3.1.44 \(\int \frac {\sqrt {g \sin (e+f x)}}{\sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))} \, dx\) [44]
Optimal. Leaf size=114 \[ \frac {2 \sqrt {-\cot ^2(e+f x)} \sqrt {\frac {b+a \csc (e+f x)}{a+b}} \Pi \left (\frac {2 c}{c+d};\sin ^{-1}\left (\frac {\sqrt {1-\csc (e+f x)}}{\sqrt {2}}\right )|\frac {2 a}{a+b}\right ) \sqrt {g \sin (e+f x)} \tan (e+f x)}{(c+d) f \sqrt {a+b \sin (e+f x)}} \]
[Out]
2*EllipticPi(1/2*(1-csc(f*x+e))^(1/2)*2^(1/2),2*c/(c+d),2^(1/2)*(a/(a+b))^(1/2))*(-cot(f*x+e)^2)^(1/2)*((b+a*c
sc(f*x+e))/(a+b))^(1/2)*(g*sin(f*x+e))^(1/2)*tan(f*x+e)/(c+d)/f/(a+b*sin(f*x+e))^(1/2)
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Rubi [A]
time = 0.14, antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.026, Rules used = {3016}
\begin {gather*} \frac {2 \tan (e+f x) \sqrt {-\cot ^2(e+f x)} \sqrt {g \sin (e+f x)} \sqrt {\frac {a \csc (e+f x)+b}{a+b}} \Pi \left (\frac {2 c}{c+d};\text {ArcSin}\left (\frac {\sqrt {1-\csc (e+f x)}}{\sqrt {2}}\right )|\frac {2 a}{a+b}\right )}{f (c+d) \sqrt {a+b \sin (e+f x)}} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[Sqrt[g*Sin[e + f*x]]/(Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])),x]
[Out]
(2*Sqrt[-Cot[e + f*x]^2]*Sqrt[(b + a*Csc[e + f*x])/(a + b)]*EllipticPi[(2*c)/(c + d), ArcSin[Sqrt[1 - Csc[e +
f*x]]/Sqrt[2]], (2*a)/(a + b)]*Sqrt[g*Sin[e + f*x]]*Tan[e + f*x])/((c + d)*f*Sqrt[a + b*Sin[e + f*x]])
Rule 3016
Int[Sqrt[(g_.)*sin[(e_.) + (f_.)*(x_)]]/(Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((c_) + (d_.)*sin[(e_.) +
(f_.)*(x_)])), x_Symbol] :> Simp[2*Sqrt[-Cot[e + f*x]^2]*(Sqrt[g*Sin[e + f*x]]/(f*(c + d)*Cot[e + f*x]*Sqrt[a
+ b*Sin[e + f*x]]))*Sqrt[(b + a*Csc[e + f*x])/(a + b)]*EllipticPi[2*(c/(c + d)), ArcSin[Sqrt[1 - Csc[e + f*x]]
/Sqrt[2]], 2*(a/(a + b))], x] /; FreeQ[{a, b, c, d, e, f, g}, 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 {g \sin (e+f x)}}{\sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))} \, dx &=\frac {2 \sqrt {-\cot ^2(e+f x)} \sqrt {\frac {b+a \csc (e+f x)}{a+b}} \Pi \left (\frac {2 c}{c+d};\sin ^{-1}\left (\frac {\sqrt {1-\csc (e+f x)}}{\sqrt {2}}\right )|\frac {2 a}{a+b}\right ) \sqrt {g \sin (e+f x)} \tan (e+f x)}{(c+d) f \sqrt {a+b \sin (e+f x)}}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(3427\) vs. \(2(114)=228\).
time = 47.73, size = 3427, normalized size = 30.06 \begin {gather*} \text {Result too large to show} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
Integrate[Sqrt[g*Sin[e + f*x]]/(Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])),x]
[Out]
(a*Sqrt[-a^2 + b^2]*((a*c + (b + Sqrt[-a^2 + b^2])*(-d + Sqrt[-c^2 + d^2]))*EllipticPi[(2*Sqrt[-a^2 + b^2]*c)/
(b*c + Sqrt[-a^2 + b^2]*c - a*d + a*Sqrt[-c^2 + d^2]), ArcSin[Sqrt[(b + Sqrt[-a^2 + b^2] + a*Tan[(e + f*x)/2])
/Sqrt[-a^2 + b^2]]/Sqrt[2]], (2*Sqrt[-a^2 + b^2])/(b + Sqrt[-a^2 + b^2])] + (-(a*c) + (b + Sqrt[-a^2 + b^2])*(
d + Sqrt[-c^2 + d^2]))*EllipticPi[(2*Sqrt[-a^2 + b^2]*c)/(b*c + Sqrt[-a^2 + b^2]*c - a*(d + Sqrt[-c^2 + d^2]))
, ArcSin[Sqrt[(b + Sqrt[-a^2 + b^2] + a*Tan[(e + f*x)/2])/Sqrt[-a^2 + b^2]]/Sqrt[2]], (2*Sqrt[-a^2 + b^2])/(b
+ Sqrt[-a^2 + b^2])])*Sqrt[Sin[e + f*x]]*Sqrt[g*Sin[e + f*x]]*Sqrt[(a*Sec[(e + f*x)/2]^2*(a + b*Sin[e + f*x]))
/(a^2 - b^2)])/((b + Sqrt[-a^2 + b^2])^2*(b*c - a*d)*Sqrt[-c^2 + d^2]*f*(a + b*Sin[e + f*x])*(c + d*Sin[e + f*
x])*Sqrt[-((a*Tan[(e + f*x)/2])/(b + Sqrt[-a^2 + b^2]))]*((a^2*Sqrt[-a^2 + b^2]*((a*c + (b + Sqrt[-a^2 + b^2])
*(-d + Sqrt[-c^2 + d^2]))*EllipticPi[(2*Sqrt[-a^2 + b^2]*c)/(b*c + Sqrt[-a^2 + b^2]*c - a*d + a*Sqrt[-c^2 + d^
2]), ArcSin[Sqrt[(b + Sqrt[-a^2 + b^2] + a*Tan[(e + f*x)/2])/Sqrt[-a^2 + b^2]]/Sqrt[2]], (2*Sqrt[-a^2 + b^2])/
(b + Sqrt[-a^2 + b^2])] + (-(a*c) + (b + Sqrt[-a^2 + b^2])*(d + Sqrt[-c^2 + d^2]))*EllipticPi[(2*Sqrt[-a^2 + b
^2]*c)/(b*c + Sqrt[-a^2 + b^2]*c - a*(d + Sqrt[-c^2 + d^2])), ArcSin[Sqrt[(b + Sqrt[-a^2 + b^2] + a*Tan[(e + f
*x)/2])/Sqrt[-a^2 + b^2]]/Sqrt[2]], (2*Sqrt[-a^2 + b^2])/(b + Sqrt[-a^2 + b^2])])*Sec[(e + f*x)/2]^2*Sqrt[Sin[
e + f*x]]*Sqrt[(a*Sec[(e + f*x)/2]^2*(a + b*Sin[e + f*x]))/(a^2 - b^2)])/(4*(b + Sqrt[-a^2 + b^2])^3*(b*c - a*
d)*Sqrt[-c^2 + d^2]*Sqrt[a + b*Sin[e + f*x]]*(-((a*Tan[(e + f*x)/2])/(b + Sqrt[-a^2 + b^2])))^(3/2)) - (a*b*Sq
rt[-a^2 + b^2]*Cos[e + f*x]*((a*c + (b + Sqrt[-a^2 + b^2])*(-d + Sqrt[-c^2 + d^2]))*EllipticPi[(2*Sqrt[-a^2 +
b^2]*c)/(b*c + Sqrt[-a^2 + b^2]*c - a*d + a*Sqrt[-c^2 + d^2]), ArcSin[Sqrt[(b + Sqrt[-a^2 + b^2] + a*Tan[(e +
f*x)/2])/Sqrt[-a^2 + b^2]]/Sqrt[2]], (2*Sqrt[-a^2 + b^2])/(b + Sqrt[-a^2 + b^2])] + (-(a*c) + (b + Sqrt[-a^2 +
b^2])*(d + Sqrt[-c^2 + d^2]))*EllipticPi[(2*Sqrt[-a^2 + b^2]*c)/(b*c + Sqrt[-a^2 + b^2]*c - a*(d + Sqrt[-c^2
+ d^2])), ArcSin[Sqrt[(b + Sqrt[-a^2 + b^2] + a*Tan[(e + f*x)/2])/Sqrt[-a^2 + b^2]]/Sqrt[2]], (2*Sqrt[-a^2 + b
^2])/(b + Sqrt[-a^2 + b^2])])*Sqrt[Sin[e + f*x]]*Sqrt[(a*Sec[(e + f*x)/2]^2*(a + b*Sin[e + f*x]))/(a^2 - b^2)]
)/(2*(b + Sqrt[-a^2 + b^2])^2*(b*c - a*d)*Sqrt[-c^2 + d^2]*(a + b*Sin[e + f*x])^(3/2)*Sqrt[-((a*Tan[(e + f*x)/
2])/(b + Sqrt[-a^2 + b^2]))]) + (a*Sqrt[-a^2 + b^2]*Cos[e + f*x]*((a*c + (b + Sqrt[-a^2 + b^2])*(-d + Sqrt[-c^
2 + d^2]))*EllipticPi[(2*Sqrt[-a^2 + b^2]*c)/(b*c + Sqrt[-a^2 + b^2]*c - a*d + a*Sqrt[-c^2 + d^2]), ArcSin[Sqr
t[(b + Sqrt[-a^2 + b^2] + a*Tan[(e + f*x)/2])/Sqrt[-a^2 + b^2]]/Sqrt[2]], (2*Sqrt[-a^2 + b^2])/(b + Sqrt[-a^2
+ b^2])] + (-(a*c) + (b + Sqrt[-a^2 + b^2])*(d + Sqrt[-c^2 + d^2]))*EllipticPi[(2*Sqrt[-a^2 + b^2]*c)/(b*c + S
qrt[-a^2 + b^2]*c - a*(d + Sqrt[-c^2 + d^2])), ArcSin[Sqrt[(b + Sqrt[-a^2 + b^2] + a*Tan[(e + f*x)/2])/Sqrt[-a
^2 + b^2]]/Sqrt[2]], (2*Sqrt[-a^2 + b^2])/(b + Sqrt[-a^2 + b^2])])*Sqrt[(a*Sec[(e + f*x)/2]^2*(a + b*Sin[e + f
*x]))/(a^2 - b^2)])/(2*(b + Sqrt[-a^2 + b^2])^2*(b*c - a*d)*Sqrt[-c^2 + d^2]*Sqrt[Sin[e + f*x]]*Sqrt[a + b*Sin
[e + f*x]]*Sqrt[-((a*Tan[(e + f*x)/2])/(b + Sqrt[-a^2 + b^2]))]) + (a*Sqrt[-a^2 + b^2]*((a*c + (b + Sqrt[-a^2
+ b^2])*(-d + Sqrt[-c^2 + d^2]))*EllipticPi[(2*Sqrt[-a^2 + b^2]*c)/(b*c + Sqrt[-a^2 + b^2]*c - a*d + a*Sqrt[-c
^2 + d^2]), ArcSin[Sqrt[(b + Sqrt[-a^2 + b^2] + a*Tan[(e + f*x)/2])/Sqrt[-a^2 + b^2]]/Sqrt[2]], (2*Sqrt[-a^2 +
b^2])/(b + Sqrt[-a^2 + b^2])] + (-(a*c) + (b + Sqrt[-a^2 + b^2])*(d + Sqrt[-c^2 + d^2]))*EllipticPi[(2*Sqrt[-
a^2 + b^2]*c)/(b*c + Sqrt[-a^2 + b^2]*c - a*(d + Sqrt[-c^2 + d^2])), ArcSin[Sqrt[(b + Sqrt[-a^2 + b^2] + a*Tan
[(e + f*x)/2])/Sqrt[-a^2 + b^2]]/Sqrt[2]], (2*Sqrt[-a^2 + b^2])/(b + Sqrt[-a^2 + b^2])])*Sqrt[Sin[e + f*x]]*((
a*b*Cos[e + f*x]*Sec[(e + f*x)/2]^2)/(a^2 - b^2) + (a*Sec[(e + f*x)/2]^2*(a + b*Sin[e + f*x])*Tan[(e + f*x)/2]
)/(a^2 - b^2)))/(2*(b + Sqrt[-a^2 + b^2])^2*(b*c - a*d)*Sqrt[-c^2 + d^2]*Sqrt[a + b*Sin[e + f*x]]*Sqrt[(a*Sec[
(e + f*x)/2]^2*(a + b*Sin[e + f*x]))/(a^2 - b^2)]*Sqrt[-((a*Tan[(e + f*x)/2])/(b + Sqrt[-a^2 + b^2]))]) + (a*S
qrt[-a^2 + b^2]*Sqrt[Sin[e + f*x]]*Sqrt[(a*Sec[(e + f*x)/2]^2*(a + b*Sin[e + f*x]))/(a^2 - b^2)]*((a*(a*c + (b
+ Sqrt[-a^2 + b^2])*(-d + Sqrt[-c^2 + d^2]))*Sec[(e + f*x)/2]^2)/(4*Sqrt[2]*Sqrt[-a^2 + b^2]*Sqrt[(b + Sqrt[-
a^2 + b^2] + a*Tan[(e + f*x)/2])/Sqrt[-a^2 + b^2]]*Sqrt[1 - (b + Sqrt[-a^2 + b^2] + a*Tan[(e + f*x)/2])/(2*Sqr
t[-a^2 + b^2])]*Sqrt[1 - (b + Sqrt[-a^2 + b^2] + a*Tan[(e + f*x)/2])/(b + Sqrt[-a^2 + b^2])]*(1 - (c*(b + Sqrt
[-a^2 + b^2] + a*Tan[(e + f*x)/2]))/(b*c + Sqrt[-a^2 + b^2]*c - a*d + a*Sqrt[-c^2 + d^2]))) + (a*(-(a*c) + (b
+ Sqrt[-a^2 + b^2])*(d + Sqrt[-c^2 + d^2]))*Sec[(e + f*x)/2]^2)/(4*Sqrt[2]*Sqrt[-a^2 + b^2]*Sqrt[(b + Sqrt[-a^
2 + b^2] + a*Tan[(e + f*x)/2])/Sqrt[-a^2 + b^2]]*Sqrt[1 - (b + Sqrt[-a^2 + b^2] + a*Tan[(e + f*x)/2])/(2*Sqrt[
-a^2 + b^2])]*Sqrt[1 - (b + Sqrt[-a^2 + b^2] + ...
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(2932\) vs.
\(2(107)=214\).
time = 0.35, size = 2933, normalized size = 25.73
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| method |
result |
size |
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| default |
\(\text {Expression too large to display}\) |
\(2933\) |
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Verification of antiderivative is not currently implemented for this CAS.
[In]
int((g*sin(f*x+e))^(1/2)/(c+d*sin(f*x+e))/(a+b*sin(f*x+e))^(1/2),x,method=_RETURNVERBOSE)
[Out]
-1/f*(2*(-c^2+d^2)^(1/2)*(-a^2+b^2)^(1/2)*EllipticPi((-(cos(f*x+e)*a-(-a^2+b^2)^(1/2)*sin(f*x+e)-b*sin(f*x+e)-
a)/(b+(-a^2+b^2)^(1/2))/sin(f*x+e))^(1/2),(b+(-a^2+b^2)^(1/2))*c/(a*(-c^2+d^2)^(1/2)+c*(-a^2+b^2)^(1/2)-a*d+b*
c),1/2*2^(1/2)*((b+(-a^2+b^2)^(1/2))/(-a^2+b^2)^(1/2))^(1/2))*b-(-c^2+d^2)^(1/2)*EllipticPi((-(cos(f*x+e)*a-(-
a^2+b^2)^(1/2)*sin(f*x+e)-b*sin(f*x+e)-a)/(b+(-a^2+b^2)^(1/2))/sin(f*x+e))^(1/2),(b+(-a^2+b^2)^(1/2))*c/(a*(-c
^2+d^2)^(1/2)+c*(-a^2+b^2)^(1/2)-a*d+b*c),1/2*2^(1/2)*((b+(-a^2+b^2)^(1/2))/(-a^2+b^2)^(1/2))^(1/2))*a^2+2*(-c
^2+d^2)^(1/2)*EllipticPi((-(cos(f*x+e)*a-(-a^2+b^2)^(1/2)*sin(f*x+e)-b*sin(f*x+e)-a)/(b+(-a^2+b^2)^(1/2))/sin(
f*x+e))^(1/2),(b+(-a^2+b^2)^(1/2))*c/(a*(-c^2+d^2)^(1/2)+c*(-a^2+b^2)^(1/2)-a*d+b*c),1/2*2^(1/2)*((b+(-a^2+b^2
)^(1/2))/(-a^2+b^2)^(1/2))^(1/2))*b^2+2*(-c^2+d^2)^(1/2)*(-a^2+b^2)^(1/2)*EllipticPi((-(cos(f*x+e)*a-(-a^2+b^2
)^(1/2)*sin(f*x+e)-b*sin(f*x+e)-a)/(b+(-a^2+b^2)^(1/2))/sin(f*x+e))^(1/2),-(b+(-a^2+b^2)^(1/2))*c/(a*(-c^2+d^2
)^(1/2)-c*(-a^2+b^2)^(1/2)+a*d-b*c),1/2*2^(1/2)*((b+(-a^2+b^2)^(1/2))/(-a^2+b^2)^(1/2))^(1/2))*b-(-c^2+d^2)^(1
/2)*EllipticPi((-(cos(f*x+e)*a-(-a^2+b^2)^(1/2)*sin(f*x+e)-b*sin(f*x+e)-a)/(b+(-a^2+b^2)^(1/2))/sin(f*x+e))^(1
/2),-(b+(-a^2+b^2)^(1/2))*c/(a*(-c^2+d^2)^(1/2)-c*(-a^2+b^2)^(1/2)+a*d-b*c),1/2*2^(1/2)*((b+(-a^2+b^2)^(1/2))/
(-a^2+b^2)^(1/2))^(1/2))*a^2+2*(-c^2+d^2)^(1/2)*EllipticPi((-(cos(f*x+e)*a-(-a^2+b^2)^(1/2)*sin(f*x+e)-b*sin(f
*x+e)-a)/(b+(-a^2+b^2)^(1/2))/sin(f*x+e))^(1/2),-(b+(-a^2+b^2)^(1/2))*c/(a*(-c^2+d^2)^(1/2)-c*(-a^2+b^2)^(1/2)
+a*d-b*c),1/2*2^(1/2)*((b+(-a^2+b^2)^(1/2))/(-a^2+b^2)^(1/2))^(1/2))*b^2+(-a^2+b^2)^(1/2)*EllipticPi((-(cos(f*
x+e)*a-(-a^2+b^2)^(1/2)*sin(f*x+e)-b*sin(f*x+e)-a)/(b+(-a^2+b^2)^(1/2))/sin(f*x+e))^(1/2),(b+(-a^2+b^2)^(1/2))
*c/(a*(-c^2+d^2)^(1/2)+c*(-a^2+b^2)^(1/2)-a*d+b*c),1/2*2^(1/2)*((b+(-a^2+b^2)^(1/2))/(-a^2+b^2)^(1/2))^(1/2))*
a*c-2*(-a^2+b^2)^(1/2)*EllipticPi((-(cos(f*x+e)*a-(-a^2+b^2)^(1/2)*sin(f*x+e)-b*sin(f*x+e)-a)/(b+(-a^2+b^2)^(1
/2))/sin(f*x+e))^(1/2),(b+(-a^2+b^2)^(1/2))*c/(a*(-c^2+d^2)^(1/2)+c*(-a^2+b^2)^(1/2)-a*d+b*c),1/2*2^(1/2)*((b+
(-a^2+b^2)^(1/2))/(-a^2+b^2)^(1/2))^(1/2))*b*d+EllipticPi((-(cos(f*x+e)*a-(-a^2+b^2)^(1/2)*sin(f*x+e)-b*sin(f*
x+e)-a)/(b+(-a^2+b^2)^(1/2))/sin(f*x+e))^(1/2),(b+(-a^2+b^2)^(1/2))*c/(a*(-c^2+d^2)^(1/2)+c*(-a^2+b^2)^(1/2)-a
*d+b*c),1/2*2^(1/2)*((b+(-a^2+b^2)^(1/2))/(-a^2+b^2)^(1/2))^(1/2))*a^2*d+EllipticPi((-(cos(f*x+e)*a-(-a^2+b^2)
^(1/2)*sin(f*x+e)-b*sin(f*x+e)-a)/(b+(-a^2+b^2)^(1/2))/sin(f*x+e))^(1/2),(b+(-a^2+b^2)^(1/2))*c/(a*(-c^2+d^2)^
(1/2)+c*(-a^2+b^2)^(1/2)-a*d+b*c),1/2*2^(1/2)*((b+(-a^2+b^2)^(1/2))/(-a^2+b^2)^(1/2))^(1/2))*a*b*c-2*EllipticP
i((-(cos(f*x+e)*a-(-a^2+b^2)^(1/2)*sin(f*x+e)-b*sin(f*x+e)-a)/(b+(-a^2+b^2)^(1/2))/sin(f*x+e))^(1/2),(b+(-a^2+
b^2)^(1/2))*c/(a*(-c^2+d^2)^(1/2)+c*(-a^2+b^2)^(1/2)-a*d+b*c),1/2*2^(1/2)*((b+(-a^2+b^2)^(1/2))/(-a^2+b^2)^(1/
2))^(1/2))*b^2*d-(-a^2+b^2)^(1/2)*EllipticPi((-(cos(f*x+e)*a-(-a^2+b^2)^(1/2)*sin(f*x+e)-b*sin(f*x+e)-a)/(b+(-
a^2+b^2)^(1/2))/sin(f*x+e))^(1/2),-(b+(-a^2+b^2)^(1/2))*c/(a*(-c^2+d^2)^(1/2)-c*(-a^2+b^2)^(1/2)+a*d-b*c),1/2*
2^(1/2)*((b+(-a^2+b^2)^(1/2))/(-a^2+b^2)^(1/2))^(1/2))*a*c+2*(-a^2+b^2)^(1/2)*EllipticPi((-(cos(f*x+e)*a-(-a^2
+b^2)^(1/2)*sin(f*x+e)-b*sin(f*x+e)-a)/(b+(-a^2+b^2)^(1/2))/sin(f*x+e))^(1/2),-(b+(-a^2+b^2)^(1/2))*c/(a*(-c^2
+d^2)^(1/2)-c*(-a^2+b^2)^(1/2)+a*d-b*c),1/2*2^(1/2)*((b+(-a^2+b^2)^(1/2))/(-a^2+b^2)^(1/2))^(1/2))*b*d-Ellipti
cPi((-(cos(f*x+e)*a-(-a^2+b^2)^(1/2)*sin(f*x+e)-b*sin(f*x+e)-a)/(b+(-a^2+b^2)^(1/2))/sin(f*x+e))^(1/2),-(b+(-a
^2+b^2)^(1/2))*c/(a*(-c^2+d^2)^(1/2)-c*(-a^2+b^2)^(1/2)+a*d-b*c),1/2*2^(1/2)*((b+(-a^2+b^2)^(1/2))/(-a^2+b^2)^
(1/2))^(1/2))*a^2*d-EllipticPi((-(cos(f*x+e)*a-(-a^2+b^2)^(1/2)*sin(f*x+e)-b*sin(f*x+e)-a)/(b+(-a^2+b^2)^(1/2)
)/sin(f*x+e))^(1/2),-(b+(-a^2+b^2)^(1/2))*c/(a*(-c^2+d^2)^(1/2)-c*(-a^2+b^2)^(1/2)+a*d-b*c),1/2*2^(1/2)*((b+(-
a^2+b^2)^(1/2))/(-a^2+b^2)^(1/2))^(1/2))*a*b*c+2*EllipticPi((-(cos(f*x+e)*a-(-a^2+b^2)^(1/2)*sin(f*x+e)-b*sin(
f*x+e)-a)/(b+(-a^2+b^2)^(1/2))/sin(f*x+e))^(1/2),-(b+(-a^2+b^2)^(1/2))*c/(a*(-c^2+d^2)^(1/2)-c*(-a^2+b^2)^(1/2
)+a*d-b*c),1/2*2^(1/2)*((b+(-a^2+b^2)^(1/2))/(-a^2+b^2)^(1/2))^(1/2))*b^2*d)*2^(1/2)*(g*sin(f*x+e))^(1/2)*sin(
f*x+e)/(a+b*sin(f*x+e))^(1/2)*(a*(-1+cos(f*x+e))/(b+(-a^2+b^2)^(1/2))/sin(f*x+e))^(1/2)*((cos(f*x+e)*a+(-a^2+b
^2)^(1/2)*sin(f*x+e)-b*sin(f*x+e)-a)/(-a^2+b^2)^(1/2)/sin(f*x+e))^(1/2)*(-(cos(f*x+e)*a-(-a^2+b^2)^(1/2)*sin(f
*x+e)-b*sin(f*x+e)-a)/(b+(-a^2+b^2)^(1/2))/sin(f*x+e))^(1/2)/(-1+cos(f*x+e))*c/(a*(-c^2+d^2)^(1/2)-c*(-a^2+b^2
)^(1/2)+a*d-b*c)/(-c^2+d^2)^(1/2)/(a*(-c^2+d^2)^(1/2)+c*(-a^2+b^2)^(1/2)-a*d+b*c)
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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((g*sin(f*x+e))^(1/2)/(c+d*sin(f*x+e))/(a+b*sin(f*x+e))^(1/2),x, algorithm="maxima")
[Out]
integrate(sqrt(g*sin(f*x + e))/(sqrt(b*sin(f*x + e) + a)*(d*sin(f*x + e) + c)), 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((g*sin(f*x+e))^(1/2)/(c+d*sin(f*x+e))/(a+b*sin(f*x+e))^(1/2),x, algorithm="fricas")
[Out]
Timed out
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {g \sin {\left (e + f x \right )}}}{\sqrt {a + b \sin {\left (e + f x \right )}} \left (c + d \sin {\left (e + f x \right )}\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((g*sin(f*x+e))**(1/2)/(c+d*sin(f*x+e))/(a+b*sin(f*x+e))**(1/2),x)
[Out]
Integral(sqrt(g*sin(e + f*x))/(sqrt(a + b*sin(e + f*x))*(c + d*sin(e + f*x))), 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((g*sin(f*x+e))^(1/2)/(c+d*sin(f*x+e))/(a+b*sin(f*x+e))^(1/2),x, algorithm="giac")
[Out]
integrate(sqrt(g*sin(f*x + e))/(sqrt(b*sin(f*x + e) + a)*(d*sin(f*x + e) + c)), x)
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\sqrt {g\,\sin \left (e+f\,x\right )}}{\sqrt {a+b\,\sin \left (e+f\,x\right )}\,\left (c+d\,\sin \left (e+f\,x\right )\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((g*sin(e + f*x))^(1/2)/((a + b*sin(e + f*x))^(1/2)*(c + d*sin(e + f*x))),x)
[Out]
int((g*sin(e + f*x))^(1/2)/((a + b*sin(e + f*x))^(1/2)*(c + d*sin(e + f*x))), x)
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