Integrand size = 25, antiderivative size = 552 \[ \int \frac {\sqrt {a+a \csc (c+d x)}}{\csc ^{\frac {4}{3}}(c+d x)} \, dx=-\frac {15 a \cot (c+d x)}{8 d \left (1+\sqrt {3}-\sqrt [3]{\csc (c+d x)}\right ) \sqrt {a+a \csc (c+d x)}}-\frac {3 a \cos (c+d x)}{4 d \sqrt [3]{\csc (c+d x)} \sqrt {a+a \csc (c+d x)}}-\frac {15 a \cos (c+d x) \csc ^{\frac {2}{3}}(c+d x)}{8 d \sqrt {a+a \csc (c+d x)}}+\frac {15 \sqrt [4]{3} \sqrt {2-\sqrt {3}} a^2 \cot (c+d x) \left (1-\sqrt [3]{\csc (c+d x)}\right ) \sqrt {\frac {1+\sqrt [3]{\csc (c+d x)}+\csc ^{\frac {2}{3}}(c+d x)}{\left (1+\sqrt {3}-\sqrt [3]{\csc (c+d x)}\right )^2}} E\left (\arcsin \left (\frac {1-\sqrt {3}-\sqrt [3]{\csc (c+d x)}}{1+\sqrt {3}-\sqrt [3]{\csc (c+d x)}}\right )|-7-4 \sqrt {3}\right )}{16 d \sqrt {\frac {1-\sqrt [3]{\csc (c+d x)}}{\left (1+\sqrt {3}-\sqrt [3]{\csc (c+d x)}\right )^2}} (a-a \csc (c+d x)) \sqrt {a+a \csc (c+d x)}}-\frac {5\ 3^{3/4} a^2 \cot (c+d x) \left (1-\sqrt [3]{\csc (c+d x)}\right ) \sqrt {\frac {1+\sqrt [3]{\csc (c+d x)}+\csc ^{\frac {2}{3}}(c+d x)}{\left (1+\sqrt {3}-\sqrt [3]{\csc (c+d x)}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1-\sqrt {3}-\sqrt [3]{\csc (c+d x)}}{1+\sqrt {3}-\sqrt [3]{\csc (c+d x)}}\right ),-7-4 \sqrt {3}\right )}{4 \sqrt {2} d \sqrt {\frac {1-\sqrt [3]{\csc (c+d x)}}{\left (1+\sqrt {3}-\sqrt [3]{\csc (c+d x)}\right )^2}} (a-a \csc (c+d x)) \sqrt {a+a \csc (c+d x)}} \]
-3/4*a*cos(d*x+c)/d/csc(d*x+c)^(1/3)/(a+a*csc(d*x+c))^(1/2)-15/8*a*cos(d*x +c)*csc(d*x+c)^(2/3)/d/(a+a*csc(d*x+c))^(1/2)-15/8*a*cot(d*x+c)/d/(1-csc(d *x+c)^(1/3)+3^(1/2))/(a+a*csc(d*x+c))^(1/2)-5/8*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)+15/16*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-c sc(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 = 15.88 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.13 \[ \int \frac {\sqrt {a+a \csc (c+d x)}}{\csc ^{\frac {4}{3}}(c+d x)} \, dx=-\frac {a \cos (c+d x) \left (3+5 \csc ^{\frac {4}{3}}(c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {4}{3},\frac {3}{2},1-\csc (c+d x)\right )\right )}{4 d \sqrt [3]{\csc (c+d x)} \sqrt {a (1+\csc (c+d x))}} \]
-1/4*(a*Cos[c + d*x]*(3 + 5*Csc[c + d*x]^(4/3)*Hypergeometric2F1[1/2, 4/3, 3/2, 1 - Csc[c + d*x]]))/(d*Csc[c + d*x]^(1/3)*Sqrt[a*(1 + Csc[c + d*x])] )
Time = 0.50 (sec) , antiderivative size = 545, normalized size of antiderivative = 0.99, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {3042, 4293, 61, 61, 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.
| \(\displaystyle \int \frac {\sqrt {a \csc (c+d x)+a}}{\csc ^{\frac {4}{3}}(c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sqrt {a \csc (c+d x)+a}}{\csc (c+d x)^{4/3}}dx\) |
| \(\Big \downarrow \) 4293 |
| \(\displaystyle \frac {a^2 \cot (c+d x) \int \frac {1}{\csc ^{\frac {7}{3}}(c+d x) \sqrt {a-a \csc (c+d x)}}d\csc (c+d x)}{d \sqrt {a-a \csc (c+d x)} \sqrt {a \csc (c+d x)+a}}\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle \frac {a^2 \cot (c+d x) \left (\frac {5}{8} \int \frac {1}{\csc ^{\frac {4}{3}}(c+d x) \sqrt {a-a \csc (c+d x)}}d\csc (c+d x)-\frac {3 \sqrt {a-a \csc (c+d x)}}{4 a \csc ^{\frac {4}{3}}(c+d x)}\right )}{d \sqrt {a-a \csc (c+d x)} \sqrt {a \csc (c+d x)+a}}\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle \frac {a^2 \cot (c+d x) \left (\frac {5}{8} \left (-\frac {1}{2} \int \frac {1}{\sqrt [3]{\csc (c+d x)} \sqrt {a-a \csc (c+d x)}}d\csc (c+d x)-\frac {3 \sqrt {a-a \csc (c+d x)}}{a \sqrt [3]{\csc (c+d x)}}\right )-\frac {3 \sqrt {a-a \csc (c+d x)}}{4 a \csc ^{\frac {4}{3}}(c+d x)}\right )}{d \sqrt {a-a \csc (c+d x)} \sqrt {a \csc (c+d x)+a}}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {a^2 \cot (c+d x) \left (\frac {5}{8} \left (-\frac {3}{2} \int \frac {\sqrt [3]{\csc (c+d x)}}{\sqrt {a-a \csc (c+d x)}}d\sqrt [3]{\csc (c+d x)}-\frac {3 \sqrt {a-a \csc (c+d x)}}{a \sqrt [3]{\csc (c+d x)}}\right )-\frac {3 \sqrt {a-a \csc (c+d x)}}{4 a \csc ^{\frac {4}{3}}(c+d x)}\right )}{d \sqrt {a-a \csc (c+d x)} \sqrt {a \csc (c+d x)+a}}\) |
| \(\Big \downarrow \) 832 |
| \(\displaystyle \frac {a^2 \cot (c+d x) \left (\frac {5}{8} \left (-\frac {3}{2} \left (\left (1-\sqrt {3}\right ) \int \frac {1}{\sqrt {a-a \csc (c+d x)}}d\sqrt [3]{\csc (c+d x)}-\int \frac {-\sqrt [3]{\csc (c+d x)}-\sqrt {3}+1}{\sqrt {a-a \csc (c+d x)}}d\sqrt [3]{\csc (c+d x)}\right )-\frac {3 \sqrt {a-a \csc (c+d x)}}{a \sqrt [3]{\csc (c+d x)}}\right )-\frac {3 \sqrt {a-a \csc (c+d x)}}{4 a \csc ^{\frac {4}{3}}(c+d x)}\right )}{d \sqrt {a-a \csc (c+d x)} \sqrt {a \csc (c+d x)+a}}\) |
| \(\Big \downarrow \) 759 |
| \(\displaystyle \frac {a^2 \cot (c+d x) \left (\frac {5}{8} \left (-\frac {3}{2} \left (-\int \frac {-\sqrt [3]{\csc (c+d x)}-\sqrt {3}+1}{\sqrt {a-a \csc (c+d x)}}d\sqrt [3]{\csc (c+d x)}-\frac {2 \left (1-\sqrt {3}\right ) \sqrt {2+\sqrt {3}} \left (1-\sqrt [3]{\csc (c+d x)}\right ) \sqrt {\frac {\csc ^{\frac {2}{3}}(c+d x)+\sqrt [3]{\csc (c+d x)}+1}{\left (-\sqrt [3]{\csc (c+d x)}+\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {-\sqrt [3]{\csc (c+d x)}-\sqrt {3}+1}{-\sqrt [3]{\csc (c+d x)}+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {\frac {1-\sqrt [3]{\csc (c+d x)}}{\left (-\sqrt [3]{\csc (c+d x)}+\sqrt {3}+1\right )^2}} \sqrt {a-a \csc (c+d x)}}\right )-\frac {3 \sqrt {a-a \csc (c+d x)}}{a \sqrt [3]{\csc (c+d x)}}\right )-\frac {3 \sqrt {a-a \csc (c+d x)}}{4 a \csc ^{\frac {4}{3}}(c+d x)}\right )}{d \sqrt {a-a \csc (c+d x)} \sqrt {a \csc (c+d x)+a}}\) |
| \(\Big \downarrow \) 2416 |
| \(\displaystyle \frac {a^2 \cot (c+d x) \left (\frac {5}{8} \left (-\frac {3}{2} \left (-\frac {2 \left (1-\sqrt {3}\right ) \sqrt {2+\sqrt {3}} \left (1-\sqrt [3]{\csc (c+d x)}\right ) \sqrt {\frac {\csc ^{\frac {2}{3}}(c+d x)+\sqrt [3]{\csc (c+d x)}+1}{\left (-\sqrt [3]{\csc (c+d x)}+\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {-\sqrt [3]{\csc (c+d x)}-\sqrt {3}+1}{-\sqrt [3]{\csc (c+d x)}+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {\frac {1-\sqrt [3]{\csc (c+d x)}}{\left (-\sqrt [3]{\csc (c+d x)}+\sqrt {3}+1\right )^2}} \sqrt {a-a \csc (c+d x)}}-\frac {\sqrt [4]{3} \sqrt {2-\sqrt {3}} \left (1-\sqrt [3]{\csc (c+d x)}\right ) \sqrt {\frac {\csc ^{\frac {2}{3}}(c+d x)+\sqrt [3]{\csc (c+d x)}+1}{\left (-\sqrt [3]{\csc (c+d x)}+\sqrt {3}+1\right )^2}} E\left (\arcsin \left (\frac {-\sqrt [3]{\csc (c+d x)}-\sqrt {3}+1}{-\sqrt [3]{\csc (c+d x)}+\sqrt {3}+1}\right )|-7-4 \sqrt {3}\right )}{\sqrt {\frac {1-\sqrt [3]{\csc (c+d x)}}{\left (-\sqrt [3]{\csc (c+d x)}+\sqrt {3}+1\right )^2}} \sqrt {a-a \csc (c+d x)}}+\frac {2 \sqrt {a-a \csc (c+d x)}}{a \left (-\sqrt [3]{\csc (c+d x)}+\sqrt {3}+1\right )}\right )-\frac {3 \sqrt {a-a \csc (c+d x)}}{a \sqrt [3]{\csc (c+d x)}}\right )-\frac {3 \sqrt {a-a \csc (c+d x)}}{4 a \csc ^{\frac {4}{3}}(c+d x)}\right )}{d \sqrt {a-a \csc (c+d x)} \sqrt {a \csc (c+d x)+a}}\) |
(a^2*Cot[c + d*x]*((-3*Sqrt[a - a*Csc[c + d*x]])/(4*a*Csc[c + d*x]^(4/3)) + (5*((-3*Sqrt[a - a*Csc[c + d*x]])/(a*Csc[c + d*x]^(1/3)) - (3*((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 - Sqrt[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]*EllipticF[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]*Sqr t[a - a*Csc[c + d*x]])))/2))/8))/(d*Sqrt[a - a*Csc[c + d*x]]*Sqrt[a + a*Cs c[c + d*x]])
3.1.27.3.1 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 + 1)/((b*c - a*d)*(m + 1))), x] - Simp[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] && 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]
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]
\[\int \frac {\sqrt {a +a \csc \left (d x +c \right )}}{\csc \left (d x +c \right )^{\frac {4}{3}}}d x\]
\[ \int \frac {\sqrt {a+a \csc (c+d x)}}{\csc ^{\frac {4}{3}}(c+d x)} \, dx=\int { \frac {\sqrt {a \csc \left (d x + c\right ) + a}}{\csc \left (d x + c\right )^{\frac {4}{3}}} \,d x } \]
\[ \int \frac {\sqrt {a+a \csc (c+d x)}}{\csc ^{\frac {4}{3}}(c+d x)} \, dx=\int \frac {\sqrt {a \left (\csc {\left (c + d x \right )} + 1\right )}}{\csc ^{\frac {4}{3}}{\left (c + d x \right )}}\, dx \]
\[ \int \frac {\sqrt {a+a \csc (c+d x)}}{\csc ^{\frac {4}{3}}(c+d x)} \, dx=\int { \frac {\sqrt {a \csc \left (d x + c\right ) + a}}{\csc \left (d x + c\right )^{\frac {4}{3}}} \,d x } \]
\[ \int \frac {\sqrt {a+a \csc (c+d x)}}{\csc ^{\frac {4}{3}}(c+d x)} \, dx=\int { \frac {\sqrt {a \csc \left (d x + c\right ) + a}}{\csc \left (d x + c\right )^{\frac {4}{3}}} \,d x } \]
Timed out. \[ \int \frac {\sqrt {a+a \csc (c+d x)}}{\csc ^{\frac {4}{3}}(c+d x)} \, dx=\int \frac {\sqrt {a+\frac {a}{\sin \left (c+d\,x\right )}}}{{\left (\frac {1}{\sin \left (c+d\,x\right )}\right )}^{4/3}} \,d x \]