\(\int \frac {(c+d \sin (e+f x))^{3/2}}{(3+b \sin (e+f x))^{5/2}} \, dx\) [796]

Optimal result

Integrand size = 29, antiderivative size = 486 \[ \int \frac {(c+d \sin (e+f x))^{3/2}}{(3+b \sin (e+f x))^{5/2}} \, dx=\frac {8 (c-d) \sqrt {c+d} (3 c-b d) E\left (\arcsin \left (\frac {\sqrt {3+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {3+b \sin (e+f x)}}\right )|\frac {(3-b) (c+d)}{(3+b) (c-d)}\right ) \sec (e+f x) \sqrt {-\frac {(b c-3 d) (1-\sin (e+f x))}{(c+d) (3+b \sin (e+f x))}} \sqrt {\frac {(b c-3 d) (1+\sin (e+f x))}{(c-d) (3+b \sin (e+f x))}} (3+b \sin (e+f x))}{3 (3-b)^2 (3+b)^{3/2} (b c-3 d) f}+\frac {2 (b c-3 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 \left (9-b^2\right ) f (3+b \sin (e+f x))^{3/2}}+\frac {2 (c-d) (9 c+b c-3 d-3 b d) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {3+b \sin (e+f x)}}{\sqrt {3+b} \sqrt {c+d \sin (e+f x)}}\right ),\frac {(3+b) (c-d)}{(3-b) (c+d)}\right ) \sec (e+f x) \sqrt {\frac {(b c-3 d) (1-\sin (e+f x))}{(3+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-3 d) (1+\sin (e+f x))}{(3-b) (c+d \sin (e+f x))}} (c+d \sin (e+f x))}{3 (3-b)^2 \sqrt {3+b} (b c-3 d) \sqrt {c+d} f} \]

[Out]

8/3*(c-d)*(a*c-b*d)*EllipticE((a+b)^(1/2)*(c+d*sin(f*x+e))^(1/2)/(c+d)^(1/2)/(a+b*sin(f*x+e))^(1/2),((a-b)*(c+
d)/(a+b)/(c-d))^(1/2))*sec(f*x+e)*(a+b*sin(f*x+e))*(c+d)^(1/2)*(-(-a*d+b*c)*(1-sin(f*x+e))/(c+d)/(a+b*sin(f*x+
e)))^(1/2)*((-a*d+b*c)*(1+sin(f*x+e))/(c-d)/(a+b*sin(f*x+e)))^(1/2)/(a-b)^2/(a+b)^(3/2)/(-a*d+b*c)/f+2/3*(c-d)
*(3*a*c-a*d+b*c-3*b*d)*EllipticF((c+d)^(1/2)*(a+b*sin(f*x+e))^(1/2)/(a+b)^(1/2)/(c+d*sin(f*x+e))^(1/2),((a+b)*
(c-d)/(a-b)/(c+d))^(1/2))*sec(f*x+e)*(c+d*sin(f*x+e))*((-a*d+b*c)*(1-sin(f*x+e))/(a+b)/(c+d*sin(f*x+e)))^(1/2)
*(-(-a*d+b*c)*(1+sin(f*x+e))/(a-b)/(c+d*sin(f*x+e)))^(1/2)/(a-b)^2/(-a*d+b*c)/f/(a+b)^(1/2)/(c+d)^(1/2)+2/3*(-
a*d+b*c)*cos(f*x+e)*(c+d*sin(f*x+e))^(1/2)/(a^2-b^2)/f/(a+b*sin(f*x+e))^(3/2)

Rubi [A] (verified)

Time = 0.67 (sec) , antiderivative size = 497, normalized size of antiderivative = 1.02, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {2878, 3077, 2897, 3075} \[ \int \frac {(c+d \sin (e+f x))^{3/2}}{(3+b \sin (e+f x))^{5/2}} \, dx=\frac {2 (b c-a d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f \left (a^2-b^2\right ) (a+b \sin (e+f x))^{3/2}}+\frac {2 (c-d) (3 a c-a d+b c-3 b d) \sec (e+f x) (c+d \sin (e+f x)) \sqrt {\frac {(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right )}{3 f (a-b)^2 \sqrt {a+b} \sqrt {c+d} (b c-a d)}+\frac {8 (c-d) \sqrt {c+d} (a c-b d) \sec (e+f x) (a+b \sin (e+f x)) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} E\left (\arcsin \left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right )|\frac {(a-b) (c+d)}{(a+b) (c-d)}\right )}{3 f (a-b)^2 (a+b)^{3/2} (b c-a d)} \]

[In]

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

[Out]

(8*(c - d)*Sqrt[c + d]*(a*c - b*d)*EllipticE[ArcSin[(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])/(Sqrt[c + d]*Sqrt[a
+ b*Sin[e + f*x]])], ((a - b)*(c + d))/((a + b)*(c - d))]*Sec[e + f*x]*Sqrt[-(((b*c - a*d)*(1 - Sin[e + f*x])
)/((c + d)*(a + b*Sin[e + f*x])))]*Sqrt[((b*c - a*d)*(1 + Sin[e + f*x]))/((c - d)*(a + b*Sin[e + f*x]))]*(a +
b*Sin[e + f*x]))/(3*(a - b)^2*(a + b)^(3/2)*(b*c - a*d)*f) + (2*(b*c - a*d)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*
x]])/(3*(a^2 - b^2)*f*(a + b*Sin[e + f*x])^(3/2)) + (2*(c - d)*(3*a*c + b*c - a*d - 3*b*d)*EllipticF[ArcSin[(S
qrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c +
d))]*Sec[e + f*x]*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1
+ Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*(c + d*Sin[e + f*x]))/(3*(a - b)^2*Sqrt[a + b]*Sqrt[c + d]*
(b*c - a*d)*f)

Rule 2878

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

Rule 2897

Int[1/(Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*Sqrt[(c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Si
mp[2*((c + d*Sin[e + f*x])/(f*(b*c - a*d)*Rt[(c + d)/(a + b), 2]*Cos[e + f*x]))*Sqrt[(b*c - a*d)*((1 - Sin[e +
f*x])/((a + b)*(c + d*Sin[e + f*x])))]*Sqrt[(-(b*c - a*d))*((1 + Sin[e + f*x])/((a - b)*(c + d*Sin[e + f*x]))
)]*EllipticF[ArcSin[Rt[(c + d)/(a + b), 2]*(Sqrt[a + b*Sin[e + f*x]]/Sqrt[c + d*Sin[e + f*x]])], (a + b)*((c -
d)/((a - b)*(c + d)))], 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] && PosQ[(c + d)/(a + b)]

Rule 3075

Int[((A_) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(3/2)*Sqrt[(c_) + (d_.)*sin
[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp[-2*A*(c - d)*((a + b*Sin[e + f*x])/(f*(b*c - a*d)^2*Rt[(a + b)/(c +
d), 2]*Cos[e + f*x]))*Sqrt[(b*c - a*d)*((1 + Sin[e + f*x])/((c - d)*(a + b*Sin[e + f*x])))]*Sqrt[(-(b*c - a*d)
)*((1 - Sin[e + f*x])/((c + d)*(a + b*Sin[e + f*x])))]*EllipticE[ArcSin[Rt[(a + b)/(c + d), 2]*(Sqrt[c + d*Sin
[e + f*x]]/Sqrt[a + b*Sin[e + f*x]])], (a - b)*((c + d)/((a + b)*(c - d)))], x] /; FreeQ[{a, b, c, d, e, f, A,
B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && EqQ[A, B] && PosQ[(a + b)/(c + d)]

Rule 3077

Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(3/2)*Sqrt[(c_) + (d_.)*s
in[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Dist[(A - B)/(a - b), Int[1/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e
+ f*x]]), x], x] - Dist[(A*b - a*B)/(a - b), Int[(1 + Sin[e + f*x])/((a + b*Sin[e + f*x])^(3/2)*Sqrt[c + d*Sin
[e + f*x]]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2
- d^2, 0] && NeQ[A, B]

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

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(1968\) vs. \(2(486)=972\).

Time = 6.40 (sec) , antiderivative size = 1968, normalized size of antiderivative = 4.05 \[ \int \frac {(c+d \sin (e+f x))^{3/2}}{(3+b \sin (e+f x))^{5/2}} \, dx=\frac {\sqrt {3+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)} \left (-\frac {2 (b c \cos (e+f x)-3 d \cos (e+f x))}{3 \left (-9+b^2\right ) (3+b \sin (e+f x))^2}-\frac {8 \left (-3 b c \cos (e+f x)+b^2 d \cos (e+f x)\right )}{3 \left (-9+b^2\right )^2 (3+b \sin (e+f x))}\right )}{f}+\frac {-\frac {4 (-b c+3 d) \left (27 c^2+b^2 c^2-12 b c d+9 d^2-b^2 d^2\right ) \sqrt {\frac {(c+d) \cot ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right )}{-c+d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {(-3-b) \csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) (c+d \sin (e+f x))}{-b c+3 d}}}{\sqrt {2}}\right ),\frac {2 (-b c+3 d)}{(3+b) (-c+d)}\right ) \sec (e+f x) \sin ^4\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) \sqrt {\frac {(c+d) \csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) (3+b \sin (e+f x))}{-b c+3 d}} \sqrt {\frac {(-3-b) \csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) (c+d \sin (e+f x))}{-b c+3 d}}}{(3+b) (c+d) \sqrt {3+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}-4 (-b c+3 d) \left (12 b c^2+36 c d-4 b^2 c d-12 b d^2\right ) \left (\frac {\sqrt {\frac {(c+d) \cot ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right )}{-c+d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {(-3-b) \csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) (c+d \sin (e+f x))}{-b c+3 d}}}{\sqrt {2}}\right ),\frac {2 (-b c+3 d)}{(3+b) (-c+d)}\right ) \sec (e+f x) \sin ^4\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) \sqrt {\frac {(c+d) \csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) (3+b \sin (e+f x))}{-b c+3 d}} \sqrt {\frac {(-3-b) \csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) (c+d \sin (e+f x))}{-b c+3 d}}}{(3+b) (c+d) \sqrt {3+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}-\frac {\sqrt {\frac {(c+d) \cot ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right )}{-c+d}} \operatorname {EllipticPi}\left (\frac {-b c+3 d}{(3+b) d},\arcsin \left (\frac {\sqrt {\frac {(-3-b) \csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) (c+d \sin (e+f x))}{-b c+3 d}}}{\sqrt {2}}\right ),\frac {2 (-b c+3 d)}{(3+b) (-c+d)}\right ) \sec (e+f x) \sin ^4\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) \sqrt {\frac {(c+d) \csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) (3+b \sin (e+f x))}{-b c+3 d}} \sqrt {\frac {(-3-b) \csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) (c+d \sin (e+f x))}{-b c+3 d}}}{(3+b) d \sqrt {3+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}\right )+2 \left (-12 b c d+4 b^2 d^2\right ) \left (\frac {\cos (e+f x) \sqrt {c+d \sin (e+f x)}}{d \sqrt {3+b \sin (e+f x)}}+\frac {\sqrt {\frac {3-b}{3+b}} (3+b) \cos \left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) E\left (\arcsin \left (\frac {\sqrt {\frac {3-b}{3+b}} \sin \left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right )}{\sqrt {\frac {3+b \sin (e+f x)}{3+b}}}\right )|\frac {2 (-b c+3 d)}{(3-b) (c+d)}\right ) \sqrt {c+d \sin (e+f x)}}{b d \sqrt {\frac {(3+b) \cos ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right )}{3+b \sin (e+f x)}} \sqrt {3+b \sin (e+f x)} \sqrt {\frac {3+b \sin (e+f x)}{3+b}} \sqrt {\frac {(3+b) (c+d \sin (e+f x))}{(c+d) (3+b \sin (e+f x))}}}-\frac {2 (-b c+3 d) \left (\frac {((3+b) c+3 d) \sqrt {\frac {(c+d) \cot ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right )}{-c+d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {(-3-b) \csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) (c+d \sin (e+f x))}{-b c+3 d}}}{\sqrt {2}}\right ),\frac {2 (-b c+3 d)}{(3+b) (-c+d)}\right ) \sec (e+f x) \sin ^4\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) \sqrt {\frac {(c+d) \csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) (3+b \sin (e+f x))}{-b c+3 d}} \sqrt {\frac {(-3-b) \csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) (c+d \sin (e+f x))}{-b c+3 d}}}{(3+b) (c+d) \sqrt {3+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}-\frac {(b c+3 d) \sqrt {\frac {(c+d) \cot ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right )}{-c+d}} \operatorname {EllipticPi}\left (\frac {-b c+3 d}{(3+b) d},\arcsin \left (\frac {\sqrt {\frac {(-3-b) \csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) (c+d \sin (e+f x))}{-b c+3 d}}}{\sqrt {2}}\right ),\frac {2 (-b c+3 d)}{(3+b) (-c+d)}\right ) \sec (e+f x) \sin ^4\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) \sqrt {\frac {(c+d) \csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) (3+b \sin (e+f x))}{-b c+3 d}} \sqrt {\frac {(-3-b) \csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) (c+d \sin (e+f x))}{-b c+3 d}}}{(3+b) d \sqrt {3+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}\right )}{b d}\right )}{3 (-3+b)^2 (3+b)^2 f} \]

[In]

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

[Out]

(Sqrt[3 + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]*((-2*(b*c*Cos[e + f*x] - 3*d*Cos[e + f*x]))/(3*(-9 + b^2)*(
3 + b*Sin[e + f*x])^2) - (8*(-3*b*c*Cos[e + f*x] + b^2*d*Cos[e + f*x]))/(3*(-9 + b^2)^2*(3 + b*Sin[e + f*x])))
)/f + ((-4*(-(b*c) + 3*d)*(27*c^2 + b^2*c^2 - 12*b*c*d + 9*d^2 - b^2*d^2)*Sqrt[((c + d)*Cot[(-e + Pi/2 - f*x)/
2]^2)/(-c + d)]*EllipticF[ArcSin[Sqrt[((-3 - b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*c) + 3*d
)]/Sqrt[2]], (2*(-(b*c) + 3*d))/((3 + b)*(-c + d))]*Sec[e + f*x]*Sin[(-e + Pi/2 - f*x)/2]^4*Sqrt[((c + d)*Csc[
(-e + Pi/2 - f*x)/2]^2*(3 + b*Sin[e + f*x]))/(-(b*c) + 3*d)]*Sqrt[((-3 - b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d*
Sin[e + f*x]))/(-(b*c) + 3*d)])/((3 + b)*(c + d)*Sqrt[3 + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]) - 4*(-(b*c
) + 3*d)*(12*b*c^2 + 36*c*d - 4*b^2*c*d - 12*b*d^2)*((Sqrt[((c + d)*Cot[(-e + Pi/2 - f*x)/2]^2)/(-c + d)]*Elli
pticF[ArcSin[Sqrt[((-3 - b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*c) + 3*d)]/Sqrt[2]], (2*(-(b
*c) + 3*d))/((3 + b)*(-c + d))]*Sec[e + f*x]*Sin[(-e + Pi/2 - f*x)/2]^4*Sqrt[((c + d)*Csc[(-e + Pi/2 - f*x)/2]
^2*(3 + b*Sin[e + f*x]))/(-(b*c) + 3*d)]*Sqrt[((-3 - b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*
c) + 3*d)])/((3 + b)*(c + d)*Sqrt[3 + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]) - (Sqrt[((c + d)*Cot[(-e + Pi/
2 - f*x)/2]^2)/(-c + d)]*EllipticPi[(-(b*c) + 3*d)/((3 + b)*d), ArcSin[Sqrt[((-3 - b)*Csc[(-e + Pi/2 - f*x)/2]
^2*(c + d*Sin[e + f*x]))/(-(b*c) + 3*d)]/Sqrt[2]], (2*(-(b*c) + 3*d))/((3 + b)*(-c + d))]*Sec[e + f*x]*Sin[(-e
+ Pi/2 - f*x)/2]^4*Sqrt[((c + d)*Csc[(-e + Pi/2 - f*x)/2]^2*(3 + b*Sin[e + f*x]))/(-(b*c) + 3*d)]*Sqrt[((-3 -
b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*c) + 3*d)])/((3 + b)*d*Sqrt[3 + b*Sin[e + f*x]]*Sqrt
[c + d*Sin[e + f*x]])) + 2*(-12*b*c*d + 4*b^2*d^2)*((Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(d*Sqrt[3 + b*Sin[
e + f*x]]) + (Sqrt[(3 - b)/(3 + b)]*(3 + b)*Cos[(-e + Pi/2 - f*x)/2]*EllipticE[ArcSin[(Sqrt[(3 - b)/(3 + b)]*S
in[(-e + Pi/2 - f*x)/2])/Sqrt[(3 + b*Sin[e + f*x])/(3 + b)]], (2*(-(b*c) + 3*d))/((3 - b)*(c + d))]*Sqrt[c + d
*Sin[e + f*x]])/(b*d*Sqrt[((3 + b)*Cos[(-e + Pi/2 - f*x)/2]^2)/(3 + b*Sin[e + f*x])]*Sqrt[3 + b*Sin[e + f*x]]*
Sqrt[(3 + b*Sin[e + f*x])/(3 + b)]*Sqrt[((3 + b)*(c + d*Sin[e + f*x]))/((c + d)*(3 + b*Sin[e + f*x]))]) - (2*(
-(b*c) + 3*d)*((((3 + b)*c + 3*d)*Sqrt[((c + d)*Cot[(-e + Pi/2 - f*x)/2]^2)/(-c + d)]*EllipticF[ArcSin[Sqrt[((
-3 - b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*c) + 3*d)]/Sqrt[2]], (2*(-(b*c) + 3*d))/((3 + b)
*(-c + d))]*Sec[e + f*x]*Sin[(-e + Pi/2 - f*x)/2]^4*Sqrt[((c + d)*Csc[(-e + Pi/2 - f*x)/2]^2*(3 + b*Sin[e + f*
x]))/(-(b*c) + 3*d)]*Sqrt[((-3 - b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*c) + 3*d)])/((3 + b)
*(c + d)*Sqrt[3 + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]) - ((b*c + 3*d)*Sqrt[((c + d)*Cot[(-e + Pi/2 - f*x)
/2]^2)/(-c + d)]*EllipticPi[(-(b*c) + 3*d)/((3 + b)*d), ArcSin[Sqrt[((-3 - b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c +
d*Sin[e + f*x]))/(-(b*c) + 3*d)]/Sqrt[2]], (2*(-(b*c) + 3*d))/((3 + b)*(-c + d))]*Sec[e + f*x]*Sin[(-e + Pi/2
- f*x)/2]^4*Sqrt[((c + d)*Csc[(-e + Pi/2 - f*x)/2]^2*(3 + b*Sin[e + f*x]))/(-(b*c) + 3*d)]*Sqrt[((-3 - b)*Csc[
(-e + Pi/2 - f*x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*c) + 3*d)])/((3 + b)*d*Sqrt[3 + b*Sin[e + f*x]]*Sqrt[c + d*S
in[e + f*x]])))/(b*d)))/(3*(-3 + b)^2*(3 + b)^2*f)

Maple [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(160230\) vs. \(2(457)=914\).

Time = 10.97 (sec) , antiderivative size = 160231, normalized size of antiderivative = 329.69

method	result	size
default	\(\text {Expression too large to display}\)	\(160231\)

[In]

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

[Out]

result too large to display

Fricas [F]

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

[In]

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

[Out]

integral(-sqrt(b*sin(f*x + e) + a)*(d*sin(f*x + e) + c)^(3/2)/(3*a*b^2*cos(f*x + e)^2 - a^3 - 3*a*b^2 + (b^3*c
os(f*x + e)^2 - 3*a^2*b - b^3)*sin(f*x + e)), x)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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