\(\int \frac {(3+b \sin (e+f x))^3}{(c+d \sin (e+f x))^{3/2}} \, dx\) [741]

Optimal result

Integrand size = 27, antiderivative size = 345 \[ \int \frac {(3+b \sin (e+f x))^3}{(c+d \sin (e+f x))^{3/2}} \, dx=\frac {2 (b c-3 d)^2 \cos (e+f x) (3+b \sin (e+f x))}{d \left (c^2-d^2\right ) f \sqrt {c+d \sin (e+f x)}}+\frac {2 b \left (18 b c d-27 d^2-b^2 \left (4 c^2-d^2\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 d^2 \left (c^2-d^2\right ) f}-\frac {2 \left (81 b c d^2-81 d^3-27 b^2 d \left (2 c^2-d^2\right )+b^3 \left (8 c^3-5 c d^2\right )\right ) E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{3 d^3 \left (c^2-d^2\right ) f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {2 b \left (54 b c d-81 d^2-b^2 \left (8 c^2+d^2\right )\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{3 d^3 f \sqrt {c+d \sin (e+f x)}} \]

[Out]

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))^(1/2)+2/3*b*(6*a*b*c*d-3*a^2*d^2-b^2
*(4*c^2-d^2))*cos(f*x+e)*(c+d*sin(f*x+e))^(1/2)/d^2/(c^2-d^2)/f+2/3*(9*a^2*b*c*d^2-3*a^3*d^3-9*a*b^2*d*(2*c^2-
d^2)+b^3*(8*c^3-5*c*d^2))*(sin(1/2*e+1/4*Pi+1/2*f*x)^2)^(1/2)/sin(1/2*e+1/4*Pi+1/2*f*x)*EllipticE(cos(1/2*e+1/
4*Pi+1/2*f*x),2^(1/2)*(d/(c+d))^(1/2))*(c+d*sin(f*x+e))^(1/2)/d^3/(c^2-d^2)/f/((c+d*sin(f*x+e))/(c+d))^(1/2)+2
/3*b*(18*a*b*c*d-9*a^2*d^2-b^2*(8*c^2+d^2))*(sin(1/2*e+1/4*Pi+1/2*f*x)^2)^(1/2)/sin(1/2*e+1/4*Pi+1/2*f*x)*Elli
pticF(cos(1/2*e+1/4*Pi+1/2*f*x),2^(1/2)*(d/(c+d))^(1/2))*((c+d*sin(f*x+e))/(c+d))^(1/2)/d^3/f/(c+d*sin(f*x+e))
^(1/2)

Rubi [A] (verified)

Time = 0.43 (sec) , antiderivative size = 361, normalized size of antiderivative = 1.05, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {2871, 3102, 2831, 2742, 2740, 2734, 2732} \[ \int \frac {(3+b \sin (e+f x))^3}{(c+d \sin (e+f x))^{3/2}} \, dx=\frac {2 b \left (-3 a^2 d^2+6 a b c d-\left (b^2 \left (4 c^2-d^2\right )\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 d^2 f \left (c^2-d^2\right )}-\frac {2 b \left (-9 a^2 d^2+18 a b c d-\left (b^2 \left (8 c^2+d^2\right )\right )\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{3 d^3 f \sqrt {c+d \sin (e+f x)}}-\frac {2 \left (-3 a^3 d^3+9 a^2 b c d^2-9 a b^2 d \left (2 c^2-d^2\right )+b^3 \left (8 c^3-5 c d^2\right )\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{3 d^3 f \left (c^2-d^2\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{d f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}} \]

[In]

Int[(a + b*Sin[e + f*x])^3/(c + d*Sin[e + f*x])^(3/2),x]

[Out]

(2*(b*c - a*d)^2*Cos[e + f*x]*(a + b*Sin[e + f*x]))/(d*(c^2 - d^2)*f*Sqrt[c + d*Sin[e + f*x]]) + (2*b*(6*a*b*c
*d - 3*a^2*d^2 - b^2*(4*c^2 - d^2))*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(3*d^2*(c^2 - d^2)*f) - (2*(9*a^2*b
*c*d^2 - 3*a^3*d^3 - 9*a*b^2*d*(2*c^2 - d^2) + b^3*(8*c^3 - 5*c*d^2))*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c +
d)]*Sqrt[c + d*Sin[e + f*x]])/(3*d^3*(c^2 - d^2)*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) - (2*b*(18*a*b*c*d - 9
*a^2*d^2 - b^2*(8*c^2 + d^2))*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])
/(3*d^3*f*Sqrt[c + d*Sin[e + f*x]])

Rule 2732

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[2*(Sqrt[a + b]/d)*EllipticE[(1/2)*(c - Pi/2
+ d*x), 2*(b/(a + b))], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]

Rule 2734

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[Sqrt[a + b*Sin[c + d*x]]/Sqrt[(a + b*Sin[c +
d*x])/(a + b)], Int[Sqrt[a/(a + b) + (b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 -
b^2, 0] &&  !GtQ[a + b, 0]

Rule 2740

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/(d*Sqrt[a + b]))*EllipticF[(1/2)*(c - P
i/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]

Rule 2742

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[Sqrt[(a + b*Sin[c + d*x])/(a + b)]/Sqrt[a
+ b*Sin[c + d*x]], Int[1/Sqrt[a/(a + b) + (b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a
^2 - b^2, 0] &&  !GtQ[a + b, 0]

Rule 2831

Int[((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])/Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]], x_Symbol] :> Dist[(b*c
- a*d)/b, Int[1/Sqrt[a + b*Sin[e + f*x]], x], x] + Dist[d/b, Int[Sqrt[a + b*Sin[e + f*x]], x], x] /; FreeQ[{a
, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 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 3102

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

Rubi steps \begin{align*} \text {integral}& = \frac {2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{d \left (c^2-d^2\right ) f \sqrt {c+d \sin (e+f x)}}-\frac {2 \int \frac {\frac {1}{2} \left (2 b (b c-a d)^2-a d \left (\left (a^2+b^2\right ) c-2 a b d\right )\right )+\frac {1}{2} \left (a^2 b c d-b^3 c d-a^3 d^2-a b^2 \left (2 c^2-3 d^2\right )\right ) \sin (e+f x)+\frac {1}{2} b \left (6 a b c d-3 a^2 d^2-b^2 \left (4 c^2-d^2\right )\right ) \sin ^2(e+f x)}{\sqrt {c+d \sin (e+f x)}} \, dx}{d \left (c^2-d^2\right )} \\ & = \frac {2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{d \left (c^2-d^2\right ) f \sqrt {c+d \sin (e+f x)}}+\frac {2 b \left (6 a b c d-3 a^2 d^2-b^2 \left (4 c^2-d^2\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 d^2 \left (c^2-d^2\right ) f}-\frac {4 \int \frac {-\frac {1}{4} d \left (3 a^3 c d+9 a b^2 c d-9 a^2 b d^2-b^3 \left (2 c^2+d^2\right )\right )+\frac {1}{4} \left (9 a^2 b c d^2-3 a^3 d^3-9 a b^2 d \left (2 c^2-d^2\right )+b^3 \left (8 c^3-5 c d^2\right )\right ) \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}} \, dx}{3 d^2 \left (c^2-d^2\right )} \\ & = \frac {2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{d \left (c^2-d^2\right ) f \sqrt {c+d \sin (e+f x)}}+\frac {2 b \left (6 a b c d-3 a^2 d^2-b^2 \left (4 c^2-d^2\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 d^2 \left (c^2-d^2\right ) f}-\frac {\left (b \left (18 a b c d-9 a^2 d^2-b^2 \left (8 c^2+d^2\right )\right )\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}} \, dx}{3 d^3}-\frac {\left (9 a^2 b c d^2-3 a^3 d^3-9 a b^2 d \left (2 c^2-d^2\right )+b^3 \left (8 c^3-5 c d^2\right )\right ) \int \sqrt {c+d \sin (e+f x)} \, dx}{3 d^3 \left (c^2-d^2\right )} \\ & = \frac {2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{d \left (c^2-d^2\right ) f \sqrt {c+d \sin (e+f x)}}+\frac {2 b \left (6 a b c d-3 a^2 d^2-b^2 \left (4 c^2-d^2\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 d^2 \left (c^2-d^2\right ) f}-\frac {\left (\left (9 a^2 b c d^2-3 a^3 d^3-9 a b^2 d \left (2 c^2-d^2\right )+b^3 \left (8 c^3-5 c d^2\right )\right ) \sqrt {c+d \sin (e+f x)}\right ) \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}} \, dx}{3 d^3 \left (c^2-d^2\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {\left (b \left (18 a b c d-9 a^2 d^2-b^2 \left (8 c^2+d^2\right )\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}\right ) \int \frac {1}{\sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}} \, dx}{3 d^3 \sqrt {c+d \sin (e+f x)}} \\ & = \frac {2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{d \left (c^2-d^2\right ) f \sqrt {c+d \sin (e+f x)}}+\frac {2 b \left (6 a b c d-3 a^2 d^2-b^2 \left (4 c^2-d^2\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 d^2 \left (c^2-d^2\right ) f}-\frac {2 \left (9 a^2 b c d^2-3 a^3 d^3-9 a b^2 d \left (2 c^2-d^2\right )+b^3 \left (8 c^3-5 c d^2\right )\right ) E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{3 d^3 \left (c^2-d^2\right ) f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {2 b \left (18 a b c d-9 a^2 d^2-b^2 \left (8 c^2+d^2\right )\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{3 d^3 f \sqrt {c+d \sin (e+f x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 2.85 (sec) , antiderivative size = 290, normalized size of antiderivative = 0.84 \[ \int \frac {(3+b \sin (e+f x))^3}{(c+d \sin (e+f x))^{3/2}} \, dx=\frac {2 \left (\frac {\left (d^2 \left (-81 c d-27 b^2 c d+81 b d^2+b^3 \left (2 c^2+d^2\right )\right ) \operatorname {EllipticF}\left (\frac {1}{4} (-2 e+\pi -2 f x),\frac {2 d}{c+d}\right )+\left (81 b c d^2-81 d^3+27 b^2 d \left (-2 c^2+d^2\right )+b^3 \left (8 c^3-5 c d^2\right )\right ) \left ((c+d) E\left (\frac {1}{4} (-2 e+\pi -2 f x)|\frac {2 d}{c+d}\right )-c \operatorname {EllipticF}\left (\frac {1}{4} (-2 e+\pi -2 f x),\frac {2 d}{c+d}\right )\right )\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{(c-d) (c+d)}-\frac {d \cos (e+f x) \left (27 b^2 c^2 d-81 b c d^2+81 d^3+b^3 \left (-4 c^3+c d^2\right )+b^3 d \left (-c^2+d^2\right ) \sin (e+f x)\right )}{-c^2+d^2}\right )}{3 d^3 f \sqrt {c+d \sin (e+f x)}} \]

[In]

Integrate[(3 + b*Sin[e + f*x])^3/(c + d*Sin[e + f*x])^(3/2),x]

[Out]

(2*(((d^2*(-81*c*d - 27*b^2*c*d + 81*b*d^2 + b^3*(2*c^2 + d^2))*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)
] + (81*b*c*d^2 - 81*d^3 + 27*b^2*d*(-2*c^2 + d^2) + b^3*(8*c^3 - 5*c*d^2))*((c + d)*EllipticE[(-2*e + Pi - 2*
f*x)/4, (2*d)/(c + d)] - c*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)]))*Sqrt[(c + d*Sin[e + f*x])/(c + d)
])/((c - d)*(c + d)) - (d*Cos[e + f*x]*(27*b^2*c^2*d - 81*b*c*d^2 + 81*d^3 + b^3*(-4*c^3 + c*d^2) + b^3*d*(-c^
2 + d^2)*Sin[e + f*x]))/(-c^2 + d^2)))/(3*d^3*f*Sqrt[c + d*Sin[e + f*x]])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1397\) vs. \(2(409)=818\).

Time = 14.57 (sec) , antiderivative size = 1398, normalized size of antiderivative = 4.05

method	result	size
default	\(\text {Expression too large to display}\)	\(1398\)
parts	\(\text {Expression too large to display}\)	\(2577\)

[In]

int((a+b*sin(f*x+e))^3/(c+d*sin(f*x+e))^(3/2),x,method=_RETURNVERBOSE)

[Out]

(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)*((a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c^3)/d^3*(2*d*cos(f*x+e)^2/(
c^2-d^2)/(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)+2*c/(c^2-d^2)*(c/d-1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(1-si
n(f*x+e))/(c+d))^(1/2)*(1/(c-d)*(-sin(f*x+e)-1)*d)^(1/2)/(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)*EllipticF(((c
+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))+2/(c^2-d^2)*d*(c/d-1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(1-si
n(f*x+e))/(c+d))^(1/2)*(1/(c-d)*(-sin(f*x+e)-1)*d)^(1/2)/(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)*((-c/d-1)*Ell
ipticE(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))+EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+
d))^(1/2))))+b/d^3*(2*b^2*c^2*(c/d-1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c+d))^(1/2)*(1/(c-d)*(
-sin(f*x+e)-1)*d)^(1/2)/(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)*EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d
)/(c+d))^(1/2))+6*d^2*a^2*(c/d-1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c+d))^(1/2)*(1/(c-d)*(-sin
(f*x+e)-1)*d)^(1/2)/(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)*EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c
+d))^(1/2))+d^2*b^2*(-2/3/d*(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)+2/3*(c/d-1)*((c+d*sin(f*x+e))/(c-d))^(1/2)
*(d*(1-sin(f*x+e))/(c+d))^(1/2)*(1/(c-d)*(-sin(f*x+e)-1)*d)^(1/2)/(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)*Elli
pticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))-4/3*c/d*(c/d-1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(1
-sin(f*x+e))/(c+d))^(1/2)*(1/(c-d)*(-sin(f*x+e)-1)*d)^(1/2)/(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)*((-c/d-1)*
EllipticE(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))+EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/
(c+d))^(1/2))))-6*a*b*c*d*(c/d-1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c+d))^(1/2)*(1/(c-d)*(-sin
(f*x+e)-1)*d)^(1/2)/(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)*EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c
+d))^(1/2))+2*(3*a*b*d^2-b^2*c*d)*(c/d-1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c+d))^(1/2)*(1/(c-
d)*(-sin(f*x+e)-1)*d)^(1/2)/(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)*((-c/d-1)*EllipticE(((c+d*sin(f*x+e))/(c-d
))^(1/2),((c-d)/(c+d))^(1/2))+EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2)))))/cos(f*x+e)/(c+d
*sin(f*x+e))^(1/2)/f

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.15 (sec) , antiderivative size = 1066, normalized size of antiderivative = 3.09 \[ \int \frac {(3+b \sin (e+f x))^3}{(c+d \sin (e+f x))^{3/2}} \, dx=\text {Too large to display} \]

[In]

integrate((a+b*sin(f*x+e))^3/(c+d*sin(f*x+e))^(3/2),x, algorithm="fricas")

[Out]

1/9*((sqrt(2)*(16*b^3*c^4*d - 36*a*b^2*c^3*d^2 + 2*(9*a^2*b - 8*b^3)*c^2*d^3 + 3*(a^3 + 15*a*b^2)*c*d^4 - 3*(9
*a^2*b + b^3)*d^5)*sin(f*x + e) + sqrt(2)*(16*b^3*c^5 - 36*a*b^2*c^4*d + 2*(9*a^2*b - 8*b^3)*c^3*d^2 + 3*(a^3
+ 15*a*b^2)*c^2*d^3 - 3*(9*a^2*b + b^3)*c*d^4))*sqrt(I*d)*weierstrassPInverse(-4/3*(4*c^2 - 3*d^2)/d^2, -8/27*
(8*I*c^3 - 9*I*c*d^2)/d^3, 1/3*(3*d*cos(f*x + e) - 3*I*d*sin(f*x + e) - 2*I*c)/d) + (sqrt(2)*(16*b^3*c^4*d - 3
6*a*b^2*c^3*d^2 + 2*(9*a^2*b - 8*b^3)*c^2*d^3 + 3*(a^3 + 15*a*b^2)*c*d^4 - 3*(9*a^2*b + b^3)*d^5)*sin(f*x + e)
+ sqrt(2)*(16*b^3*c^5 - 36*a*b^2*c^4*d + 2*(9*a^2*b - 8*b^3)*c^3*d^2 + 3*(a^3 + 15*a*b^2)*c^2*d^3 - 3*(9*a^2*
b + b^3)*c*d^4))*sqrt(-I*d)*weierstrassPInverse(-4/3*(4*c^2 - 3*d^2)/d^2, -8/27*(-8*I*c^3 + 9*I*c*d^2)/d^3, 1/
3*(3*d*cos(f*x + e) + 3*I*d*sin(f*x + e) + 2*I*c)/d) + 3*(sqrt(2)*(8*I*b^3*c^3*d^2 - 18*I*a*b^2*c^2*d^3 + I*(9
*a^2*b - 5*b^3)*c*d^4 - 3*I*(a^3 - 3*a*b^2)*d^5)*sin(f*x + e) + sqrt(2)*(8*I*b^3*c^4*d - 18*I*a*b^2*c^3*d^2 +
I*(9*a^2*b - 5*b^3)*c^2*d^3 - 3*I*(a^3 - 3*a*b^2)*c*d^4))*sqrt(I*d)*weierstrassZeta(-4/3*(4*c^2 - 3*d^2)/d^2,
-8/27*(8*I*c^3 - 9*I*c*d^2)/d^3, weierstrassPInverse(-4/3*(4*c^2 - 3*d^2)/d^2, -8/27*(8*I*c^3 - 9*I*c*d^2)/d^3
, 1/3*(3*d*cos(f*x + e) - 3*I*d*sin(f*x + e) - 2*I*c)/d)) + 3*(sqrt(2)*(-8*I*b^3*c^3*d^2 + 18*I*a*b^2*c^2*d^3
- I*(9*a^2*b - 5*b^3)*c*d^4 + 3*I*(a^3 - 3*a*b^2)*d^5)*sin(f*x + e) + sqrt(2)*(-8*I*b^3*c^4*d + 18*I*a*b^2*c^3
*d^2 - I*(9*a^2*b - 5*b^3)*c^2*d^3 + 3*I*(a^3 - 3*a*b^2)*c*d^4))*sqrt(-I*d)*weierstrassZeta(-4/3*(4*c^2 - 3*d^
2)/d^2, -8/27*(-8*I*c^3 + 9*I*c*d^2)/d^3, weierstrassPInverse(-4/3*(4*c^2 - 3*d^2)/d^2, -8/27*(-8*I*c^3 + 9*I*
c*d^2)/d^3, 1/3*(3*d*cos(f*x + e) + 3*I*d*sin(f*x + e) + 2*I*c)/d)) - 6*((b^3*c^2*d^3 - b^3*d^5)*cos(f*x + e)*
sin(f*x + e) + (4*b^3*c^3*d^2 - 9*a*b^2*c^2*d^3 - 3*a^3*d^5 + (9*a^2*b - b^3)*c*d^4)*cos(f*x + e))*sqrt(d*sin(
f*x + e) + c))/((c^2*d^5 - d^7)*f*sin(f*x + e) + (c^3*d^4 - c*d^6)*f)

Sympy [F(-1)]

Timed out. \[ \int \frac {(3+b \sin (e+f x))^3}{(c+d \sin (e+f x))^{3/2}} \, dx=\text {Timed out} \]

[In]

integrate((a+b*sin(f*x+e))**3/(c+d*sin(f*x+e))**(3/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {(3+b \sin (e+f x))^3}{(c+d \sin (e+f x))^{3/2}} \, dx=\int { \frac {{\left (b \sin \left (f x + e\right ) + a\right )}^{3}}{{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}}} \,d x } \]

[In]

integrate((a+b*sin(f*x+e))^3/(c+d*sin(f*x+e))^(3/2),x, algorithm="maxima")

[Out]

integrate((b*sin(f*x + e) + a)^3/(d*sin(f*x + e) + c)^(3/2), x)

Giac [F]

\[ \int \frac {(3+b \sin (e+f x))^3}{(c+d \sin (e+f x))^{3/2}} \, dx=\int { \frac {{\left (b \sin \left (f x + e\right ) + a\right )}^{3}}{{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}}} \,d x } \]

[In]

integrate((a+b*sin(f*x+e))^3/(c+d*sin(f*x+e))^(3/2),x, algorithm="giac")

[Out]

integrate((b*sin(f*x + e) + a)^3/(d*sin(f*x + e) + c)^(3/2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {(3+b \sin (e+f x))^3}{(c+d \sin (e+f x))^{3/2}} \, dx=\int \frac {{\left (a+b\,\sin \left (e+f\,x\right )\right )}^3}{{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{3/2}} \,d x \]

[In]

int((a + b*sin(e + f*x))^3/(c + d*sin(e + f*x))^(3/2),x)

[Out]

int((a + b*sin(e + f*x))^3/(c + d*sin(e + f*x))^(3/2), x)