3.8.90 \(\int \frac {(c+d \sin (e+f x))^{3/2}}{(a+b \sin (e+f x))^{3/2}} \, dx\) [790]

Optimal. Leaf size=600 \[ \frac {2 (c-d) \sqrt {c+d} E\left (\sin ^{-1}\left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right )|\frac {(a-b) (c+d)}{(a+b) (c-d)}\right ) \sec (e+f x) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (1+\sin (e+f x))}{(c-d) (a+b \sin (e+f x))}} (a+b \sin (e+f x))}{(a-b) b \sqrt {a+b} f}+\frac {2 d \sqrt {c+d} \Pi \left (\frac {b (c+d)}{(a+b) d};\sin ^{-1}\left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right )|\frac {(a-b) (c+d)}{(a+b) (c-d)}\right ) \sec (e+f x) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (1+\sin (e+f x))}{(c-d) (a+b \sin (e+f x))}} (a+b \sin (e+f x))}{b^2 \sqrt {a+b} f}+\frac {2 \sqrt {a+b} (b (c-2 d)+a d) F\left (\sin ^{-1}\left (\frac {\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}\right )|\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) \sec (e+f x) \sqrt {\frac {(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-a d) (1+\sin (e+f x))}{(a-b) (c+d \sin (e+f x))}} (c+d \sin (e+f x))}{(a-b) b^2 \sqrt {c+d} f} \]

[Out]

2*(c-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)/b/f/(a+b)^(1/2)+2*d*EllipticPi((a+b)^(1/2)*(c+d*
sin(f*x+e))^(1/2)/(c+d)^(1/2)/(a+b*sin(f*x+e))^(1/2),b*(c+d)/(a+b)/d,((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)/b^2/f/(a+b)^(1/2)+2*(b*(c-2*d)+a*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+b)^
(1/2)*((-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)/b^2/f/(c+d)^(1/2)

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Rubi [A]
time = 0.56, antiderivative size = 600, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {2877, 2890, 3077, 2897, 3075} \begin {gather*} \frac {2 \sqrt {a+b} (a d+b (c-2 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))}} F\left (\text {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 )}{b^2 f (a-b) \sqrt {c+d}}+\frac {2 d \sqrt {c+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))}} \Pi \left (\frac {b (c+d)}{(a+b) d};\text {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 )}{b^2 f \sqrt {a+b}}+\frac {2 (c-d) \sqrt {c+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 (\text {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 )}{b f (a-b) \sqrt {a+b}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(c - d)*Sqrt[c + 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]))/((a - b)*b*Sqrt[a + b]*f) + (2*d*Sqrt[c + d]*EllipticPi[(b*(c + d))/((a + b)*d), 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]))/(b^2*Sqrt[a + b]*f) + (2*Sqrt[a + b]*(b*(c - 2*d) +
a*d)*EllipticF[ArcSin[(Sqrt[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]))/((a - b)*b^2*
Sqrt[c + d]*f)

Rule 2877

Int[((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(3/2)/((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(3/2), x_Symbol] :> D
ist[d^2/b^2, Int[Sqrt[a + b*Sin[e + f*x]]/Sqrt[c + d*Sin[e + f*x]], x], x] + Dist[(b*c - a*d)/b^2, Int[Simp[b*
c + a*d + 2*b*d*Sin[e + f*x], 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}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]

Rule 2890

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

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*} \int \frac {(c+d \sin (e+f x))^{3/2}}{(a+b \sin (e+f x))^{3/2}} \, dx &=\frac {d^2 \int \frac {\sqrt {a+b \sin (e+f x)}}{\sqrt {c+d \sin (e+f x)}} \, dx}{b^2}+\frac {(b c-a d) \int \frac {b c+a d+2 b d \sin (e+f x)}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}} \, dx}{b^2}\\ &=\frac {2 d \sqrt {c+d} \Pi \left (\frac {b (c+d)}{(a+b) d};\sin ^{-1}\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))}{b^2 \sqrt {a+b} f}-\frac {(b c-a d)^2 \int \frac {1+\sin (e+f x)}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}} \, dx}{(a-b) b}+\frac {((b c-a d) (b (c-2 d)+a d)) \int \frac {1}{\sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}} \, dx}{(a-b) b^2}\\ &=\frac {2 (c-d) \sqrt {c+d} E\left (\sin ^{-1}\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))}{(a-b) b \sqrt {a+b} f}+\frac {2 d \sqrt {c+d} \Pi \left (\frac {b (c+d)}{(a+b) d};\sin ^{-1}\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))}{b^2 \sqrt {a+b} f}+\frac {2 \sqrt {a+b} (b (c-2 d)+a d) F\left (\sin ^{-1}\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))}{(a-b) b^2 \sqrt {c+d} f}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(1896\) vs. \(2(600)=1200\).
time = 8.95, size = 1896, normalized size = 3.16 \begin {gather*} -\frac {2 (-b c \cos (e+f x)+a d \cos (e+f x)) \sqrt {c+d \sin (e+f x)}}{\left (a^2-b^2\right ) f \sqrt {a+b \sin (e+f x)}}+\frac {-\frac {4 (-b c+a d) \left (a c^2-b c d\right ) \sqrt {\frac {(c+d) \cot ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right )}{-c+d}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {(-a-b) \csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) (c+d \sin (e+f x))}{-b c+a d}}}{\sqrt {2}}\right )|\frac {2 (-b c+a d)}{(a+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 ) (a+b \sin (e+f x))}{-b c+a d}} \sqrt {\frac {(-a-b) \csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) (c+d \sin (e+f x))}{-b c+a d}}}{(a+b) (c+d) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}-4 (-b c+a d) \left (b c^2-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}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {(-a-b) \csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) (c+d \sin (e+f x))}{-b c+a d}}}{\sqrt {2}}\right )|\frac {2 (-b c+a d)}{(a+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 ) (a+b \sin (e+f x))}{-b c+a d}} \sqrt {\frac {(-a-b) \csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) (c+d \sin (e+f x))}{-b c+a d}}}{(a+b) (c+d) \sqrt {a+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}} \Pi \left (\frac {-b c+a d}{(a+b) d};\sin ^{-1}\left (\frac {\sqrt {\frac {(-a-b) \csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) (c+d \sin (e+f x))}{-b c+a d}}}{\sqrt {2}}\right )|\frac {2 (-b c+a d)}{(a+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 ) (a+b \sin (e+f x))}{-b c+a d}} \sqrt {\frac {(-a-b) \csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) (c+d \sin (e+f x))}{-b c+a d}}}{(a+b) d \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}\right )+2 \left (-b c d+a d^2\right ) \left (\frac {\cos (e+f x) \sqrt {c+d \sin (e+f x)}}{d \sqrt {a+b \sin (e+f x)}}+\frac {\sqrt {\frac {a-b}{a+b}} (a+b) \cos \left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {a-b}{a+b}} \sin \left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right )}{\sqrt {\frac {a+b \sin (e+f x)}{a+b}}}\right )|\frac {2 (-b c+a d)}{(a-b) (c+d)}\right ) \sqrt {c+d \sin (e+f x)}}{b d \sqrt {\frac {(a+b) \cos ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right )}{a+b \sin (e+f x)}} \sqrt {a+b \sin (e+f x)} \sqrt {\frac {a+b \sin (e+f x)}{a+b}} \sqrt {\frac {(a+b) (c+d \sin (e+f x))}{(c+d) (a+b \sin (e+f x))}}}-\frac {2 (-b c+a d) \left (\frac {((a+b) c+a d) \sqrt {\frac {(c+d) \cot ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right )}{-c+d}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {(-a-b) \csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) (c+d \sin (e+f x))}{-b c+a d}}}{\sqrt {2}}\right )|\frac {2 (-b c+a d)}{(a+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 ) (a+b \sin (e+f x))}{-b c+a d}} \sqrt {\frac {(-a-b) \csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) (c+d \sin (e+f x))}{-b c+a d}}}{(a+b) (c+d) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}-\frac {(b c+a d) \sqrt {\frac {(c+d) \cot ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right )}{-c+d}} \Pi \left (\frac {-b c+a d}{(a+b) d};\sin ^{-1}\left (\frac {\sqrt {\frac {(-a-b) \csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) (c+d \sin (e+f x))}{-b c+a d}}}{\sqrt {2}}\right )|\frac {2 (-b c+a d)}{(a+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 ) (a+b \sin (e+f x))}{-b c+a d}} \sqrt {\frac {(-a-b) \csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) (c+d \sin (e+f x))}{-b c+a d}}}{(a+b) d \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}\right )}{b d}\right )}{(a-b) (a+b) f} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-2*(-(b*c*Cos[e + f*x]) + a*d*Cos[e + f*x])*Sqrt[c + d*Sin[e + f*x]])/((a^2 - b^2)*f*Sqrt[a + b*Sin[e + f*x]]
) + ((-4*(-(b*c) + a*d)*(a*c^2 - b*c*d)*Sqrt[((c + d)*Cot[(-e + Pi/2 - f*x)/2]^2)/(-c + d)]*EllipticF[ArcSin[S
qrt[((-a - b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*c) + a*d)]/Sqrt[2]], (2*(-(b*c) + a*d))/((
a + 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*(a + b*Sin[
e + f*x]))/(-(b*c) + a*d)]*Sqrt[((-a - b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*c) + a*d)])/((
a + b)*(c + d)*Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]) - 4*(-(b*c) + a*d)*(b*c^2 - b*d^2)*((Sqrt[((
c + d)*Cot[(-e + Pi/2 - f*x)/2]^2)/(-c + d)]*EllipticF[ArcSin[Sqrt[((-a - b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d
*Sin[e + f*x]))/(-(b*c) + a*d)]/Sqrt[2]], (2*(-(b*c) + a*d))/((a + 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*(a + b*Sin[e + f*x]))/(-(b*c) + a*d)]*Sqrt[((-a - b)*Csc[(
-e + Pi/2 - f*x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*c) + a*d)])/((a + b)*(c + d)*Sqrt[a + 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) + a*d)/((a + b)*d
), ArcSin[Sqrt[((-a - b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*c) + a*d)]/Sqrt[2]], (2*(-(b*c)
+ a*d))/((a + 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*
(a + b*Sin[e + f*x]))/(-(b*c) + a*d)]*Sqrt[((-a - b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*c)
+ a*d)])/((a + b)*d*Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])) + 2*(-(b*c*d) + a*d^2)*((Cos[e + f*x]*
Sqrt[c + d*Sin[e + f*x]])/(d*Sqrt[a + b*Sin[e + f*x]]) + (Sqrt[(a - b)/(a + b)]*(a + b)*Cos[(-e + Pi/2 - f*x)/
2]*EllipticE[ArcSin[(Sqrt[(a - b)/(a + b)]*Sin[(-e + Pi/2 - f*x)/2])/Sqrt[(a + b*Sin[e + f*x])/(a + b)]], (2*(
-(b*c) + a*d))/((a - b)*(c + d))]*Sqrt[c + d*Sin[e + f*x]])/(b*d*Sqrt[((a + b)*Cos[(-e + Pi/2 - f*x)/2]^2)/(a
+ b*Sin[e + f*x])]*Sqrt[a + b*Sin[e + f*x]]*Sqrt[(a + b*Sin[e + f*x])/(a + b)]*Sqrt[((a + b)*(c + d*Sin[e + f*
x]))/((c + d)*(a + b*Sin[e + f*x]))]) - (2*(-(b*c) + a*d)*((((a + b)*c + a*d)*Sqrt[((c + d)*Cot[(-e + Pi/2 - f
*x)/2]^2)/(-c + d)]*EllipticF[ArcSin[Sqrt[((-a - b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*c) +
a*d)]/Sqrt[2]], (2*(-(b*c) + a*d))/((a + 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*(a + b*Sin[e + f*x]))/(-(b*c) + a*d)]*Sqrt[((-a - b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c
+ d*Sin[e + f*x]))/(-(b*c) + a*d)])/((a + b)*(c + d)*Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]) - ((b*
c + a*d)*Sqrt[((c + d)*Cot[(-e + Pi/2 - f*x)/2]^2)/(-c + d)]*EllipticPi[(-(b*c) + a*d)/((a + b)*d), ArcSin[Sqr
t[((-a - b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*c) + a*d)]/Sqrt[2]], (2*(-(b*c) + a*d))/((a
+ 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*(a + b*Sin[e
+ f*x]))/(-(b*c) + a*d)]*Sqrt[((-a - b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*c) + a*d)])/((a
+ b)*d*Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])))/(b*d)))/((a - b)*(a + b)*f)

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Maple [B] result has leaf size over 500,000. Avoiding possible recursion issues.
time = 40.85, size = 2945157, normalized size = 4908.60

method	result	size
default	\(\text {Expression too large to display}\)	\(2945157\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 {3}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{3/2}}{{\left (a+b\,\sin \left (e+f\,x\right )\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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