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

Optimal. Leaf size=103 \[ \frac {2 \sqrt {a} \sqrt {g} \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {g} \cos (e+f x)}{\sqrt {g \sin (e+f x)} \sqrt {a+a \sin (e+f x)}}\right )}{c f}+\frac {2 \sec (e+f x) \sqrt {g \sin (e+f x)} \sqrt {a+a \sin (e+f x)}}{c f} \]

[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

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Rubi [A]
time = 0.31, antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {3007, 2854, 211, 3009, 12, 30} \begin {gather*} \frac {2 \sqrt {a} \sqrt {g} \text {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} \end {gather*}

Antiderivative was successfully verified.

[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*} \int \frac {\sqrt {g \sin (e+f x)} \sqrt {a+a \sin (e+f x)}}{c-c \sin (e+f x)} \, dx &=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} \tan ^{-1}\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} \tan ^{-1}\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*}

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Mathematica [C] Result contains complex when optimal does not.
time = 0.63, size = 194, normalized size = 1.88 \begin {gather*} \frac {2 e^{i (e+f x)} \left (2 \left (-1+e^{2 i (e+f x)}\right )-i \left (-i+e^{i (e+f x)}\right ) \sqrt {-1+e^{2 i (e+f x)}} \tan ^{-1}\left (\sqrt {-1+e^{2 i (e+f x)}}\right )-\left (-i+e^{i (e+f x)}\right ) \sqrt {-1+e^{2 i (e+f x)}} \tanh ^{-1}\left (\frac {e^{i (e+f x)}}{\sqrt {-1+e^{2 i (e+f x)}}}\right )\right ) \sqrt {g \sin (e+f x)} \sqrt {a (1+\sin (e+f x))}}{c \left (-1+e^{4 i (e+f x)}\right ) f} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*E^(I*(e + f*x))*(2*(-1 + E^((2*I)*(e + f*x))) - I*(-I + E^(I*(e + f*x)))*Sqrt[-1 + E^((2*I)*(e + f*x))]*Arc
Tan[Sqrt[-1 + E^((2*I)*(e + f*x))]] - (-I + E^(I*(e + f*x)))*Sqrt[-1 + E^((2*I)*(e + f*x))]*ArcTanh[E^(I*(e +
f*x))/Sqrt[-1 + E^((2*I)*(e + f*x))]])*Sqrt[g*Sin[e + f*x]]*Sqrt[a*(1 + Sin[e + f*x])])/(c*(-1 + E^((4*I)*(e +
f*x)))*f)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(913\) vs. \(2(87)=174\).
time = 28.70, size = 914, normalized size = 8.87

method	result	size
default	\(\frac {\left (4 \sqrt {-\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \sqrt {2}\, \sin \left (f x +e \right )-\sin \left (f x +e \right ) \ln \left (-\frac {\sqrt {-\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \sqrt {2}\, \sin \left (f x +e \right )+\sin \left (f x +e \right )-\cos \left (f x +e \right )+1}{\sqrt {-\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \sqrt {2}\, \sin \left (f x +e \right )-\sin \left (f x +e \right )+\cos \left (f x +e \right )-1}\right )-4 \sin \left (f x +e \right ) \arctan \left (\sqrt {-\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \sqrt {2}+1\right )-4 \sin \left (f x +e \right ) \arctan \left (\sqrt {-\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \sqrt {2}-1\right )-\sin \left (f x +e \right ) \ln \left (-\frac {\sqrt {-\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \sqrt {2}\, \sin \left (f x +e \right )-\sin \left (f x +e \right )+\cos \left (f x +e \right )-1}{\sqrt {-\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \sqrt {2}\, \sin \left (f x +e \right )+\sin \left (f x +e \right )-\cos \left (f x +e \right )+1}\right )-\cos \left (f x +e \right ) \ln \left (-\frac {\sqrt {-\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \sqrt {2}\, \sin \left (f x +e \right )+\sin \left (f x +e \right )-\cos \left (f x +e \right )+1}{\sqrt {-\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \sqrt {2}\, \sin \left (f x +e \right )-\sin \left (f x +e \right )+\cos \left (f x +e \right )-1}\right )-4 \cos \left (f x +e \right ) \arctan \left (\sqrt {-\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \sqrt {2}+1\right )-4 \cos \left (f x +e \right ) \arctan \left (\sqrt {-\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \sqrt {2}-1\right )-\cos \left (f x +e \right ) \ln \left (-\frac {\sqrt {-\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \sqrt {2}\, \sin \left (f x +e \right )-\sin \left (f x +e \right )+\cos \left (f x +e \right )-1}{\sqrt {-\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \sqrt {2}\, \sin \left (f x +e \right )+\sin \left (f x +e \right )-\cos \left (f x +e \right )+1}\right )+\ln \left (-\frac {\sqrt {-\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \sqrt {2}\, \sin \left (f x +e \right )+\sin \left (f x +e \right )-\cos \left (f x +e \right )+1}{\sqrt {-\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \sqrt {2}\, \sin \left (f x +e \right )-\sin \left (f x +e \right )+\cos \left (f x +e \right )-1}\right )+4 \arctan \left (\sqrt {-\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \sqrt {2}+1\right )+4 \arctan \left (\sqrt {-\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \sqrt {2}-1\right )+\ln \left (-\frac {\sqrt {-\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \sqrt {2}\, \sin \left (f x +e \right )-\sin \left (f x +e \right )+\cos \left (f x +e \right )-1}{\sqrt {-\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \sqrt {2}\, \sin \left (f x +e \right )+\sin \left (f x +e \right )-\cos \left (f x +e \right )+1}\right )\right ) \sqrt {g \sin \left (f x +e \right )}\, \sqrt {a \left (1+\sin \left (f x +e \right )\right )}\, \sqrt {2}}{4 c f \sin \left (f x +e \right ) \cos \left (f x +e \right ) \sqrt {-\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}}}\)	\(914\)

Verification of antiderivative is not currently implemented for this CAS.

[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*(4*(-(-1+cos(f*x+e))/sin(f*x+e))^(1/2)*2^(1/2)*sin(f*x+e)-sin(f*x+e)*ln(-((-(-1+cos(f*x+e))/sin(f*x+e)
)^(1/2)*2^(1/2)*sin(f*x+e)+sin(f*x+e)-cos(f*x+e)+1)/((-(-1+cos(f*x+e))/sin(f*x+e))^(1/2)*2^(1/2)*sin(f*x+e)-si
n(f*x+e)+cos(f*x+e)-1))-4*sin(f*x+e)*arctan((-(-1+cos(f*x+e))/sin(f*x+e))^(1/2)*2^(1/2)+1)-4*sin(f*x+e)*arctan
((-(-1+cos(f*x+e))/sin(f*x+e))^(1/2)*2^(1/2)-1)-sin(f*x+e)*ln(-((-(-1+cos(f*x+e))/sin(f*x+e))^(1/2)*2^(1/2)*si
n(f*x+e)-sin(f*x+e)+cos(f*x+e)-1)/((-(-1+cos(f*x+e))/sin(f*x+e))^(1/2)*2^(1/2)*sin(f*x+e)+sin(f*x+e)-cos(f*x+e
)+1))-cos(f*x+e)*ln(-((-(-1+cos(f*x+e))/sin(f*x+e))^(1/2)*2^(1/2)*sin(f*x+e)+sin(f*x+e)-cos(f*x+e)+1)/((-(-1+c
os(f*x+e))/sin(f*x+e))^(1/2)*2^(1/2)*sin(f*x+e)-sin(f*x+e)+cos(f*x+e)-1))-4*cos(f*x+e)*arctan((-(-1+cos(f*x+e)
)/sin(f*x+e))^(1/2)*2^(1/2)+1)-4*cos(f*x+e)*arctan((-(-1+cos(f*x+e))/sin(f*x+e))^(1/2)*2^(1/2)-1)-cos(f*x+e)*l
n(-((-(-1+cos(f*x+e))/sin(f*x+e))^(1/2)*2^(1/2)*sin(f*x+e)-sin(f*x+e)+cos(f*x+e)-1)/((-(-1+cos(f*x+e))/sin(f*x
+e))^(1/2)*2^(1/2)*sin(f*x+e)+sin(f*x+e)-cos(f*x+e)+1))+ln(-((-(-1+cos(f*x+e))/sin(f*x+e))^(1/2)*2^(1/2)*sin(f
*x+e)+sin(f*x+e)-cos(f*x+e)+1)/((-(-1+cos(f*x+e))/sin(f*x+e))^(1/2)*2^(1/2)*sin(f*x+e)-sin(f*x+e)+cos(f*x+e)-1
))+4*arctan((-(-1+cos(f*x+e))/sin(f*x+e))^(1/2)*2^(1/2)+1)+4*arctan((-(-1+cos(f*x+e))/sin(f*x+e))^(1/2)*2^(1/2
)-1)+ln(-((-(-1+cos(f*x+e))/sin(f*x+e))^(1/2)*2^(1/2)*sin(f*x+e)-sin(f*x+e)+cos(f*x+e)-1)/((-(-1+cos(f*x+e))/s
in(f*x+e))^(1/2)*2^(1/2)*sin(f*x+e)+sin(f*x+e)-cos(f*x+e)+1)))*(g*sin(f*x+e))^(1/2)*(a*(1+sin(f*x+e)))^(1/2)/s
in(f*x+e)/cos(f*x+e)/(-(-1+cos(f*x+e))/sin(f*x+e))^(1/2)*2^(1/2)

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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((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)

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Fricas [A]
time = 0.53, size = 480, normalized size = 4.66 \begin {gather*} \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 ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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

[Out]

Timed out

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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+a\,\sin \left (e+f\,x\right )}}{c-c\,\sin \left (e+f\,x\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 + 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)

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