3.8.54 \(\int \frac {\sqrt {c+d \sin (e+f x)}}{(a+b \sin (e+f x))^2} \, dx\) [754]

Optimal. Leaf size=307 \[ \frac {b \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{\left (a^2-b^2\right ) f (a+b \sin (e+f x))}+\frac {E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{\left (a^2-b^2\right ) f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {(b c-a d) F\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{b \left (a^2-b^2\right ) f \sqrt {c+d \sin (e+f x)}}+\frac {\left (2 a b c-a^2 d-b^2 d\right ) \Pi \left (\frac {2 b}{a+b};\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{(a-b) b (a+b)^2 f \sqrt {c+d \sin (e+f x)}} \]

[Out]

b*cos(f*x+e)*(c+d*sin(f*x+e))^(1/2)/(a^2-b^2)/f/(a+b*sin(f*x+e))-(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)/(a^2-b^2)
/f/((c+d*sin(f*x+e))/(c+d))^(1/2)+(-a*d+b*c)*(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)*Ell
ipticF(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)/b/(a^2-b^2)/f/(c+d*si
n(f*x+e))^(1/2)-(-a^2*d+2*a*b*c-b^2*d)*(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)*EllipticP
i(cos(1/2*e+1/4*Pi+1/2*f*x),2*b/(a+b),2^(1/2)*(d/(c+d))^(1/2))*((c+d*sin(f*x+e))/(c+d))^(1/2)/(a-b)/b/(a+b)^2/
f/(c+d*sin(f*x+e))^(1/2)

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Rubi [A]
time = 0.55, antiderivative size = 307, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2875, 3138, 2734, 2732, 3081, 2742, 2740, 2886, 2884} \begin {gather*} \frac {b \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \left (a^2-b^2\right ) (a+b \sin (e+f x))}-\frac {(b c-a d) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} F\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{b f \left (a^2-b^2\right ) \sqrt {c+d \sin (e+f x)}}+\frac {\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 )}{f \left (a^2-b^2\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {\left (a^2 (-d)+2 a b c-b^2 d\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \Pi \left (\frac {2 b}{a+b};\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{b f (a-b) (a+b)^2 \sqrt {c+d \sin (e+f x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(b*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/((a^2 - b^2)*f*(a + b*Sin[e + f*x])) + (EllipticE[(e - Pi/2 + f*x)/2
, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/((a^2 - b^2)*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) - ((b*c - a*d)*E
llipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(b*(a^2 - b^2)*f*Sqrt[c + d*Si
n[e + f*x]]) + ((2*a*b*c - a^2*d - b^2*d)*EllipticPi[(2*b)/(a + b), (e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c
+ d*Sin[e + f*x])/(c + d)])/((a - b)*b*(a + b)^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 2875

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

Rule 2884

Int[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp
[(2/(f*(a + b)*Sqrt[c + d]))*EllipticPi[2*(b/(a + b)), (1/2)*(e - Pi/2 + f*x), 2*(d/(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] && GtQ[c + d, 0]

Rule 2886

Int[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Dist
[Sqrt[(c + d*Sin[e + f*x])/(c + d)]/Sqrt[c + d*Sin[e + f*x]], Int[1/((a + b*Sin[e + f*x])*Sqrt[c/(c + d) + (d/
(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] && N
eQ[c^2 - d^2, 0] &&  !GtQ[c + d, 0]

Rule 3081

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

Rule 3138

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

Rubi steps

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

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Mathematica [C] Result contains complex when optimal does not.
time = 26.74, size = 846, normalized size = 2.76 \begin {gather*} -\frac {b \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{\left (-a^2+b^2\right ) f (a+b \sin (e+f x))}+\frac {-\frac {2 (4 a c-b d) \Pi \left (\frac {2 b}{a+b};\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}}}{(a+b) \sqrt {c+d \sin (e+f x)}}-\frac {8 i a \cos (e+f x) \left ((b c-a d) F\left (i \sinh ^{-1}\left (\sqrt {-\frac {1}{c+d}} \sqrt {c+d \sin (e+f x)}\right )|\frac {c+d}{c-d}\right )+a d \Pi \left (\frac {b (c+d)}{b c-a d};i \sinh ^{-1}\left (\sqrt {-\frac {1}{c+d}} \sqrt {c+d \sin (e+f x)}\right )|\frac {c+d}{c-d}\right )\right ) \sqrt {\frac {d-d \sin (e+f x)}{c+d}} \sqrt {-\frac {d+d \sin (e+f x)}{c-d}} (-b c+a d+b (c+d \sin (e+f x)))}{b d \sqrt {-\frac {1}{c+d}} (b c-a d) (a+b \sin (e+f x)) \sqrt {1-\sin ^2(e+f x)} \sqrt {-\frac {c^2-d^2-2 c (c+d \sin (e+f x))+(c+d \sin (e+f x))^2}{d^2}}}+\frac {2 i \cos (e+f x) \cos (2 (e+f x)) \left (2 b (c-d) (b c-a d) E\left (i \sinh ^{-1}\left (\sqrt {-\frac {1}{c+d}} \sqrt {c+d \sin (e+f x)}\right )|\frac {c+d}{c-d}\right )+d \left (-2 (a+b) (-b c+a d) F\left (i \sinh ^{-1}\left (\sqrt {-\frac {1}{c+d}} \sqrt {c+d \sin (e+f x)}\right )|\frac {c+d}{c-d}\right )+\left (2 a^2-b^2\right ) d \Pi \left (\frac {b (c+d)}{b c-a d};i \sinh ^{-1}\left (\sqrt {-\frac {1}{c+d}} \sqrt {c+d \sin (e+f x)}\right )|\frac {c+d}{c-d}\right )\right )\right ) \sqrt {\frac {d-d \sin (e+f x)}{c+d}} \sqrt {-\frac {d+d \sin (e+f x)}{c-d}} (-b c+a d+b (c+d \sin (e+f x)))}{b \sqrt {-\frac {1}{c+d}} (b c-a d) (a+b \sin (e+f x)) \sqrt {1-\sin ^2(e+f x)} \left (-2 c^2+d^2+4 c (c+d \sin (e+f x))-2 (c+d \sin (e+f x))^2\right ) \sqrt {-\frac {c^2-d^2-2 c (c+d \sin (e+f x))+(c+d \sin (e+f x))^2}{d^2}}}}{4 (a-b) (a+b) f} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-((b*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/((-a^2 + b^2)*f*(a + b*Sin[e + f*x]))) + ((-2*(4*a*c - b*d)*Ellipt
icPi[(2*b)/(a + b), (-e + Pi/2 - f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/((a + b)*Sqrt[c +
d*Sin[e + f*x]]) - ((8*I)*a*Cos[e + f*x]*((b*c - a*d)*EllipticF[I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e
+ f*x]]], (c + d)/(c - d)] + a*d*EllipticPi[(b*(c + d))/(b*c - a*d), I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d
*Sin[e + f*x]]], (c + d)/(c - d)])*Sqrt[(d - d*Sin[e + f*x])/(c + d)]*Sqrt[-((d + d*Sin[e + f*x])/(c - d))]*(-
(b*c) + a*d + b*(c + d*Sin[e + f*x])))/(b*d*Sqrt[-(c + d)^(-1)]*(b*c - a*d)*(a + b*Sin[e + f*x])*Sqrt[1 - Sin[
e + f*x]^2]*Sqrt[-((c^2 - d^2 - 2*c*(c + d*Sin[e + f*x]) + (c + d*Sin[e + f*x])^2)/d^2)]) + ((2*I)*Cos[e + f*x
]*Cos[2*(e + f*x)]*(2*b*(c - d)*(b*c - a*d)*EllipticE[I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]],
(c + d)/(c - d)] + d*(-2*(a + b)*(-(b*c) + a*d)*EllipticF[I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*
x]]], (c + d)/(c - d)] + (2*a^2 - b^2)*d*EllipticPi[(b*(c + d))/(b*c - a*d), I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqr
t[c + d*Sin[e + f*x]]], (c + d)/(c - d)]))*Sqrt[(d - d*Sin[e + f*x])/(c + d)]*Sqrt[-((d + d*Sin[e + f*x])/(c -
d))]*(-(b*c) + a*d + b*(c + d*Sin[e + f*x])))/(b*Sqrt[-(c + d)^(-1)]*(b*c - a*d)*(a + b*Sin[e + f*x])*Sqrt[1
- Sin[e + f*x]^2]*(-2*c^2 + d^2 + 4*c*(c + d*Sin[e + f*x]) - 2*(c + d*Sin[e + f*x])^2)*Sqrt[-((c^2 - d^2 - 2*c
*(c + d*Sin[e + f*x]) + (c + d*Sin[e + f*x])^2)/d^2)]))/(4*(a - b)*(a + b)*f)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(871\) vs. \(2(393)=786\).
time = 24.78, size = 872, normalized size = 2.84

method	result	size
default	\(\frac {\sqrt {-\left (-d \sin \left (f x +e \right )-c \right ) \left (\cos ^{2}\left (f x +e \right )\right )}\, \left (\frac {\left (-a d +b c \right ) \left (-\frac {b^{2} \sqrt {-\left (-d \sin \left (f x +e \right )-c \right ) \left (\cos ^{2}\left (f x +e \right )\right )}}{\left (a^{3} d -a^{2} b c -a \,b^{2} d +b^{3} c \right ) \left (a +b \sin \left (f x +e \right )\right )}-\frac {a d \left (\frac {c}{d}-1\right ) \sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}\, \sqrt {\frac {d \left (1-\sin \left (f x +e \right )\right )}{c +d}}\, \sqrt {\frac {\left (-\sin \left (f x +e \right )-1\right ) d}{c -d}}\, \EllipticF \left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right )}{\left (a^{3} d -a^{2} b c -a \,b^{2} d +b^{3} c \right ) \sqrt {-\left (-d \sin \left (f x +e \right )-c \right ) \left (\cos ^{2}\left (f x +e \right )\right )}}-\frac {b d \left (\frac {c}{d}-1\right ) \sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}\, \sqrt {\frac {d \left (1-\sin \left (f x +e \right )\right )}{c +d}}\, \sqrt {\frac {\left (-\sin \left (f x +e \right )-1\right ) d}{c -d}}\, \left (\left (-\frac {c}{d}-1\right ) \EllipticE \left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right )+\EllipticF \left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right )\right )}{\left (a^{3} d -a^{2} b c -a \,b^{2} d +b^{3} c \right ) \sqrt {-\left (-d \sin \left (f x +e \right )-c \right ) \left (\cos ^{2}\left (f x +e \right )\right )}}+\frac {\left (3 a^{2} d -2 a b c -b^{2} d \right ) \left (\frac {c}{d}-1\right ) \sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}\, \sqrt {\frac {d \left (1-\sin \left (f x +e \right )\right )}{c +d}}\, \sqrt {\frac {\left (-\sin \left (f x +e \right )-1\right ) d}{c -d}}\, \EllipticPi \left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \frac {-\frac {c}{d}+1}{-\frac {c}{d}+\frac {a}{b}}, \sqrt {\frac {c -d}{c +d}}\right )}{\left (a^{3} d -a^{2} b c -a \,b^{2} d +b^{3} c \right ) b \sqrt {-\left (-d \sin \left (f x +e \right )-c \right ) \left (\cos ^{2}\left (f x +e \right )\right )}\, \left (-\frac {c}{d}+\frac {a}{b}\right )}\right )}{b}+\frac {2 d \left (\frac {c}{d}-1\right ) \sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}\, \sqrt {\frac {d \left (1-\sin \left (f x +e \right )\right )}{c +d}}\, \sqrt {\frac {\left (-\sin \left (f x +e \right )-1\right ) d}{c -d}}\, \EllipticPi \left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \frac {-\frac {c}{d}+1}{-\frac {c}{d}+\frac {a}{b}}, \sqrt {\frac {c -d}{c +d}}\right )}{b^{2} \sqrt {-\left (-d \sin \left (f x +e \right )-c \right ) \left (\cos ^{2}\left (f x +e \right )\right )}\, \left (-\frac {c}{d}+\frac {a}{b}\right )}\right )}{\cos \left (f x +e \right ) \sqrt {c +d \sin \left (f x +e \right )}\, f}\)	\(872\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c+d*sin(f*x+e))^(1/2)/(a+b*sin(f*x+e))^2,x,method=_RETURNVERBOSE)

[Out]

(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)*((-a*d+b*c)/b*(-b^2/(a^3*d-a^2*b*c-a*b^2*d+b^3*c)*(-(-d*sin(f*x+e)-c)*
cos(f*x+e)^2)^(1/2)/(a+b*sin(f*x+e))-a*d/(a^3*d-a^2*b*c-a*b^2*d+b^3*c)*(1/d*c-1)*((c+d*sin(f*x+e))/(c-d))^(1/2
)*(d*(1-sin(f*x+e))/(c+d))^(1/2)*((-sin(f*x+e)-1)*d/(c-d))^(1/2)/(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)*Ellip
ticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))-b*d/(a^3*d-a^2*b*c-a*b^2*d+b^3*c)*(1/d*c-1)*((c+d*sin
(f*x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c+d))^(1/2)*((-sin(f*x+e)-1)*d/(c-d))^(1/2)/(-(-d*sin(f*x+e)-c)*cos(f
*x+e)^2)^(1/2)*((-1/d*c-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*a^2*d-2*a*b*c-b^2*d)/(a^3*d-a^2*b*c-a*b^2*d+b^3*c)/b*(1/d*c-1)*((
c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c+d))^(1/2)*((-sin(f*x+e)-1)*d/(c-d))^(1/2)/(-(-d*sin(f*x+e)-c
)*cos(f*x+e)^2)^(1/2)/(-1/d*c+a/b)*EllipticPi(((c+d*sin(f*x+e))/(c-d))^(1/2),(-1/d*c+1)/(-1/d*c+a/b),((c-d)/(c
+d))^(1/2)))+2*d/b^2*(1/d*c-1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c+d))^(1/2)*((-sin(f*x+e)-1)*
d/(c-d))^(1/2)/(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)/(-1/d*c+a/b)*EllipticPi(((c+d*sin(f*x+e))/(c-d))^(1/2),
(-1/d*c+1)/(-1/d*c+a/b),((c-d)/(c+d))^(1/2)))/cos(f*x+e)/(c+d*sin(f*x+e))^(1/2)/f

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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))^(1/2)/(a+b*sin(f*x+e))^2,x, algorithm="maxima")

[Out]

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

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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c+d*sin(f*x+e))^(1/2)/(a+b*sin(f*x+e))^2,x, algorithm="fricas")

[Out]

Timed out

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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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))^(1/2)/(a+b*sin(f*x+e))^2,x, algorithm="giac")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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