3.19 \(\int \sqrt{\csc (e+f x)} \sqrt{a+a \csc (e+f x)} \, dx\)

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

[Out]

(-2*Sqrt[a]*ArcSinh[(Sqrt[a]*Cot[e + f*x])/Sqrt[a + a*Csc[e + f*x]]])/f

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Rubi [A]  time = 0.0585123, 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 = {3801, 215} \[ -\frac{2 \sqrt{a} \sinh ^{-1}\left (\frac{\sqrt{a} \cot (e+f x)}{\sqrt{a \csc (e+f x)+a}}\right )}{f} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[Csc[e + f*x]]*Sqrt[a + a*Csc[e + f*x]],x]

[Out]

(-2*Sqrt[a]*ArcSinh[(Sqrt[a]*Cot[e + f*x])/Sqrt[a + a*Csc[e + f*x]]])/f

Rule 3801

Int[Sqrt[csc[(e_.) + (f_.)*(x_)]*(d_.)]*Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Dist[(-2*a*Sq
rt[(a*d)/b])/(b*f), Subst[Int[1/Sqrt[1 + x^2/a], x], x, (b*Cot[e + f*x])/Sqrt[a + b*Csc[e + f*x]]], x] /; Free
Q[{a, b, d, e, f}, x] && EqQ[a^2 - b^2, 0] && GtQ[(a*d)/b, 0]

Rule 215

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

Rubi steps

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

Mathematica [B]  time = 0.408282, size = 108, normalized size = 2.92 \[ \frac{2 \cot (e+f x) \sqrt{a (\csc (e+f x)+1)} \left (\log (\csc (e+f x)+1)-\log \left (\csc ^{\frac{3}{2}}(e+f x)+\sqrt{\csc (e+f x)}+\sqrt{\cot ^2(e+f x)} \sqrt{\csc (e+f x)+1}\right )\right )}{f \sqrt{\cot ^2(e+f x)} \sqrt{\csc (e+f x)+1}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[Csc[e + f*x]]*Sqrt[a + a*Csc[e + f*x]],x]

[Out]

(2*Cot[e + f*x]*Sqrt[a*(1 + Csc[e + f*x])]*(Log[1 + Csc[e + f*x]] - Log[Sqrt[Csc[e + f*x]] + Csc[e + f*x]^(3/2
) + Sqrt[Cot[e + f*x]^2]*Sqrt[1 + Csc[e + f*x]]]))/(f*Sqrt[Cot[e + f*x]^2]*Sqrt[1 + Csc[e + f*x]])

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Maple [B]  time = 0.359, size = 114, normalized size = 3.1 \begin{align*} -{\frac{\sqrt{2} \left ( -1+\cos \left ( fx+e \right ) \right ) }{f \left ( -1+\cos \left ( fx+e \right ) -\sin \left ( fx+e \right ) \right ) }\sqrt{ \left ( \sin \left ( fx+e \right ) \right ) ^{-1}}\sqrt{{\frac{a \left ( \sin \left ( fx+e \right ) +1 \right ) }{\sin \left ( fx+e \right ) }}} \left ({\it Arcsinh} \left ({\frac{-1+\cos \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }} \right ) +{\it Artanh} \left ({\frac{\sqrt{2}}{2}{\frac{1}{\sqrt{ \left ( 1+\cos \left ( fx+e \right ) \right ) ^{-1}}}}} \right ) \right ){\frac{1}{\sqrt{ \left ( 1+\cos \left ( fx+e \right ) \right ) ^{-1}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(a*csc(f*x + e) + a)*sqrt(csc(f*x + e)), x)

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Fricas [B]  time = 0.534522, size = 755, normalized size = 20.41 \begin{align*} \left [\frac{\sqrt{a} \log \left (\frac{a \cos \left (f x + e\right )^{3} - 7 \, a \cos \left (f x + e\right )^{2} - 9 \, a \cos \left (f x + e\right ) +{\left (a \cos \left (f x + e\right )^{2} + 8 \, a \cos \left (f x + e\right ) - a\right )} \sin \left (f x + e\right ) + \frac{4 \,{\left (\cos \left (f x + e\right )^{3} + 3 \, \cos \left (f x + e\right )^{2} -{\left (\cos \left (f x + e\right )^{2} - 2 \, \cos \left (f x + e\right ) - 3\right )} \sin \left (f x + e\right ) - \cos \left (f x + e\right ) - 3\right )} \sqrt{a} \sqrt{\frac{a \sin \left (f x + e\right ) + a}{\sin \left (f x + e\right )}}}{\sqrt{\sin \left (f x + e\right )}} - a}{\cos \left (f x + e\right )^{3} + \cos \left (f x + e\right )^{2} +{\left (\cos \left (f x + e\right )^{2} - 1\right )} \sin \left (f x + e\right ) - \cos \left (f x + e\right ) - 1}\right )}{2 \, f}, \frac{\sqrt{-a} \arctan \left (\frac{{\left (\cos \left (f x + e\right )^{2} + 2 \, \sin \left (f x + e\right ) - 1\right )} \sqrt{-a} \sqrt{\frac{a \sin \left (f x + e\right ) + a}{\sin \left (f x + e\right )}}}{2 \, a \cos \left (f x + e\right ) \sqrt{\sin \left (f x + e\right )}}\right )}{f}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(a*csc(f*x + e) + a)*sqrt(csc(f*x + e)), x)