3.452 \(\int \frac {(a+a \sin (e+f x))^3}{(c+d \sin (e+f x))^5} \, dx\)

Optimal. Leaf size=289 \[ \frac {5 a^3 (4 c-3 d) \tan ^{-1}\left (\frac {c \tan \left (\frac {1}{2} (e+f x)\right )+d}{\sqrt {c^2-d^2}}\right )}{4 f (c-d) (c+d)^4 \sqrt {c^2-d^2}}-\frac {a^3 \left (2 c^2+12 c d+45 d^2\right ) \cos (e+f x)}{24 d^2 f (c+d)^3 (c+d \sin (e+f x))^2}-\frac {a^3 \left (2 c^3+12 c^2 d+43 c d^2-72 d^3\right ) \cos (e+f x)}{24 d^2 f (c-d) (c+d)^4 (c+d \sin (e+f x))}+\frac {a^3 (c-d) (2 c+9 d) \cos (e+f x)}{12 d^2 f (c+d)^2 (c+d \sin (e+f x))^3}+\frac {(c-d) \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right )}{4 d f (c+d) (c+d \sin (e+f x))^4} \]

[Out]

1/4*(c-d)*cos(f*x+e)*(a^3+a^3*sin(f*x+e))/d/(c+d)/f/(c+d*sin(f*x+e))^4+1/12*a^3*(c-d)*(2*c+9*d)*cos(f*x+e)/d^2
/(c+d)^2/f/(c+d*sin(f*x+e))^3-1/24*a^3*(2*c^2+12*c*d+45*d^2)*cos(f*x+e)/d^2/(c+d)^3/f/(c+d*sin(f*x+e))^2-1/24*
a^3*(2*c^3+12*c^2*d+43*c*d^2-72*d^3)*cos(f*x+e)/(c-d)/d^2/(c+d)^4/f/(c+d*sin(f*x+e))+5/4*a^3*(4*c-3*d)*arctan(
(d+c*tan(1/2*f*x+1/2*e))/(c^2-d^2)^(1/2))/(c-d)/(c+d)^4/f/(c^2-d^2)^(1/2)

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Rubi [A]  time = 0.70, antiderivative size = 289, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {2762, 2968, 3021, 2754, 12, 2660, 618, 204} \[ \frac {5 a^3 (4 c-3 d) \tan ^{-1}\left (\frac {c \tan \left (\frac {1}{2} (e+f x)\right )+d}{\sqrt {c^2-d^2}}\right )}{4 f (c-d) (c+d)^4 \sqrt {c^2-d^2}}-\frac {a^3 \left (12 c^2 d+2 c^3+43 c d^2-72 d^3\right ) \cos (e+f x)}{24 d^2 f (c-d) (c+d)^4 (c+d \sin (e+f x))}-\frac {a^3 \left (2 c^2+12 c d+45 d^2\right ) \cos (e+f x)}{24 d^2 f (c+d)^3 (c+d \sin (e+f x))^2}+\frac {a^3 (c-d) (2 c+9 d) \cos (e+f x)}{12 d^2 f (c+d)^2 (c+d \sin (e+f x))^3}+\frac {(c-d) \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right )}{4 d f (c+d) (c+d \sin (e+f x))^4} \]

Antiderivative was successfully verified.

[In]

Int[(a + a*Sin[e + f*x])^3/(c + d*Sin[e + f*x])^5,x]

[Out]

(5*a^3*(4*c - 3*d)*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/(4*(c - d)*(c + d)^4*Sqrt[c^2 - d^2]*f) +
((c - d)*Cos[e + f*x]*(a^3 + a^3*Sin[e + f*x]))/(4*d*(c + d)*f*(c + d*Sin[e + f*x])^4) + (a^3*(c - d)*(2*c +
9*d)*Cos[e + f*x])/(12*d^2*(c + d)^2*f*(c + d*Sin[e + f*x])^3) - (a^3*(2*c^2 + 12*c*d + 45*d^2)*Cos[e + f*x])/
(24*d^2*(c + d)^3*f*(c + d*Sin[e + f*x])^2) - (a^3*(2*c^3 + 12*c^2*d + 43*c*d^2 - 72*d^3)*Cos[e + f*x])/(24*(c
- d)*d^2*(c + d)^4*f*(c + d*Sin[e + f*x]))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 618

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 2660

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 2754

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

Rule 2762

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> -Si
mp[(b^2*(b*c - a*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 2)*(c + d*Sin[e + f*x])^(n + 1))/(d*f*(n + 1)*(b*c
+ a*d)), x] + Dist[b^2/(d*(n + 1)*(b*c + a*d)), Int[(a + b*Sin[e + f*x])^(m - 2)*(c + d*Sin[e + f*x])^(n + 1)*
Simp[a*c*(m - 2) - b*d*(m - 2*n - 4) - (b*c*(m - 1) - a*d*(m + 2*n + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a,
b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 1] && LtQ[n, -1]
&& (IntegersQ[2*m, 2*n] || IntegerQ[m + 1/2] || (IntegerQ[m] && EqQ[c, 0]))

Rule 2968

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(
e_.) + (f_.)*(x_)]), x_Symbol] :> Int[(a + b*Sin[e + f*x])^m*(A*c + (B*c + A*d)*Sin[e + f*x] + B*d*Sin[e + f*x
]^2), x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0]

Rule 3021

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+a \sin (e+f x))^3}{(c+d \sin (e+f x))^5} \, dx &=\frac {(c-d) \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right )}{4 d (c+d) f (c+d \sin (e+f x))^4}-\frac {a \int \frac {(a+a \sin (e+f x)) (a (c-9 d)-2 a (c+3 d) \sin (e+f x))}{(c+d \sin (e+f x))^4} \, dx}{4 d (c+d)}\\ &=\frac {(c-d) \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right )}{4 d (c+d) f (c+d \sin (e+f x))^4}-\frac {a \int \frac {a^2 (c-9 d)+\left (a^2 (c-9 d)-2 a^2 (c+3 d)\right ) \sin (e+f x)-2 a^2 (c+3 d) \sin ^2(e+f x)}{(c+d \sin (e+f x))^4} \, dx}{4 d (c+d)}\\ &=\frac {(c-d) \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right )}{4 d (c+d) f (c+d \sin (e+f x))^4}+\frac {a^3 (c-d) (2 c+9 d) \cos (e+f x)}{12 d^2 (c+d)^2 f (c+d \sin (e+f x))^3}+\frac {a \int \frac {3 a^2 (c-d) d (c+15 d)+2 a^2 (c-d) \left (c^2+5 c d+18 d^2\right ) \sin (e+f x)}{(c+d \sin (e+f x))^3} \, dx}{12 (c-d) d^2 (c+d)^2}\\ &=\frac {(c-d) \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right )}{4 d (c+d) f (c+d \sin (e+f x))^4}+\frac {a^3 (c-d) (2 c+9 d) \cos (e+f x)}{12 d^2 (c+d)^2 f (c+d \sin (e+f x))^3}-\frac {a^3 \left (2 c^2+12 c d+45 d^2\right ) \cos (e+f x)}{24 d^2 (c+d)^3 f (c+d \sin (e+f x))^2}-\frac {a \int \frac {-2 a^2 (c-d)^2 d (c+36 d)-a^2 (c-d)^2 \left (2 c^2+12 c d+45 d^2\right ) \sin (e+f x)}{(c+d \sin (e+f x))^2} \, dx}{24 (c-d)^2 d^2 (c+d)^3}\\ &=\frac {(c-d) \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right )}{4 d (c+d) f (c+d \sin (e+f x))^4}+\frac {a^3 (c-d) (2 c+9 d) \cos (e+f x)}{12 d^2 (c+d)^2 f (c+d \sin (e+f x))^3}-\frac {a^3 \left (2 c^2+12 c d+45 d^2\right ) \cos (e+f x)}{24 d^2 (c+d)^3 f (c+d \sin (e+f x))^2}-\frac {a^3 \left (2 c^3+12 c^2 d+43 c d^2-72 d^3\right ) \cos (e+f x)}{24 (c-d) d^2 (c+d)^4 f (c+d \sin (e+f x))}+\frac {a \int \frac {15 a^2 (4 c-3 d) (c-d)^2 d^2}{c+d \sin (e+f x)} \, dx}{24 (c-d)^3 d^2 (c+d)^4}\\ &=\frac {(c-d) \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right )}{4 d (c+d) f (c+d \sin (e+f x))^4}+\frac {a^3 (c-d) (2 c+9 d) \cos (e+f x)}{12 d^2 (c+d)^2 f (c+d \sin (e+f x))^3}-\frac {a^3 \left (2 c^2+12 c d+45 d^2\right ) \cos (e+f x)}{24 d^2 (c+d)^3 f (c+d \sin (e+f x))^2}-\frac {a^3 \left (2 c^3+12 c^2 d+43 c d^2-72 d^3\right ) \cos (e+f x)}{24 (c-d) d^2 (c+d)^4 f (c+d \sin (e+f x))}+\frac {\left (5 a^3 (4 c-3 d)\right ) \int \frac {1}{c+d \sin (e+f x)} \, dx}{8 (c-d) (c+d)^4}\\ &=\frac {(c-d) \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right )}{4 d (c+d) f (c+d \sin (e+f x))^4}+\frac {a^3 (c-d) (2 c+9 d) \cos (e+f x)}{12 d^2 (c+d)^2 f (c+d \sin (e+f x))^3}-\frac {a^3 \left (2 c^2+12 c d+45 d^2\right ) \cos (e+f x)}{24 d^2 (c+d)^3 f (c+d \sin (e+f x))^2}-\frac {a^3 \left (2 c^3+12 c^2 d+43 c d^2-72 d^3\right ) \cos (e+f x)}{24 (c-d) d^2 (c+d)^4 f (c+d \sin (e+f x))}+\frac {\left (5 a^3 (4 c-3 d)\right ) \operatorname {Subst}\left (\int \frac {1}{c+2 d x+c x^2} \, dx,x,\tan \left (\frac {1}{2} (e+f x)\right )\right )}{4 (c-d) (c+d)^4 f}\\ &=\frac {(c-d) \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right )}{4 d (c+d) f (c+d \sin (e+f x))^4}+\frac {a^3 (c-d) (2 c+9 d) \cos (e+f x)}{12 d^2 (c+d)^2 f (c+d \sin (e+f x))^3}-\frac {a^3 \left (2 c^2+12 c d+45 d^2\right ) \cos (e+f x)}{24 d^2 (c+d)^3 f (c+d \sin (e+f x))^2}-\frac {a^3 \left (2 c^3+12 c^2 d+43 c d^2-72 d^3\right ) \cos (e+f x)}{24 (c-d) d^2 (c+d)^4 f (c+d \sin (e+f x))}-\frac {\left (5 a^3 (4 c-3 d)\right ) \operatorname {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 )}{2 (c-d) (c+d)^4 f}\\ &=\frac {5 a^3 (4 c-3 d) \tan ^{-1}\left (\frac {d+c \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c^2-d^2}}\right )}{4 (c-d) (c+d)^4 \sqrt {c^2-d^2} f}+\frac {(c-d) \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right )}{4 d (c+d) f (c+d \sin (e+f x))^4}+\frac {a^3 (c-d) (2 c+9 d) \cos (e+f x)}{12 d^2 (c+d)^2 f (c+d \sin (e+f x))^3}-\frac {a^3 \left (2 c^2+12 c d+45 d^2\right ) \cos (e+f x)}{24 d^2 (c+d)^3 f (c+d \sin (e+f x))^2}-\frac {a^3 \left (2 c^3+12 c^2 d+43 c d^2-72 d^3\right ) \cos (e+f x)}{24 (c-d) d^2 (c+d)^4 f (c+d \sin (e+f x))}\\ \end {align*}

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Mathematica [A]  time = 3.21, size = 240, normalized size = 0.83 \[ \frac {a^3 \cos (e+f x) \left (-\frac {(4 c-3 d) \left (-\frac {\sqrt {\cos ^2(e+f x)} \left (\left (2 c^2+9 c d+22 d^2\right ) \sin ^2(e+f x)+\left (9 c^2+48 c d+9 d^2\right ) \sin (e+f x)+22 c^2+9 c d+2 d^2\right )}{6 (c+d)^3 (c+d \sin (e+f x))^3}-\frac {5 \tanh ^{-1}\left (\frac {\sqrt {c-d} \sqrt {1-\sin (e+f x)}}{\sqrt {-c-d} \sqrt {\sin (e+f x)+1}}\right )}{(-c-d)^{7/2} \sqrt {c-d}}\right )}{\sqrt {\cos ^2(e+f x)}}-\frac {d (\sin (e+f x)+1)^3}{(c+d \sin (e+f x))^4}\right )}{4 f (d-c) (c+d)} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + a*Sin[e + f*x])^3/(c + d*Sin[e + f*x])^5,x]

[Out]

(a^3*Cos[e + f*x]*(-((d*(1 + Sin[e + f*x])^3)/(c + d*Sin[e + f*x])^4) - ((4*c - 3*d)*((-5*ArcTanh[(Sqrt[c - d]
*Sqrt[1 - Sin[e + f*x]])/(Sqrt[-c - d]*Sqrt[1 + Sin[e + f*x]])])/((-c - d)^(7/2)*Sqrt[c - d]) - (Sqrt[Cos[e +
f*x]^2]*(22*c^2 + 9*c*d + 2*d^2 + (9*c^2 + 48*c*d + 9*d^2)*Sin[e + f*x] + (2*c^2 + 9*c*d + 22*d^2)*Sin[e + f*x
]^2))/(6*(c + d)^3*(c + d*Sin[e + f*x])^3)))/Sqrt[Cos[e + f*x]^2]))/(4*(-c + d)*(c + d)*f)

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fricas [B]  time = 0.59, size = 2009, normalized size = 6.95 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(f*x+e))^3/(c+d*sin(f*x+e))^5,x, algorithm="fricas")

[Out]

[1/48*(2*(8*a^3*c^6 + 48*a^3*c^5*d + 164*a^3*c^4*d^2 - 276*a^3*c^3*d^3 - 217*a^3*c^2*d^4 + 228*a^3*c*d^5 + 45*
a^3*d^6)*cos(f*x + e)^3 - 15*(4*a^3*c^5 - 3*a^3*c^4*d + 24*a^3*c^3*d^2 - 18*a^3*c^2*d^3 + 4*a^3*c*d^4 - 3*a^3*
d^5 + (4*a^3*c*d^4 - 3*a^3*d^5)*cos(f*x + e)^4 - 2*(12*a^3*c^3*d^2 - 9*a^3*c^2*d^3 + 4*a^3*c*d^4 - 3*a^3*d^5)*
cos(f*x + e)^2 + 4*(4*a^3*c^4*d - 3*a^3*c^3*d^2 + 4*a^3*c^2*d^3 - 3*a^3*c*d^4 - (4*a^3*c^2*d^3 - 3*a^3*c*d^4)*
cos(f*x + e)^2)*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)) - 6*(32*a^3*c^6 + 4*a^3*c^5*d + 13*a^3*c^4*d^2 - 88*a^3*c^3*d^3 - 62*a^3*c^2*d^4 + 84*a^3*c*
d^5 + 17*a^3*d^6)*cos(f*x + e) + 2*((2*a^3*c^5*d + 12*a^3*c^4*d^2 + 41*a^3*c^3*d^3 - 84*a^3*c^2*d^4 - 43*a^3*c
*d^5 + 72*a^3*d^6)*cos(f*x + e)^3 - 3*(12*a^3*c^6 + 79*a^3*c^5*d - 72*a^3*c^4*d^2 - 98*a^3*c^3*d^3 + 28*a^3*c^
2*d^4 + 19*a^3*c*d^5 + 32*a^3*d^6)*cos(f*x + e))*sin(f*x + e))/((c^7*d^4 + 3*c^6*d^5 + c^5*d^6 - 5*c^4*d^7 - 5
*c^3*d^8 + c^2*d^9 + 3*c*d^10 + d^11)*f*cos(f*x + e)^4 - 2*(3*c^9*d^2 + 9*c^8*d^3 + 4*c^7*d^4 - 12*c^6*d^5 - 1
4*c^5*d^6 - 2*c^4*d^7 + 4*c^3*d^8 + 4*c^2*d^9 + 3*c*d^10 + d^11)*f*cos(f*x + e)^2 + (c^11 + 3*c^10*d + 7*c^9*d
^2 + 13*c^8*d^3 + 2*c^7*d^4 - 26*c^6*d^5 - 26*c^5*d^6 + 2*c^4*d^7 + 13*c^3*d^8 + 7*c^2*d^9 + 3*c*d^10 + d^11)*
f - 4*((c^8*d^3 + 3*c^7*d^4 + c^6*d^5 - 5*c^5*d^6 - 5*c^4*d^7 + c^3*d^8 + 3*c^2*d^9 + c*d^10)*f*cos(f*x + e)^2
- (c^10*d + 3*c^9*d^2 + 2*c^8*d^3 - 2*c^7*d^4 - 4*c^6*d^5 - 4*c^5*d^6 - 2*c^4*d^7 + 2*c^3*d^8 + 3*c^2*d^9 + c
*d^10)*f)*sin(f*x + e)), 1/24*((8*a^3*c^6 + 48*a^3*c^5*d + 164*a^3*c^4*d^2 - 276*a^3*c^3*d^3 - 217*a^3*c^2*d^4
+ 228*a^3*c*d^5 + 45*a^3*d^6)*cos(f*x + e)^3 - 15*(4*a^3*c^5 - 3*a^3*c^4*d + 24*a^3*c^3*d^2 - 18*a^3*c^2*d^3
+ 4*a^3*c*d^4 - 3*a^3*d^5 + (4*a^3*c*d^4 - 3*a^3*d^5)*cos(f*x + e)^4 - 2*(12*a^3*c^3*d^2 - 9*a^3*c^2*d^3 + 4*a
^3*c*d^4 - 3*a^3*d^5)*cos(f*x + e)^2 + 4*(4*a^3*c^4*d - 3*a^3*c^3*d^2 + 4*a^3*c^2*d^3 - 3*a^3*c*d^4 - (4*a^3*c
^2*d^3 - 3*a^3*c*d^4)*cos(f*x + e)^2)*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))) - 3*(32*a^3*c^6 + 4*a^3*c^5*d + 13*a^3*c^4*d^2 - 88*a^3*c^3*d^3 - 62*a^3*c^2*d^4 + 84*a^3*c
*d^5 + 17*a^3*d^6)*cos(f*x + e) + ((2*a^3*c^5*d + 12*a^3*c^4*d^2 + 41*a^3*c^3*d^3 - 84*a^3*c^2*d^4 - 43*a^3*c*
d^5 + 72*a^3*d^6)*cos(f*x + e)^3 - 3*(12*a^3*c^6 + 79*a^3*c^5*d - 72*a^3*c^4*d^2 - 98*a^3*c^3*d^3 + 28*a^3*c^2
*d^4 + 19*a^3*c*d^5 + 32*a^3*d^6)*cos(f*x + e))*sin(f*x + e))/((c^7*d^4 + 3*c^6*d^5 + c^5*d^6 - 5*c^4*d^7 - 5*
c^3*d^8 + c^2*d^9 + 3*c*d^10 + d^11)*f*cos(f*x + e)^4 - 2*(3*c^9*d^2 + 9*c^8*d^3 + 4*c^7*d^4 - 12*c^6*d^5 - 14
*c^5*d^6 - 2*c^4*d^7 + 4*c^3*d^8 + 4*c^2*d^9 + 3*c*d^10 + d^11)*f*cos(f*x + e)^2 + (c^11 + 3*c^10*d + 7*c^9*d^
2 + 13*c^8*d^3 + 2*c^7*d^4 - 26*c^6*d^5 - 26*c^5*d^6 + 2*c^4*d^7 + 13*c^3*d^8 + 7*c^2*d^9 + 3*c*d^10 + d^11)*f
- 4*((c^8*d^3 + 3*c^7*d^4 + c^6*d^5 - 5*c^5*d^6 - 5*c^4*d^7 + c^3*d^8 + 3*c^2*d^9 + c*d^10)*f*cos(f*x + e)^2
- (c^10*d + 3*c^9*d^2 + 2*c^8*d^3 - 2*c^7*d^4 - 4*c^6*d^5 - 4*c^5*d^6 - 2*c^4*d^7 + 2*c^3*d^8 + 3*c^2*d^9 + c*
d^10)*f)*sin(f*x + e))]

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giac [B]  time = 2.22, size = 1338, normalized size = 4.63 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(f*x+e))^3/(c+d*sin(f*x+e))^5,x, algorithm="giac")

[Out]

1/12*(15*(4*a^3*c - 3*a^3*d)*(pi*floor(1/2*(f*x + e)/pi + 1/2)*sgn(c) + arctan((c*tan(1/2*f*x + 1/2*e) + d)/sq
rt(c^2 - d^2)))/((c^5 + 3*c^4*d + 2*c^3*d^2 - 2*c^2*d^3 - 3*c*d^4 - d^5)*sqrt(c^2 - d^2)) + (36*a^3*c^8*tan(1/
2*f*x + 1/2*e)^7 - 117*a^3*c^7*d*tan(1/2*f*x + 1/2*e)^7 - 48*a^3*c^6*d^2*tan(1/2*f*x + 1/2*e)^7 + 48*a^3*c^5*d
^3*tan(1/2*f*x + 1/2*e)^7 + 72*a^3*c^4*d^4*tan(1/2*f*x + 1/2*e)^7 + 24*a^3*c^3*d^5*tan(1/2*f*x + 1/2*e)^7 - 72
*a^3*c^8*tan(1/2*f*x + 1/2*e)^6 + 132*a^3*c^7*d*tan(1/2*f*x + 1/2*e)^6 - 675*a^3*c^6*d^2*tan(1/2*f*x + 1/2*e)^
6 + 360*a^3*c^4*d^4*tan(1/2*f*x + 1/2*e)^6 + 288*a^3*c^3*d^5*tan(1/2*f*x + 1/2*e)^6 + 72*a^3*c^2*d^6*tan(1/2*f
*x + 1/2*e)^6 + 36*a^3*c^8*tan(1/2*f*x + 1/2*e)^5 - 813*a^3*c^7*d*tan(1/2*f*x + 1/2*e)^5 + 288*a^3*c^6*d^2*tan
(1/2*f*x + 1/2*e)^5 - 892*a^3*c^5*d^3*tan(1/2*f*x + 1/2*e)^5 + 552*a^3*c^4*d^4*tan(1/2*f*x + 1/2*e)^5 + 664*a^
3*c^3*d^5*tan(1/2*f*x + 1/2*e)^5 + 384*a^3*c^2*d^6*tan(1/2*f*x + 1/2*e)^5 + 96*a^3*c*d^7*tan(1/2*f*x + 1/2*e)^
5 - 264*a^3*c^8*tan(1/2*f*x + 1/2*e)^4 + 108*a^3*c^7*d*tan(1/2*f*x + 1/2*e)^4 - 2001*a^3*c^6*d^2*tan(1/2*f*x +
1/2*e)^4 + 936*a^3*c^5*d^3*tan(1/2*f*x + 1/2*e)^4 + 202*a^3*c^4*d^4*tan(1/2*f*x + 1/2*e)^4 + 864*a^3*c^3*d^5*
tan(1/2*f*x + 1/2*e)^4 + 440*a^3*c^2*d^6*tan(1/2*f*x + 1/2*e)^4 + 192*a^3*c*d^7*tan(1/2*f*x + 1/2*e)^4 + 48*a^
3*d^8*tan(1/2*f*x + 1/2*e)^4 - 36*a^3*c^8*tan(1/2*f*x + 1/2*e)^3 - 1299*a^3*c^7*d*tan(1/2*f*x + 1/2*e)^3 + 576
*a^3*c^6*d^2*tan(1/2*f*x + 1/2*e)^3 - 1036*a^3*c^5*d^3*tan(1/2*f*x + 1/2*e)^3 + 1176*a^3*c^4*d^4*tan(1/2*f*x +
1/2*e)^3 + 664*a^3*c^3*d^5*tan(1/2*f*x + 1/2*e)^3 + 384*a^3*c^2*d^6*tan(1/2*f*x + 1/2*e)^3 + 96*a^3*c*d^7*tan
(1/2*f*x + 1/2*e)^3 - 280*a^3*c^8*tan(1/2*f*x + 1/2*e)^2 + 12*a^3*c^7*d*tan(1/2*f*x + 1/2*e)^2 - 1289*a^3*c^6*
d^2*tan(1/2*f*x + 1/2*e)^2 + 960*a^3*c^5*d^3*tan(1/2*f*x + 1/2*e)^2 + 552*a^3*c^4*d^4*tan(1/2*f*x + 1/2*e)^2 +
288*a^3*c^3*d^5*tan(1/2*f*x + 1/2*e)^2 + 72*a^3*c^2*d^6*tan(1/2*f*x + 1/2*e)^2 - 36*a^3*c^8*tan(1/2*f*x + 1/2
*e) - 587*a^3*c^7*d*tan(1/2*f*x + 1/2*e) + 336*a^3*c^6*d^2*tan(1/2*f*x + 1/2*e) + 248*a^3*c^5*d^3*tan(1/2*f*x
+ 1/2*e) + 120*a^3*c^4*d^4*tan(1/2*f*x + 1/2*e) + 24*a^3*c^3*d^5*tan(1/2*f*x + 1/2*e) - 88*a^3*c^8 + 36*a^3*c^
7*d + 37*a^3*c^6*d^2 + 24*a^3*c^5*d^3 + 6*a^3*c^4*d^4)/((c^9 + 3*c^8*d + 2*c^7*d^2 - 2*c^6*d^3 - 3*c^5*d^4 - c
^4*d^5)*(c*tan(1/2*f*x + 1/2*e)^2 + 2*d*tan(1/2*f*x + 1/2*e) + c)^4))/f

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maple [B]  time = 0.42, size = 5149, normalized size = 17.82 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+a*sin(f*x+e))^3/(c+d*sin(f*x+e))^5,x)

[Out]

result too large to display

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(f*x+e))^3/(c+d*sin(f*x+e))^5,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 details)Is 4*d^2-4*c^2 positive or negative?

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mupad [B]  time = 10.10, size = 1231, normalized size = 4.26 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + a*sin(e + f*x))^3/(c + d*sin(e + f*x))^5,x)

[Out]

- ((6*a^3*d^4 - 88*a^3*c^4 + 24*a^3*c*d^3 + 36*a^3*c^3*d + 37*a^3*c^2*d^2)/(12*(3*c*d^4 - 3*c^4*d - c^5 + d^5
+ 2*c^2*d^3 - 2*c^3*d^2)) + (a^3*tan(e/2 + (f*x)/2)^7*(24*c*d^4 - 39*c^4*d + 12*c^5 + 8*d^5 + 16*c^2*d^3 - 16*
c^3*d^2))/(4*c*(3*c*d^4 - 3*c^4*d - c^5 + d^5 + 2*c^2*d^3 - 2*c^3*d^2)) + (a^3*tan(e/2 + (f*x)/2)^6*(96*c*d^5
+ 44*c^5*d - 24*c^6 + 24*d^6 + 120*c^2*d^4 - 225*c^4*d^2))/(4*c^2*(3*c*d^4 - 3*c^4*d - c^5 + d^5 + 2*c^2*d^3 -
2*c^3*d^2)) + (a^3*tan(e/2 + (f*x)/2)*(120*c*d^4 - 587*c^4*d - 36*c^5 + 24*d^5 + 248*c^2*d^3 + 336*c^3*d^2))/
(12*c*(3*c*d^4 - 3*c^4*d - c^5 + d^5 + 2*c^2*d^3 - 2*c^3*d^2)) + (a^3*tan(e/2 + (f*x)/2)^5*(384*c*d^6 - 813*c^
6*d + 36*c^7 + 96*d^7 + 664*c^2*d^5 + 552*c^3*d^4 - 892*c^4*d^3 + 288*c^5*d^2))/(12*c^3*(3*c*d^4 - 3*c^4*d - c
^5 + d^5 + 2*c^2*d^3 - 2*c^3*d^2)) + (a^3*tan(e/2 + (f*x)/2)^3*(384*c*d^6 - 1299*c^6*d - 36*c^7 + 96*d^7 + 664
*c^2*d^5 + 1176*c^3*d^4 - 1036*c^4*d^3 + 576*c^5*d^2))/(12*c^3*(3*c*d^4 - 3*c^4*d - c^5 + d^5 + 2*c^2*d^3 - 2*
c^3*d^2)) + (a^3*tan(e/2 + (f*x)/2)^2*(288*c*d^5 + 12*c^5*d - 280*c^6 + 72*d^6 + 552*c^2*d^4 + 960*c^3*d^3 - 1
289*c^4*d^2))/(12*c^2*(3*c*d^4 - 3*c^4*d - c^5 + d^5 + 2*c^2*d^3 - 2*c^3*d^2)) + (a^3*tan(e/2 + (f*x)/2)^4*(3*
c^4 + 8*d^4 + 24*c^2*d^2)*(24*c*d^3 + 36*c^3*d - 88*c^4 + 6*d^4 + 37*c^2*d^2))/(12*c^4*(3*c*d^4 - 3*c^4*d - c^
5 + d^5 + 2*c^2*d^3 - 2*c^3*d^2)))/(f*(tan(e/2 + (f*x)/2)^4*(6*c^4 + 16*d^4 + 48*c^2*d^2) + c^4*tan(e/2 + (f*x
)/2)^8 + c^4 + tan(e/2 + (f*x)/2)^2*(4*c^4 + 24*c^2*d^2) + tan(e/2 + (f*x)/2)^6*(4*c^4 + 24*c^2*d^2) + tan(e/2
+ (f*x)/2)^3*(32*c*d^3 + 24*c^3*d) + tan(e/2 + (f*x)/2)^5*(32*c*d^3 + 24*c^3*d) + 8*c^3*d*tan(e/2 + (f*x)/2)
+ 8*c^3*d*tan(e/2 + (f*x)/2)^7)) - (5*a^3*atan((4*((5*a^3*(4*c - 3*d)*(24*c*d^5 - 8*c^5*d + 8*d^6 + 16*c^2*d^4
- 16*c^3*d^3 - 24*c^4*d^2))/(32*(c + d)^(9/2)*(c - d)^(3/2)*(3*c*d^4 - 3*c^4*d - c^5 + d^5 + 2*c^2*d^3 - 2*c^
3*d^2)) + (5*a^3*c*tan(e/2 + (f*x)/2)*(4*c - 3*d))/(4*(c + d)^(9/2)*(c - d)^(3/2)))*(3*c*d^4 - 3*c^4*d - c^5 +
d^5 + 2*c^2*d^3 - 2*c^3*d^2))/(20*a^3*c - 15*a^3*d))*(4*c - 3*d))/(4*f*(c + d)^(9/2)*(c - d)^(3/2))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(f*x+e))**3/(c+d*sin(f*x+e))**5,x)

[Out]

Timed out

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