\right )^{5/2} f}-\frac {d^2 \cos (e+f x)}{2 (b c-a d) \left (c^2-d^2\right ) f (c+d \sin (e+f x))^2}-\frac {d^2 \left (5 b c^2-3 a c d-2 b d^2\right ) \cos (e+f x)}{2 (b c-a d)^2 \left (c^2-d^2\right )^2 f (c+d \sin (e+f x))} \]

[Out]

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

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Rubi [A]
time = 0.73, antiderivative size = 284, 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 {d \left (-a^2 d^2 \left (2 c^2+d^2\right )+6 a b c^3 d-b^2 \left (6 c^4-5 c^2 d^2+2 d^4\right )\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 )^{5/2} (b c-a d)^3}+\frac {2 b^3 \text {ArcTan}\left (\frac {a \tan \left (\frac {1}{2} (e+f x)\right )+b}{\sqrt {a^2-b^2}}\right )}{f \sqrt {a^2-b^2} (b c-a d)^3}-\frac {d^2 \left (-3 a c d+5 b c^2-2 b d^2\right ) \cos (e+f x)}{2 f \left (c^2-d^2\right )^2 (b c-a d)^2 (c+d \sin (e+f x))}-\frac {d^2 \cos (e+f x)}{2 f \left (c^2-d^2\right ) (b c-a d) (c+d \sin (e+f x))^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [A]
time = 2.38, size = 263, normalized size = 0.93 \begin {gather*} \frac {\frac {4 b^3 \tan ^{-1}\left (\frac {b+a \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2}}-\frac {2 d \left (-6 a b c^3 d+a^2 d^2 \left (2 c^2+d^2\right )+b^2 \left (6 c^4-5 c^2 d^2+2 d^4\right )\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 )^{5/2}}-\frac {d^2 (b c-a d)^2 \cos (e+f x)}{(c-d) (c+d) (c+d \sin (e+f x))^2}+\frac {d^2 (b c-a d) \left (-5 b c^2+3 a c d+2 b d^2\right ) \cos (e+f x)}{(c-d)^2 (c+d)^2 (c+d \sin (e+f x))}}{2 (b c-a d)^3 f} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(624\) vs. \(2(270)=540\).
time = 3.65, size = 625, normalized size = 2.20

method	result	size
derivativedivides	\(\frac {\frac {2 d \left (\frac {\frac {d^{2} \left (5 a^{2} c^{2} d^{2}-2 a^{2} d^{4}-12 a b \,c^{3} d +6 a b c \,d^{3}+7 b^{2} c^{4}-4 b^{2} c^{2} d^{2}\right ) \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 \left (c^{4}-2 c^{2} d^{2}+d^{4}\right ) c}+\frac {d \left (4 a^{2} c^{4} d^{2}+7 a^{2} c^{2} d^{4}-2 a^{2} d^{6}-10 a b \,c^{5} d -16 a b \,c^{3} d^{3}+8 a b c \,d^{5}+6 b^{2} c^{6}+9 b^{2} c^{4} d^{2}-6 b^{2} c^{2} d^{4}\right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 \left (c^{4}-2 c^{2} d^{2}+d^{4}\right ) c^{2}}+\frac {d^{2} \left (11 a^{2} c^{2} d^{2}-2 a^{2} d^{4}-28 a b \,c^{3} d +10 a b c \,d^{3}+17 b^{2} c^{4}-8 b^{2} c^{2} d^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{2 \left (c^{4}-2 c^{2} d^{2}+d^{4}\right ) c}+\frac {d \left (4 a^{2} c^{2} d^{2}-a^{2} d^{4}-10 a b \,c^{3} d +4 a b c \,d^{3}+6 b^{2} c^{4}-3 b^{2} c^{2} d^{2}\right )}{2 c^{4}-4 c^{2} d^{2}+2 d^{4}}}{\left (c \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+2 d \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+c \right )^{2}}+\frac {\left (2 a^{2} c^{2} d^{2}+a^{2} d^{4}-6 a b \,c^{3} d +6 b^{2} c^{4}-5 b^{2} c^{2} d^{2}+2 b^{2} d^{4}\right ) \arctan \left (\frac {2 c \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+2 d}{2 \sqrt {c^{2}-d^{2}}}\right )}{2 \left (c^{4}-2 c^{2} d^{2}+d^{4}\right ) \sqrt {c^{2}-d^{2}}}\right )}{\left (a d -b c \right )^{3}}-\frac {2 b^{3} \arctan \left (\frac {2 a \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \sqrt {a^{2}-b^{2}}}}{f}\)	\(625\)
default	\(\frac {\frac {2 d \left (\frac {\frac {d^{2} \left (5 a^{2} c^{2} d^{2}-2 a^{2} d^{4}-12 a b \,c^{3} d +6 a b c \,d^{3}+7 b^{2} c^{4}-4 b^{2} c^{2} d^{2}\right ) \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 \left (c^{4}-2 c^{2} d^{2}+d^{4}\right ) c}+\frac {d \left (4 a^{2} c^{4} d^{2}+7 a^{2} c^{2} d^{4}-2 a^{2} d^{6}-10 a b \,c^{5} d -16 a b \,c^{3} d^{3}+8 a b c \,d^{5}+6 b^{2} c^{6}+9 b^{2} c^{4} d^{2}-6 b^{2} c^{2} d^{4}\right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 \left (c^{4}-2 c^{2} d^{2}+d^{4}\right ) c^{2}}+\frac {d^{2} \left (11 a^{2} c^{2} d^{2}-2 a^{2} d^{4}-28 a b \,c^{3} d +10 a b c \,d^{3}+17 b^{2} c^{4}-8 b^{2} c^{2} d^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{2 \left (c^{4}-2 c^{2} d^{2}+d^{4}\right ) c}+\frac {d \left (4 a^{2} c^{2} d^{2}-a^{2} d^{4}-10 a b \,c^{3} d +4 a b c \,d^{3}+6 b^{2} c^{4}-3 b^{2} c^{2} d^{2}\right )}{2 c^{4}-4 c^{2} d^{2}+2 d^{4}}}{\left (c \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+2 d \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+c \right )^{2}}+\frac {\left (2 a^{2} c^{2} d^{2}+a^{2} d^{4}-6 a b \,c^{3} d +6 b^{2} c^{4}-5 b^{2} c^{2} d^{2}+2 b^{2} d^{4}\right ) \arctan \left (\frac {2 c \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+2 d}{2 \sqrt {c^{2}-d^{2}}}\right )}{2 \left (c^{4}-2 c^{2} d^{2}+d^{4}\right ) \sqrt {c^{2}-d^{2}}}\right )}{\left (a d -b c \right )^{3}}-\frac {2 b^{3} \arctan \left (\frac {2 a \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \sqrt {a^{2}-b^{2}}}}{f}\)	\(625\)
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))/(c+d*sin(f*x+e))^3,x,method=_RETURNVERBOSE)

[Out]

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

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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))/(c+d*sin(f*x+e))^3,x, algorithm="maxima")

[Out]

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

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

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 786 vs. \(2 (276) = 552\).
time = 0.49, size = 786, normalized size = 2.77 \begin {gather*} \frac {\frac {2 \, {\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 )} b^{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 {a^{2} - b^{2}}} - \frac {{\left (6 \, b^{2} c^{4} d - 6 \, a b c^{3} d^{2} + 2 \, a^{2} c^{2} d^{3} - 5 \, b^{2} c^{2} d^{3} + a^{2} d^{5} + 2 \, b^{2} d^{5}\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^{7} - 3 \, a b^{2} c^{6} d + 3 \, a^{2} b c^{5} d^{2} - 2 \, b^{3} c^{5} d^{2} - a^{3} c^{4} d^{3} + 6 \, a b^{2} c^{4} d^{3} - 6 \, a^{2} b c^{3} d^{4} + b^{3} c^{3} d^{4} + 2 \, a^{3} c^{2} d^{5} - 3 \, a b^{2} c^{2} d^{5} + 3 \, a^{2} b c d^{6} - a^{3} d^{7}\right )} \sqrt {c^{2} - d^{2}}} - \frac {7 \, b c^{4} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 5 \, a c^{3} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 4 \, b c^{2} d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 2 \, a c d^{6} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 6 \, b c^{5} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 4 \, a c^{4} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 9 \, b c^{3} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 7 \, a c^{2} d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 6 \, b c d^{6} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 2 \, a d^{7} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 17 \, b c^{4} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 11 \, a c^{3} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 8 \, b c^{2} d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 2 \, a c d^{6} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 6 \, b c^{5} d^{2} - 4 \, a c^{4} d^{3} - 3 \, b c^{3} d^{4} + a c^{2} d^{5}}{{\left (b^{2} c^{8} - 2 \, a b c^{7} d + a^{2} c^{6} d^{2} - 2 \, b^{2} c^{6} d^{2} + 4 \, a b c^{5} d^{3} - 2 \, a^{2} c^{4} d^{4} + b^{2} c^{4} d^{4} - 2 \, a b c^{3} d^{5} + a^{2} c^{2} d^{6}\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 )}^{2}}}{f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 30.30, size = 2500, normalized size = 8.80 \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))*(c + d*sin(e + f*x))^3),x)

[Out]

(b^3*atan(((b^3*(b^2 - a^2)^(1/2)*((8*(4*a*b^8*c^4*d^9 - 16*a*b^8*c^6*d^7 + 24*a*b^8*c^8*d^5 - 16*a*b^8*c^10*d
^3 + 4*a^4*b^5*c*d^12 + 4*a^6*b^3*c*d^12 + 4*a^8*b*c^3*d^10 + 4*a^8*b*c^5*d^8 - 4*a^2*b^7*c^3*d^10 + 12*a^2*b^
7*c^5*d^8 + a^2*b^7*c^7*d^6 - 28*a^2*b^7*c^9*d^4 + 28*a^2*b^7*c^11*d^2 - 4*a^3*b^6*c^2*d^11 + 24*a^3*b^6*c^4*d
^9 - 98*a^3*b^6*c^6*d^7 + 164*a^3*b^6*c^8*d^5 - 140*a^3*b^6*c^10*d^3 - 16*a^4*b^5*c^3*d^10 + 95*a^4*b^5*c^5*d^
8 - 188*a^4*b^5*c^7*d^6 + 240*a^4*b^5*c^9*d^4 - 8*a^5*b^4*c^2*d^11 - 20*a^5*b^4*c^4*d^9 + 64*a^5*b^4*c^6*d^7 -
216*a^5*b^4*c^8*d^5 - a^6*b^3*c^3*d^10 + 20*a^6*b^3*c^5*d^8 + 112*a^6*b^3*c^7*d^6 - 2*a^7*b^2*c^2*d^11 - 20*a
^7*b^2*c^4*d^9 - 32*a^7*b^2*c^6*d^7 + 4*a*b^8*c^12*d + a^8*b*c*d^12))/(a^6*d^14 + b^6*c^14 - 4*a^6*c^2*d^12 +
6*a^6*c^4*d^10 - 4*a^6*c^6*d^8 + a^6*c^8*d^6 + b^6*c^6*d^8 - 4*b^6*c^8*d^6 + 6*b^6*c^10*d^4 - 4*b^6*c^12*d^2 -
6*a*b^5*c^5*d^9 + 24*a*b^5*c^7*d^7 - 36*a*b^5*c^9*d^5 + 24*a*b^5*c^11*d^3 + 24*a^5*b*c^3*d^11 - 36*a^5*b*c^5*
d^9 + 24*a^5*b*c^7*d^7 - 6*a^5*b*c^9*d^5 + 15*a^2*b^4*c^4*d^10 - 60*a^2*b^4*c^6*d^8 + 90*a^2*b^4*c^8*d^6 - 60*
a^2*b^4*c^10*d^4 + 15*a^2*b^4*c^12*d^2 - 20*a^3*b^3*c^3*d^11 + 80*a^3*b^3*c^5*d^9 - 120*a^3*b^3*c^7*d^7 + 80*a
^3*b^3*c^9*d^5 - 20*a^3*b^3*c^11*d^3 + 15*a^4*b^2*c^2*d^12 - 60*a^4*b^2*c^4*d^10 + 90*a^4*b^2*c^6*d^8 - 60*a^4
*b^2*c^8*d^6 + 15*a^4*b^2*c^10*d^4 - 6*a*b^5*c^13*d - 6*a^5*b*c*d^13) - (8*tan(e/2 + (f*x)/2)*(4*a*b^8*c^13 +
a^9*c*d^12 + 4*a^9*c^3*d^10 + 4*a^9*c^5*d^8 - 16*a*b^8*c^3*d^10 + 76*a*b^8*c^5*d^8 - 162*a*b^8*c^7*d^6 + 176*a
*b^8*c^9*d^4 - 96*a*b^8*c^11*d^2 - 8*a^2*b^7*c^12*d - 16*a^3*b^6*c*d^12 - 4*a^5*b^4*c*d^12 + 2*a^7*b^2*c*d^12
- 2*a^8*b*c^2*d^11 - 20*a^8*b*c^4*d^9 - 32*a^8*b*c^6*d^7 + 32*a^2*b^7*c^2*d^11 - 152*a^2*b^7*c^4*d^9 + 372*a^2
*b^7*c^6*d^7 - 472*a^2*b^7*c^8*d^5 + 336*a^2*b^7*c^10*d^3 + 72*a^3*b^6*c^3*d^10 - 274*a^3*b^6*c^5*d^8 + 481*a^
3*b^6*c^7*d^6 - 564*a^3*b^6*c^9*d^4 + 40*a^3*b^6*c^11*d^2 + 8*a^4*b^5*c^2*d^11 + 80*a^4*b^5*c^4*d^9 - 250*a^4*
b^5*c^6*d^7 + 612*a^4*b^5*c^8*d^5 - 144*a^4*b^5*c^10*d^3 - 14*a^5*b^4*c^3*d^10 + 55*a^5*b^4*c^5*d^8 - 412*a^5*
b^4*c^7*d^6 + 240*a^5*b^4*c^9*d^4 - 4*a^6*b^3*c^2*d^11 + 20*a^6*b^3*c^4*d^9 + 128*a^6*b^3*c^6*d^7 - 216*a^6*b^
3*c^8*d^5 - 9*a^7*b^2*c^3*d^10 + 12*a^7*b^2*c^5*d^8 + 112*a^7*b^2*c^7*d^6))/(a^6*d^14 + b^6*c^14 - 4*a^6*c^2*d
^12 + 6*a^6*c^4*d^10 - 4*a^6*c^6*d^8 + a^6*c^8*d^6 + b^6*c^6*d^8 - 4*b^6*c^8*d^6 + 6*b^6*c^10*d^4 - 4*b^6*c^12
*d^2 - 6*a*b^5*c^5*d^9 + 24*a*b^5*c^7*d^7 - 36*a*b^5*c^9*d^5 + 24*a*b^5*c^11*d^3 + 24*a^5*b*c^3*d^11 - 36*a^5*
b*c^5*d^9 + 24*a^5*b*c^7*d^7 - 6*a^5*b*c^9*d^5 + 15*a^2*b^4*c^4*d^10 - 60*a^2*b^4*c^6*d^8 + 90*a^2*b^4*c^8*d^6
- 60*a^2*b^4*c^10*d^4 + 15*a^2*b^4*c^12*d^2 - 20*a^3*b^3*c^3*d^11 + 80*a^3*b^3*c^5*d^9 - 120*a^3*b^3*c^7*d^7
+ 80*a^3*b^3*c^9*d^5 - 20*a^3*b^3*c^11*d^3 + 15*a^4*b^2*c^2*d^12 - 60*a^4*b^2*c^4*d^10 + 90*a^4*b^2*c^6*d^8 -
60*a^4*b^2*c^8*d^6 + 15*a^4*b^2*c^10*d^4 - 6*a*b^5*c^13*d - 6*a^5*b*c*d^13) + (b^3*(b^2 - a^2)^(1/2)*((8*(4*a^
2*b^8*c^16 + 2*a^10*c^2*d^14 - 6*a^10*c^6*d^10 + 4*a^10*c^8*d^8 + 4*a*b^9*c^7*d^9 - 18*a*b^9*c^9*d^7 + 36*a*b^
9*c^11*d^5 - 34*a*b^9*c^13*d^3 - 32*a^3*b^7*c^15*d + 4*a^7*b^3*c*d^15 - 10*a^9*b*c^3*d^13 - 12*a^9*b*c^5*d^11
+ 54*a^9*b*c^7*d^9 - 32*a^9*b*c^9*d^7 - 24*a^2*b^8*c^6*d^10 + 110*a^2*b^8*c^8*d^8 - 232*a^2*b^8*c^10*d^6 + 234
*a^2*b^8*c^12*d^4 - 92*a^2*b^8*c^14*d^2 + 60*a^3*b^7*c^5*d^11 - 282*a^3*b^7*c^7*d^9 + 638*a^3*b^7*c^9*d^7 - 70
2*a^3*b^7*c^11*d^5 + 318*a^3*b^7*c^13*d^3 - 80*a^4*b^6*c^4*d^12 + 390*a^4*b^6*c^6*d^10 - 970*a^4*b^6*c^8*d^8 +
1202*a^4*b^6*c^10*d^6 - 654*a^4*b^6*c^12*d^4 + 112*a^4*b^6*c^14*d^2 + 60*a^5*b^5*c^3*d^13 - 310*a^5*b^5*c^5*d
^11 + 878*a^5*b^5*c^7*d^9 - 1290*a^5*b^5*c^9*d^7 + 886*a^5*b^5*c^11*d^5 - 224*a^5*b^5*c^13*d^3 - 24*a^6*b^4*c^
2*d^14 + 138*a^6*b^4*c^4*d^12 - 466*a^6*b^4*c^6*d^10 + 894*a^6*b^4*c^8*d^8 - 822*a^6*b^4*c^10*d^6 + 280*a^6*b^
4*c^12*d^4 - 30*a^7*b^3*c^3*d^13 + 122*a^7*b^3*c^5*d^11 - 394*a^7*b^3*c^7*d^9 + 522*a^7*b^3*c^9*d^7 - 224*a^7*
b^3*c^11*d^5 + 2*a^8*b^2*c^2*d^14 + 2*a^8*b^2*c^4*d^12 + 102*a^8*b^2*c^6*d^10 - 218*a^8*b^2*c^8*d^8 + 112*a^8*
b^2*c^10*d^6 + 12*a*b^9*c^15*d))/(a^6*d^14 + b^6*c^14 - 4*a^6*c^2*d^12 + 6*a^6*c^4*d^10 - 4*a^6*c^6*d^8 + a^6*
c^8*d^6 + b^6*c^6*d^8 - 4*b^6*c^8*d^6 + 6*b^6*c^10*d^4 - 4*b^6*c^12*d^2 - 6*a*b^5*c^5*d^9 + 24*a*b^5*c^7*d^7 -
36*a*b^5*c^9*d^5 + 24*a*b^5*c^11*d^3 + 24*a^5*b*c^3*d^11 - 36*a^5*b*c^5*d^9 + 24*a^5*b*c^7*d^7 - 6*a^5*b*c^9*
d^5 + 15*a^2*b^4*c^4*d^10 - 60*a^2*b^4*c^6*d^8 + 90*a^2*b^4*c^8*d^6 - 60*a^2*b^4*c^10*d^4 + 15*a^2*b^4*c^12*d^
2 - 20*a^3*b^3*c^3*d^11 + 80*a^3*b^3*c^5*d^9 - 120*a^3*b^3*c^7*d^7 + 80*a^3*b^3*c^9*d^5 - 20*a^3*b^3*c^11*d^3
+ 15*a^4*b^2*c^2*d^12 - 60*a^4*b^2*c^4*d^10 + 90*a^4*b^2*c^6*d^8 - 60*a^4*b^2*c^8*d^6 + 15*a^4*b^2*c^10*d^4 -
6*a*b^5*c^13*d - 6*a^5*b*c*d^13) + (8*tan(e/2 + (f*x)/2)*(8*a*b^9*c^16 + 4*a^10*c*d^15 - 12*a^10*c^5*d^11 + 8*
a^10*c^7*d^9 + 4*a*b^9*c^8*d^8 - 8*a*b^9*c^10*d^6 + 12*a*b^9*c^12*d^4 - 16*a*b^9*c^14*d^2 - 40*a^2*b^8*c^15*d
+ 4*a^8*b^2*c*d^15 - 20*a^9*b*c^2*d^14 - 24*a^9...

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