\(\int \frac {(3+b \sin (e+f x))^3}{(c+d \sin (e+f x))^{5/2}} \, dx\) [742]

Optimal result

Integrand size = 27, antiderivative size = 377 \[ \int \frac {(3+b \sin (e+f x))^3}{(c+d \sin (e+f x))^{5/2}} \, dx=\frac {2 (b c-3 d)^2 \cos (e+f x) (3+b \sin (e+f x))}{3 d \left (c^2-d^2\right ) f (c+d \sin (e+f x))^{3/2}}+\frac {8 (b c-3 d)^2 \left (3 c d+b \left (c^2-2 d^2\right )\right ) \cos (e+f x)}{3 d^2 \left (c^2-d^2\right )^2 f \sqrt {c+d \sin (e+f x)}}+\frac {2 \left (108 c d^3-18 b^2 c d \left (c^2-3 d^2\right )-27 b d^2 \left (c^2+3 d^2\right )+b^3 \left (8 c^4-15 c^2 d^2+3 d^4\right )\right ) E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{3 d^3 \left (c^2-d^2\right )^2 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {2 (b c-3 d) \left (6 b c d-9 d^2+b^2 \left (8 c^2-9 d^2\right )\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{3 d^3 \left (c^2-d^2\right ) f \sqrt {c+d \sin (e+f x)}} \]

[Out]

2/3*(-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))^(3/2)+8/3*(-a*d+b*c)^2*(a*c*d+b*(c
^2-2*d^2))*cos(f*x+e)/d^2/(c^2-d^2)^2/f/(c+d*sin(f*x+e))^(1/2)-2/3*(4*a^3*c*d^3-6*a*b^2*c*d*(c^2-3*d^2)-3*a^2*
b*d^2*(c^2+3*d^2)+b^3*(8*c^4-15*c^2*d^2+3*d^4))*(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)^2/f/((c+d*si
n(f*x+e))/(c+d))^(1/2)+2/3*(-a*d+b*c)*(2*a*b*c*d-a^2*d^2+b^2*(8*c^2-9*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)*EllipticF(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/(c^2-d^2)/f/(c+d*sin(f*x+e))^(1/2)

Rubi [A] (verified)

Time = 0.48 (sec) , antiderivative size = 391, normalized size of antiderivative = 1.04, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {2871, 3100, 2831, 2742, 2740, 2734, 2732} \[ \int \frac {(3+b \sin (e+f x))^3}{(c+d \sin (e+f x))^{5/2}} \, dx=-\frac {2 \left (-a^2 d^2+2 a b c d+b^2 \left (8 c^2-9 d^2\right )\right ) (b c-a d) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{3 d^3 f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}+\frac {2 \left (4 a^3 c d^3-3 a^2 b d^2 \left (c^2+3 d^2\right )-6 a b^2 c d \left (c^2-3 d^2\right )+b^3 \left (8 c^4-15 c^2 d^2+3 d^4\right )\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{3 d^3 f \left (c^2-d^2\right )^2 \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {8 \left (a c d+b \left (c^2-2 d^2\right )\right ) (b c-a d)^2 \cos (e+f x)}{3 d^2 f \left (c^2-d^2\right )^2 \sqrt {c+d \sin (e+f x)}}+\frac {2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{3 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}} \]

[In]

Int[(a + b*Sin[e + f*x])^3/(c + d*Sin[e + f*x])^(5/2),x]

[Out]

(2*(b*c - a*d)^2*Cos[e + f*x]*(a + b*Sin[e + f*x]))/(3*d*(c^2 - d^2)*f*(c + d*Sin[e + f*x])^(3/2)) + (8*(b*c -
a*d)^2*(a*c*d + b*(c^2 - 2*d^2))*Cos[e + f*x])/(3*d^2*(c^2 - d^2)^2*f*Sqrt[c + d*Sin[e + f*x]]) + (2*(4*a^3*c
*d^3 - 6*a*b^2*c*d*(c^2 - 3*d^2) - 3*a^2*b*d^2*(c^2 + 3*d^2) + b^3*(8*c^4 - 15*c^2*d^2 + 3*d^4))*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)^2*f*Sqrt[(c + d*Sin[e + f*x])/(c
+ d)]) - (2*(b*c - a*d)*(2*a*b*c*d - a^2*d^2 + b^2*(8*c^2 - 9*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*(c^2 - d^2)*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 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*} \text {integral}& = \frac {2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{3 d \left (c^2-d^2\right ) f (c+d \sin (e+f x))^{3/2}}-\frac {2 \int \frac {\frac {1}{2} \left (2 b (b c-a d)^2-3 a d \left (\left (a^2+b^2\right ) c-2 a b d\right )\right )-\frac {1}{2} \left (5 a^2 b c d+3 b^3 c d-a^3 d^2+a b^2 \left (2 c^2-9 d^2\right )\right ) \sin (e+f x)+\frac {1}{2} b \left (2 a b c d-a^2 d^2-b^2 \left (4 c^2-3 d^2\right )\right ) \sin ^2(e+f x)}{(c+d \sin (e+f x))^{3/2}} \, dx}{3 d \left (c^2-d^2\right )} \\ & = \frac {2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{3 d \left (c^2-d^2\right ) f (c+d \sin (e+f x))^{3/2}}+\frac {8 (b c-a d)^2 \left (a c d+b \left (c^2-2 d^2\right )\right ) \cos (e+f x)}{3 d^2 \left (c^2-d^2\right )^2 f \sqrt {c+d \sin (e+f x)}}+\frac {4 \int \frac {-\frac {1}{4} d \left (12 a^2 b c d^2-a^3 d \left (3 c^2+d^2\right )-3 a b^2 d \left (c^2+3 d^2\right )-2 b^3 \left (c^3-3 c d^2\right )\right )+\frac {1}{4} \left (4 a^3 c d^3-6 a b^2 c d \left (c^2-3 d^2\right )-3 a^2 b d^2 \left (c^2+3 d^2\right )+b^3 \left (8 c^4-15 c^2 d^2+3 d^4\right )\right ) \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}} \, dx}{3 d^2 \left (c^2-d^2\right )^2} \\ & = \frac {2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{3 d \left (c^2-d^2\right ) f (c+d \sin (e+f x))^{3/2}}+\frac {8 (b c-a d)^2 \left (a c d+b \left (c^2-2 d^2\right )\right ) \cos (e+f x)}{3 d^2 \left (c^2-d^2\right )^2 f \sqrt {c+d \sin (e+f x)}}+\frac {\left (4 a^3 c d^3-6 a b^2 c d \left (c^2-3 d^2\right )-3 a^2 b d^2 \left (c^2+3 d^2\right )+b^3 \left (8 c^4-15 c^2 d^2+3 d^4\right )\right ) \int \sqrt {c+d \sin (e+f x)} \, dx}{3 d^3 \left (c^2-d^2\right )^2}--\frac {\left (4 \left (-\frac {1}{4} d^2 \left (12 a^2 b c d^2-a^3 d \left (3 c^2+d^2\right )-3 a b^2 d \left (c^2+3 d^2\right )-2 b^3 \left (c^3-3 c d^2\right )\right )-\frac {1}{4} c \left (4 a^3 c d^3-6 a b^2 c d \left (c^2-3 d^2\right )-3 a^2 b d^2 \left (c^2+3 d^2\right )+b^3 \left (8 c^4-15 c^2 d^2+3 d^4\right )\right )\right )\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}} \, dx}{3 d^3 \left (c^2-d^2\right )^2} \\ & = \frac {2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{3 d \left (c^2-d^2\right ) f (c+d \sin (e+f x))^{3/2}}+\frac {8 (b c-a d)^2 \left (a c d+b \left (c^2-2 d^2\right )\right ) \cos (e+f x)}{3 d^2 \left (c^2-d^2\right )^2 f \sqrt {c+d \sin (e+f x)}}+\frac {\left (\left (4 a^3 c d^3-6 a b^2 c d \left (c^2-3 d^2\right )-3 a^2 b d^2 \left (c^2+3 d^2\right )+b^3 \left (8 c^4-15 c^2 d^2+3 d^4\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 )^2 \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}--\frac {\left (4 \left (-\frac {1}{4} d^2 \left (12 a^2 b c d^2-a^3 d \left (3 c^2+d^2\right )-3 a b^2 d \left (c^2+3 d^2\right )-2 b^3 \left (c^3-3 c d^2\right )\right )-\frac {1}{4} c \left (4 a^3 c d^3-6 a b^2 c d \left (c^2-3 d^2\right )-3 a^2 b d^2 \left (c^2+3 d^2\right )+b^3 \left (8 c^4-15 c^2 d^2+3 d^4\right )\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 \left (c^2-d^2\right )^2 \sqrt {c+d \sin (e+f x)}} \\ & = \frac {2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{3 d \left (c^2-d^2\right ) f (c+d \sin (e+f x))^{3/2}}+\frac {8 (b c-a d)^2 \left (a c d+b \left (c^2-2 d^2\right )\right ) \cos (e+f x)}{3 d^2 \left (c^2-d^2\right )^2 f \sqrt {c+d \sin (e+f x)}}+\frac {2 \left (4 a^3 c d^3-6 a b^2 c d \left (c^2-3 d^2\right )-3 a^2 b d^2 \left (c^2+3 d^2\right )+b^3 \left (8 c^4-15 c^2 d^2+3 d^4\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 )^2 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {2 (b c-a d) \left (8 b^2 c^2+2 a b c d-a^2 d^2-9 b^2 d^2\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 \left (c^2-d^2\right ) f \sqrt {c+d \sin (e+f x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 4.11 (sec) , antiderivative size = 340, normalized size of antiderivative = 0.90 \[ \int \frac {(3+b \sin (e+f x))^3}{(c+d \sin (e+f x))^{5/2}} \, dx=\frac {2 \left (\frac {\left (d^2 \left (-108 b c d^2+9 b^2 d \left (c^2+3 d^2\right )+2 b^3 \left (c^3-3 c d^2\right )+27 \left (3 c^2 d+d^3\right )\right ) \operatorname {EllipticF}\left (\frac {1}{4} (-2 e+\pi -2 f x),\frac {2 d}{c+d}\right )+\left (108 c d^3-18 b^2 c d \left (c^2-3 d^2\right )-27 b d^2 \left (c^2+3 d^2\right )+b^3 \left (8 c^4-15 c^2 d^2+3 d^4\right )\right ) \left ((c+d) E\left (\frac {1}{4} (-2 e+\pi -2 f x)|\frac {2 d}{c+d}\right )-c \operatorname {EllipticF}\left (\frac {1}{4} (-2 e+\pi -2 f x),\frac {2 d}{c+d}\right )\right )\right ) (-c-d \sin (e+f x)) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{(c-d)^2 (c+d)^2}+\frac {(b c-3 d)^2 d \cos (e+f x) \left (15 c^2 d-3 d^3+4 b \left (c^3-2 c d^2\right )+d \left (5 b c^2+12 c d-9 b d^2\right ) \sin (e+f x)\right )}{\left (c^2-d^2\right )^2}\right )}{3 d^3 f (c+d \sin (e+f x))^{3/2}} \]

[In]

Integrate[(3 + b*Sin[e + f*x])^3/(c + d*Sin[e + f*x])^(5/2),x]

[Out]

(2*(((d^2*(-108*b*c*d^2 + 9*b^2*d*(c^2 + 3*d^2) + 2*b^3*(c^3 - 3*c*d^2) + 27*(3*c^2*d + d^3))*EllipticF[(-2*e
+ Pi - 2*f*x)/4, (2*d)/(c + d)] + (108*c*d^3 - 18*b^2*c*d*(c^2 - 3*d^2) - 27*b*d^2*(c^2 + 3*d^2) + b^3*(8*c^4
- 15*c^2*d^2 + 3*d^4))*((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)]))*(-c - d*Sin[e + f*x])*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/((c - d)^2*(c + d)^2) + ((b*
c - 3*d)^2*d*Cos[e + f*x]*(15*c^2*d - 3*d^3 + 4*b*(c^3 - 2*c*d^2) + d*(5*b*c^2 + 12*c*d - 9*b*d^2)*Sin[e + f*x
]))/(c^2 - d^2)^2))/(3*d^3*f*(c + d*Sin[e + f*x])^(3/2))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1378\) vs. \(2(437)=874\).

Time = 29.94 (sec) , antiderivative size = 1379, normalized size of antiderivative = 3.66

method	result	size
default	\(\text {Expression too large to display}\)	\(1379\)
parts	\(\text {Expression too large to display}\)	\(3598\)

[In]

int((a+b*sin(f*x+e))^3/(c+d*sin(f*x+e))^(5/2),x,method=_RETURNVERBOSE)

[Out]

(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)*(b^2/d^3*(6*d*a*(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))-4*c*b*(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*b*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))))+3*b/d^3*(a
^2*d^2-2*a*b*c*d+b^2*c^2)*(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-sin(f*x+e))/(c+d))^(1/2)*(1/(c-d)*(-sin(f*x+e)-1)*d)^(1/2)/(-(-d*si
n(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-sin(f*x+e))/(c+d))^(1/2)*(1/(c-d)*(-sin(f*x+e)-1)*d)^(1/2)/(-(-d*si
n(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))+Ellipt
icF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))))+1/d^3*(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c^3)*
(2/3/(c^2-d^2)/d*(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)/(sin(f*x+e)+c/d)^2+8/3*d*cos(f*x+e)^2/(c^2-d^2)^2*c/(
-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)+2*(3*c^2+d^2)/(3*c^4-6*c^2*d^2+3*d^4)*(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))+8/3*c*d/(c^2-d^2)^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)*((-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 = 0.21 (sec) , antiderivative size = 1892, normalized size of antiderivative = 5.02 \[ \int \frac {(3+b \sin (e+f x))^3}{(c+d \sin (e+f x))^{5/2}} \, dx=\text {Too large to display} \]

[In]

integrate((a+b*sin(f*x+e))^3/(c+d*sin(f*x+e))^(5/2),x, algorithm="fricas")

[Out]

-1/9*((sqrt(2)*(16*b^3*c^5*d^2 - 12*a*b^2*c^4*d^3 - 6*(a^2*b + 6*b^3)*c^3*d^4 - (a^3 - 27*a*b^2)*c^2*d^5 + 6*(
3*a^2*b + 4*b^3)*c*d^6 - 3*(a^3 + 9*a*b^2)*d^7)*cos(f*x + e)^2 - 2*sqrt(2)*(16*b^3*c^6*d - 12*a*b^2*c^5*d^2 -
6*(a^2*b + 6*b^3)*c^4*d^3 - (a^3 - 27*a*b^2)*c^3*d^4 + 6*(3*a^2*b + 4*b^3)*c^2*d^5 - 3*(a^3 + 9*a*b^2)*c*d^6)*
sin(f*x + e) - sqrt(2)*(16*b^3*c^7 - 12*a*b^2*c^6*d - 4*a^3*c^2*d^5 - 2*(3*a^2*b + 10*b^3)*c^5*d^2 - (a^3 - 15
*a*b^2)*c^4*d^3 + 12*(a^2*b - b^3)*c^3*d^4 + 6*(3*a^2*b + 4*b^3)*c*d^6 - 3*(a^3 + 9*a*b^2)*d^7))*sqrt(I*d)*wei
erstrassPInverse(-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^5*d^2 - 12*a*b^2*c^4*d^3 - 6*(a^2*b + 6*b^3)*c^3*d^4 - (a^3 - 27*a*b
^2)*c^2*d^5 + 6*(3*a^2*b + 4*b^3)*c*d^6 - 3*(a^3 + 9*a*b^2)*d^7)*cos(f*x + e)^2 - 2*sqrt(2)*(16*b^3*c^6*d - 12
*a*b^2*c^5*d^2 - 6*(a^2*b + 6*b^3)*c^4*d^3 - (a^3 - 27*a*b^2)*c^3*d^4 + 6*(3*a^2*b + 4*b^3)*c^2*d^5 - 3*(a^3 +
9*a*b^2)*c*d^6)*sin(f*x + e) - sqrt(2)*(16*b^3*c^7 - 12*a*b^2*c^6*d - 4*a^3*c^2*d^5 - 2*(3*a^2*b + 10*b^3)*c^
5*d^2 - (a^3 - 15*a*b^2)*c^4*d^3 + 12*(a^2*b - b^3)*c^3*d^4 + 6*(3*a^2*b + 4*b^3)*c*d^6 - 3*(a^3 + 9*a*b^2)*d^
7))*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^4*d^3 + 6*I*a*b^2*c^3*d^4 + 3*I*(a^2*b + 5*b^
3)*c^2*d^5 - 2*I*(2*a^3 + 9*a*b^2)*c*d^6 + 3*I*(3*a^2*b - b^3)*d^7)*cos(f*x + e)^2 + 2*sqrt(2)*(8*I*b^3*c^5*d^
2 - 6*I*a*b^2*c^4*d^3 - 3*I*(a^2*b + 5*b^3)*c^3*d^4 + 2*I*(2*a^3 + 9*a*b^2)*c^2*d^5 - 3*I*(3*a^2*b - b^3)*c*d^
6)*sin(f*x + e) + sqrt(2)*(8*I*b^3*c^6*d - 6*I*a*b^2*c^5*d^2 - I*(3*a^2*b + 7*b^3)*c^4*d^3 + 4*I*(a^3 + 3*a*b^
2)*c^3*d^4 - 12*I*(a^2*b + b^3)*c^2*d^5 + 2*I*(2*a^3 + 9*a*b^2)*c*d^6 - 3*I*(3*a^2*b - b^3)*d^7))*sqrt(I*d)*we
ierstrassZeta(-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^4*d^3 - 6*I*a*b^2*c^3*d^4 - 3*I*(a^2*b + 5*b^3)*c^2*d^5 + 2*I*(2*a^3 + 9*a*b^2)*c*d^6 - 3*I*(3*a^2
*b - b^3)*d^7)*cos(f*x + e)^2 + 2*sqrt(2)*(-8*I*b^3*c^5*d^2 + 6*I*a*b^2*c^4*d^3 + 3*I*(a^2*b + 5*b^3)*c^3*d^4
- 2*I*(2*a^3 + 9*a*b^2)*c^2*d^5 + 3*I*(3*a^2*b - b^3)*c*d^6)*sin(f*x + e) + sqrt(2)*(-8*I*b^3*c^6*d + 6*I*a*b^
2*c^5*d^2 + I*(3*a^2*b + 7*b^3)*c^4*d^3 - 4*I*(a^3 + 3*a*b^2)*c^3*d^4 + 12*I*(a^2*b + b^3)*c^2*d^5 - 2*I*(2*a^
3 + 9*a*b^2)*c*d^6 + 3*I*(3*a^2*b - b^3)*d^7))*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*((5*b^3*c^4*d^3 - 6*a*b^2*c^3*d^4 - 9*a^2*b*d^7 - 3*(a^2
*b + 3*b^3)*c^2*d^5 + 2*(2*a^3 + 9*a*b^2)*c*d^6)*cos(f*x + e)*sin(f*x + e) + (4*b^3*c^5*d^2 - 3*a*b^2*c^4*d^3
- 6*a^2*b*c*d^6 - a^3*d^7 - 2*(3*a^2*b + 4*b^3)*c^3*d^4 + 5*(a^3 + 3*a*b^2)*c^2*d^5)*cos(f*x + e))*sqrt(d*sin(
f*x + e) + c))/((c^4*d^6 - 2*c^2*d^8 + d^10)*f*cos(f*x + e)^2 - 2*(c^5*d^5 - 2*c^3*d^7 + c*d^9)*f*sin(f*x + e)
- (c^6*d^4 - c^4*d^6 - c^2*d^8 + d^10)*f)

Sympy [F(-1)]

Timed out. \[ \int \frac {(3+b \sin (e+f x))^3}{(c+d \sin (e+f x))^{5/2}} \, dx=\text {Timed out} \]

[In]

integrate((a+b*sin(f*x+e))**3/(c+d*sin(f*x+e))**(5/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {(3+b \sin (e+f x))^3}{(c+d \sin (e+f x))^{5/2}} \, dx=\int { \frac {{\left (b \sin \left (f x + e\right ) + a\right )}^{3}}{{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}}} \,d x } \]

[In]

integrate((a+b*sin(f*x+e))^3/(c+d*sin(f*x+e))^(5/2),x, algorithm="maxima")

[Out]

integrate((b*sin(f*x + e) + a)^3/(d*sin(f*x + e) + c)^(5/2), x)

Giac [F]

\[ \int \frac {(3+b \sin (e+f x))^3}{(c+d \sin (e+f x))^{5/2}} \, dx=\int { \frac {{\left (b \sin \left (f x + e\right ) + a\right )}^{3}}{{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}}} \,d x } \]

[In]

integrate((a+b*sin(f*x+e))^3/(c+d*sin(f*x+e))^(5/2),x, algorithm="giac")

[Out]

integrate((b*sin(f*x + e) + a)^3/(d*sin(f*x + e) + c)^(5/2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {(3+b \sin (e+f x))^3}{(c+d \sin (e+f x))^{5/2}} \, dx=\int \frac {{\left (a+b\,\sin \left (e+f\,x\right )\right )}^3}{{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{5/2}} \,d x \]

[In]

int((a + b*sin(e + f*x))^3/(c + d*sin(e + f*x))^(5/2),x)

[Out]

int((a + b*sin(e + f*x))^3/(c + d*sin(e + f*x))^(5/2), x)