Integrand size = 23, antiderivative size = 766 \[ \int \frac {\sec ^4(c+d x)}{(a+a \sec (c+d x))^{5/3}} \, dx=-\frac {33 \tan (c+d x)}{28 d (a+a \sec (c+d x))^{5/3}}+\frac {3 \sec ^2(c+d x) \tan (c+d x)}{4 d (a+a \sec (c+d x))^{5/3}}+\frac {135 \tan (c+d x)}{14 a d (a+a \sec (c+d x))^{2/3}}+\frac {375 \left (1+\sqrt {3}\right ) \sqrt [3]{a+a \sec (c+d x)} \tan (c+d x)}{28 a^2 d (1+\sec (c+d x))^{2/3} \left (\sqrt [3]{2}-\left (1+\sqrt {3}\right ) \sqrt [3]{1+\sec (c+d x)}\right )}-\frac {375 \sqrt [4]{3} E\left (\arccos \left (\frac {\sqrt [3]{2}-\left (1-\sqrt {3}\right ) \sqrt [3]{1+\sec (c+d x)}}{\sqrt [3]{2}-\left (1+\sqrt {3}\right ) \sqrt [3]{1+\sec (c+d x)}}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right ) \sqrt [3]{a+a \sec (c+d x)} \left (\sqrt [3]{2}-\sqrt [3]{1+\sec (c+d x)}\right ) \sqrt {\frac {2^{2/3}+\sqrt [3]{2} \sqrt [3]{1+\sec (c+d x)}+(1+\sec (c+d x))^{2/3}}{\left (\sqrt [3]{2}-\left (1+\sqrt {3}\right ) \sqrt [3]{1+\sec (c+d x)}\right )^2}} \tan (c+d x)}{14\ 2^{2/3} a^2 d (1-\sec (c+d x)) (1+\sec (c+d x))^{2/3} \sqrt {-\frac {\sqrt [3]{1+\sec (c+d x)} \left (\sqrt [3]{2}-\sqrt [3]{1+\sec (c+d x)}\right )}{\left (\sqrt [3]{2}-\left (1+\sqrt {3}\right ) \sqrt [3]{1+\sec (c+d x)}\right )^2}}}-\frac {125\ 3^{3/4} \left (1-\sqrt {3}\right ) \operatorname {EllipticF}\left (\arccos \left (\frac {\sqrt [3]{2}-\left (1-\sqrt {3}\right ) \sqrt [3]{1+\sec (c+d x)}}{\sqrt [3]{2}-\left (1+\sqrt {3}\right ) \sqrt [3]{1+\sec (c+d x)}}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right ) \sqrt [3]{a+a \sec (c+d x)} \left (\sqrt [3]{2}-\sqrt [3]{1+\sec (c+d x)}\right ) \sqrt {\frac {2^{2/3}+\sqrt [3]{2} \sqrt [3]{1+\sec (c+d x)}+(1+\sec (c+d x))^{2/3}}{\left (\sqrt [3]{2}-\left (1+\sqrt {3}\right ) \sqrt [3]{1+\sec (c+d x)}\right )^2}} \tan (c+d x)}{28\ 2^{2/3} a^2 d (1-\sec (c+d x)) (1+\sec (c+d x))^{2/3} \sqrt {-\frac {\sqrt [3]{1+\sec (c+d x)} \left (\sqrt [3]{2}-\sqrt [3]{1+\sec (c+d x)}\right )}{\left (\sqrt [3]{2}-\left (1+\sqrt {3}\right ) \sqrt [3]{1+\sec (c+d x)}\right )^2}}} \]
-33/28*tan(d*x+c)/d/(a+a*sec(d*x+c))^(5/3)+3/4*sec(d*x+c)^2*tan(d*x+c)/d/( a+a*sec(d*x+c))^(5/3)+135/14*tan(d*x+c)/a/d/(a+a*sec(d*x+c))^(2/3)+375/28* (a+a*sec(d*x+c))^(1/3)*(1+3^(1/2))*tan(d*x+c)/a^2/d/(1+sec(d*x+c))^(2/3)/( 2^(1/3)-(1+sec(d*x+c))^(1/3)*(1+3^(1/2)))-375/28*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))*(a+a*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)*2^(1/3)/a^2/d/(1-sec(d*x+c))/(1+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)-125/56*3^(3/4)*((2^(1/3)-(1+se c(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))*(a+a*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+s ec(d*x+c))^(1/3)*(1+3^(1/2)))^2)^(1/2)*tan(d*x+c)*2^(1/3)/a^2/d/(1-sec(...
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.59 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.14 \[ \int \frac {\sec ^4(c+d x)}{(a+a \sec (c+d x))^{5/3}} \, dx=\frac {\left (-250 2^{5/6} \cos ^2\left (\frac {1}{2} (c+d x)\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{2},\frac {3}{2},\frac {1}{2} (1-\sec (c+d x))\right ) \sec (c+d x) \sqrt [6]{1+\sec (c+d x)}+3 \left (79+90 \sec (c+d x)+7 \sec ^2(c+d x)\right )\right ) \tan (c+d x)}{28 d (a (1+\sec (c+d x)))^{5/3}} \]
((-250*2^(5/6)*Cos[(c + d*x)/2]^2*Hypergeometric2F1[1/6, 1/2, 3/2, (1 - Se c[c + d*x])/2]*Sec[c + d*x]*(1 + Sec[c + d*x])^(1/6) + 3*(79 + 90*Sec[c + d*x] + 7*Sec[c + d*x]^2))*Tan[c + d*x])/(28*d*(a*(1 + Sec[c + d*x]))^(5/3) )
Time = 1.37 (sec) , antiderivative size = 781, normalized size of antiderivative = 1.02, number of steps used = 19, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.783, Rules used = {3042, 4311, 27, 3042, 4496, 27, 3042, 4488, 3042, 4315, 3042, 4314, 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.
| \(\displaystyle \int \frac {\sec ^4(c+d x)}{(a \sec (c+d x)+a)^{5/3}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^4}{\left (a \csc \left (c+d x+\frac {\pi }{2}\right )+a\right )^{5/3}}dx\) |
| \(\Big \downarrow \) 4311 |
| \(\displaystyle \frac {3 \int \frac {\sec ^2(c+d x) (6 a-5 a \sec (c+d x))}{3 (\sec (c+d x) a+a)^{5/3}}dx}{4 a}+\frac {3 \tan (c+d x) \sec ^2(c+d x)}{4 d (a \sec (c+d x)+a)^{5/3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\sec ^2(c+d x) (6 a-5 a \sec (c+d x))}{(\sec (c+d x) a+a)^{5/3}}dx}{4 a}+\frac {3 \tan (c+d x) \sec ^2(c+d x)}{4 d (a \sec (c+d x)+a)^{5/3}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^2 \left (6 a-5 a \csc \left (c+d x+\frac {\pi }{2}\right )\right )}{\left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^{5/3}}dx}{4 a}+\frac {3 \tan (c+d x) \sec ^2(c+d x)}{4 d (a \sec (c+d x)+a)^{5/3}}\) |
| \(\Big \downarrow \) 4496 |
| \(\displaystyle \frac {-\frac {3 \int -\frac {5 \sec (c+d x) \left (11 a^2-7 a^2 \sec (c+d x)\right )}{3 (\sec (c+d x) a+a)^{2/3}}dx}{7 a^2}-\frac {33 a \tan (c+d x)}{7 d (a \sec (c+d x)+a)^{5/3}}}{4 a}+\frac {3 \tan (c+d x) \sec ^2(c+d x)}{4 d (a \sec (c+d x)+a)^{5/3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {5 \int \frac {\sec (c+d x) \left (11 a^2-7 a^2 \sec (c+d x)\right )}{(\sec (c+d x) a+a)^{2/3}}dx}{7 a^2}-\frac {33 a \tan (c+d x)}{7 d (a \sec (c+d x)+a)^{5/3}}}{4 a}+\frac {3 \tan (c+d x) \sec ^2(c+d x)}{4 d (a \sec (c+d x)+a)^{5/3}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {5 \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (11 a^2-7 a^2 \csc \left (c+d x+\frac {\pi }{2}\right )\right )}{\left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^{2/3}}dx}{7 a^2}-\frac {33 a \tan (c+d x)}{7 d (a \sec (c+d x)+a)^{5/3}}}{4 a}+\frac {3 \tan (c+d x) \sec ^2(c+d x)}{4 d (a \sec (c+d x)+a)^{5/3}}\) |
| \(\Big \downarrow \) 4488 |
| \(\displaystyle \frac {\frac {5 \left (\frac {54 a^2 \tan (c+d x)}{d (a \sec (c+d x)+a)^{2/3}}-25 a \int \sec (c+d x) \sqrt [3]{\sec (c+d x) a+a}dx\right )}{7 a^2}-\frac {33 a \tan (c+d x)}{7 d (a \sec (c+d x)+a)^{5/3}}}{4 a}+\frac {3 \tan (c+d x) \sec ^2(c+d x)}{4 d (a \sec (c+d x)+a)^{5/3}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {5 \left (\frac {54 a^2 \tan (c+d x)}{d (a \sec (c+d x)+a)^{2/3}}-25 a \int \csc \left (c+d x+\frac {\pi }{2}\right ) \sqrt [3]{\csc \left (c+d x+\frac {\pi }{2}\right ) a+a}dx\right )}{7 a^2}-\frac {33 a \tan (c+d x)}{7 d (a \sec (c+d x)+a)^{5/3}}}{4 a}+\frac {3 \tan (c+d x) \sec ^2(c+d x)}{4 d (a \sec (c+d x)+a)^{5/3}}\) |
| \(\Big \downarrow \) 4315 |
| \(\displaystyle \frac {\frac {5 \left (\frac {54 a^2 \tan (c+d x)}{d (a \sec (c+d x)+a)^{2/3}}-\frac {25 a \sqrt [3]{a \sec (c+d x)+a} \int \sec (c+d x) \sqrt [3]{\sec (c+d x)+1}dx}{\sqrt [3]{\sec (c+d x)+1}}\right )}{7 a^2}-\frac {33 a \tan (c+d x)}{7 d (a \sec (c+d x)+a)^{5/3}}}{4 a}+\frac {3 \tan (c+d x) \sec ^2(c+d x)}{4 d (a \sec (c+d x)+a)^{5/3}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {5 \left (\frac {54 a^2 \tan (c+d x)}{d (a \sec (c+d x)+a)^{2/3}}-\frac {25 a \sqrt [3]{a \sec (c+d x)+a} \int \csc \left (c+d x+\frac {\pi }{2}\right ) \sqrt [3]{\csc \left (c+d x+\frac {\pi }{2}\right )+1}dx}{\sqrt [3]{\sec (c+d x)+1}}\right )}{7 a^2}-\frac {33 a \tan (c+d x)}{7 d (a \sec (c+d x)+a)^{5/3}}}{4 a}+\frac {3 \tan (c+d x) \sec ^2(c+d x)}{4 d (a \sec (c+d x)+a)^{5/3}}\) |
| \(\Big \downarrow \) 4314 |
| \(\displaystyle \frac {\frac {5 \left (\frac {25 a \tan (c+d x) \sqrt [3]{a \sec (c+d x)+a} \int \frac {1}{\sqrt {1-\sec (c+d x)} \sqrt [6]{\sec (c+d x)+1}}d\sec (c+d x)}{d \sqrt {1-\sec (c+d x)} (\sec (c+d x)+1)^{5/6}}+\frac {54 a^2 \tan (c+d x)}{d (a \sec (c+d x)+a)^{2/3}}\right )}{7 a^2}-\frac {33 a \tan (c+d x)}{7 d (a \sec (c+d x)+a)^{5/3}}}{4 a}+\frac {3 \tan (c+d x) \sec ^2(c+d x)}{4 d (a \sec (c+d x)+a)^{5/3}}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {\frac {5 \left (\frac {150 a \tan (c+d x) \sqrt [3]{a \sec (c+d x)+a} \int \frac {(\sec (c+d x)+1)^{2/3}}{\sqrt {1-\sec (c+d x)}}d\sqrt [6]{\sec (c+d x)+1}}{d \sqrt {1-\sec (c+d x)} (\sec (c+d x)+1)^{5/6}}+\frac {54 a^2 \tan (c+d x)}{d (a \sec (c+d x)+a)^{2/3}}\right )}{7 a^2}-\frac {33 a \tan (c+d x)}{7 d (a \sec (c+d x)+a)^{5/3}}}{4 a}+\frac {3 \tan (c+d x) \sec ^2(c+d x)}{4 d (a \sec (c+d x)+a)^{5/3}}\) |
| \(\Big \downarrow \) 837 |
| \(\displaystyle \frac {\frac {5 \left (\frac {150 a \tan (c+d x) \sqrt [3]{a \sec (c+d x)+a} \left (-\frac {\left (1-\sqrt {3}\right ) \int \frac {1}{\sqrt {1-\sec (c+d x)}}d\sqrt [6]{\sec (c+d x)+1}}{\sqrt [3]{2}}-\frac {1}{2} \int -\frac {2 (\sec (c+d x)+1)^{2/3}+2^{2/3} \left (1-\sqrt {3}\right )}{\sqrt {1-\sec (c+d x)}}d\sqrt [6]{\sec (c+d x)+1}\right )}{d \sqrt {1-\sec (c+d x)} (\sec (c+d x)+1)^{5/6}}+\frac {54 a^2 \tan (c+d x)}{d (a \sec (c+d x)+a)^{2/3}}\right )}{7 a^2}-\frac {33 a \tan (c+d x)}{7 d (a \sec (c+d x)+a)^{5/3}}}{4 a}+\frac {3 \tan (c+d x) \sec ^2(c+d x)}{4 d (a \sec (c+d x)+a)^{5/3}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {5 \left (\frac {150 a \tan (c+d x) \sqrt [3]{a \sec (c+d x)+a} \left (\frac {1}{2} \int \frac {2^{2/3} \left (\sqrt [3]{2} (\sec (c+d x)+1)^{2/3}-\sqrt {3}+1\right )}{\sqrt {1-\sec (c+d x)}}d\sqrt [6]{\sec (c+d x)+1}-\frac {\left (1-\sqrt {3}\right ) \int \frac {1}{\sqrt {1-\sec (c+d x)}}d\sqrt [6]{\sec (c+d x)+1}}{\sqrt [3]{2}}\right )}{d \sqrt {1-\sec (c+d x)} (\sec (c+d x)+1)^{5/6}}+\frac {54 a^2 \tan (c+d x)}{d (a \sec (c+d x)+a)^{2/3}}\right )}{7 a^2}-\frac {33 a \tan (c+d x)}{7 d (a \sec (c+d x)+a)^{5/3}}}{4 a}+\frac {3 \tan (c+d x) \sec ^2(c+d x)}{4 d (a \sec (c+d x)+a)^{5/3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {5 \left (\frac {150 a \tan (c+d x) \sqrt [3]{a \sec (c+d x)+a} \left (\frac {\int \frac {\sqrt [3]{2} (\sec (c+d x)+1)^{2/3}-\sqrt {3}+1}{\sqrt {1-\sec (c+d x)}}d\sqrt [6]{\sec (c+d x)+1}}{\sqrt [3]{2}}-\frac {\left (1-\sqrt {3}\right ) \int \frac {1}{\sqrt {1-\sec (c+d x)}}d\sqrt [6]{\sec (c+d x)+1}}{\sqrt [3]{2}}\right )}{d \sqrt {1-\sec (c+d x)} (\sec (c+d x)+1)^{5/6}}+\frac {54 a^2 \tan (c+d x)}{d (a \sec (c+d x)+a)^{2/3}}\right )}{7 a^2}-\frac {33 a \tan (c+d x)}{7 d (a \sec (c+d x)+a)^{5/3}}}{4 a}+\frac {3 \tan (c+d x) \sec ^2(c+d x)}{4 d (a \sec (c+d x)+a)^{5/3}}\) |
| \(\Big \downarrow \) 766 |
| \(\displaystyle \frac {\frac {5 \left (\frac {150 a \tan (c+d x) \sqrt [3]{a \sec (c+d x)+a} \left (\frac {\int \frac {\sqrt [3]{2} (\sec (c+d x)+1)^{2/3}-\sqrt {3}+1}{\sqrt {1-\sec (c+d x)}}d\sqrt [6]{\sec (c+d x)+1}}{\sqrt [3]{2}}-\frac {\left (1-\sqrt {3}\right ) \sqrt [6]{\sec (c+d x)+1} \left (\sqrt [3]{2}-\sqrt [3]{\sec (c+d x)+1}\right ) \sqrt {\frac {(\sec (c+d x)+1)^{2/3}+\sqrt [3]{2} \sqrt [3]{\sec (c+d x)+1}+2^{2/3}}{\left (\sqrt [3]{2}-\left (1+\sqrt {3}\right ) \sqrt [3]{\sec (c+d x)+1}\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {\sqrt [3]{2}-\left (1-\sqrt {3}\right ) \sqrt [3]{\sec (c+d x)+1}}{\sqrt [3]{2}-\left (1+\sqrt {3}\right ) \sqrt [3]{\sec (c+d x)+1}}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{2\ 2^{2/3} \sqrt [4]{3} \sqrt {1-\sec (c+d x)} \sqrt {-\frac {\sqrt [3]{\sec (c+d x)+1} \left (\sqrt [3]{2}-\sqrt [3]{\sec (c+d x)+1}\right )}{\left (\sqrt [3]{2}-\left (1+\sqrt {3}\right ) \sqrt [3]{\sec (c+d x)+1}\right )^2}}}\right )}{d \sqrt {1-\sec (c+d x)} (\sec (c+d x)+1)^{5/6}}+\frac {54 a^2 \tan (c+d x)}{d (a \sec (c+d x)+a)^{2/3}}\right )}{7 a^2}-\frac {33 a \tan (c+d x)}{7 d (a \sec (c+d x)+a)^{5/3}}}{4 a}+\frac {3 \tan (c+d x) \sec ^2(c+d x)}{4 d (a \sec (c+d x)+a)^{5/3}}\) |
| \(\Big \downarrow \) 2420 |
| \(\displaystyle \frac {\frac {5 \left (\frac {54 a^2 \tan (c+d x)}{d (a \sec (c+d x)+a)^{2/3}}+\frac {150 a \tan (c+d x) \sqrt [3]{a \sec (c+d x)+a} \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 )}{d \sqrt {1-\sec (c+d x)} (\sec (c+d x)+1)^{5/6}}\right )}{7 a^2}-\frac {33 a \tan (c+d x)}{7 d (a \sec (c+d x)+a)^{5/3}}}{4 a}+\frac {3 \tan (c+d x) \sec ^2(c+d x)}{4 d (a \sec (c+d x)+a)^{5/3}}\) |
(3*Sec[c + d*x]^2*Tan[c + d*x])/(4*d*(a + a*Sec[c + d*x])^(5/3)) + ((-33*a *Tan[c + d*x])/(7*d*(a + a*Sec[c + d*x])^(5/3)) + (5*((54*a^2*Tan[c + d*x] )/(d*(a + a*Sec[c + d*x])^(2/3)) + (150*a*(a + a*Sec[c + d*x])^(1/3)*(-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[Ar cCos[(2^(1/3) - (1 - Sqrt[3])*(1 + Sec[c + d*x])^(1/3))/(2^(1/3) - (1 + Sq rt[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 + S ec[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])/(d*Sqrt[1 - Sec [c + d*x]]*(1 + Sec[c + d*x])^(5/6))))/(7*a^2))/(4*a)
3.2.59.3.1 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] :> 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_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + ( a_))^(m_), x_Symbol] :> Simp[(-d^2)*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*((d *Csc[e + f*x])^(n - 2)/(f*(m + n - 1))), x] + Simp[d^2/(b*(m + n - 1)) In t[(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^(n - 2)*(b*(n - 2) + a*m*Csc[e + f*x]), x], x] /; FreeQ[{a, b, d, e, f, m}, x] && EqQ[a^2 - b^2, 0] && GtQ[n , 2] && NeQ[m + n - 1, 0] && IntegerQ[n]
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[csc[(e_.) + (f_.)*(x_)]*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_)*(cs c[(e_.) + (f_.)*(x_)]*(B_.) + (A_)), x_Symbol] :> Simp[(A*b - a*B)*Cot[e + f*x]*((a + b*Csc[e + f*x])^m/(a*f*(2*m + 1))), x] + Simp[(a*B*m + A*b*(m + 1))/(a*b*(2*m + 1)) Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m + 1), x], x] /; FreeQ[{a, b, A, B, e, f}, x] && NeQ[A*b - a*B, 0] && EqQ[a^2 - b^2, 0] && NeQ[a*B*m + A*b*(m + 1), 0] && LtQ[m, -2^(-1)]
Int[csc[(e_.) + (f_.)*(x_)]^2*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_)*( csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_)), x_Symbol] :> Simp[(-(A*b - a*B))*Cot [e + f*x]*((a + b*Csc[e + f*x])^m/(b*f*(2*m + 1))), x] + Simp[1/(b^2*(2*m + 1)) Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*Simp[A*b*m - a*B*m + b *B*(2*m + 1)*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B}, x] && Ne Q[A*b - a*B, 0] && EqQ[a^2 - b^2, 0] && LtQ[m, -2^(-1)]
\[\int \frac {\sec \left (d x +c \right )^{4}}{\left (a +a \sec \left (d x +c \right )\right )^{\frac {5}{3}}}d x\]
\[ \int \frac {\sec ^4(c+d x)}{(a+a \sec (c+d x))^{5/3}} \, dx=\int { \frac {\sec \left (d x + c\right )^{4}}{{\left (a \sec \left (d x + c\right ) + a\right )}^{\frac {5}{3}}} \,d x } \]
integral((a*sec(d*x + c) + a)^(1/3)*sec(d*x + c)^4/(a^2*sec(d*x + c)^2 + 2 *a^2*sec(d*x + c) + a^2), x)
\[ \int \frac {\sec ^4(c+d x)}{(a+a \sec (c+d x))^{5/3}} \, dx=\int \frac {\sec ^{4}{\left (c + d x \right )}}{\left (a \left (\sec {\left (c + d x \right )} + 1\right )\right )^{\frac {5}{3}}}\, dx \]
\[ \int \frac {\sec ^4(c+d x)}{(a+a \sec (c+d x))^{5/3}} \, dx=\int { \frac {\sec \left (d x + c\right )^{4}}{{\left (a \sec \left (d x + c\right ) + a\right )}^{\frac {5}{3}}} \,d x } \]
\[ \int \frac {\sec ^4(c+d x)}{(a+a \sec (c+d x))^{5/3}} \, dx=\int { \frac {\sec \left (d x + c\right )^{4}}{{\left (a \sec \left (d x + c\right ) + a\right )}^{\frac {5}{3}}} \,d x } \]
Timed out. \[ \int \frac {\sec ^4(c+d x)}{(a+a \sec (c+d x))^{5/3}} \, dx=\int \frac {1}{{\cos \left (c+d\,x\right )}^4\,{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^{5/3}} \,d x \]