Integrand size = 25, antiderivative size = 254 \[ \int \csc ^{\frac {4}{3}}(c+d x) \sqrt {a+a \csc (c+d x)} \, dx=-\frac {6 a \cos (c+d x) \csc ^{\frac {4}{3}}(c+d x)}{5 d \sqrt {a+a \csc (c+d x)}}-\frac {4\ 3^{3/4} \sqrt {2+\sqrt {3}} a^2 \cot (c+d x) \left (1-\sqrt [3]{\csc (c+d x)}\right ) \sqrt {\frac {1+\sqrt [3]{\csc (c+d x)}+\csc ^{\frac {2}{3}}(c+d x)}{\left (1+\sqrt {3}-\sqrt [3]{\csc (c+d x)}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1-\sqrt {3}-\sqrt [3]{\csc (c+d x)}}{1+\sqrt {3}-\sqrt [3]{\csc (c+d x)}}\right ),-7-4 \sqrt {3}\right )}{5 d \sqrt {\frac {1-\sqrt [3]{\csc (c+d x)}}{\left (1+\sqrt {3}-\sqrt [3]{\csc (c+d x)}\right )^2}} (a-a \csc (c+d x)) \sqrt {a+a \csc (c+d x)}} \]
-6/5*a*cos(d*x+c)*csc(d*x+c)^(4/3)/d/(a+a*csc(d*x+c))^(1/2)-4/5*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)*(1/2*6^(1/2)+1/2*2^(1/2))*((1+c sc(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 = 8.59 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.40 \[ \int \csc ^{\frac {4}{3}}(c+d x) \sqrt {a+a \csc (c+d x)} \, dx=-\frac {2 \sqrt {a (1+\csc (c+d x))} \left (3 \sqrt [3]{\csc (c+d x)}+2 \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2}{3},\frac {3}{2},1-\csc (c+d x)\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}{5 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )} \]
(-2*Sqrt[a*(1 + Csc[c + d*x])]*(3*Csc[c + d*x]^(1/3) + 2*Hypergeometric2F1 [1/2, 2/3, 3/2, 1 - Csc[c + d*x]])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2]))/ (5*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))
Time = 0.35 (sec) , antiderivative size = 266, normalized size of antiderivative = 1.05, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3042, 4293, 60, 73, 759}
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.
| \(\displaystyle \int \csc ^{\frac {4}{3}}(c+d x) \sqrt {a \csc (c+d x)+a} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \csc (c+d x)^{4/3} \sqrt {a \csc (c+d x)+a}dx\) |
| \(\Big \downarrow \) 4293 |
| \(\displaystyle \frac {a^2 \cot (c+d x) \int \frac {\sqrt [3]{\csc (c+d x)}}{\sqrt {a-a \csc (c+d x)}}d\csc (c+d x)}{d \sqrt {a-a \csc (c+d x)} \sqrt {a \csc (c+d x)+a}}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {a^2 \cot (c+d x) \left (\frac {2}{5} \int \frac {1}{\csc ^{\frac {2}{3}}(c+d x) \sqrt {a-a \csc (c+d x)}}d\csc (c+d x)-\frac {6 \sqrt [3]{\csc (c+d x)} \sqrt {a-a \csc (c+d x)}}{5 a}\right )}{d \sqrt {a-a \csc (c+d x)} \sqrt {a \csc (c+d x)+a}}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {a^2 \cot (c+d x) \left (\frac {6}{5} \int \frac {1}{\sqrt {a-a \csc (c+d x)}}d\sqrt [3]{\csc (c+d x)}-\frac {6 \sqrt [3]{\csc (c+d x)} \sqrt {a-a \csc (c+d x)}}{5 a}\right )}{d \sqrt {a-a \csc (c+d x)} \sqrt {a \csc (c+d x)+a}}\) |
| \(\Big \downarrow \) 759 |
| \(\displaystyle \frac {a^2 \cot (c+d x) \left (-\frac {4\ 3^{3/4} \sqrt {2+\sqrt {3}} \left (1-\sqrt [3]{\csc (c+d x)}\right ) \sqrt {\frac {\csc ^{\frac {2}{3}}(c+d x)+\sqrt [3]{\csc (c+d x)}+1}{\left (-\sqrt [3]{\csc (c+d x)}+\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {-\sqrt [3]{\csc (c+d x)}-\sqrt {3}+1}{-\sqrt [3]{\csc (c+d x)}+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )}{5 \sqrt {\frac {1-\sqrt [3]{\csc (c+d x)}}{\left (-\sqrt [3]{\csc (c+d x)}+\sqrt {3}+1\right )^2}} \sqrt {a-a \csc (c+d x)}}-\frac {6 \sqrt [3]{\csc (c+d x)} \sqrt {a-a \csc (c+d x)}}{5 a}\right )}{d \sqrt {a-a \csc (c+d x)} \sqrt {a \csc (c+d x)+a}}\) |
(a^2*Cot[c + d*x]*((-6*Csc[c + d*x]^(1/3)*Sqrt[a - a*Csc[c + d*x]])/(5*a) - (4*3^(3/4)*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]*Ell ipticF[ArcSin[(1 - Sqrt[3] - Csc[c + d*x]^(1/3))/(1 + Sqrt[3] - Csc[c + d* x]^(1/3))], -7 - 4*Sqrt[3]])/(5*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]])
3.1.21.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[a, b, c, d, m, n, x]
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[(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]
\[\int \csc \left (d x +c \right )^{\frac {4}{3}} \sqrt {a +a \csc \left (d x +c \right )}d x\]
\[ \int \csc ^{\frac {4}{3}}(c+d x) \sqrt {a+a \csc (c+d x)} \, dx=\int { \sqrt {a \csc \left (d x + c\right ) + a} \csc \left (d x + c\right )^{\frac {4}{3}} \,d x } \]
Timed out. \[ \int \csc ^{\frac {4}{3}}(c+d x) \sqrt {a+a \csc (c+d x)} \, dx=\text {Timed out} \]
\[ \int \csc ^{\frac {4}{3}}(c+d x) \sqrt {a+a \csc (c+d x)} \, dx=\int { \sqrt {a \csc \left (d x + c\right ) + a} \csc \left (d x + c\right )^{\frac {4}{3}} \,d x } \]
\[ \int \csc ^{\frac {4}{3}}(c+d x) \sqrt {a+a \csc (c+d x)} \, dx=\int { \sqrt {a \csc \left (d x + c\right ) + a} \csc \left (d x + c\right )^{\frac {4}{3}} \,d x } \]
Timed out. \[ \int \csc ^{\frac {4}{3}}(c+d x) \sqrt {a+a \csc (c+d x)} \, dx=\int \sqrt {a+\frac {a}{\sin \left (c+d\,x\right )}}\,{\left (\frac {1}{\sin \left (c+d\,x\right )}\right )}^{4/3} \,d x \]