Integrand size = 23, antiderivative size = 731 \[ \int \frac {\sec ^2(c+d x)}{(a+a \sec (c+d x))^{5/3}} \, dx=-\frac {3 \tan (c+d x)}{7 d (a+a \sec (c+d x))^{5/3}}+\frac {15 \tan (c+d x)}{7 a d (a+a \sec (c+d x))^{2/3}}+\frac {15 \left (1+\sqrt {3}\right ) \sqrt [3]{1+\sec (c+d x)} \tan (c+d x)}{7 a d (a+a \sec (c+d x))^{2/3} \left (\sqrt [3]{2}-\left (1+\sqrt {3}\right ) \sqrt [3]{1+\sec (c+d x)}\right )}-\frac {15 \sqrt [3]{2} \sqrt [4]{3} E\left (\arccos \left (\frac {\sqrt [3]{2}-\left (1-\sqrt {3}\right ) \sqrt [3]{1+\sec (c+d x)}}{\sqrt [3]{2}-\left (1+\sqrt {3}\right ) \sqrt [3]{1+\sec (c+d x)}}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right ) \sqrt [3]{1+\sec (c+d x)} \left (\sqrt [3]{2}-\sqrt [3]{1+\sec (c+d x)}\right ) \sqrt {\frac {2^{2/3}+\sqrt [3]{2} \sqrt [3]{1+\sec (c+d x)}+(1+\sec (c+d x))^{2/3}}{\left (\sqrt [3]{2}-\left (1+\sqrt {3}\right ) \sqrt [3]{1+\sec (c+d x)}\right )^2}} \tan (c+d x)}{7 a d (1-\sec (c+d x)) (a+a \sec (c+d x))^{2/3} \sqrt {-\frac {\sqrt [3]{1+\sec (c+d x)} \left (\sqrt [3]{2}-\sqrt [3]{1+\sec (c+d x)}\right )}{\left (\sqrt [3]{2}-\left (1+\sqrt {3}\right ) \sqrt [3]{1+\sec (c+d x)}\right )^2}}}-\frac {5\ 3^{3/4} \left (1-\sqrt {3}\right ) \operatorname {EllipticF}\left (\arccos \left (\frac {\sqrt [3]{2}-\left (1-\sqrt {3}\right ) \sqrt [3]{1+\sec (c+d x)}}{\sqrt [3]{2}-\left (1+\sqrt {3}\right ) \sqrt [3]{1+\sec (c+d x)}}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right ) \sqrt [3]{1+\sec (c+d x)} \left (\sqrt [3]{2}-\sqrt [3]{1+\sec (c+d x)}\right ) \sqrt {\frac {2^{2/3}+\sqrt [3]{2} \sqrt [3]{1+\sec (c+d x)}+(1+\sec (c+d x))^{2/3}}{\left (\sqrt [3]{2}-\left (1+\sqrt {3}\right ) \sqrt [3]{1+\sec (c+d x)}\right )^2}} \tan (c+d x)}{7\ 2^{2/3} a d (1-\sec (c+d x)) (a+a \sec (c+d x))^{2/3} \sqrt {-\frac {\sqrt [3]{1+\sec (c+d x)} \left (\sqrt [3]{2}-\sqrt [3]{1+\sec (c+d x)}\right )}{\left (\sqrt [3]{2}-\left (1+\sqrt {3}\right ) \sqrt [3]{1+\sec (c+d x)}\right )^2}}} \]
-3/7*tan(d*x+c)/d/(a+a*sec(d*x+c))^(5/3)+15/7*tan(d*x+c)/a/d/(a+a*sec(d*x+ c))^(2/3)+15/7*(1+sec(d*x+c))^(1/3)*(1+3^(1/2))*tan(d*x+c)/a/d/(a+a*sec(d* x+c))^(2/3)/(2^(1/3)-(1+sec(d*x+c))^(1/3)*(1+3^(1/2)))-15/7*2^(1/3)*3^(1/4 )*((2^(1/3)-(1+sec(d*x+c))^(1/3)*(1-3^(1/2)))^2/(2^(1/3)-(1+sec(d*x+c))^(1 /3)*(1+3^(1/2)))^2)^(1/2)/(2^(1/3)-(1+sec(d*x+c))^(1/3)*(1-3^(1/2)))*(2^(1 /3)-(1+sec(d*x+c))^(1/3)*(1+3^(1/2)))*EllipticE((1-(2^(1/3)-(1+sec(d*x+c)) ^(1/3)*(1-3^(1/2)))^2/(2^(1/3)-(1+sec(d*x+c))^(1/3)*(1+3^(1/2)))^2)^(1/2), 1/4*6^(1/2)+1/4*2^(1/2))*(1+sec(d*x+c))^(1/3)*(2^(1/3)-(1+sec(d*x+c))^(1/3 ))*((2^(2/3)+2^(1/3)*(1+sec(d*x+c))^(1/3)+(1+sec(d*x+c))^(2/3))/(2^(1/3)-( 1+sec(d*x+c))^(1/3)*(1+3^(1/2)))^2)^(1/2)*tan(d*x+c)/a/d/(1-sec(d*x+c))/(a +a*sec(d*x+c))^(2/3)/(-(1+sec(d*x+c))^(1/3)*(2^(1/3)-(1+sec(d*x+c))^(1/3)) /(2^(1/3)-(1+sec(d*x+c))^(1/3)*(1+3^(1/2)))^2)^(1/2)-5/14*3^(3/4)*((2^(1/3 )-(1+sec(d*x+c))^(1/3)*(1-3^(1/2)))^2/(2^(1/3)-(1+sec(d*x+c))^(1/3)*(1+3^( 1/2)))^2)^(1/2)/(2^(1/3)-(1+sec(d*x+c))^(1/3)*(1-3^(1/2)))*(2^(1/3)-(1+sec (d*x+c))^(1/3)*(1+3^(1/2)))*EllipticF((1-(2^(1/3)-(1+sec(d*x+c))^(1/3)*(1- 3^(1/2)))^2/(2^(1/3)-(1+sec(d*x+c))^(1/3)*(1+3^(1/2)))^2)^(1/2),1/4*6^(1/2 )+1/4*2^(1/2))*(1+sec(d*x+c))^(1/3)*(2^(1/3)-(1+sec(d*x+c))^(1/3))*(1-3^(1 /2))*((2^(2/3)+2^(1/3)*(1+sec(d*x+c))^(1/3)+(1+sec(d*x+c))^(2/3))/(2^(1/3) -(1+sec(d*x+c))^(1/3)*(1+3^(1/2)))^2)^(1/2)*tan(d*x+c)*2^(1/3)/a/d/(1-sec( d*x+c))/(a+a*sec(d*x+c))^(2/3)/(-(1+sec(d*x+c))^(1/3)*(2^(1/3)-(1+sec(d...
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.34 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.12 \[ \int \frac {\sec ^2(c+d x)}{(a+a \sec (c+d x))^{5/3}} \, dx=\frac {\left (-3+5\ 2^{5/6} \cos ^2\left (\frac {1}{2} (c+d x)\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {7}{6},\frac {3}{2},\frac {1}{2} (1-\sec (c+d x))\right ) \sec (c+d x) \sqrt [6]{1+\sec (c+d x)}\right ) \tan (c+d x)}{7 d (a (1+\sec (c+d x)))^{5/3}} \]
((-3 + 5*2^(5/6)*Cos[(c + d*x)/2]^2*Hypergeometric2F1[1/2, 7/6, 3/2, (1 - Sec[c + d*x])/2]*Sec[c + d*x]*(1 + Sec[c + d*x])^(1/6))*Tan[c + d*x])/(7*d *(a*(1 + Sec[c + d*x]))^(5/3))
Time = 0.81 (sec) , antiderivative size = 736, normalized size of antiderivative = 1.01, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.565, Rules used = {3042, 4284, 3042, 4315, 3042, 4314, 61, 73, 837, 25, 27, 766, 2420}
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.
| \(\displaystyle \int \frac {\sec ^2(c+d x)}{(a \sec (c+d x)+a)^{5/3}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^2}{\left (a \csc \left (c+d x+\frac {\pi }{2}\right )+a\right )^{5/3}}dx\) |
| \(\Big \downarrow \) 4284 |
| \(\displaystyle \frac {5 \int \frac {\sec (c+d x)}{(\sec (c+d x) a+a)^{2/3}}dx}{7 a}-\frac {3 \tan (c+d x)}{7 d (a \sec (c+d x)+a)^{5/3}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {5 \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )}{\left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^{2/3}}dx}{7 a}-\frac {3 \tan (c+d x)}{7 d (a \sec (c+d x)+a)^{5/3}}\) |
| \(\Big \downarrow \) 4315 |
| \(\displaystyle \frac {5 (\sec (c+d x)+1)^{2/3} \int \frac {\sec (c+d x)}{(\sec (c+d x)+1)^{2/3}}dx}{7 a (a \sec (c+d x)+a)^{2/3}}-\frac {3 \tan (c+d x)}{7 d (a \sec (c+d x)+a)^{5/3}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {5 (\sec (c+d x)+1)^{2/3} \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )}{\left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )^{2/3}}dx}{7 a (a \sec (c+d x)+a)^{2/3}}-\frac {3 \tan (c+d x)}{7 d (a \sec (c+d x)+a)^{5/3}}\) |
| \(\Big \downarrow \) 4314 |
| \(\displaystyle -\frac {5 \tan (c+d x) \sqrt [6]{\sec (c+d x)+1} \int \frac {1}{\sqrt {1-\sec (c+d x)} (\sec (c+d x)+1)^{7/6}}d\sec (c+d x)}{7 a d \sqrt {1-\sec (c+d x)} (a \sec (c+d x)+a)^{2/3}}-\frac {3 \tan (c+d x)}{7 d (a \sec (c+d x)+a)^{5/3}}\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle -\frac {5 \tan (c+d x) \sqrt [6]{\sec (c+d x)+1} \left (-\int \frac {1}{\sqrt {1-\sec (c+d x)} \sqrt [6]{\sec (c+d x)+1}}d\sec (c+d x)-\frac {3 \sqrt {1-\sec (c+d x)}}{\sqrt [6]{\sec (c+d x)+1}}\right )}{7 a d \sqrt {1-\sec (c+d x)} (a \sec (c+d x)+a)^{2/3}}-\frac {3 \tan (c+d x)}{7 d (a \sec (c+d x)+a)^{5/3}}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -\frac {5 \tan (c+d x) \sqrt [6]{\sec (c+d x)+1} \left (-6 \int \frac {(\sec (c+d x)+1)^{2/3}}{\sqrt {1-\sec (c+d x)}}d\sqrt [6]{\sec (c+d x)+1}-\frac {3 \sqrt {1-\sec (c+d x)}}{\sqrt [6]{\sec (c+d x)+1}}\right )}{7 a d \sqrt {1-\sec (c+d x)} (a \sec (c+d x)+a)^{2/3}}-\frac {3 \tan (c+d x)}{7 d (a \sec (c+d x)+a)^{5/3}}\) |
| \(\Big \downarrow \) 837 |
| \(\displaystyle -\frac {5 \tan (c+d x) \sqrt [6]{\sec (c+d x)+1} \left (-6 \left (-\frac {\left (1-\sqrt {3}\right ) \int \frac {1}{\sqrt {1-\sec (c+d x)}}d\sqrt [6]{\sec (c+d x)+1}}{\sqrt [3]{2}}-\frac {1}{2} \int -\frac {2 (\sec (c+d x)+1)^{2/3}+2^{2/3} \left (1-\sqrt {3}\right )}{\sqrt {1-\sec (c+d x)}}d\sqrt [6]{\sec (c+d x)+1}\right )-\frac {3 \sqrt {1-\sec (c+d x)}}{\sqrt [6]{\sec (c+d x)+1}}\right )}{7 a d \sqrt {1-\sec (c+d x)} (a \sec (c+d x)+a)^{2/3}}-\frac {3 \tan (c+d x)}{7 d (a \sec (c+d x)+a)^{5/3}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {5 \tan (c+d x) \sqrt [6]{\sec (c+d x)+1} \left (-6 \left (\frac {1}{2} \int \frac {2^{2/3} \left (\sqrt [3]{2} (\sec (c+d x)+1)^{2/3}-\sqrt {3}+1\right )}{\sqrt {1-\sec (c+d x)}}d\sqrt [6]{\sec (c+d x)+1}-\frac {\left (1-\sqrt {3}\right ) \int \frac {1}{\sqrt {1-\sec (c+d x)}}d\sqrt [6]{\sec (c+d x)+1}}{\sqrt [3]{2}}\right )-\frac {3 \sqrt {1-\sec (c+d x)}}{\sqrt [6]{\sec (c+d x)+1}}\right )}{7 a d \sqrt {1-\sec (c+d x)} (a \sec (c+d x)+a)^{2/3}}-\frac {3 \tan (c+d x)}{7 d (a \sec (c+d x)+a)^{5/3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {5 \tan (c+d x) \sqrt [6]{\sec (c+d x)+1} \left (-6 \left (\frac {\int \frac {\sqrt [3]{2} (\sec (c+d x)+1)^{2/3}-\sqrt {3}+1}{\sqrt {1-\sec (c+d x)}}d\sqrt [6]{\sec (c+d x)+1}}{\sqrt [3]{2}}-\frac {\left (1-\sqrt {3}\right ) \int \frac {1}{\sqrt {1-\sec (c+d x)}}d\sqrt [6]{\sec (c+d x)+1}}{\sqrt [3]{2}}\right )-\frac {3 \sqrt {1-\sec (c+d x)}}{\sqrt [6]{\sec (c+d x)+1}}\right )}{7 a d \sqrt {1-\sec (c+d x)} (a \sec (c+d x)+a)^{2/3}}-\frac {3 \tan (c+d x)}{7 d (a \sec (c+d x)+a)^{5/3}}\) |
| \(\Big \downarrow \) 766 |
| \(\displaystyle -\frac {5 \tan (c+d x) \sqrt [6]{\sec (c+d x)+1} \left (-6 \left (\frac {\int \frac {\sqrt [3]{2} (\sec (c+d x)+1)^{2/3}-\sqrt {3}+1}{\sqrt {1-\sec (c+d x)}}d\sqrt [6]{\sec (c+d x)+1}}{\sqrt [3]{2}}-\frac {\left (1-\sqrt {3}\right ) \sqrt [6]{\sec (c+d x)+1} \left (\sqrt [3]{2}-\sqrt [3]{\sec (c+d x)+1}\right ) \sqrt {\frac {(\sec (c+d x)+1)^{2/3}+\sqrt [3]{2} \sqrt [3]{\sec (c+d x)+1}+2^{2/3}}{\left (\sqrt [3]{2}-\left (1+\sqrt {3}\right ) \sqrt [3]{\sec (c+d x)+1}\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {\sqrt [3]{2}-\left (1-\sqrt {3}\right ) \sqrt [3]{\sec (c+d x)+1}}{\sqrt [3]{2}-\left (1+\sqrt {3}\right ) \sqrt [3]{\sec (c+d x)+1}}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{2\ 2^{2/3} \sqrt [4]{3} \sqrt {1-\sec (c+d x)} \sqrt {-\frac {\sqrt [3]{\sec (c+d x)+1} \left (\sqrt [3]{2}-\sqrt [3]{\sec (c+d x)+1}\right )}{\left (\sqrt [3]{2}-\left (1+\sqrt {3}\right ) \sqrt [3]{\sec (c+d x)+1}\right )^2}}}\right )-\frac {3 \sqrt {1-\sec (c+d x)}}{\sqrt [6]{\sec (c+d x)+1}}\right )}{7 a d \sqrt {1-\sec (c+d x)} (a \sec (c+d x)+a)^{2/3}}-\frac {3 \tan (c+d x)}{7 d (a \sec (c+d x)+a)^{5/3}}\) |
| \(\Big \downarrow \) 2420 |
| \(\displaystyle -\frac {5 \tan (c+d x) \sqrt [6]{\sec (c+d x)+1} \left (-6 \left (\frac {\frac {\left (1+\sqrt {3}\right ) \sqrt {1-\sec (c+d x)} \sqrt [6]{\sec (c+d x)+1}}{2^{2/3} \left (\sqrt [3]{2}-\left (1+\sqrt {3}\right ) \sqrt [3]{\sec (c+d x)+1}\right )}-\frac {\sqrt [4]{3} \sqrt [6]{\sec (c+d x)+1} \left (\sqrt [3]{2}-\sqrt [3]{\sec (c+d x)+1}\right ) \sqrt {\frac {(\sec (c+d x)+1)^{2/3}+\sqrt [3]{2} \sqrt [3]{\sec (c+d x)+1}+2^{2/3}}{\left (\sqrt [3]{2}-\left (1+\sqrt {3}\right ) \sqrt [3]{\sec (c+d x)+1}\right )^2}} E\left (\arccos \left (\frac {\sqrt [3]{2}-\left (1-\sqrt {3}\right ) \sqrt [3]{\sec (c+d x)+1}}{\sqrt [3]{2}-\left (1+\sqrt {3}\right ) \sqrt [3]{\sec (c+d x)+1}}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{\sqrt [3]{2} \sqrt {1-\sec (c+d x)} \sqrt {-\frac {\sqrt [3]{\sec (c+d x)+1} \left (\sqrt [3]{2}-\sqrt [3]{\sec (c+d x)+1}\right )}{\left (\sqrt [3]{2}-\left (1+\sqrt {3}\right ) \sqrt [3]{\sec (c+d x)+1}\right )^2}}}}{\sqrt [3]{2}}-\frac {\left (1-\sqrt {3}\right ) \sqrt [6]{\sec (c+d x)+1} \left (\sqrt [3]{2}-\sqrt [3]{\sec (c+d x)+1}\right ) \sqrt {\frac {(\sec (c+d x)+1)^{2/3}+\sqrt [3]{2} \sqrt [3]{\sec (c+d x)+1}+2^{2/3}}{\left (\sqrt [3]{2}-\left (1+\sqrt {3}\right ) \sqrt [3]{\sec (c+d x)+1}\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {\sqrt [3]{2}-\left (1-\sqrt {3}\right ) \sqrt [3]{\sec (c+d x)+1}}{\sqrt [3]{2}-\left (1+\sqrt {3}\right ) \sqrt [3]{\sec (c+d x)+1}}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{2\ 2^{2/3} \sqrt [4]{3} \sqrt {1-\sec (c+d x)} \sqrt {-\frac {\sqrt [3]{\sec (c+d x)+1} \left (\sqrt [3]{2}-\sqrt [3]{\sec (c+d x)+1}\right )}{\left (\sqrt [3]{2}-\left (1+\sqrt {3}\right ) \sqrt [3]{\sec (c+d x)+1}\right )^2}}}\right )-\frac {3 \sqrt {1-\sec (c+d x)}}{\sqrt [6]{\sec (c+d x)+1}}\right )}{7 a d \sqrt {1-\sec (c+d x)} (a \sec (c+d x)+a)^{2/3}}-\frac {3 \tan (c+d x)}{7 d (a \sec (c+d x)+a)^{5/3}}\) |
(-3*Tan[c + d*x])/(7*d*(a + a*Sec[c + d*x])^(5/3)) - (5*(1 + Sec[c + d*x]) ^(1/6)*((-3*Sqrt[1 - Sec[c + d*x]])/(1 + Sec[c + d*x])^(1/6) - 6*(-1/2*((1 - Sqrt[3])*EllipticF[ArcCos[(2^(1/3) - (1 - Sqrt[3])*(1 + Sec[c + d*x])^( 1/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))], (2 + Sqrt[3])/4 ]*(1 + Sec[c + d*x])^(1/6)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3))*Sqrt[(2^(2 /3) + 2^(1/3)*(1 + Sec[c + d*x])^(1/3) + (1 + Sec[c + d*x])^(2/3))/(2^(1/3 ) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2])/(2^(2/3)*3^(1/4)*Sqrt[1 - Sec[c + d*x]]*Sqrt[-(((1 + Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x ])^(1/3)))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2)]) + (((1 + Sqrt[3])*Sqrt[1 - Sec[c + d*x]]*(1 + Sec[c + d*x])^(1/6))/(2^(2/3)*(2^(1 /3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))) - (3^(1/4)*EllipticE[ArcCos [(2^(1/3) - (1 - Sqrt[3])*(1 + Sec[c + d*x])^(1/3))/(2^(1/3) - (1 + Sqrt[3 ])*(1 + Sec[c + d*x])^(1/3))], (2 + Sqrt[3])/4]*(1 + Sec[c + d*x])^(1/6)*( 2^(1/3) - (1 + Sec[c + d*x])^(1/3))*Sqrt[(2^(2/3) + 2^(1/3)*(1 + Sec[c + d *x])^(1/3) + (1 + Sec[c + d*x])^(2/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2])/(2^(1/3)*Sqrt[1 - Sec[c + d*x]]*Sqrt[-(((1 + Sec[c + d *x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3)))/(2^(1/3) - (1 + Sqrt[3])* (1 + Sec[c + d*x])^(1/3))^2)]))/2^(1/3)))*Tan[c + d*x])/(7*a*d*Sqrt[1 - Se c[c + d*x]]*(a + a*Sec[c + d*x])^(2/3))
3.2.61.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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_)^6], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[x*(s + r*x^2)*(Sqrt[(s^2 - r*s*x^2 + r^2*x^4)/ (s + (1 + Sqrt[3])*r*x^2)^2]/(2*3^(1/4)*s*Sqrt[a + b*x^6]*Sqrt[r*x^2*((s + r*x^2)/(s + (1 + Sqrt[3])*r*x^2)^2)]))*EllipticF[ArcCos[(s + (1 - Sqrt[3])* r*x^2)/(s + (1 + Sqrt[3])*r*x^2)], (2 + Sqrt[3])/4], x]] /; FreeQ[{a, b}, x ]
Int[(x_)^4/Sqrt[(a_) + (b_.)*(x_)^6], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[(Sqrt[3] - 1)*(s^2/(2*r^2)) Int[1/Sqrt[ a + b*x^6], x], x] - Simp[1/(2*r^2) Int[((Sqrt[3] - 1)*s^2 - 2*r^2*x^4)/S qrt[a + b*x^6], x], x]] /; FreeQ[{a, b}, x]
Int[((c_) + (d_.)*(x_)^4)/Sqrt[(a_) + (b_.)*(x_)^6], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[(1 + Sqrt[3])*d*s^3*x*(Sqr t[a + b*x^6]/(2*a*r^2*(s + (1 + Sqrt[3])*r*x^2))), x] - Simp[3^(1/4)*d*s*x* (s + r*x^2)*(Sqrt[(s^2 - r*s*x^2 + r^2*x^4)/(s + (1 + Sqrt[3])*r*x^2)^2]/(2 *r^2*Sqrt[(r*x^2*(s + r*x^2))/(s + (1 + Sqrt[3])*r*x^2)^2]*Sqrt[a + b*x^6]) )*EllipticE[ArcCos[(s + (1 - Sqrt[3])*r*x^2)/(s + (1 + Sqrt[3])*r*x^2)], (2 + Sqrt[3])/4], x]] /; FreeQ[{a, b, c, d}, x] && EqQ[2*Rt[b/a, 3]^2*c - (1 - Sqrt[3])*d, 0]
Int[csc[(e_.) + (f_.)*(x_)]^2*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> Simp[(-Cot[e + f*x])*((a + b*Csc[e + f*x])^m/(f*(2*m + 1))), x ] + Simp[m/(b*(2*m + 1)) Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m + 1), x ], x] /; FreeQ[{a, b, e, f}, x] && EqQ[a^2 - b^2, 0] && LtQ[m, -2^(-1)]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), 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)*((a + b*x)^(m - 1/2 )/Sqrt[a - b*x]), x], x, Csc[e + f*x]], x] /; FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 - b^2, 0] && !IntegerQ[m] && GtQ[a, 0]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> Simp[a^IntPart[m]*((a + b*Csc[e + f*x])^FracPart[m ]/(1 + (b/a)*Csc[e + f*x])^FracPart[m]) Int[(1 + (b/a)*Csc[e + f*x])^m*(d *Csc[e + f*x])^n, x], x] /; FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 - b^ 2, 0] && !IntegerQ[m] && !GtQ[a, 0]
\[\int \frac {\sec \left (d x +c \right )^{2}}{\left (a +a \sec \left (d x +c \right )\right )^{\frac {5}{3}}}d x\]
\[ \int \frac {\sec ^2(c+d x)}{(a+a \sec (c+d x))^{5/3}} \, dx=\int { \frac {\sec \left (d x + c\right )^{2}}{{\left (a \sec \left (d x + c\right ) + a\right )}^{\frac {5}{3}}} \,d x } \]
integral((a*sec(d*x + c) + a)^(1/3)*sec(d*x + c)^2/(a^2*sec(d*x + c)^2 + 2 *a^2*sec(d*x + c) + a^2), x)
\[ \int \frac {\sec ^2(c+d x)}{(a+a \sec (c+d x))^{5/3}} \, dx=\int \frac {\sec ^{2}{\left (c + d x \right )}}{\left (a \left (\sec {\left (c + d x \right )} + 1\right )\right )^{\frac {5}{3}}}\, dx \]
\[ \int \frac {\sec ^2(c+d x)}{(a+a \sec (c+d x))^{5/3}} \, dx=\int { \frac {\sec \left (d x + c\right )^{2}}{{\left (a \sec \left (d x + c\right ) + a\right )}^{\frac {5}{3}}} \,d x } \]
\[ \int \frac {\sec ^2(c+d x)}{(a+a \sec (c+d x))^{5/3}} \, dx=\int { \frac {\sec \left (d x + c\right )^{2}}{{\left (a \sec \left (d x + c\right ) + a\right )}^{\frac {5}{3}}} \,d x } \]
Timed out. \[ \int \frac {\sec ^2(c+d x)}{(a+a \sec (c+d x))^{5/3}} \, dx=\int \frac {1}{{\cos \left (c+d\,x\right )}^2\,{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^{5/3}} \,d x \]