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

Optimal. Leaf size=61 \[ -\frac{2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{c+d} \sqrt{a \sin (e+f x)+a}}\right )}{\sqrt{d} f \sqrt{c+d}} \]

[Out]

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

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Rubi [A]  time = 0.115007, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074, Rules used = {2773, 208} \[ -\frac{2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{c+d} \sqrt{a \sin (e+f x)+a}}\right )}{\sqrt{d} f \sqrt{c+d}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2773

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]/((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[(-2*
b)/f, Subst[Int[1/(b*c + a*d - d*x^2), x], x, (b*Cos[e + f*x])/Sqrt[a + b*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 208

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

Rubi steps

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

Mathematica [C]  time = 5.1734, size = 657, normalized size = 10.77 \[ \frac{\left (\frac{1}{8}+\frac{i}{8}\right ) \left (\cos \left (\frac{e}{2}\right )+i \sin \left (\frac{e}{2}\right )\right ) \sqrt{a (\sin (e+f x)+1)} \left (\text{RootSum}\left [2 i \text{$\#$1}^2 c e^{i e}+\text{$\#$1}^4 d e^{2 i e}-d\& ,\frac{\text{$\#$1}^3 \left (-\sqrt{d}\right ) e^{i e} f x \sqrt{c+d}-2 i \text{$\#$1}^3 \sqrt{d} e^{i e} \sqrt{c+d} \log \left (-\text{$\#$1}+e^{\frac{i f x}{2}}\right )+\frac{(1-i) \text{$\#$1}^2 c f x}{\sqrt{e^{-i e}}}+\frac{(2+2 i) \text{$\#$1}^2 c \log \left (-\text{$\#$1}+e^{\frac{i f x}{2}}\right )}{\sqrt{e^{-i e}}}-i \text{$\#$1} \sqrt{d} f x \sqrt{c+d}+2 \text{$\#$1} \sqrt{d} \sqrt{c+d} \log \left (-\text{$\#$1}+e^{\frac{i f x}{2}}\right )-(2-2 i) d \sqrt{e^{-i e}} \log \left (-\text{$\#$1}+e^{\frac{i f x}{2}}\right )+(1+i) d \sqrt{e^{-i e}} f x}{d-i \text{$\#$1}^2 c e^{i e}}\& \right ]-i \text{RootSum}\left [2 i \text{$\#$1}^2 c e^{i e}+\text{$\#$1}^4 d e^{2 i e}-d\& ,\frac{-i \text{$\#$1}^3 \sqrt{d} e^{i e} f x \sqrt{c+d}+2 \text{$\#$1}^3 \sqrt{d} e^{i e} \sqrt{c+d} \log \left (-\text{$\#$1}+e^{\frac{i f x}{2}}\right )-\frac{(1+i) \text{$\#$1}^2 c f x}{\sqrt{e^{-i e}}}+\frac{(2-2 i) \text{$\#$1}^2 c \log \left (-\text{$\#$1}+e^{\frac{i f x}{2}}\right )}{\sqrt{e^{-i e}}}+\text{$\#$1} \sqrt{d} f x \sqrt{c+d}+2 i \text{$\#$1} \sqrt{d} \sqrt{c+d} \log \left (-\text{$\#$1}+e^{\frac{i f x}{2}}\right )+(2+2 i) d \sqrt{e^{-i e}} \log \left (-\text{$\#$1}+e^{\frac{i f x}{2}}\right )+(1-i) d \sqrt{e^{-i e}} f x}{d-i \text{$\#$1}^2 c e^{i e}}\& \right ]\right )}{\sqrt{d} f \sqrt{c+d} \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((1/8 + I/8)*(RootSum[-d + (2*I)*c*E^(I*e)*#1^2 + d*E^((2*I)*e)*#1^4 & , ((1 + I)*d*Sqrt[E^((-I)*e)]*f*x - (2
- 2*I)*d*Sqrt[E^((-I)*e)]*Log[E^((I/2)*f*x) - #1] - I*Sqrt[d]*Sqrt[c + d]*f*x*#1 + 2*Sqrt[d]*Sqrt[c + d]*Log[E
^((I/2)*f*x) - #1]*#1 + ((1 - I)*c*f*x*#1^2)/Sqrt[E^((-I)*e)] + ((2 + 2*I)*c*Log[E^((I/2)*f*x) - #1]*#1^2)/Sqr
t[E^((-I)*e)] - Sqrt[d]*Sqrt[c + d]*E^(I*e)*f*x*#1^3 - (2*I)*Sqrt[d]*Sqrt[c + d]*E^(I*e)*Log[E^((I/2)*f*x) - #
1]*#1^3)/(d - I*c*E^(I*e)*#1^2) & ] - I*RootSum[-d + (2*I)*c*E^(I*e)*#1^2 + d*E^((2*I)*e)*#1^4 & , ((1 - I)*d*
Sqrt[E^((-I)*e)]*f*x + (2 + 2*I)*d*Sqrt[E^((-I)*e)]*Log[E^((I/2)*f*x) - #1] + Sqrt[d]*Sqrt[c + d]*f*x*#1 + (2*
I)*Sqrt[d]*Sqrt[c + d]*Log[E^((I/2)*f*x) - #1]*#1 - ((1 + I)*c*f*x*#1^2)/Sqrt[E^((-I)*e)] + ((2 - 2*I)*c*Log[E
^((I/2)*f*x) - #1]*#1^2)/Sqrt[E^((-I)*e)] - I*Sqrt[d]*Sqrt[c + d]*E^(I*e)*f*x*#1^3 + 2*Sqrt[d]*Sqrt[c + d]*E^(
I*e)*Log[E^((I/2)*f*x) - #1]*#1^3)/(d - I*c*E^(I*e)*#1^2) & ])*(Cos[e/2] + I*Sin[e/2])*Sqrt[a*(1 + Sin[e + f*x
])])/(Sqrt[d]*Sqrt[c + d]*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]))

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Maple [A]  time = 0.654, size = 80, normalized size = 1.3 \begin{align*} -2\,{\frac{a \left ( 1+\sin \left ( fx+e \right ) \right ) \sqrt{-a \left ( -1+\sin \left ( fx+e \right ) \right ) }}{\sqrt{a \left ( c+d \right ) d}\cos \left ( fx+e \right ) \sqrt{a+a\sin \left ( fx+e \right ) }f}{\it Artanh} \left ({\frac{\sqrt{-a \left ( -1+\sin \left ( fx+e \right ) \right ) }d}{\sqrt{a \left ( c+d \right ) d}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*a*(1+sin(f*x+e))*(-a*(-1+sin(f*x+e)))^(1/2)/(a*(c+d)*d)^(1/2)*arctanh((-a*(-1+sin(f*x+e)))^(1/2)*d/(a*(c+d)
*d)^(1/2))/cos(f*x+e)/(a+a*sin(f*x+e))^(1/2)/f

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.50714, size = 1061, normalized size = 17.39 \begin{align*} \left [\frac{\sqrt{\frac{a}{c d + d^{2}}} \log \left (\frac{a d^{2} \cos \left (f x + e\right )^{3} - a c^{2} - 2 \, a c d - a d^{2} -{\left (6 \, a c d + 7 \, a d^{2}\right )} \cos \left (f x + e\right )^{2} + 4 \,{\left (c^{2} d + 4 \, c d^{2} + 3 \, d^{3} -{\left (c d^{2} + d^{3}\right )} \cos \left (f x + e\right )^{2} +{\left (c^{2} d + 3 \, c d^{2} + 2 \, d^{3}\right )} \cos \left (f x + e\right ) -{\left (c^{2} d + 4 \, c d^{2} + 3 \, d^{3} +{\left (c d^{2} + d^{3}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )} \sqrt{a \sin \left (f x + e\right ) + a} \sqrt{\frac{a}{c d + d^{2}}} -{\left (a c^{2} + 8 \, a c d + 9 \, a d^{2}\right )} \cos \left (f x + e\right ) +{\left (a d^{2} \cos \left (f x + e\right )^{2} - a c^{2} - 2 \, a c d - a d^{2} + 2 \,{\left (3 \, a c d + 4 \, a d^{2}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{d^{2} \cos \left (f x + e\right )^{3} +{\left (2 \, c d + d^{2}\right )} \cos \left (f x + e\right )^{2} - c^{2} - 2 \, c d - d^{2} -{\left (c^{2} + d^{2}\right )} \cos \left (f x + e\right ) +{\left (d^{2} \cos \left (f x + e\right )^{2} - 2 \, c d \cos \left (f x + e\right ) - c^{2} - 2 \, c d - d^{2}\right )} \sin \left (f x + e\right )}\right )}{2 \, f}, -\frac{\sqrt{-\frac{a}{c d + d^{2}}} \arctan \left (\frac{\sqrt{a \sin \left (f x + e\right ) + a}{\left (d \sin \left (f x + e\right ) - c - 2 \, d\right )} \sqrt{-\frac{a}{c d + d^{2}}}}{2 \, a \cos \left (f x + e\right )}\right )}{f}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*sqrt(a/(c*d + d^2))*log((a*d^2*cos(f*x + e)^3 - a*c^2 - 2*a*c*d - a*d^2 - (6*a*c*d + 7*a*d^2)*cos(f*x + e
)^2 + 4*(c^2*d + 4*c*d^2 + 3*d^3 - (c*d^2 + d^3)*cos(f*x + e)^2 + (c^2*d + 3*c*d^2 + 2*d^3)*cos(f*x + e) - (c^
2*d + 4*c*d^2 + 3*d^3 + (c*d^2 + d^3)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(a/(c*d + d^2))
- (a*c^2 + 8*a*c*d + 9*a*d^2)*cos(f*x + e) + (a*d^2*cos(f*x + e)^2 - a*c^2 - 2*a*c*d - a*d^2 + 2*(3*a*c*d + 4
*a*d^2)*cos(f*x + e))*sin(f*x + e))/(d^2*cos(f*x + e)^3 + (2*c*d + d^2)*cos(f*x + e)^2 - c^2 - 2*c*d - d^2 - (
c^2 + d^2)*cos(f*x + e) + (d^2*cos(f*x + e)^2 - 2*c*d*cos(f*x + e) - c^2 - 2*c*d - d^2)*sin(f*x + e)))/f, -sqr
t(-a/(c*d + d^2))*arctan(1/2*sqrt(a*sin(f*x + e) + a)*(d*sin(f*x + e) - c - 2*d)*sqrt(-a/(c*d + d^2))/(a*cos(f
*x + e)))/f]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out