∫ ∘ _______ csc (e + fx) ∘ __________

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*arcsinh(cot(f*x+e)*a^(1/2)/(a+a*csc(f*x+e))^(1/2))*a^(1/2)/f

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Rubi [A] time = 0.06, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, 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 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]

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]

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*}

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Mathematica [B] time = 0.40, 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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fricas [B] time = 0.80, size = 283, normalized size = 7.65 \[ \left [\frac {\sqrt {a} \log \left (\frac {a \cos \left (f x + e\right )^{3} - 7 \, a \cos \left (f x + e\right )^{2} - 9 \, a \cos \left (f x + e\right ) + {\left (a \cos \left (f x + e\right )^{2} + 8 \, a \cos \left (f x + e\right ) - a\right )} \sin \left (f x + e\right ) + \frac {4 \, {\left (\cos \left (f x + e\right )^{3} + 3 \, \cos \left (f x + e\right )^{2} - {\left (\cos \left (f x + e\right )^{2} - 2 \, \cos \left (f x + e\right ) - 3\right )} \sin \left (f x + e\right ) - \cos \left (f x + e\right ) - 3\right )} \sqrt {a} \sqrt {\frac {a \sin \left (f x + e\right ) + a}{\sin \left (f x + e\right )}}}{\sqrt {\sin \left (f x + e\right )}} - a}{\cos \left (f x + e\right )^{3} + \cos \left (f x + e\right )^{2} + {\left (\cos \left (f x + e\right )^{2} - 1\right )} \sin \left (f x + e\right ) - \cos \left (f x + e\right ) - 1}\right )}{2 \, f}, \frac {\sqrt {-a} \arctan \left (\frac {{\left (\cos \left (f x + e\right )^{2} + 2 \, \sin \left (f x + e\right ) - 1\right )} \sqrt {-a} \sqrt {\frac {a \sin \left (f x + e\right ) + a}{\sin \left (f x + e\right )}}}{2 \, a \cos \left (f x + e\right ) \sqrt {\sin \left (f x + e\right )}}\right )}{f}\right ] \]

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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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]

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]

Exception raised: NotImplementedError >> Unable to parse Giac output: Unable to check sign: (4*pi/x/2)>(-4*pi/