3.8.16 \(\int \frac {(c+d \sin (e+f x))^3}{(a+b \sin (e+f x))^3} \, dx\) [716]

Optimal. Leaf size=248 \[ \frac {d^3 x}{b^3}+\frac {(b c-a d) \left (2 a^3 b c d-8 a b^3 c d+2 a^4 d^2+a^2 b^2 \left (2 c^2-5 d^2\right )+b^4 \left (c^2+6 d^2\right )\right ) \tan ^{-1}\left (\frac {b+a \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {a^2-b^2}}\right )}{b^3 \left (a^2-b^2\right )^{5/2} f}+\frac {(b c-a d)^2 \left (3 a b c+2 a^2 d-5 b^2 d\right ) \cos (e+f x)}{2 b^2 \left (a^2-b^2\right )^2 f (a+b \sin (e+f x))}+\frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))}{2 b \left (a^2-b^2\right ) f (a+b \sin (e+f x))^2} \]

[Out]

d^3*x/b^3+(-a*d+b*c)*(2*a^3*b*c*d-8*a*b^3*c*d+2*a^4*d^2+a^2*b^2*(2*c^2-5*d^2)+b^4*(c^2+6*d^2))*arctan((b+a*tan
(1/2*f*x+1/2*e))/(a^2-b^2)^(1/2))/b^3/(a^2-b^2)^(5/2)/f+1/2*(-a*d+b*c)^2*(2*a^2*d+3*a*b*c-5*b^2*d)*cos(f*x+e)/
b^2/(a^2-b^2)^2/f/(a+b*sin(f*x+e))+1/2*(-a*d+b*c)^2*cos(f*x+e)*(c+d*sin(f*x+e))/b/(a^2-b^2)/f/(a+b*sin(f*x+e))
^2

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Rubi [A]
time = 0.54, antiderivative size = 248, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {2871, 3100, 2814, 2739, 632, 210} \begin {gather*} \frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))}{2 b f \left (a^2-b^2\right ) (a+b \sin (e+f x))^2}+\frac {(b c-a d)^2 \left (2 a^2 d+3 a b c-5 b^2 d\right ) \cos (e+f x)}{2 b^2 f \left (a^2-b^2\right )^2 (a+b \sin (e+f x))}+\frac {(b c-a d) \left (2 a^4 d^2+2 a^3 b c d+a^2 b^2 \left (2 c^2-5 d^2\right )-8 a b^3 c d+b^4 \left (c^2+6 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 )}{b^3 f \left (a^2-b^2\right )^{5/2}}+\frac {d^3 x}{b^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(d^3*x)/b^3 + ((b*c - a*d)*(2*a^3*b*c*d - 8*a*b^3*c*d + 2*a^4*d^2 + a^2*b^2*(2*c^2 - 5*d^2) + b^4*(c^2 + 6*d^2
))*ArcTan[(b + a*Tan[(e + f*x)/2])/Sqrt[a^2 - b^2]])/(b^3*(a^2 - b^2)^(5/2)*f) + ((b*c - a*d)^2*(3*a*b*c + 2*a
^2*d - 5*b^2*d)*Cos[e + f*x])/(2*b^2*(a^2 - b^2)^2*f*(a + b*Sin[e + f*x])) + ((b*c - a*d)^2*Cos[e + f*x]*(c +
d*Sin[e + f*x]))/(2*b*(a^2 - b^2)*f*(a + b*Sin[e + f*x])^2)

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 2871

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

Rule 3100

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f
_.)*(x_)]^2), x_Symbol] :> Simp[(-(A*b^2 - a*b*B + a^2*C))*Cos[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m
+ 1)*(a^2 - b^2))), x] + Dist[1/(b*(m + 1)*(a^2 - b^2)), Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[b*(a*A - b*B +
a*C)*(m + 1) - (A*b^2 - a*b*B + a^2*C + b*(A*b - a*B + b*C)*(m + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b,
e, f, A, B, C}, x] && LtQ[m, -1] && NeQ[a^2 - b^2, 0]

Rubi steps

\begin {align*} \int \frac {(c+d \sin (e+f x))^3}{(a+b \sin (e+f x))^3} \, dx &=\frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))}{2 b \left (a^2-b^2\right ) f (a+b \sin (e+f x))^2}-\frac {\int \frac {5 b^2 c^2 d+a^2 d^3-2 a b c \left (c^2+2 d^2\right )-\left (a^2 c d^2+2 a b d \left (2 c^2+d^2\right )-b^2 \left (c^3+6 c d^2\right )\right ) \sin (e+f x)-2 \left (a^2-b^2\right ) d^3 \sin ^2(e+f x)}{(a+b \sin (e+f x))^2} \, dx}{2 b \left (a^2-b^2\right )}\\ &=\frac {(b c-a d)^2 \left (3 a b c+2 a^2 d-5 b^2 d\right ) \cos (e+f x)}{2 b^2 \left (a^2-b^2\right )^2 f (a+b \sin (e+f x))}+\frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))}{2 b \left (a^2-b^2\right ) f (a+b \sin (e+f x))^2}+\frac {\int \frac {b \left (a^3 d^3+a^2 b c \left (2 c^2+3 d^2\right )-a b^2 d \left (9 c^2+4 d^2\right )+b^3 c \left (c^2+6 d^2\right )\right )+2 \left (a^2-b^2\right )^2 d^3 \sin (e+f x)}{a+b \sin (e+f x)} \, dx}{2 b^2 \left (a^2-b^2\right )^2}\\ &=\frac {d^3 x}{b^3}+\frac {(b c-a d)^2 \left (3 a b c+2 a^2 d-5 b^2 d\right ) \cos (e+f x)}{2 b^2 \left (a^2-b^2\right )^2 f (a+b \sin (e+f x))}+\frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))}{2 b \left (a^2-b^2\right ) f (a+b \sin (e+f x))^2}-\frac {\left (2 a^5 d^3-5 a^3 b^2 d^3+3 a b^4 d \left (3 c^2+2 d^2\right )-a^2 b^3 c \left (2 c^2+3 d^2\right )-b^5 c \left (c^2+6 d^2\right )\right ) \int \frac {1}{a+b \sin (e+f x)} \, dx}{2 b^3 \left (a^2-b^2\right )^2}\\ &=\frac {d^3 x}{b^3}+\frac {(b c-a d)^2 \left (3 a b c+2 a^2 d-5 b^2 d\right ) \cos (e+f x)}{2 b^2 \left (a^2-b^2\right )^2 f (a+b \sin (e+f x))}+\frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))}{2 b \left (a^2-b^2\right ) f (a+b \sin (e+f x))^2}-\frac {\left (2 a^5 d^3-5 a^3 b^2 d^3+3 a b^4 d \left (3 c^2+2 d^2\right )-a^2 b^3 c \left (2 c^2+3 d^2\right )-b^5 c \left (c^2+6 d^2\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 )}{b^3 \left (a^2-b^2\right )^2 f}\\ &=\frac {d^3 x}{b^3}+\frac {(b c-a d)^2 \left (3 a b c+2 a^2 d-5 b^2 d\right ) \cos (e+f x)}{2 b^2 \left (a^2-b^2\right )^2 f (a+b \sin (e+f x))}+\frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))}{2 b \left (a^2-b^2\right ) f (a+b \sin (e+f x))^2}+\frac {\left (2 \left (2 a^5 d^3-5 a^3 b^2 d^3+3 a b^4 d \left (3 c^2+2 d^2\right )-a^2 b^3 c \left (2 c^2+3 d^2\right )-b^5 c \left (c^2+6 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 )}{b^3 \left (a^2-b^2\right )^2 f}\\ &=\frac {d^3 x}{b^3}+\frac {(b c-a d) \left (2 a^2 b^2 c^2+b^4 c^2+2 a^3 b c d-8 a b^3 c d+2 a^4 d^2-5 a^2 b^2 d^2+6 b^4 d^2\right ) \tan ^{-1}\left (\frac {b+a \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {a^2-b^2}}\right )}{b^3 \left (a^2-b^2\right )^{5/2} f}+\frac {(b c-a d)^2 \left (3 a b c+2 a^2 d-5 b^2 d\right ) \cos (e+f x)}{2 b^2 \left (a^2-b^2\right )^2 f (a+b \sin (e+f x))}+\frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))}{2 b \left (a^2-b^2\right ) f (a+b \sin (e+f x))^2}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(524\) vs. \(2(248)=496\).
time = 2.52, size = 524, normalized size = 2.11 \begin {gather*} \frac {-\frac {4 \left (2 a^5 d^3-5 a^3 b^2 d^3+3 a b^4 d \left (3 c^2+2 d^2\right )-a^2 b^3 c \left (2 c^2+3 d^2\right )-b^5 c \left (c^2+6 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}}+\frac {4 a^6 d^3 e-6 a^4 b^2 d^3 e+2 b^6 d^3 e+4 a^6 d^3 f x-6 a^4 b^2 d^3 f x+2 b^6 d^3 f x-2 b (b c-a d)^2 \left (-4 a^2 b c+b^3 c-2 a^3 d+5 a b^2 d\right ) \cos (e+f x)-2 \left (-a^2 b+b^3\right )^2 d^3 (e+f x) \cos (2 (e+f x))+8 a^5 b d^3 e \sin (e+f x)-16 a^3 b^3 d^3 e \sin (e+f x)+8 a b^5 d^3 e \sin (e+f x)+8 a^5 b d^3 f x \sin (e+f x)-16 a^3 b^3 d^3 f x \sin (e+f x)+8 a b^5 d^3 f x \sin (e+f x)+3 a b^5 c^3 \sin (2 (e+f x))-3 a^2 b^4 c^2 d \sin (2 (e+f x))-6 b^6 c^2 d \sin (2 (e+f x))-3 a^3 b^3 c d^2 \sin (2 (e+f x))+12 a b^5 c d^2 \sin (2 (e+f x))+3 a^4 b^2 d^3 \sin (2 (e+f x))-6 a^2 b^4 d^3 \sin (2 (e+f x))}{\left (a^2-b^2\right )^2 (a+b \sin (e+f x))^2}}{4 b^3 f} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-4*(2*a^5*d^3 - 5*a^3*b^2*d^3 + 3*a*b^4*d*(3*c^2 + 2*d^2) - a^2*b^3*c*(2*c^2 + 3*d^2) - b^5*c*(c^2 + 6*d^2))
*ArcTan[(b + a*Tan[(e + f*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(5/2) + (4*a^6*d^3*e - 6*a^4*b^2*d^3*e + 2*b^6*
d^3*e + 4*a^6*d^3*f*x - 6*a^4*b^2*d^3*f*x + 2*b^6*d^3*f*x - 2*b*(b*c - a*d)^2*(-4*a^2*b*c + b^3*c - 2*a^3*d +
5*a*b^2*d)*Cos[e + f*x] - 2*(-(a^2*b) + b^3)^2*d^3*(e + f*x)*Cos[2*(e + f*x)] + 8*a^5*b*d^3*e*Sin[e + f*x] - 1
6*a^3*b^3*d^3*e*Sin[e + f*x] + 8*a*b^5*d^3*e*Sin[e + f*x] + 8*a^5*b*d^3*f*x*Sin[e + f*x] - 16*a^3*b^3*d^3*f*x*
Sin[e + f*x] + 8*a*b^5*d^3*f*x*Sin[e + f*x] + 3*a*b^5*c^3*Sin[2*(e + f*x)] - 3*a^2*b^4*c^2*d*Sin[2*(e + f*x)]
- 6*b^6*c^2*d*Sin[2*(e + f*x)] - 3*a^3*b^3*c*d^2*Sin[2*(e + f*x)] + 12*a*b^5*c*d^2*Sin[2*(e + f*x)] + 3*a^4*b^
2*d^3*Sin[2*(e + f*x)] - 6*a^2*b^4*d^3*Sin[2*(e + f*x)])/((a^2 - b^2)^2*(a + b*Sin[e + f*x])^2))/(4*b^3*f)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(670\) vs. \(2(239)=478\).
time = 0.68, size = 671, normalized size = 2.71

method	result	size
derivativedivides	\(\frac {\frac {2 d^{3} \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{b^{3}}-\frac {2 \left (\frac {-\frac {b^{2} \left (a^{5} d^{3}+3 a^{4} b c \,d^{2}-9 a^{3} b^{2} c^{2} d -4 a^{3} b^{2} d^{3}+5 a^{2} b^{3} c^{3}+6 a^{2} b^{3} c \,d^{2}-2 b^{5} c^{3}\right ) \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) a}-\frac {b \left (2 a^{7} d^{3}-6 a^{5} b^{2} c^{2} d -a^{5} b^{2} d^{3}+4 a^{4} b^{3} c^{3}+9 a^{4} b^{3} c \,d^{2}-15 a^{3} b^{4} c^{2} d -10 a^{3} b^{4} d^{3}+7 a^{2} b^{5} c^{3}+18 a^{2} b^{5} c \,d^{2}-6 a \,b^{6} c^{2} d -2 b^{7} c^{3}\right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) a^{2}}-\frac {b^{2} \left (7 a^{5} d^{3}-3 a^{4} b c \,d^{2}-15 a^{3} b^{2} c^{2} d -16 a^{3} b^{2} d^{3}+11 a^{2} b^{3} c^{3}+30 a^{2} b^{3} c \,d^{2}-12 a \,b^{4} c^{2} d -2 b^{5} c^{3}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{2 a \left (a^{4}-2 a^{2} b^{2}+b^{4}\right )}-\frac {b \left (2 a^{5} d^{3}-6 a^{3} b^{2} c^{2} d -5 a^{3} b^{2} d^{3}+4 a^{2} b^{3} c^{3}+9 a^{2} b^{3} c \,d^{2}-3 a \,b^{4} c^{2} d -b^{5} c^{3}\right )}{2 \left (a^{4}-2 a^{2} b^{2}+b^{4}\right )}}{\left (a \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+2 b \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+a \right )^{2}}+\frac {\left (2 a^{5} d^{3}-5 a^{3} b^{2} d^{3}-2 a^{2} b^{3} c^{3}-3 a^{2} b^{3} c \,d^{2}+9 a \,b^{4} c^{2} d +6 a \,b^{4} d^{3}-b^{5} c^{3}-6 b^{5} c \,d^{2}\right ) \arctan \left (\frac {2 a \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{2 \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \sqrt {a^{2}-b^{2}}}\right )}{b^{3}}}{f}\)	\(671\)
default	\(\frac {\frac {2 d^{3} \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{b^{3}}-\frac {2 \left (\frac {-\frac {b^{2} \left (a^{5} d^{3}+3 a^{4} b c \,d^{2}-9 a^{3} b^{2} c^{2} d -4 a^{3} b^{2} d^{3}+5 a^{2} b^{3} c^{3}+6 a^{2} b^{3} c \,d^{2}-2 b^{5} c^{3}\right ) \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) a}-\frac {b \left (2 a^{7} d^{3}-6 a^{5} b^{2} c^{2} d -a^{5} b^{2} d^{3}+4 a^{4} b^{3} c^{3}+9 a^{4} b^{3} c \,d^{2}-15 a^{3} b^{4} c^{2} d -10 a^{3} b^{4} d^{3}+7 a^{2} b^{5} c^{3}+18 a^{2} b^{5} c \,d^{2}-6 a \,b^{6} c^{2} d -2 b^{7} c^{3}\right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) a^{2}}-\frac {b^{2} \left (7 a^{5} d^{3}-3 a^{4} b c \,d^{2}-15 a^{3} b^{2} c^{2} d -16 a^{3} b^{2} d^{3}+11 a^{2} b^{3} c^{3}+30 a^{2} b^{3} c \,d^{2}-12 a \,b^{4} c^{2} d -2 b^{5} c^{3}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{2 a \left (a^{4}-2 a^{2} b^{2}+b^{4}\right )}-\frac {b \left (2 a^{5} d^{3}-6 a^{3} b^{2} c^{2} d -5 a^{3} b^{2} d^{3}+4 a^{2} b^{3} c^{3}+9 a^{2} b^{3} c \,d^{2}-3 a \,b^{4} c^{2} d -b^{5} c^{3}\right )}{2 \left (a^{4}-2 a^{2} b^{2}+b^{4}\right )}}{\left (a \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+2 b \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+a \right )^{2}}+\frac {\left (2 a^{5} d^{3}-5 a^{3} b^{2} d^{3}-2 a^{2} b^{3} c^{3}-3 a^{2} b^{3} c \,d^{2}+9 a \,b^{4} c^{2} d +6 a \,b^{4} d^{3}-b^{5} c^{3}-6 b^{5} c \,d^{2}\right ) \arctan \left (\frac {2 a \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{2 \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \sqrt {a^{2}-b^{2}}}\right )}{b^{3}}}{f}\)	\(671\)
risch	\(\text {Expression too large to display}\)	\(2015\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/f*(2*d^3/b^3*arctan(tan(1/2*f*x+1/2*e))-2/b^3*((-1/2*b^2*(a^5*d^3+3*a^4*b*c*d^2-9*a^3*b^2*c^2*d-4*a^3*b^2*d^
3+5*a^2*b^3*c^3+6*a^2*b^3*c*d^2-2*b^5*c^3)/(a^4-2*a^2*b^2+b^4)/a*tan(1/2*f*x+1/2*e)^3-1/2*b*(2*a^7*d^3-6*a^5*b
^2*c^2*d-a^5*b^2*d^3+4*a^4*b^3*c^3+9*a^4*b^3*c*d^2-15*a^3*b^4*c^2*d-10*a^3*b^4*d^3+7*a^2*b^5*c^3+18*a^2*b^5*c*
d^2-6*a*b^6*c^2*d-2*b^7*c^3)/(a^4-2*a^2*b^2+b^4)/a^2*tan(1/2*f*x+1/2*e)^2-1/2*b^2*(7*a^5*d^3-3*a^4*b*c*d^2-15*
a^3*b^2*c^2*d-16*a^3*b^2*d^3+11*a^2*b^3*c^3+30*a^2*b^3*c*d^2-12*a*b^4*c^2*d-2*b^5*c^3)/a/(a^4-2*a^2*b^2+b^4)*t
an(1/2*f*x+1/2*e)-1/2*b*(2*a^5*d^3-6*a^3*b^2*c^2*d-5*a^3*b^2*d^3+4*a^2*b^3*c^3+9*a^2*b^3*c*d^2-3*a*b^4*c^2*d-b
^5*c^3)/(a^4-2*a^2*b^2+b^4))/(a*tan(1/2*f*x+1/2*e)^2+2*b*tan(1/2*f*x+1/2*e)+a)^2+1/2*(2*a^5*d^3-5*a^3*b^2*d^3-
2*a^2*b^3*c^3-3*a^2*b^3*c*d^2+9*a*b^4*c^2*d+6*a*b^4*d^3-b^5*c^3-6*b^5*c*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))))

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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((c+d*sin(f*x+e))^3/(a+b*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 [B] Leaf count of result is larger than twice the leaf count of optimal. 783 vs. \(2 (245) = 490\).
time = 0.43, size = 1656, normalized size = 6.68 \begin {gather*} \left [\frac {4 \, {\left (a^{6} b^{2} - 3 \, a^{4} b^{4} + 3 \, a^{2} b^{6} - b^{8}\right )} d^{3} f x \cos \left (f x + e\right )^{2} - 4 \, {\left (a^{8} - 2 \, a^{6} b^{2} + 2 \, a^{2} b^{6} - b^{8}\right )} d^{3} f x - {\left ({\left (2 \, a^{4} b^{3} + 3 \, a^{2} b^{5} + b^{7}\right )} c^{3} - 9 \, {\left (a^{3} b^{4} + a b^{6}\right )} c^{2} d + 3 \, {\left (a^{4} b^{3} + 3 \, a^{2} b^{5} + 2 \, b^{7}\right )} c d^{2} - {\left (2 \, a^{7} - 3 \, a^{5} b^{2} + a^{3} b^{4} + 6 \, a b^{6}\right )} d^{3} + {\left (9 \, a b^{6} c^{2} d - {\left (2 \, a^{2} b^{5} + b^{7}\right )} c^{3} - 3 \, {\left (a^{2} b^{5} + 2 \, b^{7}\right )} c d^{2} + {\left (2 \, a^{5} b^{2} - 5 \, a^{3} b^{4} + 6 \, a b^{6}\right )} d^{3}\right )} \cos \left (f x + e\right )^{2} - 2 \, {\left (9 \, a^{2} b^{5} c^{2} d - {\left (2 \, a^{3} b^{4} + a b^{6}\right )} c^{3} - 3 \, {\left (a^{3} b^{4} + 2 \, a b^{6}\right )} c d^{2} + {\left (2 \, a^{6} b - 5 \, a^{4} b^{3} + 6 \, a^{2} b^{5}\right )} d^{3}\right )} \sin \left (f x + e\right )\right )} \sqrt {-a^{2} + b^{2}} \log \left (-\frac {{\left (2 \, a^{2} - b^{2}\right )} \cos \left (f x + e\right )^{2} - 2 \, a b \sin \left (f x + e\right ) - a^{2} - b^{2} - 2 \, {\left (a \cos \left (f x + e\right ) \sin \left (f x + e\right ) + b \cos \left (f x + e\right )\right )} \sqrt {-a^{2} + b^{2}}}{b^{2} \cos \left (f x + e\right )^{2} - 2 \, a b \sin \left (f x + e\right ) - a^{2} - b^{2}}\right ) - 2 \, {\left ({\left (4 \, a^{4} b^{4} - 5 \, a^{2} b^{6} + b^{8}\right )} c^{3} - 3 \, {\left (2 \, a^{5} b^{3} - a^{3} b^{5} - a b^{7}\right )} c^{2} d + 9 \, {\left (a^{4} b^{4} - a^{2} b^{6}\right )} c d^{2} + {\left (2 \, a^{7} b - 7 \, a^{5} b^{3} + 5 \, a^{3} b^{5}\right )} d^{3}\right )} \cos \left (f x + e\right ) - 2 \, {\left (4 \, {\left (a^{7} b - 3 \, a^{5} b^{3} + 3 \, a^{3} b^{5} - a b^{7}\right )} d^{3} f x + 3 \, {\left ({\left (a^{3} b^{5} - a b^{7}\right )} c^{3} - {\left (a^{4} b^{4} + a^{2} b^{6} - 2 \, b^{8}\right )} c^{2} d - {\left (a^{5} b^{3} - 5 \, a^{3} b^{5} + 4 \, a b^{7}\right )} c d^{2} + {\left (a^{6} b^{2} - 3 \, a^{4} b^{4} + 2 \, a^{2} b^{6}\right )} d^{3}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{4 \, {\left ({\left (a^{6} b^{5} - 3 \, a^{4} b^{7} + 3 \, a^{2} b^{9} - b^{11}\right )} f \cos \left (f x + e\right )^{2} - 2 \, {\left (a^{7} b^{4} - 3 \, a^{5} b^{6} + 3 \, a^{3} b^{8} - a b^{10}\right )} f \sin \left (f x + e\right ) - {\left (a^{8} b^{3} - 2 \, a^{6} b^{5} + 2 \, a^{2} b^{9} - b^{11}\right )} f\right )}}, \frac {2 \, {\left (a^{6} b^{2} - 3 \, a^{4} b^{4} + 3 \, a^{2} b^{6} - b^{8}\right )} d^{3} f x \cos \left (f x + e\right )^{2} - 2 \, {\left (a^{8} - 2 \, a^{6} b^{2} + 2 \, a^{2} b^{6} - b^{8}\right )} d^{3} f x + {\left ({\left (2 \, a^{4} b^{3} + 3 \, a^{2} b^{5} + b^{7}\right )} c^{3} - 9 \, {\left (a^{3} b^{4} + a b^{6}\right )} c^{2} d + 3 \, {\left (a^{4} b^{3} + 3 \, a^{2} b^{5} + 2 \, b^{7}\right )} c d^{2} - {\left (2 \, a^{7} - 3 \, a^{5} b^{2} + a^{3} b^{4} + 6 \, a b^{6}\right )} d^{3} + {\left (9 \, a b^{6} c^{2} d - {\left (2 \, a^{2} b^{5} + b^{7}\right )} c^{3} - 3 \, {\left (a^{2} b^{5} + 2 \, b^{7}\right )} c d^{2} + {\left (2 \, a^{5} b^{2} - 5 \, a^{3} b^{4} + 6 \, a b^{6}\right )} d^{3}\right )} \cos \left (f x + e\right )^{2} - 2 \, {\left (9 \, a^{2} b^{5} c^{2} d - {\left (2 \, a^{3} b^{4} + a b^{6}\right )} c^{3} - 3 \, {\left (a^{3} b^{4} + 2 \, a b^{6}\right )} c d^{2} + {\left (2 \, a^{6} b - 5 \, a^{4} b^{3} + 6 \, a^{2} b^{5}\right )} d^{3}\right )} \sin \left (f x + e\right )\right )} \sqrt {a^{2} - b^{2}} \arctan \left (-\frac {a \sin \left (f x + e\right ) + b}{\sqrt {a^{2} - b^{2}} \cos \left (f x + e\right )}\right ) - {\left ({\left (4 \, a^{4} b^{4} - 5 \, a^{2} b^{6} + b^{8}\right )} c^{3} - 3 \, {\left (2 \, a^{5} b^{3} - a^{3} b^{5} - a b^{7}\right )} c^{2} d + 9 \, {\left (a^{4} b^{4} - a^{2} b^{6}\right )} c d^{2} + {\left (2 \, a^{7} b - 7 \, a^{5} b^{3} + 5 \, a^{3} b^{5}\right )} d^{3}\right )} \cos \left (f x + e\right ) - {\left (4 \, {\left (a^{7} b - 3 \, a^{5} b^{3} + 3 \, a^{3} b^{5} - a b^{7}\right )} d^{3} f x + 3 \, {\left ({\left (a^{3} b^{5} - a b^{7}\right )} c^{3} - {\left (a^{4} b^{4} + a^{2} b^{6} - 2 \, b^{8}\right )} c^{2} d - {\left (a^{5} b^{3} - 5 \, a^{3} b^{5} + 4 \, a b^{7}\right )} c d^{2} + {\left (a^{6} b^{2} - 3 \, a^{4} b^{4} + 2 \, a^{2} b^{6}\right )} d^{3}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{2 \, {\left ({\left (a^{6} b^{5} - 3 \, a^{4} b^{7} + 3 \, a^{2} b^{9} - b^{11}\right )} f \cos \left (f x + e\right )^{2} - 2 \, {\left (a^{7} b^{4} - 3 \, a^{5} b^{6} + 3 \, a^{3} b^{8} - a b^{10}\right )} f \sin \left (f x + e\right ) - {\left (a^{8} b^{3} - 2 \, a^{6} b^{5} + 2 \, a^{2} b^{9} - b^{11}\right )} f\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/4*(4*(a^6*b^2 - 3*a^4*b^4 + 3*a^2*b^6 - b^8)*d^3*f*x*cos(f*x + e)^2 - 4*(a^8 - 2*a^6*b^2 + 2*a^2*b^6 - b^8)
*d^3*f*x - ((2*a^4*b^3 + 3*a^2*b^5 + b^7)*c^3 - 9*(a^3*b^4 + a*b^6)*c^2*d + 3*(a^4*b^3 + 3*a^2*b^5 + 2*b^7)*c*
d^2 - (2*a^7 - 3*a^5*b^2 + a^3*b^4 + 6*a*b^6)*d^3 + (9*a*b^6*c^2*d - (2*a^2*b^5 + b^7)*c^3 - 3*(a^2*b^5 + 2*b^
7)*c*d^2 + (2*a^5*b^2 - 5*a^3*b^4 + 6*a*b^6)*d^3)*cos(f*x + e)^2 - 2*(9*a^2*b^5*c^2*d - (2*a^3*b^4 + a*b^6)*c^
3 - 3*(a^3*b^4 + 2*a*b^6)*c*d^2 + (2*a^6*b - 5*a^4*b^3 + 6*a^2*b^5)*d^3)*sin(f*x + e))*sqrt(-a^2 + b^2)*log(-(
(2*a^2 - b^2)*cos(f*x + e)^2 - 2*a*b*sin(f*x + e) - a^2 - b^2 - 2*(a*cos(f*x + e)*sin(f*x + e) + b*cos(f*x + e
))*sqrt(-a^2 + b^2))/(b^2*cos(f*x + e)^2 - 2*a*b*sin(f*x + e) - a^2 - b^2)) - 2*((4*a^4*b^4 - 5*a^2*b^6 + b^8)
*c^3 - 3*(2*a^5*b^3 - a^3*b^5 - a*b^7)*c^2*d + 9*(a^4*b^4 - a^2*b^6)*c*d^2 + (2*a^7*b - 7*a^5*b^3 + 5*a^3*b^5)
*d^3)*cos(f*x + e) - 2*(4*(a^7*b - 3*a^5*b^3 + 3*a^3*b^5 - a*b^7)*d^3*f*x + 3*((a^3*b^5 - a*b^7)*c^3 - (a^4*b^
4 + a^2*b^6 - 2*b^8)*c^2*d - (a^5*b^3 - 5*a^3*b^5 + 4*a*b^7)*c*d^2 + (a^6*b^2 - 3*a^4*b^4 + 2*a^2*b^6)*d^3)*co
s(f*x + e))*sin(f*x + e))/((a^6*b^5 - 3*a^4*b^7 + 3*a^2*b^9 - b^11)*f*cos(f*x + e)^2 - 2*(a^7*b^4 - 3*a^5*b^6
+ 3*a^3*b^8 - a*b^10)*f*sin(f*x + e) - (a^8*b^3 - 2*a^6*b^5 + 2*a^2*b^9 - b^11)*f), 1/2*(2*(a^6*b^2 - 3*a^4*b^
4 + 3*a^2*b^6 - b^8)*d^3*f*x*cos(f*x + e)^2 - 2*(a^8 - 2*a^6*b^2 + 2*a^2*b^6 - b^8)*d^3*f*x + ((2*a^4*b^3 + 3*
a^2*b^5 + b^7)*c^3 - 9*(a^3*b^4 + a*b^6)*c^2*d + 3*(a^4*b^3 + 3*a^2*b^5 + 2*b^7)*c*d^2 - (2*a^7 - 3*a^5*b^2 +
a^3*b^4 + 6*a*b^6)*d^3 + (9*a*b^6*c^2*d - (2*a^2*b^5 + b^7)*c^3 - 3*(a^2*b^5 + 2*b^7)*c*d^2 + (2*a^5*b^2 - 5*a
^3*b^4 + 6*a*b^6)*d^3)*cos(f*x + e)^2 - 2*(9*a^2*b^5*c^2*d - (2*a^3*b^4 + a*b^6)*c^3 - 3*(a^3*b^4 + 2*a*b^6)*c
*d^2 + (2*a^6*b - 5*a^4*b^3 + 6*a^2*b^5)*d^3)*sin(f*x + e))*sqrt(a^2 - b^2)*arctan(-(a*sin(f*x + e) + b)/(sqrt
(a^2 - b^2)*cos(f*x + e))) - ((4*a^4*b^4 - 5*a^2*b^6 + b^8)*c^3 - 3*(2*a^5*b^3 - a^3*b^5 - a*b^7)*c^2*d + 9*(a
^4*b^4 - a^2*b^6)*c*d^2 + (2*a^7*b - 7*a^5*b^3 + 5*a^3*b^5)*d^3)*cos(f*x + e) - (4*(a^7*b - 3*a^5*b^3 + 3*a^3*
b^5 - a*b^7)*d^3*f*x + 3*((a^3*b^5 - a*b^7)*c^3 - (a^4*b^4 + a^2*b^6 - 2*b^8)*c^2*d - (a^5*b^3 - 5*a^3*b^5 + 4
*a*b^7)*c*d^2 + (a^6*b^2 - 3*a^4*b^4 + 2*a^2*b^6)*d^3)*cos(f*x + e))*sin(f*x + e))/((a^6*b^5 - 3*a^4*b^7 + 3*a
^2*b^9 - b^11)*f*cos(f*x + e)^2 - 2*(a^7*b^4 - 3*a^5*b^6 + 3*a^3*b^8 - a*b^10)*f*sin(f*x + e) - (a^8*b^3 - 2*a
^6*b^5 + 2*a^2*b^9 - b^11)*f)]

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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((c+d*sin(f*x+e))**3/(a+b*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. 887 vs. \(2 (245) = 490\).
time = 0.52, size = 887, normalized size = 3.58 \begin {gather*} \frac {\frac {{\left (f x + e\right )} d^{3}}{b^{3}} + \frac {{\left (2 \, a^{2} b^{3} c^{3} + b^{5} c^{3} - 9 \, a b^{4} c^{2} d + 3 \, a^{2} b^{3} c d^{2} + 6 \, b^{5} c d^{2} - 2 \, a^{5} d^{3} + 5 \, a^{3} b^{2} d^{3} - 6 \, a b^{4} d^{3}\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} - 2 \, a^{2} b^{5} + b^{7}\right )} \sqrt {a^{2} - b^{2}}} + \frac {5 \, a^{3} b^{4} c^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 2 \, a b^{6} c^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 9 \, a^{4} b^{3} c^{2} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 3 \, a^{5} b^{2} c d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 6 \, a^{3} b^{4} c d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + a^{6} b d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 4 \, a^{4} b^{3} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 4 \, a^{4} b^{3} c^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 7 \, a^{2} b^{5} c^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 2 \, b^{7} c^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 6 \, a^{5} b^{2} c^{2} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 15 \, a^{3} b^{4} c^{2} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 6 \, a b^{6} c^{2} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 9 \, a^{4} b^{3} c d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 18 \, a^{2} b^{5} c d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 2 \, a^{7} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - a^{5} b^{2} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 10 \, a^{3} b^{4} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 11 \, a^{3} b^{4} c^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 2 \, a b^{6} c^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 15 \, a^{4} b^{3} c^{2} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 12 \, a^{2} b^{5} c^{2} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 3 \, a^{5} b^{2} c d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 30 \, a^{3} b^{4} c d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 7 \, a^{6} b d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 16 \, a^{4} b^{3} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 4 \, a^{4} b^{3} c^{3} - a^{2} b^{5} c^{3} - 6 \, a^{5} b^{2} c^{2} d - 3 \, a^{3} b^{4} c^{2} d + 9 \, a^{4} b^{3} c d^{2} + 2 \, a^{7} d^{3} - 5 \, a^{5} b^{2} d^{3}}{{\left (a^{6} b^{2} - 2 \, a^{4} b^{4} + a^{2} b^{6}\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((c+d*sin(f*x+e))^3/(a+b*sin(f*x+e))^3,x, algorithm="giac")

[Out]

((f*x + e)*d^3/b^3 + (2*a^2*b^3*c^3 + b^5*c^3 - 9*a*b^4*c^2*d + 3*a^2*b^3*c*d^2 + 6*b^5*c*d^2 - 2*a^5*d^3 + 5*
a^3*b^2*d^3 - 6*a*b^4*d^3)*(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 - 2*a^2*b^5 + b^7)*sqrt(a^2 - b^2)) + (5*a^3*b^4*c^3*tan(1/2*f*x + 1/2*e)^3 - 2*a*b^6*
c^3*tan(1/2*f*x + 1/2*e)^3 - 9*a^4*b^3*c^2*d*tan(1/2*f*x + 1/2*e)^3 + 3*a^5*b^2*c*d^2*tan(1/2*f*x + 1/2*e)^3 +
6*a^3*b^4*c*d^2*tan(1/2*f*x + 1/2*e)^3 + a^6*b*d^3*tan(1/2*f*x + 1/2*e)^3 - 4*a^4*b^3*d^3*tan(1/2*f*x + 1/2*e
)^3 + 4*a^4*b^3*c^3*tan(1/2*f*x + 1/2*e)^2 + 7*a^2*b^5*c^3*tan(1/2*f*x + 1/2*e)^2 - 2*b^7*c^3*tan(1/2*f*x + 1/
2*e)^2 - 6*a^5*b^2*c^2*d*tan(1/2*f*x + 1/2*e)^2 - 15*a^3*b^4*c^2*d*tan(1/2*f*x + 1/2*e)^2 - 6*a*b^6*c^2*d*tan(
1/2*f*x + 1/2*e)^2 + 9*a^4*b^3*c*d^2*tan(1/2*f*x + 1/2*e)^2 + 18*a^2*b^5*c*d^2*tan(1/2*f*x + 1/2*e)^2 + 2*a^7*
d^3*tan(1/2*f*x + 1/2*e)^2 - a^5*b^2*d^3*tan(1/2*f*x + 1/2*e)^2 - 10*a^3*b^4*d^3*tan(1/2*f*x + 1/2*e)^2 + 11*a
^3*b^4*c^3*tan(1/2*f*x + 1/2*e) - 2*a*b^6*c^3*tan(1/2*f*x + 1/2*e) - 15*a^4*b^3*c^2*d*tan(1/2*f*x + 1/2*e) - 1
2*a^2*b^5*c^2*d*tan(1/2*f*x + 1/2*e) - 3*a^5*b^2*c*d^2*tan(1/2*f*x + 1/2*e) + 30*a^3*b^4*c*d^2*tan(1/2*f*x + 1
/2*e) + 7*a^6*b*d^3*tan(1/2*f*x + 1/2*e) - 16*a^4*b^3*d^3*tan(1/2*f*x + 1/2*e) + 4*a^4*b^3*c^3 - a^2*b^5*c^3 -
6*a^5*b^2*c^2*d - 3*a^3*b^4*c^2*d + 9*a^4*b^3*c*d^2 + 2*a^7*d^3 - 5*a^5*b^2*d^3)/((a^6*b^2 - 2*a^4*b^4 + a^2*
b^6)*(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 = 20.93, size = 2500, normalized size = 10.08 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

- ((b^5*c^3 - 2*a^5*d^3 - 4*a^2*b^3*c^3 + 5*a^3*b^2*d^3 - 9*a^2*b^3*c*d^2 + 6*a^3*b^2*c^2*d + 3*a*b^4*c^2*d)/(
b^2*(a^4 + b^4 - 2*a^2*b^2)) - (tan(e/2 + (f*x)/2)^3*(a^5*d^3 - 2*b^5*c^3 + 5*a^2*b^3*c^3 - 4*a^3*b^2*d^3 + 6*
a^2*b^3*c*d^2 - 9*a^3*b^2*c^2*d + 3*a^4*b*c*d^2))/(a*b*(a^4 + b^4 - 2*a^2*b^2)) + (tan(e/2 + (f*x)/2)*(2*b^5*c
^3 - 7*a^5*d^3 - 11*a^2*b^3*c^3 + 16*a^3*b^2*d^3 - 30*a^2*b^3*c*d^2 + 15*a^3*b^2*c^2*d + 12*a*b^4*c^2*d + 3*a^
4*b*c*d^2))/(a*b*(a^4 + b^4 - 2*a^2*b^2)) + (tan(e/2 + (f*x)/2)^2*(a^2 + 2*b^2)*(b^5*c^3 - 2*a^5*d^3 - 4*a^2*b
^3*c^3 + 5*a^3*b^2*d^3 - 9*a^2*b^3*c*d^2 + 6*a^3*b^2*c^2*d + 3*a*b^4*c^2*d))/(a^2*b^2*(a^4 + b^4 - 2*a^2*b^2))
)/(f*(tan(e/2 + (f*x)/2)^2*(2*a^2 + 4*b^2) + a^2*tan(e/2 + (f*x)/2)^4 + a^2 + 4*a*b*tan(e/2 + (f*x)/2)^3 + 4*a
*b*tan(e/2 + (f*x)/2))) - (2*d^3*atan(((d^3*((8*(4*a^2*b^10*d^6 - 16*a^4*b^8*d^6 + 24*a^6*b^6*d^6 - 16*a^8*b^4
*d^6 + 4*a^10*b^2*d^6))/(b^13 - 4*a^2*b^11 + 6*a^4*b^9 - 4*a^6*b^7 + a^8*b^5) + (d^3*((8*tan(e/2 + (f*x)/2)*(4
*a*b^15*c^3 - 12*a^5*b^11*c^3 + 8*a^7*b^9*c^3 - 24*a^2*b^14*d^3 + 68*a^4*b^12*d^3 - 72*a^6*b^10*d^3 + 36*a^8*b
^8*d^3 - 8*a^10*b^6*d^3 - 36*a^2*b^14*c^2*d - 36*a^3*b^13*c*d^2 + 72*a^4*b^12*c^2*d - 36*a^6*b^10*c^2*d + 12*a
^7*b^9*c*d^2 + 24*a*b^15*c*d^2))/(b^14 - 4*a^2*b^12 + 6*a^4*b^10 - 4*a^6*b^8 + a^8*b^6) - (8*(4*a*b^14*d^3 - 2
*a^2*b^13*c^3 + 6*a^6*b^9*c^3 - 4*a^8*b^7*c^3 - 8*a^3*b^12*d^3 + 6*a^5*b^10*d^3 - 4*a^7*b^8*d^3 + 2*a^9*b^6*d^
3 - 12*a^2*b^13*c*d^2 + 18*a^3*b^12*c^2*d + 18*a^4*b^11*c*d^2 - 36*a^5*b^10*c^2*d + 18*a^7*b^8*c^2*d - 6*a^8*b
^7*c*d^2))/(b^13 - 4*a^2*b^11 + 6*a^4*b^9 - 4*a^6*b^7 + a^8*b^5) + (d^3*((8*(4*a^2*b^16 - 16*a^4*b^14 + 24*a^6
*b^12 - 16*a^8*b^10 + 4*a^10*b^8))/(b^13 - 4*a^2*b^11 + 6*a^4*b^9 - 4*a^6*b^7 + a^8*b^5) + (8*tan(e/2 + (f*x)/
2)*(12*a*b^18 - 56*a^3*b^16 + 104*a^5*b^14 - 96*a^7*b^12 + 44*a^9*b^10 - 8*a^11*b^8))/(b^14 - 4*a^2*b^12 + 6*a
^4*b^10 - 4*a^6*b^8 + a^8*b^6))*1i)/b^3)*1i)/b^3 - (8*tan(e/2 + (f*x)/2)*(a*b^12*c^6 - 8*a*b^12*d^6 + 4*a^3*b^
10*c^6 + 4*a^5*b^8*c^6 + 72*a^3*b^10*d^6 - 124*a^5*b^8*d^6 + 105*a^7*b^6*d^6 - 44*a^9*b^4*d^6 + 8*a^11*b^2*d^6
+ 36*a*b^12*c^2*d^4 + 12*a*b^12*c^4*d^2 - 72*a^2*b^11*c*d^5 - 18*a^2*b^11*c^5*d + 24*a^4*b^9*c*d^5 - 36*a^4*b
^9*c^5*d + 6*a^6*b^7*c*d^5 - 12*a^8*b^5*c*d^5 - 120*a^2*b^11*c^3*d^3 + 144*a^3*b^10*c^2*d^4 + 111*a^3*b^10*c^4
*d^2 - 68*a^4*b^9*c^3*d^3 - 81*a^5*b^8*c^2*d^4 + 12*a^5*b^8*c^4*d^2 + 16*a^6*b^7*c^3*d^3 + 36*a^7*b^6*c^2*d^4
- 8*a^8*b^5*c^3*d^3))/(b^14 - 4*a^2*b^12 + 6*a^4*b^10 - 4*a^6*b^8 + a^8*b^6)))/b^3 + (d^3*((8*(4*a^2*b^10*d^6
- 16*a^4*b^8*d^6 + 24*a^6*b^6*d^6 - 16*a^8*b^4*d^6 + 4*a^10*b^2*d^6))/(b^13 - 4*a^2*b^11 + 6*a^4*b^9 - 4*a^6*b
^7 + a^8*b^5) + (d^3*((8*(4*a*b^14*d^3 - 2*a^2*b^13*c^3 + 6*a^6*b^9*c^3 - 4*a^8*b^7*c^3 - 8*a^3*b^12*d^3 + 6*a
^5*b^10*d^3 - 4*a^7*b^8*d^3 + 2*a^9*b^6*d^3 - 12*a^2*b^13*c*d^2 + 18*a^3*b^12*c^2*d + 18*a^4*b^11*c*d^2 - 36*a
^5*b^10*c^2*d + 18*a^7*b^8*c^2*d - 6*a^8*b^7*c*d^2))/(b^13 - 4*a^2*b^11 + 6*a^4*b^9 - 4*a^6*b^7 + a^8*b^5) - (
8*tan(e/2 + (f*x)/2)*(4*a*b^15*c^3 - 12*a^5*b^11*c^3 + 8*a^7*b^9*c^3 - 24*a^2*b^14*d^3 + 68*a^4*b^12*d^3 - 72*
a^6*b^10*d^3 + 36*a^8*b^8*d^3 - 8*a^10*b^6*d^3 - 36*a^2*b^14*c^2*d - 36*a^3*b^13*c*d^2 + 72*a^4*b^12*c^2*d - 3
6*a^6*b^10*c^2*d + 12*a^7*b^9*c*d^2 + 24*a*b^15*c*d^2))/(b^14 - 4*a^2*b^12 + 6*a^4*b^10 - 4*a^6*b^8 + a^8*b^6)
+ (d^3*((8*(4*a^2*b^16 - 16*a^4*b^14 + 24*a^6*b^12 - 16*a^8*b^10 + 4*a^10*b^8))/(b^13 - 4*a^2*b^11 + 6*a^4*b^
9 - 4*a^6*b^7 + a^8*b^5) + (8*tan(e/2 + (f*x)/2)*(12*a*b^18 - 56*a^3*b^16 + 104*a^5*b^14 - 96*a^7*b^12 + 44*a^
9*b^10 - 8*a^11*b^8))/(b^14 - 4*a^2*b^12 + 6*a^4*b^10 - 4*a^6*b^8 + a^8*b^6))*1i)/b^3)*1i)/b^3 - (8*tan(e/2 +
(f*x)/2)*(a*b^12*c^6 - 8*a*b^12*d^6 + 4*a^3*b^10*c^6 + 4*a^5*b^8*c^6 + 72*a^3*b^10*d^6 - 124*a^5*b^8*d^6 + 105
*a^7*b^6*d^6 - 44*a^9*b^4*d^6 + 8*a^11*b^2*d^6 + 36*a*b^12*c^2*d^4 + 12*a*b^12*c^4*d^2 - 72*a^2*b^11*c*d^5 - 1
8*a^2*b^11*c^5*d + 24*a^4*b^9*c*d^5 - 36*a^4*b^9*c^5*d + 6*a^6*b^7*c*d^5 - 12*a^8*b^5*c*d^5 - 120*a^2*b^11*c^3
*d^3 + 144*a^3*b^10*c^2*d^4 + 111*a^3*b^10*c^4*d^2 - 68*a^4*b^9*c^3*d^3 - 81*a^5*b^8*c^2*d^4 + 12*a^5*b^8*c^4*
d^2 + 16*a^6*b^7*c^3*d^3 + 36*a^7*b^6*c^2*d^4 - 8*a^8*b^5*c^3*d^3))/(b^14 - 4*a^2*b^12 + 6*a^4*b^10 - 4*a^6*b^
8 + a^8*b^6)))/b^3)/((16*(24*a^3*b^6*d^9 - 2*a^9*d^9 - 26*a^5*b^4*d^9 + 13*a^7*b^2*d^9 + 36*a*b^8*c^2*d^7 + 12
*a*b^8*c^4*d^5 + a*b^8*c^6*d^3 - 60*a^2*b^7*c*d^8 + 6*a^4*b^5*c*d^8 + 6*a^6*b^3*c*d^8 - 4*a^8*b*c^3*d^6 - 118*
a^2*b^7*c^3*d^6 - 18*a^2*b^7*c^5*d^4 + 126*a^3*b^6*c^2*d^7 + 111*a^3*b^6*c^4*d^5 + 4*a^3*b^6*c^6*d^3 - 68*a^4*
b^5*c^3*d^6 - 36*a^4*b^5*c^5*d^4 - 45*a^5*b^4*c^2*d^7 + 12*a^5*b^4*c^4*d^5 + 4*a^5*b^4*c^6*d^3 + 10*a^6*b^3*c^
3*d^6 + 18*a^7*b^2*c^2*d^7 - 6*a^8*b*c*d^8))/(b^13 - 4*a^2*b^11 + 6*a^4*b^9 - 4*a^6*b^7 + a^8*b^5) - (16*tan(e
/2 + (f*x)/2)*(8*a^10*d^9 + 24*a^2*b^8*d^9 - 68*a^4*b^6*d^9 + 72*a^6*b^4*d^9 - 36*a^8*b^2*d^9 - 4*a*b^9*c^3*d^
6 + 36*a^3*b^7*c*d^8 - 12*a^7*b^3*c*d^8 + 36*a^2*b^8*c^2*d^7 - 72*a^4*b^6*c^2*d^7 + 12*a^5*b^5*c^3*d^6 + 36*a^
6*b^4*c^2*d^7 - 8*a^7*b^3*c^3*d^6 - 24*a*b^9*c*...

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