Integrand size = 23, antiderivative size = 982 \[ \int \sqrt [3]{a+a \sin (e+f x)} \tan ^4(e+f x) \, dx=-\frac {361 \sec (e+f x) \sqrt [3]{a+a \sin (e+f x)}}{126 f}+\frac {361 \sec (e+f x) (1-\sin (e+f x)) \sqrt [3]{a+a \sin (e+f x)}}{63 f}-\frac {\sec (e+f x) \left (65 a^2-142 a^2 \sin (e+f x)\right )}{42 f (a-a \sin (e+f x)) (a+a \sin (e+f x))^{2/3}}+\frac {361 \left (1+\sqrt {3}\right ) \sec (e+f x) (1-\sin (e+f x)) (a+a \sin (e+f x))^{2/3}}{63 f \left (\sqrt [3]{2} \sqrt [3]{a}-\left (1+\sqrt {3}\right ) \sqrt [3]{a+a \sin (e+f x)}\right )}-\frac {361 \sqrt [3]{2} E\left (\arccos \left (\frac {\sqrt [3]{2} \sqrt [3]{a}-\left (1-\sqrt {3}\right ) \sqrt [3]{a+a \sin (e+f x)}}{\sqrt [3]{2} \sqrt [3]{a}-\left (1+\sqrt {3}\right ) \sqrt [3]{a+a \sin (e+f x)}}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right ) \sec (e+f x) (a+a \sin (e+f x))^{2/3} \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+a \sin (e+f x)}\right ) \sqrt {\frac {2^{2/3} a^{2/3}+\sqrt [3]{2} \sqrt [3]{a} \sqrt [3]{a+a \sin (e+f x)}+(a+a \sin (e+f x))^{2/3}}{\left (\sqrt [3]{2} \sqrt [3]{a}-\left (1+\sqrt {3}\right ) \sqrt [3]{a+a \sin (e+f x)}\right )^2}}}{21\ 3^{3/4} a^{2/3} f \sqrt {-\frac {\sqrt [3]{a+a \sin (e+f x)} \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+a \sin (e+f x)}\right )}{\left (\sqrt [3]{2} \sqrt [3]{a}-\left (1+\sqrt {3}\right ) \sqrt [3]{a+a \sin (e+f x)}\right )^2}}}-\frac {361 \left (1-\sqrt {3}\right ) \operatorname {EllipticF}\left (\arccos \left (\frac {\sqrt [3]{2} \sqrt [3]{a}-\left (1-\sqrt {3}\right ) \sqrt [3]{a+a \sin (e+f x)}}{\sqrt [3]{2} \sqrt [3]{a}-\left (1+\sqrt {3}\right ) \sqrt [3]{a+a \sin (e+f x)}}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right ) \sec (e+f x) (a+a \sin (e+f x))^{2/3} \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+a \sin (e+f x)}\right ) \sqrt {\frac {2^{2/3} a^{2/3}+\sqrt [3]{2} \sqrt [3]{a} \sqrt [3]{a+a \sin (e+f x)}+(a+a \sin (e+f x))^{2/3}}{\left (\sqrt [3]{2} \sqrt [3]{a}-\left (1+\sqrt {3}\right ) \sqrt [3]{a+a \sin (e+f x)}\right )^2}}}{63\ 2^{2/3} \sqrt [4]{3} a^{2/3} f \sqrt {-\frac {\sqrt [3]{a+a \sin (e+f x)} \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+a \sin (e+f x)}\right )}{\left (\sqrt [3]{2} \sqrt [3]{a}-\left (1+\sqrt {3}\right ) \sqrt [3]{a+a \sin (e+f x)}\right )^2}}}+\frac {3 a^2 \sin (e+f x) \tan (e+f x)}{2 f (a-a \sin (e+f x)) (a+a \sin (e+f x))^{2/3}}-\frac {3 a^2 \sin ^2(e+f x) \tan (e+f x)}{f (a-a \sin (e+f x)) (a+a \sin (e+f x))^{2/3}} \]
-361/126*sec(f*x+e)*(a+a*sin(f*x+e))^(1/3)/f+361/63*sec(f*x+e)*(1-sin(f*x+ e))*(a+a*sin(f*x+e))^(1/3)/f-1/42*sec(f*x+e)*(65*a^2-142*a^2*sin(f*x+e))/f /(a-a*sin(f*x+e))/(a+a*sin(f*x+e))^(2/3)+361/63*sec(f*x+e)*(1-sin(f*x+e))* (a+a*sin(f*x+e))^(2/3)*(1+3^(1/2))/f/(2^(1/3)*a^(1/3)-(a+a*sin(f*x+e))^(1/ 3)*(1+3^(1/2)))-361/63*2^(1/3)*((2^(1/3)*a^(1/3)-(a+a*sin(f*x+e))^(1/3)*(1 -3^(1/2)))^2/(2^(1/3)*a^(1/3)-(a+a*sin(f*x+e))^(1/3)*(1+3^(1/2)))^2)^(1/2) /(2^(1/3)*a^(1/3)-(a+a*sin(f*x+e))^(1/3)*(1-3^(1/2)))*(2^(1/3)*a^(1/3)-(a+ a*sin(f*x+e))^(1/3)*(1+3^(1/2)))*EllipticE((1-(2^(1/3)*a^(1/3)-(a+a*sin(f* x+e))^(1/3)*(1-3^(1/2)))^2/(2^(1/3)*a^(1/3)-(a+a*sin(f*x+e))^(1/3)*(1+3^(1 /2)))^2)^(1/2),1/4*6^(1/2)+1/4*2^(1/2))*sec(f*x+e)*(a+a*sin(f*x+e))^(2/3)* (2^(1/3)*a^(1/3)-(a+a*sin(f*x+e))^(1/3))*((2^(2/3)*a^(2/3)+2^(1/3)*a^(1/3) *(a+a*sin(f*x+e))^(1/3)+(a+a*sin(f*x+e))^(2/3))/(2^(1/3)*a^(1/3)-(a+a*sin( f*x+e))^(1/3)*(1+3^(1/2)))^2)^(1/2)*3^(1/4)/a^(2/3)/f/(-(a+a*sin(f*x+e))^( 1/3)*(2^(1/3)*a^(1/3)-(a+a*sin(f*x+e))^(1/3))/(2^(1/3)*a^(1/3)-(a+a*sin(f* x+e))^(1/3)*(1+3^(1/2)))^2)^(1/2)-361/378*((2^(1/3)*a^(1/3)-(a+a*sin(f*x+e ))^(1/3)*(1-3^(1/2)))^2/(2^(1/3)*a^(1/3)-(a+a*sin(f*x+e))^(1/3)*(1+3^(1/2) ))^2)^(1/2)/(2^(1/3)*a^(1/3)-(a+a*sin(f*x+e))^(1/3)*(1-3^(1/2)))*(2^(1/3)* a^(1/3)-(a+a*sin(f*x+e))^(1/3)*(1+3^(1/2)))*EllipticF((1-(2^(1/3)*a^(1/3)- (a+a*sin(f*x+e))^(1/3)*(1-3^(1/2)))^2/(2^(1/3)*a^(1/3)-(a+a*sin(f*x+e))^(1 /3)*(1+3^(1/2)))^2)^(1/2),1/4*6^(1/2)+1/4*2^(1/2))*sec(f*x+e)*(a+a*sin(...
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 3.33 (sec) , antiderivative size = 232, normalized size of antiderivative = 0.24 \[ \int \sqrt [3]{a+a \sin (e+f x)} \tan ^4(e+f x) \, dx=\frac {\sqrt [3]{a (1+\sin (e+f x))} \left (-722 \sqrt [3]{2} \cos \left (\frac {1}{4} (2 e+\pi +2 f x)\right )+361\ 2^{5/6} \, _2F_1\left (-\frac {1}{2},-\frac {1}{6};\frac {5}{6};\sin ^2\left (\frac {1}{4} (2 e+\pi +2 f x)\right )\right ) \sec \left (\frac {1}{4} (2 e+\pi +2 f x)\right ) \sqrt {1-\sin (e+f x)}+3 \sec ^3(e+f x) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^{2/3} (40+43 \cos (2 (e+f x))-19 \sin (e+f x)-43 \sin (3 (e+f x))) \sqrt [3]{\sin \left (\frac {1}{4} (2 e+\pi +2 f x)\right )}\right )}{189 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^{2/3} \sqrt [3]{\sin \left (\frac {1}{4} (2 e+\pi +2 f x)\right )}} \]
((a*(1 + Sin[e + f*x]))^(1/3)*(-722*2^(1/3)*Cos[(2*e + Pi + 2*f*x)/4] + 36 1*2^(5/6)*HypergeometricPFQ[{-1/2, -1/6}, {5/6}, Sin[(2*e + Pi + 2*f*x)/4] ^2]*Sec[(2*e + Pi + 2*f*x)/4]*Sqrt[1 - Sin[e + f*x]] + 3*Sec[e + f*x]^3*(C os[(e + f*x)/2] + Sin[(e + f*x)/2])^(2/3)*(40 + 43*Cos[2*(e + f*x)] - 19*S in[e + f*x] - 43*Sin[3*(e + f*x)])*Sin[(2*e + Pi + 2*f*x)/4]^(1/3)))/(189* f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^(2/3)*Sin[(2*e + Pi + 2*f*x)/4]^(1 /3))
Time = 0.83 (sec) , antiderivative size = 988, normalized size of antiderivative = 1.01, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.652, Rules used = {3042, 3198, 111, 27, 170, 27, 161, 61, 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.
| \(\displaystyle \int \tan ^4(e+f x) \sqrt [3]{a \sin (e+f x)+a} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \tan (e+f x)^4 \sqrt [3]{a \sin (e+f x)+a}dx\) |
| \(\Big \downarrow \) 3198 |
| \(\displaystyle \frac {\sec (e+f x) \sqrt {a-a \sin (e+f x)} \sqrt {a \sin (e+f x)+a} \int \frac {a^4 \sin ^4(e+f x)}{(a-a \sin (e+f x))^{5/2} (\sin (e+f x) a+a)^{13/6}}d(a \sin (e+f x))}{a f}\) |
| \(\Big \downarrow \) 111 |
| \(\displaystyle \frac {\sec (e+f x) \sqrt {a-a \sin (e+f x)} \sqrt {a \sin (e+f x)+a} \left (-3 \int -\frac {a^3 \sin ^2(e+f x) (\sin (e+f x) a+9 a)}{3 (a-a \sin (e+f x))^{5/2} (\sin (e+f x) a+a)^{13/6}}d(a \sin (e+f x))-\frac {3 a^3 \sin ^3(e+f x)}{(a-a \sin (e+f x))^{3/2} (a \sin (e+f x)+a)^{7/6}}\right )}{a f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sec (e+f x) \sqrt {a-a \sin (e+f x)} \sqrt {a \sin (e+f x)+a} \left (a \int \frac {a^2 \sin ^2(e+f x) (\sin (e+f x) a+9 a)}{(a-a \sin (e+f x))^{5/2} (\sin (e+f x) a+a)^{13/6}}d(a \sin (e+f x))-\frac {3 a^3 \sin ^3(e+f x)}{(a-a \sin (e+f x))^{3/2} (a \sin (e+f x)+a)^{7/6}}\right )}{a f}\) |
| \(\Big \downarrow \) 170 |
| \(\displaystyle \frac {\sec (e+f x) \sqrt {a-a \sin (e+f x)} \sqrt {a \sin (e+f x)+a} \left (a \left (\frac {3}{2} \int -\frac {a^2 \sin (e+f x) (6 a-17 a \sin (e+f x))}{3 (a-a \sin (e+f x))^{5/2} (\sin (e+f x) a+a)^{13/6}}d(a \sin (e+f x))+\frac {3 a^2 \sin ^2(e+f x)}{2 (a-a \sin (e+f x))^{3/2} (a \sin (e+f x)+a)^{7/6}}\right )-\frac {3 a^3 \sin ^3(e+f x)}{(a-a \sin (e+f x))^{3/2} (a \sin (e+f x)+a)^{7/6}}\right )}{a f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sec (e+f x) \sqrt {a-a \sin (e+f x)} \sqrt {a \sin (e+f x)+a} \left (a \left (\frac {3 a^2 \sin ^2(e+f x)}{2 (a-a \sin (e+f x))^{3/2} (a \sin (e+f x)+a)^{7/6}}-\frac {1}{2} a \int \frac {a \sin (e+f x) (6 a-17 a \sin (e+f x))}{(a-a \sin (e+f x))^{5/2} (\sin (e+f x) a+a)^{13/6}}d(a \sin (e+f x))\right )-\frac {3 a^3 \sin ^3(e+f x)}{(a-a \sin (e+f x))^{3/2} (a \sin (e+f x)+a)^{7/6}}\right )}{a f}\) |
| \(\Big \downarrow \) 161 |
| \(\displaystyle \frac {\sec (e+f x) \sqrt {a-a \sin (e+f x)} \sqrt {a \sin (e+f x)+a} \left (a \left (\frac {3 a^2 \sin ^2(e+f x)}{2 (a-a \sin (e+f x))^{3/2} (a \sin (e+f x)+a)^{7/6}}-\frac {1}{2} a \left (\frac {361}{63} \int \frac {1}{(a-a \sin (e+f x))^{3/2} (\sin (e+f x) a+a)^{7/6}}d(a \sin (e+f x))+\frac {65 a-142 a \sin (e+f x)}{21 (a-a \sin (e+f x))^{3/2} (a \sin (e+f x)+a)^{7/6}}\right )\right )-\frac {3 a^3 \sin ^3(e+f x)}{(a-a \sin (e+f x))^{3/2} (a \sin (e+f x)+a)^{7/6}}\right )}{a f}\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle \frac {\sec (e+f x) \sqrt {a-a \sin (e+f x)} \sqrt {a \sin (e+f x)+a} \left (a \left (\frac {3 a^2 \sin ^2(e+f x)}{2 (a-a \sin (e+f x))^{3/2} (a \sin (e+f x)+a)^{7/6}}-\frac {1}{2} a \left (\frac {361}{63} \left (\frac {2 \int \frac {1}{\sqrt {a-a \sin (e+f x)} (\sin (e+f x) a+a)^{7/6}}d(a \sin (e+f x))}{3 a}+\frac {1}{a \sqrt {a-a \sin (e+f x)} \sqrt [6]{a \sin (e+f x)+a}}\right )+\frac {65 a-142 a \sin (e+f x)}{21 (a-a \sin (e+f x))^{3/2} (a \sin (e+f x)+a)^{7/6}}\right )\right )-\frac {3 a^3 \sin ^3(e+f x)}{(a-a \sin (e+f x))^{3/2} (a \sin (e+f x)+a)^{7/6}}\right )}{a f}\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle \frac {\sec (e+f x) \sqrt {a-a \sin (e+f x)} \sqrt {a \sin (e+f x)+a} \left (a \left (\frac {3 a^2 \sin ^2(e+f x)}{2 (a-a \sin (e+f x))^{3/2} (a \sin (e+f x)+a)^{7/6}}-\frac {1}{2} a \left (\frac {361}{63} \left (\frac {2 \left (-\frac {\int \frac {1}{\sqrt {a-a \sin (e+f x)} \sqrt [6]{\sin (e+f x) a+a}}d(a \sin (e+f x))}{a}-\frac {3 \sqrt {a-a \sin (e+f x)}}{a \sqrt [6]{a \sin (e+f x)+a}}\right )}{3 a}+\frac {1}{a \sqrt {a-a \sin (e+f x)} \sqrt [6]{a \sin (e+f x)+a}}\right )+\frac {65 a-142 a \sin (e+f x)}{21 (a-a \sin (e+f x))^{3/2} (a \sin (e+f x)+a)^{7/6}}\right )\right )-\frac {3 a^3 \sin ^3(e+f x)}{(a-a \sin (e+f x))^{3/2} (a \sin (e+f x)+a)^{7/6}}\right )}{a f}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {\sec (e+f x) \sqrt {a-a \sin (e+f x)} \sqrt {a \sin (e+f x)+a} \left (a \left (\frac {3 a^2 \sin ^2(e+f x)}{2 (a-a \sin (e+f x))^{3/2} (a \sin (e+f x)+a)^{7/6}}-\frac {1}{2} a \left (\frac {361}{63} \left (\frac {2 \left (-\frac {6 \int \frac {a^4 \sin ^4(e+f x)}{\sqrt {2 a-a^6 \sin ^6(e+f x)}}d\sqrt [6]{\sin (e+f x) a+a}}{a}-\frac {3 \sqrt {a-a \sin (e+f x)}}{a \sqrt [6]{a \sin (e+f x)+a}}\right )}{3 a}+\frac {1}{a \sqrt {a-a \sin (e+f x)} \sqrt [6]{a \sin (e+f x)+a}}\right )+\frac {65 a-142 a \sin (e+f x)}{21 (a-a \sin (e+f x))^{3/2} (a \sin (e+f x)+a)^{7/6}}\right )\right )-\frac {3 a^3 \sin ^3(e+f x)}{(a-a \sin (e+f x))^{3/2} (a \sin (e+f x)+a)^{7/6}}\right )}{a f}\) |
| \(\Big \downarrow \) 837 |
| \(\displaystyle \frac {\sec (e+f x) \sqrt {a-a \sin (e+f x)} \sqrt {a \sin (e+f x)+a} \left (a \left (\frac {3 a^2 \sin ^2(e+f x)}{2 (a-a \sin (e+f x))^{3/2} (a \sin (e+f x)+a)^{7/6}}-\frac {1}{2} a \left (\frac {361}{63} \left (\frac {2 \left (-\frac {6 \left (-\frac {\left (1-\sqrt {3}\right ) a^{2/3} \int \frac {1}{\sqrt {2 a-a^6 \sin ^6(e+f x)}}d\sqrt [6]{\sin (e+f x) a+a}}{\sqrt [3]{2}}-\frac {1}{2} \int -\frac {2 a^4 \sin ^4(e+f x)+2^{2/3} \left (1-\sqrt {3}\right ) a^{2/3}}{\sqrt {2 a-a^6 \sin ^6(e+f x)}}d\sqrt [6]{\sin (e+f x) a+a}\right )}{a}-\frac {3 \sqrt {a-a \sin (e+f x)}}{a \sqrt [6]{a \sin (e+f x)+a}}\right )}{3 a}+\frac {1}{a \sqrt {a-a \sin (e+f x)} \sqrt [6]{a \sin (e+f x)+a}}\right )+\frac {65 a-142 a \sin (e+f x)}{21 (a-a \sin (e+f x))^{3/2} (a \sin (e+f x)+a)^{7/6}}\right )\right )-\frac {3 a^3 \sin ^3(e+f x)}{(a-a \sin (e+f x))^{3/2} (a \sin (e+f x)+a)^{7/6}}\right )}{a f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sec (e+f x) \sqrt {a-a \sin (e+f x)} \sqrt {a \sin (e+f x)+a} \left (a \left (\frac {3 a^2 \sin ^2(e+f x)}{2 (a-a \sin (e+f x))^{3/2} (a \sin (e+f x)+a)^{7/6}}-\frac {1}{2} a \left (\frac {361}{63} \left (\frac {2 \left (-\frac {6 \left (\frac {1}{2} \int \frac {2^{2/3} \left (\sqrt [3]{2} a^4 \sin ^4(e+f x)+\left (1-\sqrt {3}\right ) a^{2/3}\right )}{\sqrt {2 a-a^6 \sin ^6(e+f x)}}d\sqrt [6]{\sin (e+f x) a+a}-\frac {\left (1-\sqrt {3}\right ) a^{2/3} \int \frac {1}{\sqrt {2 a-a^6 \sin ^6(e+f x)}}d\sqrt [6]{\sin (e+f x) a+a}}{\sqrt [3]{2}}\right )}{a}-\frac {3 \sqrt {a-a \sin (e+f x)}}{a \sqrt [6]{a \sin (e+f x)+a}}\right )}{3 a}+\frac {1}{a \sqrt {a-a \sin (e+f x)} \sqrt [6]{a \sin (e+f x)+a}}\right )+\frac {65 a-142 a \sin (e+f x)}{21 (a-a \sin (e+f x))^{3/2} (a \sin (e+f x)+a)^{7/6}}\right )\right )-\frac {3 a^3 \sin ^3(e+f x)}{(a-a \sin (e+f x))^{3/2} (a \sin (e+f x)+a)^{7/6}}\right )}{a f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sec (e+f x) \sqrt {a-a \sin (e+f x)} \sqrt {a \sin (e+f x)+a} \left (a \left (\frac {3 a^2 \sin ^2(e+f x)}{2 (a-a \sin (e+f x))^{3/2} (a \sin (e+f x)+a)^{7/6}}-\frac {1}{2} a \left (\frac {361}{63} \left (\frac {2 \left (-\frac {6 \left (\frac {\int \frac {\sqrt [3]{2} a^4 \sin ^4(e+f x)+\left (1-\sqrt {3}\right ) a^{2/3}}{\sqrt {2 a-a^6 \sin ^6(e+f x)}}d\sqrt [6]{\sin (e+f x) a+a}}{\sqrt [3]{2}}-\frac {\left (1-\sqrt {3}\right ) a^{2/3} \int \frac {1}{\sqrt {2 a-a^6 \sin ^6(e+f x)}}d\sqrt [6]{\sin (e+f x) a+a}}{\sqrt [3]{2}}\right )}{a}-\frac {3 \sqrt {a-a \sin (e+f x)}}{a \sqrt [6]{a \sin (e+f x)+a}}\right )}{3 a}+\frac {1}{a \sqrt {a-a \sin (e+f x)} \sqrt [6]{a \sin (e+f x)+a}}\right )+\frac {65 a-142 a \sin (e+f x)}{21 (a-a \sin (e+f x))^{3/2} (a \sin (e+f x)+a)^{7/6}}\right )\right )-\frac {3 a^3 \sin ^3(e+f x)}{(a-a \sin (e+f x))^{3/2} (a \sin (e+f x)+a)^{7/6}}\right )}{a f}\) |
| \(\Big \downarrow \) 766 |
| \(\displaystyle \frac {\sec (e+f x) \sqrt {a-a \sin (e+f x)} \sqrt {a \sin (e+f x)+a} \left (a \left (\frac {3 a^2 \sin ^2(e+f x)}{2 (a-a \sin (e+f x))^{3/2} (a \sin (e+f x)+a)^{7/6}}-\frac {1}{2} a \left (\frac {361}{63} \left (\frac {2 \left (-\frac {6 \left (\frac {\int \frac {\sqrt [3]{2} a^4 \sin ^4(e+f x)+\left (1-\sqrt {3}\right ) a^{2/3}}{\sqrt {2 a-a^6 \sin ^6(e+f x)}}d\sqrt [6]{\sin (e+f x) a+a}}{\sqrt [3]{2}}-\frac {\left (1-\sqrt {3}\right ) a^{4/3} \sin (e+f x) \left (\sqrt [3]{2} \sqrt [3]{a}-a^2 \sin ^2(e+f x)\right ) \sqrt {\frac {\sqrt [3]{2} a^{7/3} \sin ^2(e+f x)+2^{2/3} a^{2/3}+a^4 \sin ^4(e+f x)}{\left (\sqrt [3]{2} \sqrt [3]{a}-\left (1+\sqrt {3}\right ) a^2 \sin ^2(e+f x)\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {\sqrt [3]{2} \sqrt [3]{a}-\left (1-\sqrt {3}\right ) a^2 \sin ^2(e+f x)}{\sqrt [3]{2} \sqrt [3]{a}-\left (1+\sqrt {3}\right ) a^2 \sin ^2(e+f x)}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{2\ 2^{2/3} \sqrt [4]{3} \sqrt {-\frac {a^2 \sin ^2(e+f x) \left (\sqrt [3]{2} \sqrt [3]{a}-a^2 \sin ^2(e+f x)\right )}{\left (\sqrt [3]{2} \sqrt [3]{a}-\left (1+\sqrt {3}\right ) a^2 \sin ^2(e+f x)\right )^2}} \sqrt {2 a-a^6 \sin ^6(e+f x)}}\right )}{a}-\frac {3 \sqrt {a-a \sin (e+f x)}}{a \sqrt [6]{a \sin (e+f x)+a}}\right )}{3 a}+\frac {1}{a \sqrt {a-a \sin (e+f x)} \sqrt [6]{a \sin (e+f x)+a}}\right )+\frac {65 a-142 a \sin (e+f x)}{21 (a-a \sin (e+f x))^{3/2} (a \sin (e+f x)+a)^{7/6}}\right )\right )-\frac {3 a^3 \sin ^3(e+f x)}{(a-a \sin (e+f x))^{3/2} (a \sin (e+f x)+a)^{7/6}}\right )}{a f}\) |
| \(\Big \downarrow \) 2420 |
| \(\displaystyle \frac {\sec (e+f x) \sqrt {a-a \sin (e+f x)} \sqrt {\sin (e+f x) a+a} \left (a \left (\frac {3 a^2 \sin ^2(e+f x)}{2 (a-a \sin (e+f x))^{3/2} (\sin (e+f x) a+a)^{7/6}}-\frac {1}{2} a \left (\frac {65 a-142 a \sin (e+f x)}{21 (a-a \sin (e+f x))^{3/2} (\sin (e+f x) a+a)^{7/6}}+\frac {361}{63} \left (\frac {2 \left (-\frac {6 \left (\frac {\frac {\left (1+\sqrt {3}\right ) \sqrt [6]{\sin (e+f x) a+a} \sqrt {2 a-a^6 \sin ^6(e+f x)}}{2^{2/3} \left (\sqrt [3]{2} \sqrt [3]{a}-\left (1+\sqrt {3}\right ) a^2 \sin ^2(e+f x)\right )}-\frac {\sqrt [4]{3} a^{4/3} E\left (\arccos \left (\frac {\sqrt [3]{2} \sqrt [3]{a}-\left (1-\sqrt {3}\right ) a^2 \sin ^2(e+f x)}{\sqrt [3]{2} \sqrt [3]{a}-\left (1+\sqrt {3}\right ) a^2 \sin ^2(e+f x)}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right ) \sin (e+f x) \left (\sqrt [3]{2} \sqrt [3]{a}-a^2 \sin ^2(e+f x)\right ) \sqrt {\frac {a^4 \sin ^4(e+f x)+\sqrt [3]{2} a^{7/3} \sin ^2(e+f x)+2^{2/3} a^{2/3}}{\left (\sqrt [3]{2} \sqrt [3]{a}-\left (1+\sqrt {3}\right ) a^2 \sin ^2(e+f x)\right )^2}}}{\sqrt [3]{2} \sqrt {-\frac {a^2 \sin ^2(e+f x) \left (\sqrt [3]{2} \sqrt [3]{a}-a^2 \sin ^2(e+f x)\right )}{\left (\sqrt [3]{2} \sqrt [3]{a}-\left (1+\sqrt {3}\right ) a^2 \sin ^2(e+f x)\right )^2}} \sqrt {2 a-a^6 \sin ^6(e+f x)}}}{\sqrt [3]{2}}-\frac {\left (1-\sqrt {3}\right ) a^{4/3} \operatorname {EllipticF}\left (\arccos \left (\frac {\sqrt [3]{2} \sqrt [3]{a}-\left (1-\sqrt {3}\right ) a^2 \sin ^2(e+f x)}{\sqrt [3]{2} \sqrt [3]{a}-\left (1+\sqrt {3}\right ) a^2 \sin ^2(e+f x)}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right ) \sin (e+f x) \left (\sqrt [3]{2} \sqrt [3]{a}-a^2 \sin ^2(e+f x)\right ) \sqrt {\frac {a^4 \sin ^4(e+f x)+\sqrt [3]{2} a^{7/3} \sin ^2(e+f x)+2^{2/3} a^{2/3}}{\left (\sqrt [3]{2} \sqrt [3]{a}-\left (1+\sqrt {3}\right ) a^2 \sin ^2(e+f x)\right )^2}}}{2\ 2^{2/3} \sqrt [4]{3} \sqrt {-\frac {a^2 \sin ^2(e+f x) \left (\sqrt [3]{2} \sqrt [3]{a}-a^2 \sin ^2(e+f x)\right )}{\left (\sqrt [3]{2} \sqrt [3]{a}-\left (1+\sqrt {3}\right ) a^2 \sin ^2(e+f x)\right )^2}} \sqrt {2 a-a^6 \sin ^6(e+f x)}}\right )}{a}-\frac {3 \sqrt {a-a \sin (e+f x)}}{a \sqrt [6]{\sin (e+f x) a+a}}\right )}{3 a}+\frac {1}{a \sqrt {a-a \sin (e+f x)} \sqrt [6]{\sin (e+f x) a+a}}\right )\right )\right )-\frac {3 a^3 \sin ^3(e+f x)}{(a-a \sin (e+f x))^{3/2} (\sin (e+f x) a+a)^{7/6}}\right )}{a f}\) |
(Sec[e + f*x]*Sqrt[a - a*Sin[e + f*x]]*Sqrt[a + a*Sin[e + f*x]]*((-3*a^3*S in[e + f*x]^3)/((a - a*Sin[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^(7/6)) + a *((3*a^2*Sin[e + f*x]^2)/(2*(a - a*Sin[e + f*x])^(3/2)*(a + a*Sin[e + f*x] )^(7/6)) - (a*((65*a - 142*a*Sin[e + f*x])/(21*(a - a*Sin[e + f*x])^(3/2)* (a + a*Sin[e + f*x])^(7/6)) + (361*(1/(a*Sqrt[a - a*Sin[e + f*x]]*(a + a*S in[e + f*x])^(1/6)) + (2*((-3*Sqrt[a - a*Sin[e + f*x]])/(a*(a + a*Sin[e + f*x])^(1/6)) - (6*(-1/2*((1 - Sqrt[3])*a^(4/3)*EllipticF[ArcCos[(2^(1/3)*a ^(1/3) - (1 - Sqrt[3])*a^2*Sin[e + f*x]^2)/(2^(1/3)*a^(1/3) - (1 + Sqrt[3] )*a^2*Sin[e + f*x]^2)], (2 + Sqrt[3])/4]*Sin[e + f*x]*(2^(1/3)*a^(1/3) - a ^2*Sin[e + f*x]^2)*Sqrt[(2^(2/3)*a^(2/3) + 2^(1/3)*a^(7/3)*Sin[e + f*x]^2 + a^4*Sin[e + f*x]^4)/(2^(1/3)*a^(1/3) - (1 + Sqrt[3])*a^2*Sin[e + f*x]^2) ^2])/(2^(2/3)*3^(1/4)*Sqrt[-((a^2*Sin[e + f*x]^2*(2^(1/3)*a^(1/3) - a^2*Si n[e + f*x]^2))/(2^(1/3)*a^(1/3) - (1 + Sqrt[3])*a^2*Sin[e + f*x]^2)^2)]*Sq rt[2*a - a^6*Sin[e + f*x]^6]) + (-((3^(1/4)*a^(4/3)*EllipticE[ArcCos[(2^(1 /3)*a^(1/3) - (1 - Sqrt[3])*a^2*Sin[e + f*x]^2)/(2^(1/3)*a^(1/3) - (1 + Sq rt[3])*a^2*Sin[e + f*x]^2)], (2 + Sqrt[3])/4]*Sin[e + f*x]*(2^(1/3)*a^(1/3 ) - a^2*Sin[e + f*x]^2)*Sqrt[(2^(2/3)*a^(2/3) + 2^(1/3)*a^(7/3)*Sin[e + f* x]^2 + a^4*Sin[e + f*x]^4)/(2^(1/3)*a^(1/3) - (1 + Sqrt[3])*a^2*Sin[e + f* x]^2)^2])/(2^(1/3)*Sqrt[-((a^2*Sin[e + f*x]^2*(2^(1/3)*a^(1/3) - a^2*Sin[e + f*x]^2))/(2^(1/3)*a^(1/3) - (1 + Sqrt[3])*a^2*Sin[e + f*x]^2)^2)]*Sq...
3.2.15.3.1 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[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[b*(a + b*x)^(m - 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 )/(d*f*(m + n + p + 1))), x] + Simp[1/(d*f*(m + n + p + 1)) Int[(a + b*x) ^(m - 2)*(c + d*x)^n*(e + f*x)^p*Simp[a^2*d*f*(m + n + p + 1) - b*(b*c*e*(m - 1) + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*(2*m + n + p) - b*(d*e*(m + n) + c*f*(m + p)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] & & GtQ[m, 1] && NeQ[m + n + p + 1, 0] && IntegerQ[m]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_) + (f_.)*(x_)) *((g_.) + (h_.)*(x_)), x_] :> Simp[((b^2*c*d*e*g*(n + 1) + a^2*c*d*f*h*(n + 1) + a*b*(d^2*e*g*(m + 1) + c^2*f*h*(m + 1) - c*d*(f*g + e*h)*(m + n + 2)) + (a^2*d^2*f*h*(n + 1) - a*b*d^2*(f*g + e*h)*(n + 1) + b^2*(c^2*f*h*(m + 1 ) - c*d*(f*g + e*h)*(m + 1) + d^2*e*g*(m + n + 2)))*x)/(b*d*(b*c - a*d)^2*( m + 1)*(n + 1)))*(a + b*x)^(m + 1)*(c + d*x)^(n + 1), x] - Simp[(a^2*d^2*f* h*(2 + 3*n + n^2) + a*b*d*(n + 1)*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b^2*(c^2*f*h*(2 + 3*m + m^2) - c*d*(f*g + e*h)*(m + 1)*(m + n + 3) + d^2*e*g*(6 + m^2 + 5*n + n^2 + m*(2*n + 5))))/(b*d*(b*c - a*d)^2*(m + 1)*( n + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n + 1), x], x] /; FreeQ[{a, b, c , d, e, f, g, h}, x] && LtQ[m, -1] && LtQ[n, -1]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[h*(a + b*x)^m*(c + d*x)^(n + 1)*(( e + f*x)^(p + 1)/(d*f*(m + n + p + 2))), x] + Simp[1/(d*f*(m + n + p + 2)) Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2 ) - h*(b*c*e*m + a*(d*e*(n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) + h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x], x], x] /; Fre eQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] && IntegerQ[m]
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[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*tan[(e_.) + (f_.)*(x_)]^(p_ ), x_Symbol] :> Simp[Sqrt[a + b*Sin[e + f*x]]*(Sqrt[a - b*Sin[e + f*x]]/(b* f*Cos[e + f*x])) Subst[Int[x^p*((a + x)^(m - (p + 1)/2)/(a - x)^((p + 1)/ 2)), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && EqQ[a^2 - b ^2, 0] && !IntegerQ[m] && IntegerQ[p/2]
\[\int \left (a +a \sin \left (f x +e \right )\right )^{\frac {1}{3}} \left (\tan ^{4}\left (f x +e \right )\right )d x\]
\[ \int \sqrt [3]{a+a \sin (e+f x)} \tan ^4(e+f x) \, dx=\int { {\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {1}{3}} \tan \left (f x + e\right )^{4} \,d x } \]
\[ \int \sqrt [3]{a+a \sin (e+f x)} \tan ^4(e+f x) \, dx=\int \sqrt [3]{a \left (\sin {\left (e + f x \right )} + 1\right )} \tan ^{4}{\left (e + f x \right )}\, dx \]
\[ \int \sqrt [3]{a+a \sin (e+f x)} \tan ^4(e+f x) \, dx=\int { {\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {1}{3}} \tan \left (f x + e\right )^{4} \,d x } \]
\[ \int \sqrt [3]{a+a \sin (e+f x)} \tan ^4(e+f x) \, dx=\int { {\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {1}{3}} \tan \left (f x + e\right )^{4} \,d x } \]
Timed out. \[ \int \sqrt [3]{a+a \sin (e+f x)} \tan ^4(e+f x) \, dx=\int {\mathrm {tan}\left (e+f\,x\right )}^4\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{1/3} \,d x \]