3.7.92 \(\int \frac {(a+b \sin (e+f x))^3}{(c+d \sin (e+f x))^3} \, dx\) [692]

Optimal. Leaf size=255 \[ \frac {b^3 x}{d^3}-\frac {\left (9 a^2 b c d^4-a^3 d^3 \left (2 c^2+d^2\right )-3 a b^2 d^3 \left (c^2+2 d^2\right )+b^3 \left (2 c^5-5 c^3 d^2+6 c d^4\right )\right ) \tan ^{-1}\left (\frac {d+c \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c^2-d^2}}\right )}{d^3 \left (c^2-d^2\right )^{5/2} f}+\frac {(b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{2 d \left (c^2-d^2\right ) f (c+d \sin (e+f x))^2}+\frac {(b c-a d)^2 \left (2 b c^2+3 a c d-5 b d^2\right ) \cos (e+f x)}{2 d^2 \left (c^2-d^2\right )^2 f (c+d \sin (e+f x))} \]

[Out]

b^3*x/d^3-(9*a^2*b*c*d^4-a^3*d^3*(2*c^2+d^2)-3*a*b^2*d^3*(c^2+2*d^2)+b^3*(2*c^5-5*c^3*d^2+6*c*d^4))*arctan((d+
c*tan(1/2*f*x+1/2*e))/(c^2-d^2)^(1/2))/d^3/(c^2-d^2)^(5/2)/f+1/2*(-a*d+b*c)^2*cos(f*x+e)*(a+b*sin(f*x+e))/d/(c
^2-d^2)/f/(c+d*sin(f*x+e))^2+1/2*(-a*d+b*c)^2*(3*a*c*d+2*b*c^2-5*b*d^2)*cos(f*x+e)/d^2/(c^2-d^2)^2/f/(c+d*sin(
f*x+e))

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Rubi [A]
time = 0.46, antiderivative size = 255, 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 {\left (-a^3 d^3 \left (2 c^2+d^2\right )+9 a^2 b c d^4-3 a b^2 d^3 \left (c^2+2 d^2\right )+b^3 \left (2 c^5-5 c^3 d^2+6 c 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 )}{d^3 f \left (c^2-d^2\right )^{5/2}}+\frac {(b c-a d)^2 \left (3 a c d+2 b c^2-5 b d^2\right ) \cos (e+f x)}{2 d^2 f \left (c^2-d^2\right )^2 (c+d \sin (e+f x))}+\frac {(b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{2 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^2}+\frac {b^3 x}{d^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(b^3*x)/d^3 - ((9*a^2*b*c*d^4 - a^3*d^3*(2*c^2 + d^2) - 3*a*b^2*d^3*(c^2 + 2*d^2) + b^3*(2*c^5 - 5*c^3*d^2 + 6
*c*d^4))*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/(d^3*(c^2 - d^2)^(5/2)*f) + ((b*c - a*d)^2*Cos[e +
f*x]*(a + b*Sin[e + f*x]))/(2*d*(c^2 - d^2)*f*(c + d*Sin[e + f*x])^2) + ((b*c - a*d)^2*(2*b*c^2 + 3*a*c*d - 5*
b*d^2)*Cos[e + f*x])/(2*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 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 {(a+b \sin (e+f x))^3}{(c+d \sin (e+f x))^3} \, dx &=\frac {(b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{2 d \left (c^2-d^2\right ) f (c+d \sin (e+f x))^2}-\frac {\int \frac {b^3 c^2-2 a^3 c d-4 a b^2 c d+5 a^2 b d^2-\left (4 a^2 b c d+2 b^3 c d-a^3 d^2+a b^2 \left (c^2-6 d^2\right )\right ) \sin (e+f x)-2 b^3 \left (c^2-d^2\right ) \sin ^2(e+f x)}{(c+d \sin (e+f x))^2} \, dx}{2 d \left (c^2-d^2\right )}\\ &=\frac {(b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{2 d \left (c^2-d^2\right ) f (c+d \sin (e+f x))^2}+\frac {(b c-a d)^2 \left (2 b c^2+3 a c d-5 b d^2\right ) \cos (e+f x)}{2 d^2 \left (c^2-d^2\right )^2 f (c+d \sin (e+f x))}+\frac {\int \frac {-d \left (9 a^2 b c d^2-a^3 d \left (2 c^2+d^2\right )-3 a b^2 d \left (c^2+2 d^2\right )-b^3 \left (c^3-4 c d^2\right )\right )+2 b^3 \left (c^2-d^2\right )^2 \sin (e+f x)}{c+d \sin (e+f x)} \, dx}{2 d^2 \left (c^2-d^2\right )^2}\\ &=\frac {b^3 x}{d^3}+\frac {(b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{2 d \left (c^2-d^2\right ) f (c+d \sin (e+f x))^2}+\frac {(b c-a d)^2 \left (2 b c^2+3 a c d-5 b d^2\right ) \cos (e+f x)}{2 d^2 \left (c^2-d^2\right )^2 f (c+d \sin (e+f x))}-\frac {\left (9 a^2 b c d^4-a^3 d^3 \left (2 c^2+d^2\right )-3 a b^2 d^3 \left (c^2+2 d^2\right )+b^3 \left (2 c^5-5 c^3 d^2+6 c d^4\right )\right ) \int \frac {1}{c+d \sin (e+f x)} \, dx}{2 d^3 \left (c^2-d^2\right )^2}\\ &=\frac {b^3 x}{d^3}+\frac {(b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{2 d \left (c^2-d^2\right ) f (c+d \sin (e+f x))^2}+\frac {(b c-a d)^2 \left (2 b c^2+3 a c d-5 b d^2\right ) \cos (e+f x)}{2 d^2 \left (c^2-d^2\right )^2 f (c+d \sin (e+f x))}-\frac {\left (9 a^2 b c d^4-a^3 d^3 \left (2 c^2+d^2\right )-3 a b^2 d^3 \left (c^2+2 d^2\right )+b^3 \left (2 c^5-5 c^3 d^2+6 c d^4\right )\right ) \text {Subst}\left (\int \frac {1}{c+2 d x+c x^2} \, dx,x,\tan \left (\frac {1}{2} (e+f x)\right )\right )}{d^3 \left (c^2-d^2\right )^2 f}\\ &=\frac {b^3 x}{d^3}+\frac {(b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{2 d \left (c^2-d^2\right ) f (c+d \sin (e+f x))^2}+\frac {(b c-a d)^2 \left (2 b c^2+3 a c d-5 b d^2\right ) \cos (e+f x)}{2 d^2 \left (c^2-d^2\right )^2 f (c+d \sin (e+f x))}+\frac {\left (2 \left (9 a^2 b c d^4-a^3 d^3 \left (2 c^2+d^2\right )-3 a b^2 d^3 \left (c^2+2 d^2\right )+b^3 \left (2 c^5-5 c^3 d^2+6 c 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 )}{d^3 \left (c^2-d^2\right )^2 f}\\ &=\frac {b^3 x}{d^3}-\frac {\left (9 a^2 b c d^4-a^3 d^3 \left (2 c^2+d^2\right )-3 a b^2 d^3 \left (c^2+2 d^2\right )+b^3 \left (2 c^5-5 c^3 d^2+6 c 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 )}{d^3 \left (c^2-d^2\right )^{5/2} f}+\frac {(b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{2 d \left (c^2-d^2\right ) f (c+d \sin (e+f x))^2}+\frac {(b c-a d)^2 \left (2 b c^2+3 a c d-5 b d^2\right ) \cos (e+f x)}{2 d^2 \left (c^2-d^2\right )^2 f (c+d \sin (e+f x))}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(521\) vs. \(2(255)=510\).
time = 2.50, size = 521, normalized size = 2.04 \begin {gather*} \frac {-\frac {4 \left (9 a^2 b c d^4-a^3 d^3 \left (2 c^2+d^2\right )-3 a b^2 d^3 \left (c^2+2 d^2\right )+b^3 \left (2 c^5-5 c^3 d^2+6 c 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 {4 b^3 c^6 e-6 b^3 c^4 d^2 e+2 b^3 d^6 e+4 b^3 c^6 f x-6 b^3 c^4 d^2 f x+2 b^3 d^6 f x-2 d (b c-a d)^2 \left (-2 b c^3-4 a c^2 d+5 b c d^2+a d^3\right ) \cos (e+f x)-2 b^3 \left (-c^2 d+d^3\right )^2 (e+f x) \cos (2 (e+f x))+8 b^3 c^5 d e \sin (e+f x)-16 b^3 c^3 d^3 e \sin (e+f x)+8 b^3 c d^5 e \sin (e+f x)+8 b^3 c^5 d f x \sin (e+f x)-16 b^3 c^3 d^3 f x \sin (e+f x)+8 b^3 c d^5 f x \sin (e+f x)+3 b^3 c^4 d^2 \sin (2 (e+f x))-3 a b^2 c^3 d^3 \sin (2 (e+f x))-3 a^2 b c^2 d^4 \sin (2 (e+f x))-6 b^3 c^2 d^4 \sin (2 (e+f x))+3 a^3 c d^5 \sin (2 (e+f x))+12 a b^2 c d^5 \sin (2 (e+f x))-6 a^2 b d^6 \sin (2 (e+f x))}{\left (c^2-d^2\right )^2 (c+d \sin (e+f x))^2}}{4 d^3 f} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-4*(9*a^2*b*c*d^4 - a^3*d^3*(2*c^2 + d^2) - 3*a*b^2*d^3*(c^2 + 2*d^2) + b^3*(2*c^5 - 5*c^3*d^2 + 6*c*d^4))*A
rcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/(c^2 - d^2)^(5/2) + (4*b^3*c^6*e - 6*b^3*c^4*d^2*e + 2*b^3*d^
6*e + 4*b^3*c^6*f*x - 6*b^3*c^4*d^2*f*x + 2*b^3*d^6*f*x - 2*d*(b*c - a*d)^2*(-2*b*c^3 - 4*a*c^2*d + 5*b*c*d^2
+ a*d^3)*Cos[e + f*x] - 2*b^3*(-(c^2*d) + d^3)^2*(e + f*x)*Cos[2*(e + f*x)] + 8*b^3*c^5*d*e*Sin[e + f*x] - 16*
b^3*c^3*d^3*e*Sin[e + f*x] + 8*b^3*c*d^5*e*Sin[e + f*x] + 8*b^3*c^5*d*f*x*Sin[e + f*x] - 16*b^3*c^3*d^3*f*x*Si
n[e + f*x] + 8*b^3*c*d^5*f*x*Sin[e + f*x] + 3*b^3*c^4*d^2*Sin[2*(e + f*x)] - 3*a*b^2*c^3*d^3*Sin[2*(e + f*x)]
- 3*a^2*b*c^2*d^4*Sin[2*(e + f*x)] - 6*b^3*c^2*d^4*Sin[2*(e + f*x)] + 3*a^3*c*d^5*Sin[2*(e + f*x)] + 12*a*b^2*
c*d^5*Sin[2*(e + f*x)] - 6*a^2*b*d^6*Sin[2*(e + f*x)])/((c^2 - d^2)^2*(c + d*Sin[e + f*x])^2))/(4*d^3*f)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(669\) vs. \(2(246)=492\).
time = 0.76, size = 670, normalized size = 2.63

method	result	size
derivativedivides	\(\frac {\frac {\frac {2 \left (\frac {d^{2} \left (5 a^{3} c^{2} d^{3}-2 a^{3} d^{5}-9 a^{2} b \,c^{3} d^{2}+3 a \,b^{2} c^{4} d +6 a \,b^{2} c^{2} d^{3}+b^{3} c^{5}-4 b^{3} c^{3} d^{2}\right ) \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 \left (c^{4}-2 c^{2} d^{2}+d^{4}\right ) c}+\frac {d \left (4 a^{3} c^{4} d^{3}+7 a^{3} c^{2} d^{5}-2 a^{3} d^{7}-6 a^{2} b \,c^{5} d^{2}-15 a^{2} b \,c^{3} d^{4}-6 a^{2} b c \,d^{6}+9 a \,b^{2} c^{4} d^{3}+18 a \,b^{2} c^{2} d^{5}+2 b^{3} c^{7}-b^{3} c^{5} d^{2}-10 b^{3} c^{3} d^{4}\right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 \left (c^{4}-2 c^{2} d^{2}+d^{4}\right ) c^{2}}+\frac {d^{2} \left (11 a^{3} c^{2} d^{3}-2 a^{3} d^{5}-15 a^{2} b \,c^{3} d^{2}-12 a^{2} b c \,d^{4}-3 a \,b^{2} c^{4} d +30 a \,b^{2} c^{2} d^{3}+7 b^{3} c^{5}-16 b^{3} c^{3} d^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{2 \left (c^{4}-2 c^{2} d^{2}+d^{4}\right ) c}+\frac {d \left (4 a^{3} c^{2} d^{3}-a^{3} d^{5}-6 a^{2} b \,c^{3} d^{2}-3 a^{2} b c \,d^{4}+9 a \,b^{2} c^{2} d^{3}+2 b^{3} c^{5}-5 b^{3} c^{3} d^{2}\right )}{2 c^{4}-4 c^{2} d^{2}+2 d^{4}}\right )}{\left (c \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+2 d \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+c \right )^{2}}+\frac {\left (2 a^{3} c^{2} d^{3}+a^{3} d^{5}-9 a^{2} b c \,d^{4}+3 a \,b^{2} c^{2} d^{3}+6 a \,b^{2} d^{5}-2 b^{3} c^{5}+5 b^{3} c^{3} d^{2}-6 d^{4} b^{3} c \right ) \arctan \left (\frac {2 c \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+2 d}{2 \sqrt {c^{2}-d^{2}}}\right )}{\left (c^{4}-2 c^{2} d^{2}+d^{4}\right ) \sqrt {c^{2}-d^{2}}}}{d^{3}}+\frac {2 b^{3} \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{d^{3}}}{f}\)	\(670\)
default	\(\frac {\frac {\frac {2 \left (\frac {d^{2} \left (5 a^{3} c^{2} d^{3}-2 a^{3} d^{5}-9 a^{2} b \,c^{3} d^{2}+3 a \,b^{2} c^{4} d +6 a \,b^{2} c^{2} d^{3}+b^{3} c^{5}-4 b^{3} c^{3} d^{2}\right ) \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 \left (c^{4}-2 c^{2} d^{2}+d^{4}\right ) c}+\frac {d \left (4 a^{3} c^{4} d^{3}+7 a^{3} c^{2} d^{5}-2 a^{3} d^{7}-6 a^{2} b \,c^{5} d^{2}-15 a^{2} b \,c^{3} d^{4}-6 a^{2} b c \,d^{6}+9 a \,b^{2} c^{4} d^{3}+18 a \,b^{2} c^{2} d^{5}+2 b^{3} c^{7}-b^{3} c^{5} d^{2}-10 b^{3} c^{3} d^{4}\right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 \left (c^{4}-2 c^{2} d^{2}+d^{4}\right ) c^{2}}+\frac {d^{2} \left (11 a^{3} c^{2} d^{3}-2 a^{3} d^{5}-15 a^{2} b \,c^{3} d^{2}-12 a^{2} b c \,d^{4}-3 a \,b^{2} c^{4} d +30 a \,b^{2} c^{2} d^{3}+7 b^{3} c^{5}-16 b^{3} c^{3} d^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{2 \left (c^{4}-2 c^{2} d^{2}+d^{4}\right ) c}+\frac {d \left (4 a^{3} c^{2} d^{3}-a^{3} d^{5}-6 a^{2} b \,c^{3} d^{2}-3 a^{2} b c \,d^{4}+9 a \,b^{2} c^{2} d^{3}+2 b^{3} c^{5}-5 b^{3} c^{3} d^{2}\right )}{2 c^{4}-4 c^{2} d^{2}+2 d^{4}}\right )}{\left (c \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+2 d \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+c \right )^{2}}+\frac {\left (2 a^{3} c^{2} d^{3}+a^{3} d^{5}-9 a^{2} b c \,d^{4}+3 a \,b^{2} c^{2} d^{3}+6 a \,b^{2} d^{5}-2 b^{3} c^{5}+5 b^{3} c^{3} d^{2}-6 d^{4} b^{3} c \right ) \arctan \left (\frac {2 c \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+2 d}{2 \sqrt {c^{2}-d^{2}}}\right )}{\left (c^{4}-2 c^{2} d^{2}+d^{4}\right ) \sqrt {c^{2}-d^{2}}}}{d^{3}}+\frac {2 b^{3} \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{d^{3}}}{f}\)	\(670\)
risch	\(\text {Expression too large to display}\)	\(2015\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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((a+b*sin(f*x+e))^3/(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*d^2-4*c^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. 822 vs. \(2 (252) = 504\).
time = 0.48, size = 1732, normalized size = 6.79 \begin {gather*} \left [\frac {4 \, {\left (b^{3} c^{6} d^{2} - 3 \, b^{3} c^{4} d^{4} + 3 \, b^{3} c^{2} d^{6} - b^{3} d^{8}\right )} f x \cos \left (f x + e\right )^{2} - 4 \, {\left (b^{3} c^{8} - 2 \, b^{3} c^{6} d^{2} + 2 \, b^{3} c^{2} d^{6} - b^{3} d^{8}\right )} f x - {\left (2 \, b^{3} c^{7} - 3 \, b^{3} c^{5} d^{2} - {\left (2 \, a^{3} + 3 \, a b^{2}\right )} c^{4} d^{3} + {\left (9 \, a^{2} b + b^{3}\right )} c^{3} d^{4} - 3 \, {\left (a^{3} + 3 \, a b^{2}\right )} c^{2} d^{5} + 3 \, {\left (3 \, a^{2} b + 2 \, b^{3}\right )} c d^{6} - {\left (a^{3} + 6 \, a b^{2}\right )} d^{7} - {\left (2 \, b^{3} c^{5} d^{2} - 5 \, b^{3} c^{3} d^{4} - {\left (2 \, a^{3} + 3 \, a b^{2}\right )} c^{2} d^{5} + 3 \, {\left (3 \, a^{2} b + 2 \, b^{3}\right )} c d^{6} - {\left (a^{3} + 6 \, a b^{2}\right )} d^{7}\right )} \cos \left (f x + e\right )^{2} + 2 \, {\left (2 \, b^{3} c^{6} d - 5 \, b^{3} c^{4} d^{3} - {\left (2 \, a^{3} + 3 \, a b^{2}\right )} c^{3} d^{4} + 3 \, {\left (3 \, a^{2} b + 2 \, b^{3}\right )} c^{2} d^{5} - {\left (a^{3} + 6 \, a b^{2}\right )} c d^{6}\right )} \sin \left (f x + e\right )\right )} \sqrt {-c^{2} + d^{2}} \log \left (\frac {{\left (2 \, c^{2} - d^{2}\right )} \cos \left (f x + e\right )^{2} - 2 \, c d \sin \left (f x + e\right ) - c^{2} - d^{2} + 2 \, {\left (c \cos \left (f x + e\right ) \sin \left (f x + e\right ) + d \cos \left (f x + e\right )\right )} \sqrt {-c^{2} + d^{2}}}{d^{2} \cos \left (f x + e\right )^{2} - 2 \, c d \sin \left (f x + e\right ) - c^{2} - d^{2}}\right ) - 2 \, {\left (2 \, b^{3} c^{7} d + 3 \, a^{2} b c d^{7} + a^{3} d^{8} - {\left (6 \, a^{2} b + 7 \, b^{3}\right )} c^{5} d^{3} + {\left (4 \, a^{3} + 9 \, a b^{2}\right )} c^{4} d^{4} + {\left (3 \, a^{2} b + 5 \, b^{3}\right )} c^{3} d^{5} - {\left (5 \, a^{3} + 9 \, a b^{2}\right )} c^{2} d^{6}\right )} \cos \left (f x + e\right ) - 2 \, {\left (4 \, {\left (b^{3} c^{7} d - 3 \, b^{3} c^{5} d^{3} + 3 \, b^{3} c^{3} d^{5} - b^{3} c d^{7}\right )} f x + 3 \, {\left (b^{3} c^{6} d^{2} - a b^{2} c^{5} d^{3} + 2 \, a^{2} b d^{8} - {\left (a^{2} b + 3 \, b^{3}\right )} c^{4} d^{4} + {\left (a^{3} + 5 \, a b^{2}\right )} c^{3} d^{5} - {\left (a^{2} b - 2 \, b^{3}\right )} c^{2} d^{6} - {\left (a^{3} + 4 \, a b^{2}\right )} c d^{7}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{4 \, {\left ({\left (c^{6} d^{5} - 3 \, c^{4} d^{7} + 3 \, c^{2} d^{9} - d^{11}\right )} f \cos \left (f x + e\right )^{2} - 2 \, {\left (c^{7} d^{4} - 3 \, c^{5} d^{6} + 3 \, c^{3} d^{8} - c d^{10}\right )} f \sin \left (f x + e\right ) - {\left (c^{8} d^{3} - 2 \, c^{6} d^{5} + 2 \, c^{2} d^{9} - d^{11}\right )} f\right )}}, \frac {2 \, {\left (b^{3} c^{6} d^{2} - 3 \, b^{3} c^{4} d^{4} + 3 \, b^{3} c^{2} d^{6} - b^{3} d^{8}\right )} f x \cos \left (f x + e\right )^{2} - 2 \, {\left (b^{3} c^{8} - 2 \, b^{3} c^{6} d^{2} + 2 \, b^{3} c^{2} d^{6} - b^{3} d^{8}\right )} f x - {\left (2 \, b^{3} c^{7} - 3 \, b^{3} c^{5} d^{2} - {\left (2 \, a^{3} + 3 \, a b^{2}\right )} c^{4} d^{3} + {\left (9 \, a^{2} b + b^{3}\right )} c^{3} d^{4} - 3 \, {\left (a^{3} + 3 \, a b^{2}\right )} c^{2} d^{5} + 3 \, {\left (3 \, a^{2} b + 2 \, b^{3}\right )} c d^{6} - {\left (a^{3} + 6 \, a b^{2}\right )} d^{7} - {\left (2 \, b^{3} c^{5} d^{2} - 5 \, b^{3} c^{3} d^{4} - {\left (2 \, a^{3} + 3 \, a b^{2}\right )} c^{2} d^{5} + 3 \, {\left (3 \, a^{2} b + 2 \, b^{3}\right )} c d^{6} - {\left (a^{3} + 6 \, a b^{2}\right )} d^{7}\right )} \cos \left (f x + e\right )^{2} + 2 \, {\left (2 \, b^{3} c^{6} d - 5 \, b^{3} c^{4} d^{3} - {\left (2 \, a^{3} + 3 \, a b^{2}\right )} c^{3} d^{4} + 3 \, {\left (3 \, a^{2} b + 2 \, b^{3}\right )} c^{2} d^{5} - {\left (a^{3} + 6 \, a b^{2}\right )} c d^{6}\right )} \sin \left (f x + e\right )\right )} \sqrt {c^{2} - d^{2}} \arctan \left (-\frac {c \sin \left (f x + e\right ) + d}{\sqrt {c^{2} - d^{2}} \cos \left (f x + e\right )}\right ) - {\left (2 \, b^{3} c^{7} d + 3 \, a^{2} b c d^{7} + a^{3} d^{8} - {\left (6 \, a^{2} b + 7 \, b^{3}\right )} c^{5} d^{3} + {\left (4 \, a^{3} + 9 \, a b^{2}\right )} c^{4} d^{4} + {\left (3 \, a^{2} b + 5 \, b^{3}\right )} c^{3} d^{5} - {\left (5 \, a^{3} + 9 \, a b^{2}\right )} c^{2} d^{6}\right )} \cos \left (f x + e\right ) - {\left (4 \, {\left (b^{3} c^{7} d - 3 \, b^{3} c^{5} d^{3} + 3 \, b^{3} c^{3} d^{5} - b^{3} c d^{7}\right )} f x + 3 \, {\left (b^{3} c^{6} d^{2} - a b^{2} c^{5} d^{3} + 2 \, a^{2} b d^{8} - {\left (a^{2} b + 3 \, b^{3}\right )} c^{4} d^{4} + {\left (a^{3} + 5 \, a b^{2}\right )} c^{3} d^{5} - {\left (a^{2} b - 2 \, b^{3}\right )} c^{2} d^{6} - {\left (a^{3} + 4 \, a b^{2}\right )} c d^{7}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{2 \, {\left ({\left (c^{6} d^{5} - 3 \, c^{4} d^{7} + 3 \, c^{2} d^{9} - d^{11}\right )} f \cos \left (f x + e\right )^{2} - 2 \, {\left (c^{7} d^{4} - 3 \, c^{5} d^{6} + 3 \, c^{3} d^{8} - c d^{10}\right )} f \sin \left (f x + e\right ) - {\left (c^{8} d^{3} - 2 \, c^{6} d^{5} + 2 \, c^{2} d^{9} - d^{11}\right )} f\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/4*(4*(b^3*c^6*d^2 - 3*b^3*c^4*d^4 + 3*b^3*c^2*d^6 - b^3*d^8)*f*x*cos(f*x + e)^2 - 4*(b^3*c^8 - 2*b^3*c^6*d^
2 + 2*b^3*c^2*d^6 - b^3*d^8)*f*x - (2*b^3*c^7 - 3*b^3*c^5*d^2 - (2*a^3 + 3*a*b^2)*c^4*d^3 + (9*a^2*b + b^3)*c^
3*d^4 - 3*(a^3 + 3*a*b^2)*c^2*d^5 + 3*(3*a^2*b + 2*b^3)*c*d^6 - (a^3 + 6*a*b^2)*d^7 - (2*b^3*c^5*d^2 - 5*b^3*c
^3*d^4 - (2*a^3 + 3*a*b^2)*c^2*d^5 + 3*(3*a^2*b + 2*b^3)*c*d^6 - (a^3 + 6*a*b^2)*d^7)*cos(f*x + e)^2 + 2*(2*b^
3*c^6*d - 5*b^3*c^4*d^3 - (2*a^3 + 3*a*b^2)*c^3*d^4 + 3*(3*a^2*b + 2*b^3)*c^2*d^5 - (a^3 + 6*a*b^2)*c*d^6)*sin
(f*x + e))*sqrt(-c^2 + d^2)*log(((2*c^2 - d^2)*cos(f*x + e)^2 - 2*c*d*sin(f*x + e) - c^2 - d^2 + 2*(c*cos(f*x
+ e)*sin(f*x + e) + d*cos(f*x + e))*sqrt(-c^2 + d^2))/(d^2*cos(f*x + e)^2 - 2*c*d*sin(f*x + e) - c^2 - d^2)) -
2*(2*b^3*c^7*d + 3*a^2*b*c*d^7 + a^3*d^8 - (6*a^2*b + 7*b^3)*c^5*d^3 + (4*a^3 + 9*a*b^2)*c^4*d^4 + (3*a^2*b +
5*b^3)*c^3*d^5 - (5*a^3 + 9*a*b^2)*c^2*d^6)*cos(f*x + e) - 2*(4*(b^3*c^7*d - 3*b^3*c^5*d^3 + 3*b^3*c^3*d^5 -
b^3*c*d^7)*f*x + 3*(b^3*c^6*d^2 - a*b^2*c^5*d^3 + 2*a^2*b*d^8 - (a^2*b + 3*b^3)*c^4*d^4 + (a^3 + 5*a*b^2)*c^3*
d^5 - (a^2*b - 2*b^3)*c^2*d^6 - (a^3 + 4*a*b^2)*c*d^7)*cos(f*x + e))*sin(f*x + e))/((c^6*d^5 - 3*c^4*d^7 + 3*c
^2*d^9 - d^11)*f*cos(f*x + e)^2 - 2*(c^7*d^4 - 3*c^5*d^6 + 3*c^3*d^8 - c*d^10)*f*sin(f*x + e) - (c^8*d^3 - 2*c
^6*d^5 + 2*c^2*d^9 - d^11)*f), 1/2*(2*(b^3*c^6*d^2 - 3*b^3*c^4*d^4 + 3*b^3*c^2*d^6 - b^3*d^8)*f*x*cos(f*x + e)
^2 - 2*(b^3*c^8 - 2*b^3*c^6*d^2 + 2*b^3*c^2*d^6 - b^3*d^8)*f*x - (2*b^3*c^7 - 3*b^3*c^5*d^2 - (2*a^3 + 3*a*b^2
)*c^4*d^3 + (9*a^2*b + b^3)*c^3*d^4 - 3*(a^3 + 3*a*b^2)*c^2*d^5 + 3*(3*a^2*b + 2*b^3)*c*d^6 - (a^3 + 6*a*b^2)*
d^7 - (2*b^3*c^5*d^2 - 5*b^3*c^3*d^4 - (2*a^3 + 3*a*b^2)*c^2*d^5 + 3*(3*a^2*b + 2*b^3)*c*d^6 - (a^3 + 6*a*b^2)
*d^7)*cos(f*x + e)^2 + 2*(2*b^3*c^6*d - 5*b^3*c^4*d^3 - (2*a^3 + 3*a*b^2)*c^3*d^4 + 3*(3*a^2*b + 2*b^3)*c^2*d^
5 - (a^3 + 6*a*b^2)*c*d^6)*sin(f*x + e))*sqrt(c^2 - d^2)*arctan(-(c*sin(f*x + e) + d)/(sqrt(c^2 - d^2)*cos(f*x
+ e))) - (2*b^3*c^7*d + 3*a^2*b*c*d^7 + a^3*d^8 - (6*a^2*b + 7*b^3)*c^5*d^3 + (4*a^3 + 9*a*b^2)*c^4*d^4 + (3*
a^2*b + 5*b^3)*c^3*d^5 - (5*a^3 + 9*a*b^2)*c^2*d^6)*cos(f*x + e) - (4*(b^3*c^7*d - 3*b^3*c^5*d^3 + 3*b^3*c^3*d
^5 - b^3*c*d^7)*f*x + 3*(b^3*c^6*d^2 - a*b^2*c^5*d^3 + 2*a^2*b*d^8 - (a^2*b + 3*b^3)*c^4*d^4 + (a^3 + 5*a*b^2)
*c^3*d^5 - (a^2*b - 2*b^3)*c^2*d^6 - (a^3 + 4*a*b^2)*c*d^7)*cos(f*x + e))*sin(f*x + e))/((c^6*d^5 - 3*c^4*d^7
+ 3*c^2*d^9 - d^11)*f*cos(f*x + e)^2 - 2*(c^7*d^4 - 3*c^5*d^6 + 3*c^3*d^8 - c*d^10)*f*sin(f*x + e) - (c^8*d^3
- 2*c^6*d^5 + 2*c^2*d^9 - d^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((a+b*sin(f*x+e))**3/(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. 889 vs. \(2 (252) = 504\).
time = 0.54, size = 889, normalized size = 3.49 \begin {gather*} \frac {\frac {{\left (f x + e\right )} b^{3}}{d^{3}} - \frac {{\left (2 \, b^{3} c^{5} - 5 \, b^{3} c^{3} d^{2} - 2 \, a^{3} c^{2} d^{3} - 3 \, a b^{2} c^{2} d^{3} + 9 \, a^{2} b c d^{4} + 6 \, b^{3} c d^{4} - a^{3} d^{5} - 6 \, a 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 (c^{4} d^{3} - 2 \, c^{2} d^{5} + d^{7}\right )} \sqrt {c^{2} - d^{2}}} + \frac {b^{3} c^{6} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 3 \, a b^{2} c^{5} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 9 \, a^{2} b c^{4} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 4 \, b^{3} c^{4} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 5 \, a^{3} c^{3} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 6 \, a b^{2} c^{3} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 2 \, a^{3} c d^{6} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 2 \, b^{3} c^{7} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 6 \, a^{2} b c^{5} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - b^{3} c^{5} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 4 \, a^{3} c^{4} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 9 \, a b^{2} c^{4} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 15 \, a^{2} b c^{3} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 10 \, b^{3} c^{3} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 7 \, a^{3} c^{2} d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 18 \, a b^{2} c^{2} d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 6 \, a^{2} b c d^{6} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 2 \, a^{3} d^{7} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 7 \, b^{3} c^{6} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 3 \, a b^{2} c^{5} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 15 \, a^{2} b c^{4} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 16 \, b^{3} c^{4} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 11 \, a^{3} c^{3} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 30 \, a b^{2} c^{3} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 12 \, a^{2} b c^{2} d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 2 \, a^{3} c d^{6} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 2 \, b^{3} c^{7} - 6 \, a^{2} b c^{5} d^{2} - 5 \, b^{3} c^{5} d^{2} + 4 \, a^{3} c^{4} d^{3} + 9 \, a b^{2} c^{4} d^{3} - 3 \, a^{2} b c^{3} d^{4} - a^{3} c^{2} d^{5}}{{\left (c^{6} d^{2} - 2 \, c^{4} d^{4} + 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((a+b*sin(f*x+e))^3/(c+d*sin(f*x+e))^3,x, algorithm="giac")

[Out]

((f*x + e)*b^3/d^3 - (2*b^3*c^5 - 5*b^3*c^3*d^2 - 2*a^3*c^2*d^3 - 3*a*b^2*c^2*d^3 + 9*a^2*b*c*d^4 + 6*b^3*c*d^
4 - a^3*d^5 - 6*a*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)))/((c^4*d^3 - 2*c^2*d^5 + d^7)*sqrt(c^2 - d^2)) + (b^3*c^6*d*tan(1/2*f*x + 1/2*e)^3 + 3*a*b^2*c^5*
d^2*tan(1/2*f*x + 1/2*e)^3 - 9*a^2*b*c^4*d^3*tan(1/2*f*x + 1/2*e)^3 - 4*b^3*c^4*d^3*tan(1/2*f*x + 1/2*e)^3 + 5
*a^3*c^3*d^4*tan(1/2*f*x + 1/2*e)^3 + 6*a*b^2*c^3*d^4*tan(1/2*f*x + 1/2*e)^3 - 2*a^3*c*d^6*tan(1/2*f*x + 1/2*e
)^3 + 2*b^3*c^7*tan(1/2*f*x + 1/2*e)^2 - 6*a^2*b*c^5*d^2*tan(1/2*f*x + 1/2*e)^2 - b^3*c^5*d^2*tan(1/2*f*x + 1/
2*e)^2 + 4*a^3*c^4*d^3*tan(1/2*f*x + 1/2*e)^2 + 9*a*b^2*c^4*d^3*tan(1/2*f*x + 1/2*e)^2 - 15*a^2*b*c^3*d^4*tan(
1/2*f*x + 1/2*e)^2 - 10*b^3*c^3*d^4*tan(1/2*f*x + 1/2*e)^2 + 7*a^3*c^2*d^5*tan(1/2*f*x + 1/2*e)^2 + 18*a*b^2*c
^2*d^5*tan(1/2*f*x + 1/2*e)^2 - 6*a^2*b*c*d^6*tan(1/2*f*x + 1/2*e)^2 - 2*a^3*d^7*tan(1/2*f*x + 1/2*e)^2 + 7*b^
3*c^6*d*tan(1/2*f*x + 1/2*e) - 3*a*b^2*c^5*d^2*tan(1/2*f*x + 1/2*e) - 15*a^2*b*c^4*d^3*tan(1/2*f*x + 1/2*e) -
16*b^3*c^4*d^3*tan(1/2*f*x + 1/2*e) + 11*a^3*c^3*d^4*tan(1/2*f*x + 1/2*e) + 30*a*b^2*c^3*d^4*tan(1/2*f*x + 1/2
*e) - 12*a^2*b*c^2*d^5*tan(1/2*f*x + 1/2*e) - 2*a^3*c*d^6*tan(1/2*f*x + 1/2*e) + 2*b^3*c^7 - 6*a^2*b*c^5*d^2 -
5*b^3*c^5*d^2 + 4*a^3*c^4*d^3 + 9*a*b^2*c^4*d^3 - 3*a^2*b*c^3*d^4 - a^3*c^2*d^5)/((c^6*d^2 - 2*c^4*d^4 + 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 = 20.46, size = 2500, normalized size = 9.80 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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