Integrand size = 25, antiderivative size = 470 \[ \int \csc ^{\frac {2}{3}}(c+d x) \sqrt {a+a \csc (c+d x)} \, dx=\frac {6 a \cot (c+d x)}{d \left (1+\sqrt {3}-\sqrt [3]{\csc (c+d x)}\right ) \sqrt {a+a \csc (c+d x)}}-\frac {3 \sqrt [4]{3} \sqrt {2-\sqrt {3}} a^2 \cot (c+d x) \left (1-\sqrt [3]{\csc (c+d x)}\right ) \sqrt {\frac {1+\sqrt [3]{\csc (c+d x)}+\csc ^{\frac {2}{3}}(c+d x)}{\left (1+\sqrt {3}-\sqrt [3]{\csc (c+d x)}\right )^2}} E\left (\arcsin \left (\frac {1-\sqrt {3}-\sqrt [3]{\csc (c+d x)}}{1+\sqrt {3}-\sqrt [3]{\csc (c+d x)}}\right )|-7-4 \sqrt {3}\right )}{d \sqrt {\frac {1-\sqrt [3]{\csc (c+d x)}}{\left (1+\sqrt {3}-\sqrt [3]{\csc (c+d x)}\right )^2}} (a-a \csc (c+d x)) \sqrt {a+a \csc (c+d x)}}+\frac {2 \sqrt {2} 3^{3/4} a^2 \cot (c+d x) \left (1-\sqrt [3]{\csc (c+d x)}\right ) \sqrt {\frac {1+\sqrt [3]{\csc (c+d x)}+\csc ^{\frac {2}{3}}(c+d x)}{\left (1+\sqrt {3}-\sqrt [3]{\csc (c+d x)}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1-\sqrt {3}-\sqrt [3]{\csc (c+d x)}}{1+\sqrt {3}-\sqrt [3]{\csc (c+d x)}}\right ),-7-4 \sqrt {3}\right )}{d \sqrt {\frac {1-\sqrt [3]{\csc (c+d x)}}{\left (1+\sqrt {3}-\sqrt [3]{\csc (c+d x)}\right )^2}} (a-a \csc (c+d x)) \sqrt {a+a \csc (c+d x)}} \]
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-csc(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)
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 \[ \int \csc ^{\frac {2}{3}}(c+d x) \sqrt {a+a \csc (c+d x)} \, dx=\frac {2 \sqrt {a (1+\csc (c+d x))} \left (-3+2 \sqrt [3]{\csc (c+d x)} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {4}{3},\frac {3}{2},1-\csc (c+d x)\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}{d \sqrt [3]{\csc (c+d x)} \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )} \]
(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]))
Time = 0.48 (sec) , antiderivative size = 474, normalized size of antiderivative = 1.01, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {3042, 4293, 73, 832, 759, 2416}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \csc ^{\frac {2}{3}}(c+d x) \sqrt {a \csc (c+d x)+a} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \csc (c+d x)^{2/3} \sqrt {a \csc (c+d x)+a}dx\) |
| \(\Big \downarrow \) 4293 |
| \(\displaystyle \frac {a^2 \cot (c+d x) \int \frac {1}{\sqrt [3]{\csc (c+d x)} \sqrt {a-a \csc (c+d x)}}d\csc (c+d x)}{d \sqrt {a-a \csc (c+d x)} \sqrt {a \csc (c+d x)+a}}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {3 a^2 \cot (c+d x) \int \frac {\sqrt [3]{\csc (c+d x)}}{\sqrt {a-a \csc (c+d x)}}d\sqrt [3]{\csc (c+d x)}}{d \sqrt {a-a \csc (c+d x)} \sqrt {a \csc (c+d x)+a}}\) |
| \(\Big \downarrow \) 832 |
| \(\displaystyle \frac {3 a^2 \cot (c+d x) \left (\left (1-\sqrt {3}\right ) \int \frac {1}{\sqrt {a-a \csc (c+d x)}}d\sqrt [3]{\csc (c+d x)}-\int \frac {-\sqrt [3]{\csc (c+d x)}-\sqrt {3}+1}{\sqrt {a-a \csc (c+d x)}}d\sqrt [3]{\csc (c+d x)}\right )}{d \sqrt {a-a \csc (c+d x)} \sqrt {a \csc (c+d x)+a}}\) |
| \(\Big \downarrow \) 759 |
| \(\displaystyle \frac {3 a^2 \cot (c+d x) \left (-\int \frac {-\sqrt [3]{\csc (c+d x)}-\sqrt {3}+1}{\sqrt {a-a \csc (c+d x)}}d\sqrt [3]{\csc (c+d x)}-\frac {2 \left (1-\sqrt {3}\right ) \sqrt {2+\sqrt {3}} \left (1-\sqrt [3]{\csc (c+d x)}\right ) \sqrt {\frac {\csc ^{\frac {2}{3}}(c+d x)+\sqrt [3]{\csc (c+d x)}+1}{\left (-\sqrt [3]{\csc (c+d x)}+\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {-\sqrt [3]{\csc (c+d x)}-\sqrt {3}+1}{-\sqrt [3]{\csc (c+d x)}+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {\frac {1-\sqrt [3]{\csc (c+d x)}}{\left (-\sqrt [3]{\csc (c+d x)}+\sqrt {3}+1\right )^2}} \sqrt {a-a \csc (c+d x)}}\right )}{d \sqrt {a-a \csc (c+d x)} \sqrt {a \csc (c+d x)+a}}\) |
| \(\Big \downarrow \) 2416 |
| \(\displaystyle \frac {3 a^2 \cot (c+d x) \left (-\frac {2 \left (1-\sqrt {3}\right ) \sqrt {2+\sqrt {3}} \left (1-\sqrt [3]{\csc (c+d x)}\right ) \sqrt {\frac {\csc ^{\frac {2}{3}}(c+d x)+\sqrt [3]{\csc (c+d x)}+1}{\left (-\sqrt [3]{\csc (c+d x)}+\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {-\sqrt [3]{\csc (c+d x)}-\sqrt {3}+1}{-\sqrt [3]{\csc (c+d x)}+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {\frac {1-\sqrt [3]{\csc (c+d x)}}{\left (-\sqrt [3]{\csc (c+d x)}+\sqrt {3}+1\right )^2}} \sqrt {a-a \csc (c+d x)}}-\frac {\sqrt [4]{3} \sqrt {2-\sqrt {3}} \left (1-\sqrt [3]{\csc (c+d x)}\right ) \sqrt {\frac {\csc ^{\frac {2}{3}}(c+d x)+\sqrt [3]{\csc (c+d x)}+1}{\left (-\sqrt [3]{\csc (c+d x)}+\sqrt {3}+1\right )^2}} E\left (\arcsin \left (\frac {-\sqrt [3]{\csc (c+d x)}-\sqrt {3}+1}{-\sqrt [3]{\csc (c+d x)}+\sqrt {3}+1}\right )|-7-4 \sqrt {3}\right )}{\sqrt {\frac {1-\sqrt [3]{\csc (c+d x)}}{\left (-\sqrt [3]{\csc (c+d x)}+\sqrt {3}+1\right )^2}} \sqrt {a-a \csc (c+d x)}}+\frac {2 \sqrt {a-a \csc (c+d x)}}{a \left (-\sqrt [3]{\csc (c+d x)}+\sqrt {3}+1\right )}\right )}{d \sqrt {a-a \csc (c+d x)} \sqrt {a \csc (c+d x)+a}}\) |
(3*a^2*Cot[c + d*x]*((2*Sqrt[a - a*Csc[c + d*x]])/(a*(1 + Sqrt[3] - Csc[c + d*x]^(1/3))) - (3^(1/4)*Sqrt[2 - Sqrt[3]]*(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]])/(Sqrt[(1 - Csc[c + d*x]^(1/3))/( 1 + Sqrt[3] - Csc[c + d*x]^(1/3))^2]*Sqrt[a - a*Csc[c + d*x]]) - (2*(1 - S qrt[3])*Sqrt[2 + Sqrt[3]]*(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]*Elliptic F[ArcSin[(1 - Sqrt[3] - Csc[c + d*x]^(1/3))/(1 + Sqrt[3] - Csc[c + d*x]^(1 /3))], -7 - 4*Sqrt[3]])/(3^(1/4)*Sqrt[(1 - Csc[c + d*x]^(1/3))/(1 + Sqrt[3 ] - Csc[c + d*x]^(1/3))^2]*Sqrt[a - a*Csc[c + d*x]])))/(d*Sqrt[a - a*Csc[c + d*x]]*Sqrt[a + a*Csc[c + d*x]])
3.1.25.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[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] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
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]*Sqrt[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]
Int[(x_)/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3] ], s = Denom[Rt[b/a, 3]]}, Simp[(-(1 - Sqrt[3]))*(s/r) Int[1/Sqrt[a + b*x ^3], x], x] + Simp[1/r Int[((1 - Sqrt[3])*s + r*x)/Sqrt[a + b*x^3], x], x ]] /; FreeQ[{a, b}, x] && PosQ[a]
Int[((c_) + (d_.)*(x_))/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = N umer[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] - S imp[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] && Eq Q[b*c^3 - 2*(5 - 3*Sqrt[3])*a*d^3, 0]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[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]
\[\int \csc \left (d x +c \right )^{\frac {2}{3}} \sqrt {a +a \csc \left (d x +c \right )}d x\]
\[ \int \csc ^{\frac {2}{3}}(c+d x) \sqrt {a+a \csc (c+d x)} \, dx=\int { \sqrt {a \csc \left (d x + c\right ) + a} \csc \left (d x + c\right )^{\frac {2}{3}} \,d x } \]
\[ \int \csc ^{\frac {2}{3}}(c+d x) \sqrt {a+a \csc (c+d x)} \, dx=\int \sqrt {a \left (\csc {\left (c + d x \right )} + 1\right )} \csc ^{\frac {2}{3}}{\left (c + d x \right )}\, dx \]
\[ \int \csc ^{\frac {2}{3}}(c+d x) \sqrt {a+a \csc (c+d x)} \, dx=\int { \sqrt {a \csc \left (d x + c\right ) + a} \csc \left (d x + c\right )^{\frac {2}{3}} \,d x } \]
\[ \int \csc ^{\frac {2}{3}}(c+d x) \sqrt {a+a \csc (c+d x)} \, dx=\int { \sqrt {a \csc \left (d x + c\right ) + a} \csc \left (d x + c\right )^{\frac {2}{3}} \,d x } \]
Timed out. \[ \int \csc ^{\frac {2}{3}}(c+d x) \sqrt {a+a \csc (c+d x)} \, dx=\int \sqrt {a+\frac {a}{\sin \left (c+d\,x\right )}}\,{\left (\frac {1}{\sin \left (c+d\,x\right )}\right )}^{2/3} \,d x \]