3.23 \(\int \frac{\sqrt{a+a \csc (c+d x)}}{\csc ^{\frac{2}{3}}(c+d x)} \, dx\)

Optimal. Leaf size=254 \[ -\frac{3^{3/4} \sqrt{2+\sqrt{3}} a^2 \cot (c+d x) \left (1-\sqrt [3]{\csc (c+d x)}\right ) \sqrt{\frac{\csc ^{\frac{2}{3}}(c+d x)+\sqrt [3]{\csc (c+d x)}+1}{\left (-\sqrt [3]{\csc (c+d x)}+\sqrt{3}+1\right )^2}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{-\sqrt [3]{\csc (c+d x)}-\sqrt{3}+1}{-\sqrt [3]{\csc (c+d x)}+\sqrt{3}+1}\right ),-7-4 \sqrt{3}\right )}{2 d \sqrt{\frac{1-\sqrt [3]{\csc (c+d x)}}{\left (-\sqrt [3]{\csc (c+d x)}+\sqrt{3}+1\right )^2}} (a-a \csc (c+d x)) \sqrt{a \csc (c+d x)+a}}-\frac{3 a \cos (c+d x) \sqrt [3]{\csc (c+d x)}}{2 d \sqrt{a \csc (c+d x)+a}} \]

[Out]

(-3*a*Cos[c + d*x]*Csc[c + d*x]^(1/3))/(2*d*Sqrt[a + a*Csc[c + d*x]]) - (3^(3/4)*Sqrt[2 + Sqrt[3]]*a^2*Cot[c +
d*x]*(1 - Csc[c + d*x]^(1/3))*Sqrt[(1 + Csc[c + d*x]^(1/3) + Csc[c + d*x]^(2/3))/(1 + Sqrt[3] - Csc[c + d*x]^
(1/3))^2]*EllipticF[ArcSin[(1 - Sqrt[3] - Csc[c + d*x]^(1/3))/(1 + Sqrt[3] - Csc[c + d*x]^(1/3))], -7 - 4*Sqrt
[3]])/(2*d*Sqrt[(1 - Csc[c + d*x]^(1/3))/(1 + Sqrt[3] - Csc[c + d*x]^(1/3))^2]*(a - a*Csc[c + d*x])*Sqrt[a + a
*Csc[c + d*x]])

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Rubi [A]  time = 0.144931, antiderivative size = 254, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {3806, 51, 63, 218} \[ -\frac{3^{3/4} \sqrt{2+\sqrt{3}} a^2 \cot (c+d x) \left (1-\sqrt [3]{\csc (c+d x)}\right ) \sqrt{\frac{\csc ^{\frac{2}{3}}(c+d x)+\sqrt [3]{\csc (c+d x)}+1}{\left (-\sqrt [3]{\csc (c+d x)}+\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{-\sqrt [3]{\csc (c+d x)}-\sqrt{3}+1}{-\sqrt [3]{\csc (c+d x)}+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{2 d \sqrt{\frac{1-\sqrt [3]{\csc (c+d x)}}{\left (-\sqrt [3]{\csc (c+d x)}+\sqrt{3}+1\right )^2}} (a-a \csc (c+d x)) \sqrt{a \csc (c+d x)+a}}-\frac{3 a \cos (c+d x) \sqrt [3]{\csc (c+d x)}}{2 d \sqrt{a \csc (c+d x)+a}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + a*Csc[c + d*x]]/Csc[c + d*x]^(2/3),x]

[Out]

(-3*a*Cos[c + d*x]*Csc[c + d*x]^(1/3))/(2*d*Sqrt[a + a*Csc[c + d*x]]) - (3^(3/4)*Sqrt[2 + Sqrt[3]]*a^2*Cot[c +
d*x]*(1 - Csc[c + d*x]^(1/3))*Sqrt[(1 + Csc[c + d*x]^(1/3) + Csc[c + d*x]^(2/3))/(1 + Sqrt[3] - Csc[c + d*x]^
(1/3))^2]*EllipticF[ArcSin[(1 - Sqrt[3] - Csc[c + d*x]^(1/3))/(1 + Sqrt[3] - Csc[c + d*x]^(1/3))], -7 - 4*Sqrt
[3]])/(2*d*Sqrt[(1 - Csc[c + d*x]^(1/3))/(1 + Sqrt[3] - Csc[c + d*x]^(1/3))^2]*(a - a*Csc[c + d*x])*Sqrt[a + a
*Csc[c + d*x]])

Rule 3806

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

Rule 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*(m + n + 2))/((b*c - a*d)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 218

Int[1/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[(2*Sqr
t[2 + Sqrt[3]]*(s + r*x)*Sqrt[(s^2 - r*s*x + r^2*x^2)/((1 + Sqrt[3])*s + r*x)^2]*EllipticF[ArcSin[((1 - Sqrt[3
])*s + r*x)/((1 + Sqrt[3])*s + r*x)], -7 - 4*Sqrt[3]])/(3^(1/4)*r*Sqrt[a + b*x^3]*Sqrt[(s*(s + r*x))/((1 + Sqr
t[3])*s + r*x)^2]), x]] /; FreeQ[{a, b}, x] && PosQ[a]

Rubi steps

\begin{align*} \int \frac{\sqrt{a+a \csc (c+d x)}}{\csc ^{\frac{2}{3}}(c+d x)} \, dx &=\frac{\left (a^2 \cot (c+d x)\right ) \operatorname{Subst}\left (\int \frac{1}{x^{5/3} \sqrt{a-a x}} \, dx,x,\csc (c+d x)\right )}{d \sqrt{a-a \csc (c+d x)} \sqrt{a+a \csc (c+d x)}}\\ &=-\frac{3 a \cos (c+d x) \sqrt [3]{\csc (c+d x)}}{2 d \sqrt{a+a \csc (c+d x)}}+\frac{\left (a^2 \cot (c+d x)\right ) \operatorname{Subst}\left (\int \frac{1}{x^{2/3} \sqrt{a-a x}} \, dx,x,\csc (c+d x)\right )}{4 d \sqrt{a-a \csc (c+d x)} \sqrt{a+a \csc (c+d x)}}\\ &=-\frac{3 a \cos (c+d x) \sqrt [3]{\csc (c+d x)}}{2 d \sqrt{a+a \csc (c+d x)}}+\frac{\left (3 a^2 \cot (c+d x)\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a-a x^3}} \, dx,x,\sqrt [3]{\csc (c+d x)}\right )}{4 d \sqrt{a-a \csc (c+d x)} \sqrt{a+a \csc (c+d x)}}\\ &=-\frac{3 a \cos (c+d x) \sqrt [3]{\csc (c+d x)}}{2 d \sqrt{a+a \csc (c+d x)}}-\frac{3^{3/4} \sqrt{2+\sqrt{3}} a^2 \cot (c+d x) \left (1-\sqrt [3]{\csc (c+d x)}\right ) \sqrt{\frac{1+\sqrt [3]{\csc (c+d x)}+\csc ^{\frac{2}{3}}(c+d x)}{\left (1+\sqrt{3}-\sqrt [3]{\csc (c+d x)}\right )^2}} F\left (\sin ^{-1}\left (\frac{1-\sqrt{3}-\sqrt [3]{\csc (c+d x)}}{1+\sqrt{3}-\sqrt [3]{\csc (c+d x)}}\right )|-7-4 \sqrt{3}\right )}{2 d \sqrt{\frac{1-\sqrt [3]{\csc (c+d x)}}{\left (1+\sqrt{3}-\sqrt [3]{\csc (c+d x)}\right )^2}} (a-a \csc (c+d x)) \sqrt{a+a \csc (c+d x)}}\\ \end{align*}

Mathematica [C]  time = 0.44718, size = 110, normalized size = 0.43 \[ -\frac{\sqrt{a (\csc (c+d x)+1)} \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right ) \left (\csc ^{\frac{2}{3}}(c+d x) \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{2}{3},\frac{3}{2},1-\csc (c+d x)\right )+3\right )}{2 d \csc ^{\frac{2}{3}}(c+d x) \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a + a*Csc[c + d*x]]/Csc[c + d*x]^(2/3),x]

[Out]

-(Sqrt[a*(1 + Csc[c + d*x])]*(3 + Csc[c + d*x]^(2/3)*Hypergeometric2F1[1/2, 2/3, 3/2, 1 - Csc[c + d*x]])*(Cos[
(c + d*x)/2] - Sin[(c + d*x)/2]))/(2*d*Csc[c + d*x]^(2/3)*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))

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Maple [F]  time = 0.448, size = 0, normalized size = 0. \begin{align*} \int{\sqrt{a+a\csc \left ( dx+c \right ) } \left ( \csc \left ( dx+c \right ) \right ) ^{-{\frac{2}{3}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+a*csc(d*x+c))^(1/2)/csc(d*x+c)^(2/3),x)

[Out]

int((a+a*csc(d*x+c))^(1/2)/csc(d*x+c)^(2/3),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{a \csc \left (d x + c\right ) + a}}{\csc \left (d x + c\right )^{\frac{2}{3}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*csc(d*x+c))^(1/2)/csc(d*x+c)^(2/3),x, algorithm="maxima")

[Out]

integrate(sqrt(a*csc(d*x + c) + a)/csc(d*x + c)^(2/3), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{a \csc \left (d x + c\right ) + a}}{\csc \left (d x + c\right )^{\frac{2}{3}}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*csc(d*x+c))^(1/2)/csc(d*x+c)^(2/3),x, algorithm="fricas")

[Out]

integral(sqrt(a*csc(d*x + c) + a)/csc(d*x + c)^(2/3), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{a \left (\csc{\left (c + d x \right )} + 1\right )}}{\csc ^{\frac{2}{3}}{\left (c + d x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*csc(d*x+c))**(1/2)/csc(d*x+c)**(2/3),x)

[Out]

Integral(sqrt(a*(csc(c + d*x) + 1))/csc(c + d*x)**(2/3), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{a \csc \left (d x + c\right ) + a}}{\csc \left (d x + c\right )^{\frac{2}{3}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*csc(d*x+c))^(1/2)/csc(d*x+c)^(2/3),x, algorithm="giac")

[Out]

integrate(sqrt(a*csc(d*x + c) + a)/csc(d*x + c)^(2/3), x)