3.85 \(\int \frac{\sqrt{a+a \sin (e+f x)}}{\sqrt{\sin (e+f x)}} \, dx\)

Optimal. Leaf size=37 \[ -\frac{2 \sqrt{a} \sin ^{-1}\left (\frac{\sqrt{a} \cos (e+f x)}{\sqrt{a \sin (e+f x)+a}}\right )}{f} \]

[Out]

(-2*Sqrt[a]*ArcSin[(Sqrt[a]*Cos[e + f*x])/Sqrt[a + a*Sin[e + f*x]]])/f

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Rubi [A]  time = 0.0555735, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, Rules used = {2774, 216} \[ -\frac{2 \sqrt{a} \sin ^{-1}\left (\frac{\sqrt{a} \cos (e+f x)}{\sqrt{a \sin (e+f x)+a}}\right )}{f} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + a*Sin[e + f*x]]/Sqrt[Sin[e + f*x]],x]

[Out]

(-2*Sqrt[a]*ArcSin[(Sqrt[a]*Cos[e + f*x])/Sqrt[a + a*Sin[e + f*x]]])/f

Rule 2774

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

Rule 216

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[(Rt[-b, 2]*x)/Sqrt[a]]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rubi steps

\begin{align*} \int \frac{\sqrt{a+a \sin (e+f x)}}{\sqrt{\sin (e+f x)}} \, dx &=-\frac{2 \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{x^2}{a}}} \, dx,x,\frac{a \cos (e+f x)}{\sqrt{a+a \sin (e+f x)}}\right )}{f}\\ &=-\frac{2 \sqrt{a} \sin ^{-1}\left (\frac{\sqrt{a} \cos (e+f x)}{\sqrt{a+a \sin (e+f x)}}\right )}{f}\\ \end{align*}

Mathematica [C]  time = 0.51352, size = 164, normalized size = 4.43 \[ \frac{(1+i) e^{\frac{1}{2} i (e+f x)} \sqrt{-i e^{-i (e+f x)} \left (-1+e^{2 i (e+f x)}\right )} \sqrt{a (\sin (e+f x)+1)} \left (\tan ^{-1}\left (\sqrt{-1+e^{2 i (e+f x)}}\right )-i \tanh ^{-1}\left (\frac{e^{i (e+f x)}}{\sqrt{-1+e^{2 i (e+f x)}}}\right )\right )}{\sqrt{2} f \sqrt{-1+e^{2 i (e+f x)}} \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a + a*Sin[e + f*x]]/Sqrt[Sin[e + f*x]],x]

[Out]

((1 + I)*E^((I/2)*(e + f*x))*Sqrt[((-I)*(-1 + E^((2*I)*(e + f*x))))/E^(I*(e + f*x))]*(ArcTan[Sqrt[-1 + E^((2*I
)*(e + f*x))]] - I*ArcTanh[E^(I*(e + f*x))/Sqrt[-1 + E^((2*I)*(e + f*x))]])*Sqrt[a*(1 + Sin[e + f*x])])/(Sqrt[
2]*Sqrt[-1 + E^((2*I)*(e + f*x))]*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]))

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Maple [B]  time = 0.141, size = 320, normalized size = 8.7 \begin{align*}{\frac{\sqrt{2}}{2\,f \left ( 1-\cos \left ( fx+e \right ) +\sin \left ( fx+e \right ) \right ) }\sqrt{-{\frac{-1+\cos \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }}}\sqrt{a \left ( 1+\sin \left ( fx+e \right ) \right ) }\sqrt{\sin \left ( fx+e \right ) } \left ( \ln \left ( -{ \left ( \sqrt{2}\sqrt{-{\frac{-1+\cos \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }}}\sin \left ( fx+e \right ) +\sin \left ( fx+e \right ) -\cos \left ( fx+e \right ) +1 \right ) \left ( \sqrt{2}\sqrt{-{\frac{-1+\cos \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }}}\sin \left ( fx+e \right ) -\sin \left ( fx+e \right ) +\cos \left ( fx+e \right ) -1 \right ) ^{-1}} \right ) +4\,\arctan \left ( \sqrt{2}\sqrt{-{\frac{-1+\cos \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }}}+1 \right ) +4\,\arctan \left ( \sqrt{2}\sqrt{-{\frac{-1+\cos \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }}}-1 \right ) +\ln \left ( -{ \left ( \sqrt{2}\sqrt{-{\frac{-1+\cos \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }}}\sin \left ( fx+e \right ) -\sin \left ( fx+e \right ) +\cos \left ( fx+e \right ) -1 \right ) \left ( \sqrt{2}\sqrt{-{\frac{-1+\cos \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }}}\sin \left ( fx+e \right ) +\sin \left ( fx+e \right ) -\cos \left ( fx+e \right ) +1 \right ) ^{-1}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2/f*(-(-1+cos(f*x+e))/sin(f*x+e))^(1/2)*(a*(1+sin(f*x+e)))^(1/2)*sin(f*x+e)^(1/2)*(ln(-(2^(1/2)*(-(-1+cos(f*
x+e))/sin(f*x+e))^(1/2)*sin(f*x+e)+sin(f*x+e)-cos(f*x+e)+1)/(2^(1/2)*(-(-1+cos(f*x+e))/sin(f*x+e))^(1/2)*sin(f
*x+e)-sin(f*x+e)+cos(f*x+e)-1))+4*arctan(2^(1/2)*(-(-1+cos(f*x+e))/sin(f*x+e))^(1/2)+1)+4*arctan(2^(1/2)*(-(-1
+cos(f*x+e))/sin(f*x+e))^(1/2)-1)+ln(-(2^(1/2)*(-(-1+cos(f*x+e))/sin(f*x+e))^(1/2)*sin(f*x+e)-sin(f*x+e)+cos(f
*x+e)-1)/(2^(1/2)*(-(-1+cos(f*x+e))/sin(f*x+e))^(1/2)*sin(f*x+e)+sin(f*x+e)-cos(f*x+e)+1)))*2^(1/2)/(1-cos(f*x
+e)+sin(f*x+e))

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Maxima [B]  time = 2.09046, size = 284, normalized size = 7.68 \begin{align*} -\frac{2 \, \sqrt{2} \sqrt{a} \left (\frac{\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )^{\frac{3}{2}} - 3 \, \sqrt{2}{\left (\sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} + 2 \, \sqrt{\frac{\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}}\right )}\right ) + \sqrt{2} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} - 2 \, \sqrt{\frac{\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}}\right )}\right )\right )} \sqrt{a} + 6 \, \sqrt{2} \sqrt{a} \sqrt{\frac{\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}} - \frac{2 \,{\left (\frac{3 \, \sqrt{2} \sqrt{a} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac{\sqrt{2} \sqrt{a} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}}\right )}}{\sqrt{\frac{\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}}}}{3 \, f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(2*sqrt(2)*sqrt(a)*(sin(f*x + e)/(cos(f*x + e) + 1))^(3/2) - 3*sqrt(2)*(sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(
2) + 2*sqrt(sin(f*x + e)/(cos(f*x + e) + 1)))) + sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2) - 2*sqrt(sin(f*x + e)/(c
os(f*x + e) + 1)))))*sqrt(a) + 6*sqrt(2)*sqrt(a)*sqrt(sin(f*x + e)/(cos(f*x + e) + 1)) - 2*(3*sqrt(2)*sqrt(a)*
sin(f*x + e)/(cos(f*x + e) + 1) + sqrt(2)*sqrt(a)*sin(f*x + e)^2/(cos(f*x + e) + 1)^2)/sqrt(sin(f*x + e)/(cos(
f*x + e) + 1)))/f

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Fricas [B]  time = 2.57388, size = 934, normalized size = 25.24 \begin{align*} \left [\frac{\sqrt{-a} \log \left (\frac{128 \, a \cos \left (f x + e\right )^{5} - 128 \, a \cos \left (f x + e\right )^{4} - 416 \, a \cos \left (f x + e\right )^{3} + 128 \, a \cos \left (f x + e\right )^{2} - 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 \sin \left (f x + e\right ) + a} \sqrt{-a} \sqrt{\sin \left (f x + e\right )} + 289 \, a \cos \left (f x + e\right ) +{\left (128 \, a \cos \left (f x + e\right )^{4} + 256 \, a \cos \left (f x + e\right )^{3} - 160 \, a \cos \left (f x + e\right )^{2} - 288 \, a \cos \left (f x + e\right ) + a\right )} \sin \left (f x + e\right ) + a}{\cos \left (f x + e\right ) + \sin \left (f x + e\right ) + 1}\right )}{4 \, f}, \frac{\sqrt{a} \arctan \left (\frac{{\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{a} \sqrt{\sin \left (f x + e\right )}}{4 \,{\left (2 \, a \cos \left (f x + e\right )^{3} + a \cos \left (f x + e\right ) \sin \left (f x + e\right ) - 2 \, a \cos \left (f x + e\right )\right )}}\right )}{2 \, f}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/4*sqrt(-a)*log((128*a*cos(f*x + e)^5 - 128*a*cos(f*x + e)^4 - 416*a*cos(f*x + e)^3 + 128*a*cos(f*x + e)^2 -
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*co
s(f*x + e) - 51)*sin(f*x + e) + 25*cos(f*x + e) + 51)*sqrt(a*sin(f*x + e) + a)*sqrt(-a)*sqrt(sin(f*x + e)) + 2
89*a*cos(f*x + e) + (128*a*cos(f*x + e)^4 + 256*a*cos(f*x + e)^3 - 160*a*cos(f*x + e)^2 - 288*a*cos(f*x + e) +
a)*sin(f*x + e) + a)/(cos(f*x + e) + sin(f*x + e) + 1))/f, 1/2*sqrt(a)*arctan(1/4*(8*cos(f*x + e)^2 + 8*sin(f
*x + e) - 9)*sqrt(a*sin(f*x + e) + a)*sqrt(a)*sqrt(sin(f*x + e))/(2*a*cos(f*x + e)^3 + a*cos(f*x + e)*sin(f*x
+ e) - 2*a*cos(f*x + e)))/f]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{a \left (\sin{\left (e + f x \right )} + 1\right )}}{\sqrt{\sin{\left (e + f x \right )}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{a \sin \left (f x + e\right ) + a}}{\sqrt{\sin \left (f x + e\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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