\(\int \frac {1}{\sqrt {g \sin (e+f x)} \sqrt {a+b \sin (e+f x)} (c+c \sin (e+f x))} \, dx\) [34]

Optimal result

Integrand size = 39, antiderivative size = 256 \[ \int \frac {1}{\sqrt {g \sin (e+f x)} \sqrt {a+b \sin (e+f x)} (c+c \sin (e+f x))} \, dx=-\frac {E\left (\arcsin \left (\frac {\cos (e+f x)}{1+\sin (e+f x)}\right )|-\frac {a-b}{a+b}\right ) \sqrt {\frac {\sin (e+f x)}{1+\sin (e+f x)}} \sqrt {a+b \sin (e+f x)}}{(a-b) c f \sqrt {g \sin (e+f x)} \sqrt {\frac {a+b \sin (e+f x)}{(a+b) (1+\sin (e+f x))}}}+\frac {2 b \sqrt {a+b} \sqrt {\frac {a (1-\csc (e+f x))}{a+b}} \sqrt {\frac {a (1+\csc (e+f x))}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {g} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {g \sin (e+f x)}}\right ),-\frac {a+b}{a-b}\right ) \tan (e+f x)}{a (a-b) c f \sqrt {g}} \]

[Out]

-EllipticE(cos(f*x+e)/(1+sin(f*x+e)),((-a+b)/(a+b))^(1/2))*(sin(f*x+e)/(1+sin(f*x+e)))^(1/2)*(a+b*sin(f*x+e))^
(1/2)/(a-b)/c/f/(g*sin(f*x+e))^(1/2)/((a+b*sin(f*x+e))/(a+b)/(1+sin(f*x+e)))^(1/2)+2*b*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/(a-b)/c/f/g^(1/2)

Rubi [A] (verified)

Time = 0.36 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {3017, 2895, 3011} \[ \int \frac {1}{\sqrt {g \sin (e+f x)} \sqrt {a+b \sin (e+f x)} (c+c \sin (e+f x))} \, dx=\frac {2 b \sqrt {a+b} \tan (e+f x) \sqrt {\frac {a (1-\csc (e+f x))}{a+b}} \sqrt {\frac {a (\csc (e+f x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {g} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {g \sin (e+f x)}}\right ),-\frac {a+b}{a-b}\right )}{a c f \sqrt {g} (a-b)}-\frac {\sqrt {\frac {\sin (e+f x)}{\sin (e+f x)+1}} \sqrt {a+b \sin (e+f x)} E\left (\arcsin \left (\frac {\cos (e+f x)}{\sin (e+f x)+1}\right )|-\frac {a-b}{a+b}\right )}{c f (a-b) \sqrt {g \sin (e+f x)} \sqrt {\frac {a+b \sin (e+f x)}{(a+b) (\sin (e+f x)+1)}}} \]

[In]

Int[1/(Sqrt[g*Sin[e + f*x]]*Sqrt[a + b*Sin[e + f*x]]*(c + c*Sin[e + f*x])),x]

[Out]

-((EllipticE[ArcSin[Cos[e + f*x]/(1 + Sin[e + f*x])], -((a - b)/(a + b))]*Sqrt[Sin[e + f*x]/(1 + Sin[e + f*x])
]*Sqrt[a + b*Sin[e + f*x]])/((a - b)*c*f*Sqrt[g*Sin[e + f*x]]*Sqrt[(a + b*Sin[e + f*x])/((a + b)*(1 + Sin[e +
f*x]))])) + (2*b*Sqrt[a + b]*Sqrt[(a*(1 - Csc[e + f*x]))/(a + b)]*Sqrt[(a*(1 + Csc[e + f*x]))/(a - b)]*Ellipti
cF[ArcSin[(Sqrt[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*(a - b)*c*f*Sqrt[g])

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 3011

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

Rule 3017

Int[1/(Sqrt[(g_.)*sin[(e_.) + (f_.)*(x_)]]*Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((c_) + (d_.)*sin[(e_.)
+ (f_.)*(x_)])), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(Sqrt[g*Sin[e + f*x]]*Sqrt[a + b*Sin[e + f*x]]), x], x
] - Dist[d/(b*c - a*d), Int[Sqrt[a + b*Sin[e + f*x]]/(Sqrt[g*Sin[e + f*x]]*(c + d*Sin[e + f*x])), x], x] /; Fr
eeQ[{a, b, c, d, e, f, g}, x] && NeQ[b*c - a*d, 0] && (EqQ[a^2 - b^2, 0] || EqQ[c^2 - d^2, 0])

Rubi steps \begin{align*} \text {integral}& = -\frac {b \int \frac {1}{\sqrt {g \sin (e+f x)} \sqrt {a+b \sin (e+f x)}} \, dx}{(a-b) c}-\frac {c \int \frac {\sqrt {a+b \sin (e+f x)}}{\sqrt {g \sin (e+f x)} (c+c \sin (e+f x))} \, dx}{-a c+b c} \\ & = -\frac {E\left (\arcsin \left (\frac {\cos (e+f x)}{1+\sin (e+f x)}\right )|-\frac {a-b}{a+b}\right ) \sqrt {\frac {\sin (e+f x)}{1+\sin (e+f x)}} \sqrt {a+b \sin (e+f x)}}{(a-b) c f \sqrt {g \sin (e+f x)} \sqrt {\frac {a+b \sin (e+f x)}{(a+b) (1+\sin (e+f x))}}}+\frac {2 b \sqrt {a+b} \sqrt {\frac {a (1-\csc (e+f x))}{a+b}} \sqrt {\frac {a (1+\csc (e+f x))}{a-b}} \operatorname {EllipticF}\left (\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 ) \tan (e+f x)}{a (a-b) c f \sqrt {g}} \\ \end{align*}

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 19.75 (sec) , antiderivative size = 1667, normalized size of antiderivative = 6.51 \[ \int \frac {1}{\sqrt {g \sin (e+f x)} \sqrt {a+b \sin (e+f x)} (c+c \sin (e+f x))} \, dx=-\frac {2 \sin \left (\frac {1}{2} (e+f x)\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) \sin (e+f x) \sqrt {a+b \sin (e+f x)}}{(a-b) f \sqrt {g \sin (e+f x)} (c+c \sin (e+f x))}+\frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^2 \sqrt {\sin (e+f x)} \left (\frac {4 a (a-b) \sqrt {\frac {(a+b) \cot ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right )}{-a+b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {\csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) (a+b \sin (e+f x))}{a}}}{\sqrt {2}}\right ),-\frac {2 a}{-a+b}\right ) \sec (e+f x) \sin ^4\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) \sqrt {-\frac {(a+b) \csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) \sin (e+f x)}{a}} \sqrt {\frac {\csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) (a+b \sin (e+f x))}{a}}}{(a+b) \sqrt {\sin (e+f x)} \sqrt {a+b \sin (e+f x)}}+\frac {2 a \text {arctanh}\left (\frac {\sqrt {b} \sqrt {\sin (e+f x)}}{\sqrt {a+b \sin (e+f x)}}\right ) \cos ^2(e+f x)}{\sqrt {b} \left (1-\sin ^2(e+f x)\right )}+4 a^2 \left (\frac {\sqrt {\frac {(a+b) \cot ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right )}{-a+b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {\csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) (a+b \sin (e+f x))}{a}}}{\sqrt {2}}\right ),-\frac {2 a}{-a+b}\right ) \sec (e+f x) \sin ^4\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) \sqrt {-\frac {(a+b) \csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) \sin (e+f x)}{a}} \sqrt {\frac {\csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) (a+b \sin (e+f x))}{a}}}{(a+b) \sqrt {\sin (e+f x)} \sqrt {a+b \sin (e+f x)}}-\frac {\sqrt {\frac {(a+b) \cot ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right )}{-a+b}} \operatorname {EllipticPi}\left (-\frac {a}{b},\arcsin \left (\frac {\sqrt {\frac {\csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) (a+b \sin (e+f x))}{a}}}{\sqrt {2}}\right ),-\frac {2 a}{-a+b}\right ) \sec (e+f x) \sin ^4\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) \sqrt {-\frac {(a+b) \csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) \sin (e+f x)}{a}} \sqrt {\frac {\csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) (a+b \sin (e+f x))}{a}}}{b \sqrt {\sin (e+f x)} \sqrt {a+b \sin (e+f x)}}\right )-2 b \left (\frac {\cos (e+f x) \sqrt {a+b \sin (e+f x)}}{b \sqrt {\sin (e+f x)}}+\frac {i \cos \left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) \csc (e+f x) E\left (i \text {arcsinh}\left (\frac {\sin \left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right )}{\sqrt {\sin (e+f x)}}\right )|-\frac {2 a}{-a-b}\right ) \sqrt {a+b \sin (e+f x)}}{b \sqrt {\cos ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) \csc (e+f x)} \sqrt {\frac {\csc (e+f x) (a+b \sin (e+f x))}{a+b}}}+\frac {2 a \left (\frac {a \sqrt {\frac {(a+b) \cot ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right )}{-a+b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {\csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) (a+b \sin (e+f x))}{a}}}{\sqrt {2}}\right ),-\frac {2 a}{-a+b}\right ) \sec (e+f x) \sin ^4\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) \sqrt {-\frac {(a+b) \csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) \sin (e+f x)}{a}} \sqrt {\frac {\csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) (a+b \sin (e+f x))}{a}}}{(a+b) \sqrt {\sin (e+f x)} \sqrt {a+b \sin (e+f x)}}-\frac {a \sqrt {\frac {(a+b) \cot ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right )}{-a+b}} \operatorname {EllipticPi}\left (-\frac {a}{b},\arcsin \left (\frac {\sqrt {\frac {\csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) (a+b \sin (e+f x))}{a}}}{\sqrt {2}}\right ),-\frac {2 a}{-a+b}\right ) \sec (e+f x) \sin ^4\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) \sqrt {-\frac {(a+b) \csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) \sin (e+f x)}{a}} \sqrt {\frac {\csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) (a+b \sin (e+f x))}{a}}}{b \sqrt {\sin (e+f x)} \sqrt {a+b \sin (e+f x)}}\right )}{b}\right )+\frac {2 b \cot (e+f x) \left (-\frac {a \text {arctanh}\left (\frac {\sqrt {b} \sqrt {\sin (e+f x)}}{-\sqrt {a}+\sqrt {a+b \sin (e+f x)}}\right )}{b^{3/2}}+\frac {\sqrt {\sin (e+f x)} \sqrt {a+b \sin (e+f x)}}{2 b}\right ) \sin (2 (e+f x))}{1-\sin ^2(e+f x)}\right )}{2 (a-b) f \sqrt {g \sin (e+f x)} (c+c \sin (e+f x))} \]

[In]

Integrate[1/(Sqrt[g*Sin[e + f*x]]*Sqrt[a + b*Sin[e + f*x]]*(c + c*Sin[e + f*x])),x]

[Out]

(-2*Sin[(e + f*x)/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*Sin[e + f*x]*Sqrt[a + b*Sin[e + f*x]])/((a - b)*f*S
qrt[g*Sin[e + f*x]]*(c + c*Sin[e + f*x])) + ((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2*Sqrt[Sin[e + f*x]]*((4*a*
(a - b)*Sqrt[((a + b)*Cot[(-e + Pi/2 - f*x)/2]^2)/(-a + b)]*EllipticF[ArcSin[Sqrt[(Csc[(-e + Pi/2 - f*x)/2]^2*
(a + b*Sin[e + f*x]))/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sec[e + f*x]*Sin[(-e + Pi/2 - f*x)/2]^4*Sqrt[-(((a + b)*Cs
c[(-e + Pi/2 - f*x)/2]^2*Sin[e + f*x])/a)]*Sqrt[(Csc[(-e + Pi/2 - f*x)/2]^2*(a + b*Sin[e + f*x]))/a])/((a + b)
*Sqrt[Sin[e + f*x]]*Sqrt[a + b*Sin[e + f*x]]) + (2*a*ArcTanh[(Sqrt[b]*Sqrt[Sin[e + f*x]])/Sqrt[a + b*Sin[e + f
*x]]]*Cos[e + f*x]^2)/(Sqrt[b]*(1 - Sin[e + f*x]^2)) + 4*a^2*((Sqrt[((a + b)*Cot[(-e + Pi/2 - f*x)/2]^2)/(-a +
b)]*EllipticF[ArcSin[Sqrt[(Csc[(-e + Pi/2 - f*x)/2]^2*(a + b*Sin[e + f*x]))/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sec
[e + f*x]*Sin[(-e + Pi/2 - f*x)/2]^4*Sqrt[-(((a + b)*Csc[(-e + Pi/2 - f*x)/2]^2*Sin[e + f*x])/a)]*Sqrt[(Csc[(-
e + Pi/2 - f*x)/2]^2*(a + b*Sin[e + f*x]))/a])/((a + b)*Sqrt[Sin[e + f*x]]*Sqrt[a + b*Sin[e + f*x]]) - (Sqrt[(
(a + b)*Cot[(-e + Pi/2 - f*x)/2]^2)/(-a + b)]*EllipticPi[-(a/b), ArcSin[Sqrt[(Csc[(-e + Pi/2 - f*x)/2]^2*(a +
b*Sin[e + f*x]))/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sec[e + f*x]*Sin[(-e + Pi/2 - f*x)/2]^4*Sqrt[-(((a + b)*Csc[(-e
+ Pi/2 - f*x)/2]^2*Sin[e + f*x])/a)]*Sqrt[(Csc[(-e + Pi/2 - f*x)/2]^2*(a + b*Sin[e + f*x]))/a])/(b*Sqrt[Sin[e
+ f*x]]*Sqrt[a + b*Sin[e + f*x]])) - 2*b*((Cos[e + f*x]*Sqrt[a + b*Sin[e + f*x]])/(b*Sqrt[Sin[e + f*x]]) + (I
*Cos[(-e + Pi/2 - f*x)/2]*Csc[e + f*x]*EllipticE[I*ArcSinh[Sin[(-e + Pi/2 - f*x)/2]/Sqrt[Sin[e + f*x]]], (-2*a
)/(-a - b)]*Sqrt[a + b*Sin[e + f*x]])/(b*Sqrt[Cos[(-e + Pi/2 - f*x)/2]^2*Csc[e + f*x]]*Sqrt[(Csc[e + f*x]*(a +
b*Sin[e + f*x]))/(a + b)]) + (2*a*((a*Sqrt[((a + b)*Cot[(-e + Pi/2 - f*x)/2]^2)/(-a + b)]*EllipticF[ArcSin[Sq
rt[(Csc[(-e + Pi/2 - f*x)/2]^2*(a + b*Sin[e + f*x]))/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sec[e + f*x]*Sin[(-e + Pi/2
- f*x)/2]^4*Sqrt[-(((a + b)*Csc[(-e + Pi/2 - f*x)/2]^2*Sin[e + f*x])/a)]*Sqrt[(Csc[(-e + Pi/2 - f*x)/2]^2*(a
+ b*Sin[e + f*x]))/a])/((a + b)*Sqrt[Sin[e + f*x]]*Sqrt[a + b*Sin[e + f*x]]) - (a*Sqrt[((a + b)*Cot[(-e + Pi/2
- f*x)/2]^2)/(-a + b)]*EllipticPi[-(a/b), ArcSin[Sqrt[(Csc[(-e + Pi/2 - f*x)/2]^2*(a + b*Sin[e + f*x]))/a]/Sq
rt[2]], (-2*a)/(-a + b)]*Sec[e + f*x]*Sin[(-e + Pi/2 - f*x)/2]^4*Sqrt[-(((a + b)*Csc[(-e + Pi/2 - f*x)/2]^2*Si
n[e + f*x])/a)]*Sqrt[(Csc[(-e + Pi/2 - f*x)/2]^2*(a + b*Sin[e + f*x]))/a])/(b*Sqrt[Sin[e + f*x]]*Sqrt[a + b*Si
n[e + f*x]])))/b) + (2*b*Cot[e + f*x]*(-((a*ArcTanh[(Sqrt[b]*Sqrt[Sin[e + f*x]])/(-Sqrt[a] + Sqrt[a + b*Sin[e
+ f*x]])])/b^(3/2)) + (Sqrt[Sin[e + f*x]]*Sqrt[a + b*Sin[e + f*x]])/(2*b))*Sin[2*(e + f*x)])/(1 - Sin[e + f*x]
^2)))/(2*(a - b)*f*Sqrt[g*Sin[e + f*x]]*(c + c*Sin[e + f*x]))

Maple [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(3937\) vs. \(2(236)=472\).

Time = 2.61 (sec) , antiderivative size = 3938, normalized size of antiderivative = 15.38

method	result	size
default	\(\text {Expression too large to display}\)	\(3938\)

[In]

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

[Out]

1/2/c/f/(g/((1-cos(f*x+e))^2*csc(f*x+e)^2+1)*(csc(f*x+e)-cot(f*x+e)))^(1/2)*((a*(1-cos(f*x+e))^2*csc(f*x+e)^2+
2*b*(csc(f*x+e)-cot(f*x+e))+a)/((1-cos(f*x+e))^2*csc(f*x+e)^2+1))^(1/2)*((1/(b+(-a^2+b^2)^(1/2))*(a*(csc(f*x+e
)-cot(f*x+e))+(-a^2+b^2)^(1/2)+b))^(1/2)*(1/(-a^2+b^2)^(1/2)*(-a*(csc(f*x+e)-cot(f*x+e))+(-a^2+b^2)^(1/2)-b))^
(1/2)*(-a/(b+(-a^2+b^2)^(1/2))*(csc(f*x+e)-cot(f*x+e)))^(1/2)*EllipticF((1/(b+(-a^2+b^2)^(1/2))*(a*(csc(f*x+e)
-cot(f*x+e))+(-a^2+b^2)^(1/2)+b))^(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^2)^
(1/2)*2^(1/2)*a*(csc(f*x+e)-cot(f*x+e))-2*2^(1/2)*(-a^2+b^2)^(1/2)*(1/(b+(-a^2+b^2)^(1/2))*(a*(csc(f*x+e)-cot(
f*x+e))+(-a^2+b^2)^(1/2)+b))^(1/2)*(1/(-a^2+b^2)^(1/2)*(-a*(csc(f*x+e)-cot(f*x+e))+(-a^2+b^2)^(1/2)-b))^(1/2)*
(-a/(b+(-a^2+b^2)^(1/2))*(csc(f*x+e)-cot(f*x+e)))^(1/2)*EllipticF((1/(b+(-a^2+b^2)^(1/2))*(a*(csc(f*x+e)-cot(f
*x+e))+(-a^2+b^2)^(1/2)+b))^(1/2),1/2*2^(1/2)*((b+(-a^2+b^2)^(1/2))/(-a^2+b^2)^(1/2))^(1/2))*b*(csc(f*x+e)-cot
(f*x+e))+2*(1/(b+(-a^2+b^2)^(1/2))*(a*(csc(f*x+e)-cot(f*x+e))+(-a^2+b^2)^(1/2)+b))^(1/2)*(1/(-a^2+b^2)^(1/2)*(
-a*(csc(f*x+e)-cot(f*x+e))+(-a^2+b^2)^(1/2)-b))^(1/2)*(-a/(b+(-a^2+b^2)^(1/2))*(csc(f*x+e)-cot(f*x+e)))^(1/2)*
EllipticE((1/(b+(-a^2+b^2)^(1/2))*(a*(csc(f*x+e)-cot(f*x+e))+(-a^2+b^2)^(1/2)+b))^(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^2)^(1/2)*2^(1/2)*b*(csc(f*x+e)-cot(f*x+e))+(1/(b+(-a^2+b^2)^(1/2)
)*(a*(csc(f*x+e)-cot(f*x+e))+(-a^2+b^2)^(1/2)+b))^(1/2)*(1/(-a^2+b^2)^(1/2)*(-a*(csc(f*x+e)-cot(f*x+e))+(-a^2+
b^2)^(1/2)-b))^(1/2)*(-a/(b+(-a^2+b^2)^(1/2))*(csc(f*x+e)-cot(f*x+e)))^(1/2)*EllipticF((1/(b+(-a^2+b^2)^(1/2))
*(a*(csc(f*x+e)-cot(f*x+e))+(-a^2+b^2)^(1/2)+b))^(1/2),1/2*2^(1/2)*((b+(-a^2+b^2)^(1/2))/(-a^2+b^2)^(1/2))^(1/
2))*2^(1/2)*a^2*(csc(f*x+e)-cot(f*x+e))+(1/(b+(-a^2+b^2)^(1/2))*(a*(csc(f*x+e)-cot(f*x+e))+(-a^2+b^2)^(1/2)+b)
)^(1/2)*(1/(-a^2+b^2)^(1/2)*(-a*(csc(f*x+e)-cot(f*x+e))+(-a^2+b^2)^(1/2)-b))^(1/2)*(-a/(b+(-a^2+b^2)^(1/2))*(c
sc(f*x+e)-cot(f*x+e)))^(1/2)*EllipticF((1/(b+(-a^2+b^2)^(1/2))*(a*(csc(f*x+e)-cot(f*x+e))+(-a^2+b^2)^(1/2)+b))
^(1/2),1/2*2^(1/2)*((b+(-a^2+b^2)^(1/2))/(-a^2+b^2)^(1/2))^(1/2))*2^(1/2)*a*b*(csc(f*x+e)-cot(f*x+e))-2*2^(1/2
)*(1/(b+(-a^2+b^2)^(1/2))*(a*(csc(f*x+e)-cot(f*x+e))+(-a^2+b^2)^(1/2)+b))^(1/2)*(1/(-a^2+b^2)^(1/2)*(-a*(csc(f
*x+e)-cot(f*x+e))+(-a^2+b^2)^(1/2)-b))^(1/2)*(-a/(b+(-a^2+b^2)^(1/2))*(csc(f*x+e)-cot(f*x+e)))^(1/2)*EllipticF
((1/(b+(-a^2+b^2)^(1/2))*(a*(csc(f*x+e)-cot(f*x+e))+(-a^2+b^2)^(1/2)+b))^(1/2),1/2*2^(1/2)*((b+(-a^2+b^2)^(1/2
))/(-a^2+b^2)^(1/2))^(1/2))*b^2*(csc(f*x+e)-cot(f*x+e))-2*(1/(b+(-a^2+b^2)^(1/2))*(a*(csc(f*x+e)-cot(f*x+e))+(
-a^2+b^2)^(1/2)+b))^(1/2)*(1/(-a^2+b^2)^(1/2)*(-a*(csc(f*x+e)-cot(f*x+e))+(-a^2+b^2)^(1/2)-b))^(1/2)*(-a/(b+(-
a^2+b^2)^(1/2))*(csc(f*x+e)-cot(f*x+e)))^(1/2)*EllipticE((1/(b+(-a^2+b^2)^(1/2))*(a*(csc(f*x+e)-cot(f*x+e))+(-
a^2+b^2)^(1/2)+b))^(1/2),1/2*2^(1/2)*((b+(-a^2+b^2)^(1/2))/(-a^2+b^2)^(1/2))^(1/2))*2^(1/2)*a^2*(csc(f*x+e)-co
t(f*x+e))+2*(1/(b+(-a^2+b^2)^(1/2))*(a*(csc(f*x+e)-cot(f*x+e))+(-a^2+b^2)^(1/2)+b))^(1/2)*(1/(-a^2+b^2)^(1/2)*
(-a*(csc(f*x+e)-cot(f*x+e))+(-a^2+b^2)^(1/2)-b))^(1/2)*(-a/(b+(-a^2+b^2)^(1/2))*(csc(f*x+e)-cot(f*x+e)))^(1/2)
*EllipticE((1/(b+(-a^2+b^2)^(1/2))*(a*(csc(f*x+e)-cot(f*x+e))+(-a^2+b^2)^(1/2)+b))^(1/2),1/2*2^(1/2)*((b+(-a^2
+b^2)^(1/2))/(-a^2+b^2)^(1/2))^(1/2))*2^(1/2)*b^2*(csc(f*x+e)-cot(f*x+e))+(1/(b+(-a^2+b^2)^(1/2))*(a*(csc(f*x+
e)-cot(f*x+e))+(-a^2+b^2)^(1/2)+b))^(1/2)*(1/(-a^2+b^2)^(1/2)*(-a*(csc(f*x+e)-cot(f*x+e))+(-a^2+b^2)^(1/2)-b))
^(1/2)*(-a/(b+(-a^2+b^2)^(1/2))*(csc(f*x+e)-cot(f*x+e)))^(1/2)*EllipticF((1/(b+(-a^2+b^2)^(1/2))*(a*(csc(f*x+e
)-cot(f*x+e))+(-a^2+b^2)^(1/2)+b))^(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^2)
^(1/2)*2^(1/2)*a-2*2^(1/2)*(-a^2+b^2)^(1/2)*(1/(b+(-a^2+b^2)^(1/2))*(a*(csc(f*x+e)-cot(f*x+e))+(-a^2+b^2)^(1/2
)+b))^(1/2)*(1/(-a^2+b^2)^(1/2)*(-a*(csc(f*x+e)-cot(f*x+e))+(-a^2+b^2)^(1/2)-b))^(1/2)*(-a/(b+(-a^2+b^2)^(1/2)
)*(csc(f*x+e)-cot(f*x+e)))^(1/2)*EllipticF((1/(b+(-a^2+b^2)^(1/2))*(a*(csc(f*x+e)-cot(f*x+e))+(-a^2+b^2)^(1/2)
+b))^(1/2),1/2*2^(1/2)*((b+(-a^2+b^2)^(1/2))/(-a^2+b^2)^(1/2))^(1/2))*b+2*(1/(b+(-a^2+b^2)^(1/2))*(a*(csc(f*x+
e)-cot(f*x+e))+(-a^2+b^2)^(1/2)+b))^(1/2)*(1/(-a^2+b^2)^(1/2)*(-a*(csc(f*x+e)-cot(f*x+e))+(-a^2+b^2)^(1/2)-b))
^(1/2)*(-a/(b+(-a^2+b^2)^(1/2))*(csc(f*x+e)-cot(f*x+e)))^(1/2)*EllipticE((1/(b+(-a^2+b^2)^(1/2))*(a*(csc(f*x+e
)-cot(f*x+e))+(-a^2+b^2)^(1/2)+b))^(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^2)
^(1/2)*2^(1/2)*b+(1/(b+(-a^2+b^2)^(1/2))*(a*(csc(f*x+e)-cot(f*x+e))+(-a^2+b^2)^(1/2)+b))^(1/2)*2^(1/2)*(1/(-a^
2+b^2)^(1/2)*(-a*(csc(f*x+e)-cot(f*x+e))+(-a^2+b^2)^(1/2)-b))^(1/2)*(-a/(b+(-a^2+b^2)^(1/2))*(csc(f*x+e)-cot(f
*x+e)))^(1/2)*EllipticF((1/(b+(-a^2+b^2)^(1/2))*(a*(csc(f*x+e)-cot(f*x+e))+(-a^2+b^2)^(1/2)+b))^(1/2),1/2*2^(1
/2)*((b+(-a^2+b^2)^(1/2))/(-a^2+b^2)^(1/2))^(1/2))*a^2+(1/(b+(-a^2+b^2)^(1/2))*(a*(csc(f*x+e)-cot(f*x+e))+(-a^
2+b^2)^(1/2)+b))^(1/2)*(1/(-a^2+b^2)^(1/2)*(-a*(csc(f*x+e)-cot(f*x+e))+(-a^2+b^2)^(1/2)-b))^(1/2)*(-a/(b+(-a^2
+b^2)^(1/2))*(csc(f*x+e)-cot(f*x+e)))^(1/2)*EllipticF((1/(b+(-a^2+b^2)^(1/2))*(a*(csc(f*x+e)-cot(f*x+e))+(-a^2
+b^2)^(1/2)+b))^(1/2),1/2*2^(1/2)*((b+(-a^2+b^2)^(1/2))/(-a^2+b^2)^(1/2))^(1/2))*2^(1/2)*a*b-2*2^(1/2)*(1/(b+(
-a^2+b^2)^(1/2))*(a*(csc(f*x+e)-cot(f*x+e))+(-a^2+b^2)^(1/2)+b))^(1/2)*(1/(-a^2+b^2)^(1/2)*(-a*(csc(f*x+e)-cot
(f*x+e))+(-a^2+b^2)^(1/2)-b))^(1/2)*(-a/(b+(-a^2+b^2)^(1/2))*(csc(f*x+e)-cot(f*x+e)))^(1/2)*EllipticF((1/(b+(-
a^2+b^2)^(1/2))*(a*(csc(f*x+e)-cot(f*x+e))+(-a^2+b^2)^(1/2)+b))^(1/2),1/2*2^(1/2)*((b+(-a^2+b^2)^(1/2))/(-a^2+
b^2)^(1/2))^(1/2))*b^2-2*(1/(b+(-a^2+b^2)^(1/2))*(a*(csc(f*x+e)-cot(f*x+e))+(-a^2+b^2)^(1/2)+b))^(1/2)*(1/(-a^
2+b^2)^(1/2)*(-a*(csc(f*x+e)-cot(f*x+e))+(-a^2+b^2)^(1/2)-b))^(1/2)*(-a/(b+(-a^2+b^2)^(1/2))*(csc(f*x+e)-cot(f
*x+e)))^(1/2)*EllipticE((1/(b+(-a^2+b^2)^(1/2))*(a*(csc(f*x+e)-cot(f*x+e))+(-a^2+b^2)^(1/2)+b))^(1/2),1/2*2^(1
/2)*((b+(-a^2+b^2)^(1/2))/(-a^2+b^2)^(1/2))^(1/2))*2^(1/2)*a^2+2*(1/(b+(-a^2+b^2)^(1/2))*(a*(csc(f*x+e)-cot(f*
x+e))+(-a^2+b^2)^(1/2)+b))^(1/2)*(1/(-a^2+b^2)^(1/2)*(-a*(csc(f*x+e)-cot(f*x+e))+(-a^2+b^2)^(1/2)-b))^(1/2)*(-
a/(b+(-a^2+b^2)^(1/2))*(csc(f*x+e)-cot(f*x+e)))^(1/2)*EllipticE((1/(b+(-a^2+b^2)^(1/2))*(a*(csc(f*x+e)-cot(f*x
+e))+(-a^2+b^2)^(1/2)+b))^(1/2),1/2*2^(1/2)*((b+(-a^2+b^2)^(1/2))/(-a^2+b^2)^(1/2))^(1/2))*2^(1/2)*b^2+2*csc(f
*x+e)^3*a^2*(1-cos(f*x+e))^3+4*csc(f*x+e)^2*a*b*(1-cos(f*x+e))^2+2*a^2*(csc(f*x+e)-cot(f*x+e)))/(-cot(f*x+e)+c
sc(f*x+e)+1)/(a*(1-cos(f*x+e))^2*csc(f*x+e)^2+2*b*(csc(f*x+e)-cot(f*x+e))+a)*2^(1/2)/a/(a-b)

Fricas [F]

\[ \int \frac {1}{\sqrt {g \sin (e+f x)} \sqrt {a+b \sin (e+f x)} (c+c \sin (e+f x))} \, dx=\int { \frac {1}{\sqrt {b \sin \left (f x + e\right ) + a} {\left (c \sin \left (f x + e\right ) + c\right )} \sqrt {g \sin \left (f x + e\right )}} \,d x } \]

[In]

integrate(1/(c+c*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))/((a + b)*c*g*cos(f*x + e)^2 - (a + b)*c*g + (b*c*g*cos
(f*x + e)^2 - (a + b)*c*g)*sin(f*x + e)), x)

Sympy [F]

\[ \int \frac {1}{\sqrt {g \sin (e+f x)} \sqrt {a+b \sin (e+f x)} (c+c \sin (e+f x))} \, dx=\frac {\int \frac {1}{\sqrt {g \sin {\left (e + f x \right )}} \sqrt {a + b \sin {\left (e + f x \right )}} \sin {\left (e + f x \right )} + \sqrt {g \sin {\left (e + f x \right )}} \sqrt {a + b \sin {\left (e + f x \right )}}}\, dx}{c} \]

[In]

integrate(1/(c+c*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))*sin(e + f*x) + sqrt(g*sin(e + f*x))*sqrt(a + b*sin(e
+ f*x))), x)/c

Maxima [F]

\[ \int \frac {1}{\sqrt {g \sin (e+f x)} \sqrt {a+b \sin (e+f x)} (c+c \sin (e+f x))} \, dx=\int { \frac {1}{\sqrt {b \sin \left (f x + e\right ) + a} {\left (c \sin \left (f x + e\right ) + c\right )} \sqrt {g \sin \left (f x + e\right )}} \,d x } \]

[In]

integrate(1/(c+c*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)*(c*sin(f*x + e) + c)*sqrt(g*sin(f*x + e))), x)

Giac [F]

\[ \int \frac {1}{\sqrt {g \sin (e+f x)} \sqrt {a+b \sin (e+f x)} (c+c \sin (e+f x))} \, dx=\int { \frac {1}{\sqrt {b \sin \left (f x + e\right ) + a} {\left (c \sin \left (f x + e\right ) + c\right )} \sqrt {g \sin \left (f x + e\right )}} \,d x } \]

[In]

integrate(1/(c+c*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)*(c*sin(f*x + e) + c)*sqrt(g*sin(f*x + e))), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {g \sin (e+f x)} \sqrt {a+b \sin (e+f x)} (c+c \sin (e+f x))} \, dx=\int \frac {1}{\sqrt {g\,\sin \left (e+f\,x\right )}\,\sqrt {a+b\,\sin \left (e+f\,x\right )}\,\left (c+c\,\sin \left (e+f\,x\right )\right )} \,d x \]

[In]

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

[Out]

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