\(\int \frac {\sqrt {g \sin (e+f x)} \sqrt {a+a \sin (e+f x)}}{c-c \sin (e+f x)} \, dx\) [15]

Optimal result

Integrand size = 40, antiderivative size = 103 \[ \int \frac {\sqrt {g \sin (e+f x)} \sqrt {a+a \sin (e+f x)}}{c-c \sin (e+f x)} \, dx=\frac {2 \sqrt {a} \sqrt {g} \arctan \left (\frac {\sqrt {a} \sqrt {g} \cos (e+f x)}{\sqrt {g \sin (e+f x)} \sqrt {a+a \sin (e+f x)}}\right )}{c f}+\frac {2 \sec (e+f x) \sqrt {g \sin (e+f x)} \sqrt {a+a \sin (e+f x)}}{c f} \]

[Out]

2*arctan(cos(f*x+e)*a^(1/2)*g^(1/2)/(g*sin(f*x+e))^(1/2)/(a+a*sin(f*x+e))^(1/2))*a^(1/2)*g^(1/2)/c/f+2*sec(f*x
+e)*(g*sin(f*x+e))^(1/2)*(a+a*sin(f*x+e))^(1/2)/c/f

Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {3007, 2854, 211, 3009, 12, 30} \[ \int \frac {\sqrt {g \sin (e+f x)} \sqrt {a+a \sin (e+f x)}}{c-c \sin (e+f x)} \, dx=\frac {2 \sqrt {a} \sqrt {g} \arctan \left (\frac {\sqrt {a} \sqrt {g} \cos (e+f x)}{\sqrt {a \sin (e+f x)+a} \sqrt {g \sin (e+f x)}}\right )}{c f}+\frac {2 \sec (e+f x) \sqrt {a \sin (e+f x)+a} \sqrt {g \sin (e+f x)}}{c f} \]

[In]

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

[Out]

(2*Sqrt[a]*Sqrt[g]*ArcTan[(Sqrt[a]*Sqrt[g]*Cos[e + f*x])/(Sqrt[g*Sin[e + f*x]]*Sqrt[a + a*Sin[e + f*x]])])/(c*
f) + (2*Sec[e + f*x]*Sqrt[g*Sin[e + f*x]]*Sqrt[a + a*Sin[e + f*x]])/(c*f)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 2854

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

Rule 3007

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

Rule 3009

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

Rubi steps \begin{align*} \text {integral}& = g \int \frac {\sqrt {a+a \sin (e+f x)}}{\sqrt {g \sin (e+f x)} (c-c \sin (e+f x))} \, dx-\frac {g \int \frac {\sqrt {a+a \sin (e+f x)}}{\sqrt {g \sin (e+f x)}} \, dx}{c} \\ & = -\frac {(2 a g) \text {Subst}\left (\int \frac {1}{c g x^2} \, dx,x,\frac {a \cos (e+f x)}{\sqrt {g \sin (e+f x)} \sqrt {a+a \sin (e+f x)}}\right )}{f}+\frac {(2 a g) \text {Subst}\left (\int \frac {1}{a+g x^2} \, dx,x,\frac {a \cos (e+f x)}{\sqrt {g \sin (e+f x)} \sqrt {a+a \sin (e+f x)}}\right )}{c f} \\ & = \frac {2 \sqrt {a} \sqrt {g} \arctan \left (\frac {\sqrt {a} \sqrt {g} \cos (e+f x)}{\sqrt {g \sin (e+f x)} \sqrt {a+a \sin (e+f x)}}\right )}{c f}-\frac {(2 a) \text {Subst}\left (\int \frac {1}{x^2} \, dx,x,\frac {a \cos (e+f x)}{\sqrt {g \sin (e+f x)} \sqrt {a+a \sin (e+f x)}}\right )}{c f} \\ & = \frac {2 \sqrt {a} \sqrt {g} \arctan \left (\frac {\sqrt {a} \sqrt {g} \cos (e+f x)}{\sqrt {g \sin (e+f x)} \sqrt {a+a \sin (e+f x)}}\right )}{c f}+\frac {2 \sec (e+f x) \sqrt {g \sin (e+f x)} \sqrt {a+a \sin (e+f x)}}{c f} \\ \end{align*}

Mathematica [A] (verified)

Time = 2.59 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.88 \[ \int \frac {\sqrt {g \sin (e+f x)} \sqrt {a+a \sin (e+f x)}}{c-c \sin (e+f x)} \, dx=\frac {2 \sec (e+f x) \left (\arcsin \left (\sqrt {1-\sin (e+f x)}\right ) \sqrt {1-\sin (e+f x)}+\sqrt {\sin (e+f x)}\right ) \sqrt {g \sin (e+f x)} \sqrt {a (1+\sin (e+f x))}}{c f \sqrt {\sin (e+f x)}} \]

[In]

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

[Out]

(2*Sec[e + f*x]*(ArcSin[Sqrt[1 - Sin[e + f*x]]]*Sqrt[1 - Sin[e + f*x]] + Sqrt[Sin[e + f*x]])*Sqrt[g*Sin[e + f*
x]]*Sqrt[a*(1 + Sin[e + f*x])])/(c*f*Sqrt[Sin[e + f*x]])

Maple [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(753\) vs. \(2(87)=174\).

Time = 3.21 (sec) , antiderivative size = 754, normalized size of antiderivative = 7.32

method	result	size
default	\(-\frac {\sqrt {\frac {g \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right )}{\left (1-\cos \left (f x +e \right )\right )^{2} \left (\csc ^{2}\left (f x +e \right )\right )+1}}\, \left (\left (1-\cos \left (f x +e \right )\right )^{2} \left (\csc ^{2}\left (f x +e \right )\right )+1\right ) \sqrt {\frac {a \left (\left (1-\cos \left (f x +e \right )\right )^{2} \left (\csc ^{2}\left (f x +e \right )\right )+2 \csc \left (f x +e \right )-2 \cot \left (f x +e \right )+1\right )}{\left (1-\cos \left (f x +e \right )\right )^{2} \left (\csc ^{2}\left (f x +e \right )\right )+1}}\, \left (\sqrt {2}\, \ln \left (-\frac {\csc \left (f x +e \right )-\cot \left (f x +e \right )+\sqrt {\csc \left (f x +e \right )-\cot \left (f x +e \right )}\, \sqrt {2}+1}{\sqrt {\csc \left (f x +e \right )-\cot \left (f x +e \right )}\, \sqrt {2}-\csc \left (f x +e \right )+\cot \left (f x +e \right )-1}\right ) \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right )+4 \sqrt {2}\, \arctan \left (\sqrt {\csc \left (f x +e \right )-\cot \left (f x +e \right )}\, \sqrt {2}+1\right ) \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right )+4 \sqrt {2}\, \arctan \left (\sqrt {\csc \left (f x +e \right )-\cot \left (f x +e \right )}\, \sqrt {2}-1\right ) \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right )+\sqrt {2}\, \ln \left (-\frac {\sqrt {\csc \left (f x +e \right )-\cot \left (f x +e \right )}\, \sqrt {2}-\csc \left (f x +e \right )+\cot \left (f x +e \right )-1}{\csc \left (f x +e \right )-\cot \left (f x +e \right )+\sqrt {\csc \left (f x +e \right )-\cot \left (f x +e \right )}\, \sqrt {2}+1}\right ) \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right )-\sqrt {2}\, \ln \left (-\frac {\csc \left (f x +e \right )-\cot \left (f x +e \right )+\sqrt {\csc \left (f x +e \right )-\cot \left (f x +e \right )}\, \sqrt {2}+1}{\sqrt {\csc \left (f x +e \right )-\cot \left (f x +e \right )}\, \sqrt {2}-\csc \left (f x +e \right )+\cot \left (f x +e \right )-1}\right )-4 \sqrt {2}\, \arctan \left (\sqrt {\csc \left (f x +e \right )-\cot \left (f x +e \right )}\, \sqrt {2}+1\right )-4 \sqrt {2}\, \arctan \left (\sqrt {\csc \left (f x +e \right )-\cot \left (f x +e \right )}\, \sqrt {2}-1\right )-\sqrt {2}\, \ln \left (-\frac {\sqrt {\csc \left (f x +e \right )-\cot \left (f x +e \right )}\, \sqrt {2}-\csc \left (f x +e \right )+\cot \left (f x +e \right )-1}{\csc \left (f x +e \right )-\cot \left (f x +e \right )+\sqrt {\csc \left (f x +e \right )-\cot \left (f x +e \right )}\, \sqrt {2}+1}\right )+8 \sqrt {\csc \left (f x +e \right )-\cot \left (f x +e \right )}\right ) \sqrt {2}}{4 c f \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1\right ) \sqrt {\csc \left (f x +e \right )-\cot \left (f x +e \right )}\, \left (\sqrt {\csc \left (f x +e \right )-\cot \left (f x +e \right )}+1\right ) \left (\sqrt {\csc \left (f x +e \right )-\cot \left (f x +e \right )}-1\right )}\)	\(754\)

[In]

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

[Out]

-1/4/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)*((1-cos(f*x+e))^2*csc(f*x+e)^2+1)
*(a*((1-cos(f*x+e))^2*csc(f*x+e)^2+2*csc(f*x+e)-2*cot(f*x+e)+1)/((1-cos(f*x+e))^2*csc(f*x+e)^2+1))^(1/2)/(-cot
(f*x+e)+csc(f*x+e)+1)*(2^(1/2)*ln(-(csc(f*x+e)-cot(f*x+e)+(csc(f*x+e)-cot(f*x+e))^(1/2)*2^(1/2)+1)/((csc(f*x+e
)-cot(f*x+e))^(1/2)*2^(1/2)-csc(f*x+e)+cot(f*x+e)-1))*(csc(f*x+e)-cot(f*x+e))+4*2^(1/2)*arctan((csc(f*x+e)-cot
(f*x+e))^(1/2)*2^(1/2)+1)*(csc(f*x+e)-cot(f*x+e))+4*2^(1/2)*arctan((csc(f*x+e)-cot(f*x+e))^(1/2)*2^(1/2)-1)*(c
sc(f*x+e)-cot(f*x+e))+2^(1/2)*ln(-((csc(f*x+e)-cot(f*x+e))^(1/2)*2^(1/2)-csc(f*x+e)+cot(f*x+e)-1)/(csc(f*x+e)-
cot(f*x+e)+(csc(f*x+e)-cot(f*x+e))^(1/2)*2^(1/2)+1))*(csc(f*x+e)-cot(f*x+e))-2^(1/2)*ln(-(csc(f*x+e)-cot(f*x+e
)+(csc(f*x+e)-cot(f*x+e))^(1/2)*2^(1/2)+1)/((csc(f*x+e)-cot(f*x+e))^(1/2)*2^(1/2)-csc(f*x+e)+cot(f*x+e)-1))-4*
2^(1/2)*arctan((csc(f*x+e)-cot(f*x+e))^(1/2)*2^(1/2)+1)-4*2^(1/2)*arctan((csc(f*x+e)-cot(f*x+e))^(1/2)*2^(1/2)
-1)-2^(1/2)*ln(-((csc(f*x+e)-cot(f*x+e))^(1/2)*2^(1/2)-csc(f*x+e)+cot(f*x+e)-1)/(csc(f*x+e)-cot(f*x+e)+(csc(f*
x+e)-cot(f*x+e))^(1/2)*2^(1/2)+1))+8*(csc(f*x+e)-cot(f*x+e))^(1/2))/(csc(f*x+e)-cot(f*x+e))^(1/2)/((csc(f*x+e)
-cot(f*x+e))^(1/2)+1)/((csc(f*x+e)-cot(f*x+e))^(1/2)-1)*2^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.41 (sec) , antiderivative size = 442, normalized size of antiderivative = 4.29 \[ \int \frac {\sqrt {g \sin (e+f x)} \sqrt {a+a \sin (e+f x)}}{c-c \sin (e+f x)} \, dx=\left [\frac {\sqrt {-a g} \cos \left (f x + e\right ) \log \left (\frac {128 \, a g \cos \left (f x + e\right )^{5} - 128 \, a g \cos \left (f x + e\right )^{4} - 416 \, a g \cos \left (f x + e\right )^{3} + 128 \, a g \cos \left (f x + e\right )^{2} + 289 \, a g \cos \left (f x + e\right ) + 8 \, {\left (16 \, \cos \left (f x + e\right )^{4} - 24 \, \cos \left (f x + e\right )^{3} - 66 \, \cos \left (f x + e\right )^{2} + {\left (16 \, \cos \left (f x + e\right )^{3} + 40 \, \cos \left (f x + e\right )^{2} - 26 \, \cos \left (f x + e\right ) - 51\right )} \sin \left (f x + e\right ) + 25 \, \cos \left (f x + e\right ) + 51\right )} \sqrt {-a g} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {g \sin \left (f x + e\right )} + a g + {\left (128 \, a g \cos \left (f x + e\right )^{4} + 256 \, a g \cos \left (f x + e\right )^{3} - 160 \, a g \cos \left (f x + e\right )^{2} - 288 \, a g \cos \left (f x + e\right ) + a g\right )} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + \sin \left (f x + e\right ) + 1}\right ) + 8 \, \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {g \sin \left (f x + e\right )}}{4 \, c f \cos \left (f x + e\right )}, -\frac {\sqrt {a g} \arctan \left (\frac {\sqrt {a g} {\left (8 \, \cos \left (f x + e\right )^{2} + 8 \, \sin \left (f x + e\right ) - 9\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {g \sin \left (f x + e\right )}}{4 \, {\left (2 \, a g \cos \left (f x + e\right )^{3} + a g \cos \left (f x + e\right ) \sin \left (f x + e\right ) - 2 \, a g \cos \left (f x + e\right )\right )}}\right ) \cos \left (f x + e\right ) - 4 \, \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {g \sin \left (f x + e\right )}}{2 \, c f \cos \left (f x + e\right )}\right ] \]

[In]

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

[Out]

[1/4*(sqrt(-a*g)*cos(f*x + e)*log((128*a*g*cos(f*x + e)^5 - 128*a*g*cos(f*x + e)^4 - 416*a*g*cos(f*x + e)^3 +
128*a*g*cos(f*x + e)^2 + 289*a*g*cos(f*x + e) + 8*(16*cos(f*x + e)^4 - 24*cos(f*x + e)^3 - 66*cos(f*x + e)^2 +
(16*cos(f*x + e)^3 + 40*cos(f*x + e)^2 - 26*cos(f*x + e) - 51)*sin(f*x + e) + 25*cos(f*x + e) + 51)*sqrt(-a*g
)*sqrt(a*sin(f*x + e) + a)*sqrt(g*sin(f*x + e)) + a*g + (128*a*g*cos(f*x + e)^4 + 256*a*g*cos(f*x + e)^3 - 160
*a*g*cos(f*x + e)^2 - 288*a*g*cos(f*x + e) + a*g)*sin(f*x + e))/(cos(f*x + e) + sin(f*x + e) + 1)) + 8*sqrt(a*
sin(f*x + e) + a)*sqrt(g*sin(f*x + e)))/(c*f*cos(f*x + e)), -1/2*(sqrt(a*g)*arctan(1/4*sqrt(a*g)*(8*cos(f*x +
e)^2 + 8*sin(f*x + e) - 9)*sqrt(a*sin(f*x + e) + a)*sqrt(g*sin(f*x + e))/(2*a*g*cos(f*x + e)^3 + a*g*cos(f*x +
e)*sin(f*x + e) - 2*a*g*cos(f*x + e)))*cos(f*x + e) - 4*sqrt(a*sin(f*x + e) + a)*sqrt(g*sin(f*x + e)))/(c*f*c
os(f*x + e))]

Sympy [F]

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

[In]

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

[Out]

-Integral(sqrt(g*sin(e + f*x))*sqrt(a*sin(e + f*x) + a)/(sin(e + f*x) - 1), x)/c

Maxima [F]

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

[In]

integrate((g*sin(f*x+e))^(1/2)*(a+a*sin(f*x+e))^(1/2)/(c-c*sin(f*x+e)),x, algorithm="maxima")

[Out]

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

Giac [F(-1)]

Timed out. \[ \int \frac {\sqrt {g \sin (e+f x)} \sqrt {a+a \sin (e+f x)}}{c-c \sin (e+f x)} \, dx=\text {Timed out} \]

[In]

integrate((g*sin(f*x+e))^(1/2)*(a+a*sin(f*x+e))^(1/2)/(c-c*sin(f*x+e)),x, algorithm="giac")

[Out]

Timed out

Mupad [F(-1)]

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

[In]

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

[Out]

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