\(\int \csc ^{\frac {2}{3}}(c+d x) \sqrt {a+a \csc (c+d x)} \, dx\) [25]

Optimal result

Integrand size = 25, antiderivative size = 470 \[ \int \csc ^{\frac {2}{3}}(c+d x) \sqrt {a+a \csc (c+d x)} \, dx=\frac {6 a \cot (c+d x)}{d \left (1+\sqrt {3}-\sqrt [3]{\csc (c+d x)}\right ) \sqrt {a+a \csc (c+d x)}}-\frac {3 \sqrt [4]{3} \sqrt {2-\sqrt {3}} a^2 \cot (c+d x) \left (1-\sqrt [3]{\csc (c+d x)}\right ) \sqrt {\frac {1+\sqrt [3]{\csc (c+d x)}+\csc ^{\frac {2}{3}}(c+d x)}{\left (1+\sqrt {3}-\sqrt [3]{\csc (c+d x)}\right )^2}} E\left (\arcsin \left (\frac {1-\sqrt {3}-\sqrt [3]{\csc (c+d x)}}{1+\sqrt {3}-\sqrt [3]{\csc (c+d x)}}\right )|-7-4 \sqrt {3}\right )}{d \sqrt {\frac {1-\sqrt [3]{\csc (c+d x)}}{\left (1+\sqrt {3}-\sqrt [3]{\csc (c+d x)}\right )^2}} (a-a \csc (c+d x)) \sqrt {a+a \csc (c+d x)}}+\frac {2 \sqrt {2} 3^{3/4} a^2 \cot (c+d x) \left (1-\sqrt [3]{\csc (c+d x)}\right ) \sqrt {\frac {1+\sqrt [3]{\csc (c+d x)}+\csc ^{\frac {2}{3}}(c+d x)}{\left (1+\sqrt {3}-\sqrt [3]{\csc (c+d x)}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1-\sqrt {3}-\sqrt [3]{\csc (c+d x)}}{1+\sqrt {3}-\sqrt [3]{\csc (c+d x)}}\right ),-7-4 \sqrt {3}\right )}{d \sqrt {\frac {1-\sqrt [3]{\csc (c+d x)}}{\left (1+\sqrt {3}-\sqrt [3]{\csc (c+d x)}\right )^2}} (a-a \csc (c+d x)) \sqrt {a+a \csc (c+d x)}} \]

[Out]

6*a*cot(d*x+c)/d/(1-csc(d*x+c)^(1/3)+3^(1/2))/(a+a*csc(d*x+c))^(1/2)+2*3^(3/4)*a^2*cot(d*x+c)*(1-csc(d*x+c)^(1
/3))*EllipticF((1-csc(d*x+c)^(1/3)-3^(1/2))/(1-csc(d*x+c)^(1/3)+3^(1/2)),I*3^(1/2)+2*I)*2^(1/2)*((1+csc(d*x+c)
^(1/3)+csc(d*x+c)^(2/3))/(1-csc(d*x+c)^(1/3)+3^(1/2))^2)^(1/2)/d/(a-a*csc(d*x+c))/(a+a*csc(d*x+c))^(1/2)/((1-c
sc(d*x+c)^(1/3))/(1-csc(d*x+c)^(1/3)+3^(1/2))^2)^(1/2)-3*3^(1/4)*a^2*cot(d*x+c)*(1-csc(d*x+c)^(1/3))*EllipticE
((1-csc(d*x+c)^(1/3)-3^(1/2))/(1-csc(d*x+c)^(1/3)+3^(1/2)),I*3^(1/2)+2*I)*(1/2*6^(1/2)-1/2*2^(1/2))*((1+csc(d*
x+c)^(1/3)+csc(d*x+c)^(2/3))/(1-csc(d*x+c)^(1/3)+3^(1/2))^2)^(1/2)/d/(a-a*csc(d*x+c))/(a+a*csc(d*x+c))^(1/2)/(
(1-csc(d*x+c)^(1/3))/(1-csc(d*x+c)^(1/3)+3^(1/2))^2)^(1/2)

Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 470, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3891, 65, 309, 224, 1891} \[ \int \csc ^{\frac {2}{3}}(c+d x) \sqrt {a+a \csc (c+d x)} \, dx=\frac {2 \sqrt {2} 3^{3/4} a^2 \cot (c+d x) \left (1-\sqrt [3]{\csc (c+d x)}\right ) \sqrt {\frac {\csc ^{\frac {2}{3}}(c+d x)+\sqrt [3]{\csc (c+d x)}+1}{\left (-\sqrt [3]{\csc (c+d x)}+\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {-\sqrt [3]{\csc (c+d x)}-\sqrt {3}+1}{-\sqrt [3]{\csc (c+d x)}+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )}{d \sqrt {\frac {1-\sqrt [3]{\csc (c+d x)}}{\left (-\sqrt [3]{\csc (c+d x)}+\sqrt {3}+1\right )^2}} (a-a \csc (c+d x)) \sqrt {a \csc (c+d x)+a}}-\frac {3 \sqrt [4]{3} \sqrt {2-\sqrt {3}} a^2 \cot (c+d x) \left (1-\sqrt [3]{\csc (c+d x)}\right ) \sqrt {\frac {\csc ^{\frac {2}{3}}(c+d x)+\sqrt [3]{\csc (c+d x)}+1}{\left (-\sqrt [3]{\csc (c+d x)}+\sqrt {3}+1\right )^2}} E\left (\arcsin \left (\frac {-\sqrt [3]{\csc (c+d x)}-\sqrt {3}+1}{-\sqrt [3]{\csc (c+d x)}+\sqrt {3}+1}\right )|-7-4 \sqrt {3}\right )}{d \sqrt {\frac {1-\sqrt [3]{\csc (c+d x)}}{\left (-\sqrt [3]{\csc (c+d x)}+\sqrt {3}+1\right )^2}} (a-a \csc (c+d x)) \sqrt {a \csc (c+d x)+a}}+\frac {6 a \cot (c+d x)}{d \left (-\sqrt [3]{\csc (c+d x)}+\sqrt {3}+1\right ) \sqrt {a \csc (c+d x)+a}} \]

[In]

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

[Out]

(6*a*Cot[c + d*x])/(d*(1 + Sqrt[3] - Csc[c + d*x]^(1/3))*Sqrt[a + a*Csc[c + d*x]]) - (3*3^(1/4)*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]*EllipticE[ArcSin[(1 - Sqrt[3] - Csc[c + d*x]^(1/3))/(1 + Sqrt[3] - Csc[c + d*x]^(1/3)
)], -7 - 4*Sqrt[3]])/(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]]) + (2*Sqrt[2]*3^(3/4)*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]])/(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 65

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 224

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

Rule 309

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

Rule 1891

Int[((c_) + (d_.)*(x_))/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Simplify[(1 - Sqrt[3])*(d/c)]]
, s = Denom[Simplify[(1 - Sqrt[3])*(d/c)]]}, Simp[2*d*s^3*(Sqrt[a + b*x^3]/(a*r^2*((1 + Sqrt[3])*s + r*x))), x
] - Simp[3^(1/4)*Sqrt[2 - Sqrt[3]]*d*s*(s + r*x)*(Sqrt[(s^2 - r*s*x + r^2*x^2)/((1 + Sqrt[3])*s + r*x)^2]/(r^2
*Sqrt[a + b*x^3]*Sqrt[s*((s + r*x)/((1 + Sqrt[3])*s + r*x)^2)]))*EllipticE[ArcSin[((1 - Sqrt[3])*s + r*x)/((1
+ Sqrt[3])*s + r*x)], -7 - 4*Sqrt[3]], x]] /; FreeQ[{a, b, c, d}, x] && PosQ[a] && EqQ[b*c^3 - 2*(5 - 3*Sqrt[3
])*a*d^3, 0]

Rule 3891

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]

Rubi steps \begin{align*} \text {integral}& = \frac {\left (a^2 \cot (c+d x)\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{x} \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 {\left (3 a^2 \cot (c+d x)\right ) \text {Subst}\left (\int \frac {x}{\sqrt {a-a x^3}} \, dx,x,\sqrt [3]{\csc (c+d x)}\right )}{d \sqrt {a-a \csc (c+d x)} \sqrt {a+a \csc (c+d x)}} \\ & = -\frac {\left (3 a^2 \cot (c+d x)\right ) \text {Subst}\left (\int \frac {1-\sqrt {3}-x}{\sqrt {a-a x^3}} \, dx,x,\sqrt [3]{\csc (c+d x)}\right )}{d \sqrt {a-a \csc (c+d x)} \sqrt {a+a \csc (c+d x)}}+\frac {\left (3 \left (1-\sqrt {3}\right ) a^2 \cot (c+d x)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a-a x^3}} \, dx,x,\sqrt [3]{\csc (c+d x)}\right )}{d \sqrt {a-a \csc (c+d x)} \sqrt {a+a \csc (c+d x)}} \\ & = \frac {6 a \cot (c+d x)}{d \left (1+\sqrt {3}-\sqrt [3]{\csc (c+d x)}\right ) \sqrt {a+a \csc (c+d x)}}-\frac {3 \sqrt [4]{3} \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}} E\left (\arcsin \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 )}{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)}}+\frac {2 \sqrt {2} 3^{3/4} 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}} \operatorname {EllipticF}\left (\arcsin \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 )}{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] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 15.58 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.23 \[ \int \csc ^{\frac {2}{3}}(c+d x) \sqrt {a+a \csc (c+d x)} \, dx=\frac {2 \sqrt {a (1+\csc (c+d x))} \left (-3+2 \sqrt [3]{\csc (c+d x)} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {4}{3},\frac {3}{2},1-\csc (c+d x)\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}{d \sqrt [3]{\csc (c+d x)} \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )} \]

[In]

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

[Out]

(2*Sqrt[a*(1 + Csc[c + d*x])]*(-3 + 2*Csc[c + d*x]^(1/3)*Hypergeometric2F1[1/2, 4/3, 3/2, 1 - Csc[c + d*x]])*(
Cos[(c + d*x)/2] - Sin[(c + d*x)/2]))/(d*Csc[c + d*x]^(1/3)*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))

Maple [F]

\[\int \csc \left (d x +c \right )^{\frac {2}{3}} \sqrt {a +a \csc \left (d x +c \right )}d x\]

[In]

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

[Out]

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

Fricas [F]

\[ \int \csc ^{\frac {2}{3}}(c+d x) \sqrt {a+a \csc (c+d x)} \, dx=\int { \sqrt {a \csc \left (d x + c\right ) + a} \csc \left (d x + c\right )^{\frac {2}{3}} \,d x } \]

[In]

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

[Out]

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

Sympy [F]

\[ \int \csc ^{\frac {2}{3}}(c+d x) \sqrt {a+a \csc (c+d x)} \, dx=\int \sqrt {a \left (\csc {\left (c + d x \right )} + 1\right )} \csc ^{\frac {2}{3}}{\left (c + d x \right )}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \csc ^{\frac {2}{3}}(c+d x) \sqrt {a+a \csc (c+d x)} \, dx=\int { \sqrt {a \csc \left (d x + c\right ) + a} \csc \left (d x + c\right )^{\frac {2}{3}} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \csc ^{\frac {2}{3}}(c+d x) \sqrt {a+a \csc (c+d x)} \, dx=\int { \sqrt {a \csc \left (d x + c\right ) + a} \csc \left (d x + c\right )^{\frac {2}{3}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \csc ^{\frac {2}{3}}(c+d x) \sqrt {a+a \csc (c+d x)} \, dx=\int \sqrt {a+\frac {a}{\sin \left (c+d\,x\right )}}\,{\left (\frac {1}{\sin \left (c+d\,x\right )}\right )}^{2/3} \,d x \]

[In]

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

[Out]

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