3.1.45 \(\int \frac {1}{\sqrt {g \sin (e+f x)} \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))} \, dx\) [45]
Optimal. Leaf size=246 \[ -\frac {2 \sqrt {a+b} \sqrt {\frac {a (1-\csc (e+f x))}{a+b}} \sqrt {\frac {a (1+\csc (e+f x))}{a-b}} F\left (\sin ^{-1}\left (\frac {\sqrt {g} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {g \sin (e+f x)}}\right )|-\frac {a+b}{a-b}\right ) \tan (e+f x)}{a c f \sqrt {g}}-\frac {2 d \sqrt {-\cot ^2(e+f x)} \sqrt {\frac {b+a \csc (e+f x)}{a+b}} \Pi \left (\frac {2 c}{c+d};\sin ^{-1}\left (\frac {\sqrt {1-\csc (e+f x)}}{\sqrt {2}}\right )|\frac {2 a}{a+b}\right ) \sqrt {g \sin (e+f x)} \tan (e+f x)}{c (c+d) f g \sqrt {a+b \sin (e+f x)}} \]
[Out]
-2*EllipticF(g^(1/2)*(a+b*sin(f*x+e))^(1/2)/(a+b)^(1/2)/(g*sin(f*x+e))^(1/2),((-a-b)/(a-b))^(1/2))*(a+b)^(1/2)
*(a*(1-csc(f*x+e))/(a+b))^(1/2)*(a*(1+csc(f*x+e))/(a-b))^(1/2)*tan(f*x+e)/a/c/f/g^(1/2)-2*d*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*csc(f*x+e))/(a+b))^(1
/2)*(g*sin(f*x+e))^(1/2)*tan(f*x+e)/c/(c+d)/f/g/(a+b*sin(f*x+e))^(1/2)
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Rubi [A]
time = 0.37, antiderivative size = 246, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {3018, 2895,
3016} \begin {gather*} -\frac {2 d \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 )}{c f g (c+d) \sqrt {a+b \sin (e+f x)}}-\frac {2 \sqrt {a+b} \tan (e+f x) \sqrt {\frac {a (1-\csc (e+f x))}{a+b}} \sqrt {\frac {a (\csc (e+f x)+1)}{a-b}} F\left (\text {ArcSin}\left (\frac {\sqrt {g} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {g \sin (e+f x)}}\right )|-\frac {a+b}{a-b}\right )}{a c f \sqrt {g}} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[1/(Sqrt[g*Sin[e + f*x]]*Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])),x]
[Out]
(-2*Sqrt[a + b]*Sqrt[(a*(1 - Csc[e + f*x]))/(a + b)]*Sqrt[(a*(1 + Csc[e + f*x]))/(a - b)]*EllipticF[ArcSin[(Sq
rt[g]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[g*Sin[e + f*x]])], -((a + b)/(a - b))]*Tan[e + f*x])/(a*c*f*
Sqrt[g]) - (2*d*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*(c + d)*f*g*Sqrt[a + b*Sin[e
+ f*x]])
Rule 2895
Int[1/(Sqrt[(d_.)*sin[(e_.) + (f_.)*(x_)]]*Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp[-2*(
Tan[e + f*x]/(a*f))*Rt[(a + b)/d, 2]*Sqrt[a*((1 - Csc[e + f*x])/(a + b))]*Sqrt[a*((1 + Csc[e + f*x])/(a - b))]
*EllipticF[ArcSin[Sqrt[a + b*Sin[e + f*x]]/Sqrt[d*Sin[e + f*x]]/Rt[(a + b)/d, 2]], -(a + b)/(a - b)], x] /; Fr
eeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0] && PosQ[(a + b)/d]
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]
Rule 3018
Int[1/(Sqrt[(g_.)*sin[(e_.) + (f_.)*(x_)]]*Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((c_) + (d_.)*sin[(e_.)
+ (f_.)*(x_)])), x_Symbol] :> Dist[1/c, Int[1/(Sqrt[g*Sin[e + f*x]]*Sqrt[a + b*Sin[e + f*x]]), x], x] - Dist[d
/(c*g), Int[Sqrt[g*Sin[e + f*x]]/(Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])), x], 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 {1}{\sqrt {g \sin (e+f x)} \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))} \, dx &=\frac {\int \frac {1}{\sqrt {g \sin (e+f x)} \sqrt {a+b \sin (e+f x)}} \, dx}{c}-\frac {d \int \frac {\sqrt {g \sin (e+f x)}}{\sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))} \, dx}{c g}\\ &=-\frac {2 \sqrt {a+b} \sqrt {\frac {a (1-\csc (e+f x))}{a+b}} \sqrt {\frac {a (1+\csc (e+f x))}{a-b}} F\left (\sin ^{-1}\left (\frac {\sqrt {g} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {g \sin (e+f x)}}\right )|-\frac {a+b}{a-b}\right ) \tan (e+f x)}{a c f \sqrt {g}}-\frac {2 d \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 (c+d) f g \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. \(4935\) vs. \(2(246)=492\).
time = 49.28, size = 4935, normalized size = 20.06 \begin {gather*} \text {Result too large to show} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
Integrate[1/(Sqrt[g*Sin[e + f*x]]*Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])),x]
[Out]
(-4*Sqrt[-a^2 + b^2]*Cos[(e + f*x)/2]^4*(-2*(b + Sqrt[-a^2 + b^2])*(b*c - a*d)*Sqrt[-c^2 + d^2]*EllipticF[ArcS
in[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*d*((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[(a*Sec[(e + f*x)/2]^2*(a + b*Sin[e + f*x]))/(a^2 - b^2)]*(-((a*Tan[(e + f*x)/2])/
(b + Sqrt[-a^2 + b^2])))^(3/2))/(a^2*c*(-(b*c) + a*d)*Sqrt[-c^2 + d^2]*f*Sin[e + f*x]^(3/2)*Sqrt[g*Sin[e + f*x
]]*(a + b*Sin[e + f*x])*(c + d*Sin[e + f*x])*((3*Sqrt[-a^2 + b^2]*Cos[(e + f*x)/2]^2*(-2*(b + Sqrt[-a^2 + b^2]
)*(b*c - a*d)*Sqrt[-c^2 + d^2]*EllipticF[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*d*((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[Sq
rt[(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[(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*(b + Sqrt[-a^2 + b^2])*c*(-(b*c)
+ a*d)*Sqrt[-c^2 + d^2]*Sin[e + f*x]^(3/2)*Sqrt[a + b*Sin[e + f*x]]) + (2*b*Sqrt[-a^2 + b^2]*Cos[(e + f*x)/2]
^4*Cos[e + f*x]*(-2*(b + Sqrt[-a^2 + b^2])*(b*c - a*d)*Sqrt[-c^2 + d^2]*EllipticF[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*d*((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[(a*Sec[(e + f*x)/2]^2*(a + b*Sin[e + f*x]))/(a^2 - b^2)]*(-((a*Tan[(e + f*x)/2])/(b + Sqrt[-a^2 + b^2])))
^(3/2))/(a^2*c*(-(b*c) + a*d)*Sqrt[-c^2 + d^2]*Sin[e + f*x]^(3/2)*(a + b*Sin[e + f*x])^(3/2)) + (6*Sqrt[-a^2 +
b^2]*Cos[(e + f*x)/2]^4*Cos[e + f*x]*(-2*(b + Sqrt[-a^2 + b^2])*(b*c - a*d)*Sqrt[-c^2 + d^2]*EllipticF[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*d*((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[(a*Sec[(e + f*x)/2]^2*(a + b*Sin[e + f*x]))/(a^2 - b^2)]*(-((a*Tan[(e + f*x)/2])/(b
+ Sqrt[-a^2 + b^2])))^(3/2))/(a^2*c*(-(b*c) + a*d)*Sqrt[-c^2 + d^2]*Sin[e + f*x]^(5/2)*Sqrt[a + b*Sin[e + f*x
]]) + (8*Sqrt[-a^2 + b^2]*Cos[(e + f*x)/2]^3*(-2*(b + Sqrt[-a^2 + b^2])*(b*c - a*d)*Sqrt[-c^2 + d^2]*EllipticF
[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*d*((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])]))*Sin[(e + f*x)/2]*Sqrt[(a*Sec[(e + f*x)/2]^2*(a + b*Sin[e + f*x]))/(a^2 - b^2)]*(-
((a*Tan[(e + f*x)/2])/(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. \(3697\) vs.
\(2(227)=454\).
time = 0.29, size = 3698, normalized size = 15.03
| | |
method |
result |
size |
| | |
default |
\(\text {Expression too large to display}\) |
\(3698\) |
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|
|
|
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(c+d*sin(f*x+e))/(g*sin(f*x+e))^(1/2)/(a+b*sin(f*x+e))^(1/2),x,method=_RETURNVERBOSE)
[Out]
-1/f/(a+b*sin(f*x+e))^(1/2)*(4*EllipticF((-(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),1/2*2^(1/2)*((b+(-a^2+b^2)^(1/2))/(-a^2+b^2)^(1/2))^(1/2))*a*b*d*(-a^2+b^2)^(1/2
)*(-c^2+d^2)^(1/2)-4*EllipticF((-(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),1/2*2^(1/2)*((b+(-a^2+b^2)^(1/2))/(-a^2+b^2)^(1/2))^(1/2))*b^2*c*(-a^2+b^2)^(1/2)*(-c^2+d^
2)^(1/2)-2*EllipticF((-(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),1/2*2^(1/2)*((b+(-a^2+b^2)^(1/2))/(-a^2+b^2)^(1/2))^(1/2))*a^3*d*(-c^2+d^2)^(1/2)+2*EllipticF((-(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),1/2*2^(1/2)*((b+(
-a^2+b^2)^(1/2))/(-a^2+b^2)^(1/2))^(1/2))*a^2*b*c*(-c^2+d^2)^(1/2)+4*EllipticF((-(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),1/2*2^(1/2)*((b+(-a^2+b^2)^(1/2))/(-a^2+b^
2)^(1/2))^(1/2))*a*b^2*d*(-c^2+d^2)^(1/2)-4*EllipticF((-(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),1/2*2^(1/2)*((b+(-a^2+b^2)^(1/2))/(-a^2+b^2)^(1/2))^(1/2))*b^3*c*(-
c^2+d^2)^(1/2)-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))/s
in(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*d*(-a^2+b^2)^(1/2)*(-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))*a^3*d*(-c^2+d
^2)^(1/2)-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*b^2*d*(-c^2+d^2)^(1/2)-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*b*d*(-a^2+b^2)^(1/2)*(-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))*a^3*d*(-c^2+d^2)^(1/2)-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*b^2*d*(-c^2+d^2)^(1/2)-Elliptic
Pi((-(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*c*d*(-a^2+b^2)^(1/2)+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*b*d^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^3*d^
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*b*c*d+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*b^2*d^2+EllipticPi((-(cos(f*x+e)*a-(-a^2+b^2)^(1/2)*si
n(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*c*d*(-a^2+b^2)^(1/2)-
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*b*d^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^3*d^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+...
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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(1/(c+d*sin(f*x+e))/(g*sin(f*x+e))^(1/2)/(a+b*sin(f*x+e))^(1/2),x, algorithm="maxima")
[Out]
integrate(1/(sqrt(b*sin(f*x + e) + a)*(d*sin(f*x + e) + c)*sqrt(g*sin(f*x + e))), 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(1/(c+d*sin(f*x+e))/(g*sin(f*x+e))^(1/2)/(a+b*sin(f*x+e))^(1/2),x, algorithm="fricas")
[Out]
integral(-sqrt(b*sin(f*x + e) + a)*sqrt(g*sin(f*x + e))/((b*c + a*d)*g*cos(f*x + e)^2 - (b*c + a*d)*g + (b*d*g
*cos(f*x + e)^2 - (a*c + b*d)*g)*sin(f*x + e)), x)
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\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(1/(c+d*sin(f*x+e))/(g*sin(f*x+e))**(1/2)/(a+b*sin(f*x+e))**(1/2),x)
[Out]
Integral(1/(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(1/(c+d*sin(f*x+e))/(g*sin(f*x+e))^(1/2)/(a+b*sin(f*x+e))^(1/2),x, algorithm="giac")
[Out]
integrate(1/(sqrt(b*sin(f*x + e) + a)*(d*sin(f*x + e) + c)*sqrt(g*sin(f*x + e))), x)
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {1}{\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(1/((g*sin(e + f*x))^(1/2)*(a + b*sin(e + f*x))^(1/2)*(c + d*sin(e + f*x))),x)
[Out]
int(1/((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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