Integrand size = 25, antiderivative size = 787 \[ \int (a+a \sec (c+d x))^{4/3} (A+B \sec (c+d x)) \, dx=\frac {3 a B \sqrt [3]{a+a \sec (c+d x)} \tan (c+d x)}{4 d}+\frac {3 \sqrt {2} a A \operatorname {AppellF1}\left (\frac {11}{6},\frac {1}{2},1,\frac {17}{6},\frac {1}{2} (1+\sec (c+d x)),1+\sec (c+d x)\right ) (1+\sec (c+d x)) \sqrt [3]{a+a \sec (c+d x)} \tan (c+d x)}{11 d \sqrt {1-\sec (c+d x)}}-\frac {15 \left (1+\sqrt {3}\right ) a B \sqrt [3]{a+a \sec (c+d x)} \tan (c+d x)}{4 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 {15 \sqrt [4]{3} a B 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)}{2\ 2^{2/3} 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 {5\ 3^{3/4} \left (1-\sqrt {3}\right ) a B \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)}{4\ 2^{2/3} 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}}} \] Output:
3/4*a*B*(a+a*sec(d*x+c))^(1/3)*tan(d*x+c)/d+3/11*2^(1/2)*a*A*AppellF1(11/6 ,1,1/2,17/6,1+sec(d*x+c),1/2+1/2*sec(d*x+c))*(1+sec(d*x+c))*(a+a*sec(d*x+c ))^(1/3)*tan(d*x+c)/d/(1-sec(d*x+c))^(1/2)-15/4*(1+3^(1/2))*a*B*(a+a*sec(d *x+c))^(1/3)*tan(d*x+c)/d/(1+sec(d*x+c))^(2/3)/(2^(1/3)-(1+3^(1/2))*(1+sec (d*x+c))^(1/3))+15/4*3^(1/4)*a*B*EllipticE((1-(2^(1/3)-(1-3^(1/2))*(1+sec( d*x+c))^(1/3))^2/(2^(1/3)-(1+3^(1/2))*(1+sec(d*x+c))^(1/3))^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+3 ^(1/2))*(1+sec(d*x+c))^(1/3))^2)^(1/2)*tan(d*x+c)*2^(1/3)/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+3^(1/2))*(1+sec(d*x+c))^(1/3))^2)^(1/2)+5/8*3^(3/4)*(1-3^(1/ 2))*a*B*InverseJacobiAM(arccos((2^(1/3)-(1-3^(1/2))*(1+sec(d*x+c))^(1/3))/ (2^(1/3)-(1+3^(1/2))*(1+sec(d*x+c))^(1/3))),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+3^(1/2))*(1+sec(d*x+c))^(1 /3))^2)^(1/2)*tan(d*x+c)*2^(1/3)/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+3^(1/2))*(1 +sec(d*x+c))^(1/3))^2)^(1/2)
Leaf count is larger than twice the leaf count of optimal. \(4110\) vs. \(2(787)=1574\).
Time = 18.16 (sec) , antiderivative size = 4110, normalized size of antiderivative = 5.22 \[ \int (a+a \sec (c+d x))^{4/3} (A+B \sec (c+d x)) \, dx=\text {Result too large to show} \] Input:
Integrate[(a + a*Sec[c + d*x])^(4/3)*(A + B*Sec[c + d*x]),x]
Output:
(Cos[c + d*x]*((1 + Cos[c + d*x])*Sec[c + d*x])^(1/3)*(a*(1 + Sec[c + d*x] ))^(4/3)*(A + B*Sec[c + d*x])*((3*(4*A + 5*B)*Sin[c + d*x])/4 + (3*B*Tan[c + d*x])/4))/(d*(B + A*Cos[c + d*x])*(1 + Sec[c + d*x])^(4/3)) + (Cos[c + d*x]*(a*(1 + Sec[c + d*x]))^(4/3)*(A + B*Sec[c + d*x])*(2*A*(1 + Sec[c + d *x])^(1/3) + (5*B*(1 + Sec[c + d*x])^(1/3))/4 + Cos[c + d*x]*(-3*A*(1 + Se c[c + d*x])^(1/3) - (15*B*(1 + Sec[c + d*x])^(1/3))/4))*Tan[(c + d*x)/2]*( -(((4*A + 5*B)*AppellF1[3/2, 1/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d* x)/2]^2]*Tan[(c + d*x)/2]^2)/(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(2/3)) - (9 *(3*AppellF1[1/2, 1/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(- 4*A + 5*B + 5*(4*A + 7*B)*Cos[c + d*x]) - 4*(4*A + 5*B)*(3*AppellF1[3/2, 1 /3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - AppellF1[3/2, 4/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Cos[c + d*x]*Tan[(c + d* x)/2]^2))/(2*(-1 + Tan[(c + d*x)/2]^2)*(-9*AppellF1[1/2, 1/3, 1, 3/2, Tan[ (c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*(3*AppellF1[3/2, 1/3, 2, 5/2, Tan [(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - AppellF1[3/2, 4/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2))))/(6*2^(2/3)*d*(B + A*Cos[c + d*x])*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(2/3)*(1 + Sec[c + d*x ])^(4/3)*((Sec[(c + d*x)/2]^2*(-(((4*A + 5*B)*AppellF1[3/2, 1/3, 1, 5/2, T an[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Tan[(c + d*x)/2]^2)/(Cos[c + d*x]* Sec[(c + d*x)/2]^2)^(2/3)) - (9*(3*AppellF1[1/2, 1/3, 1, 3/2, Tan[(c + ...
Time = 1.18 (sec) , antiderivative size = 797, normalized size of antiderivative = 1.01, number of steps used = 20, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.760, Rules used = {3042, 4412, 3042, 4266, 3042, 4265, 149, 25, 1012, 4315, 3042, 4314, 60, 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.
| \(\displaystyle \int (a \sec (c+d x)+a)^{4/3} (A+B \sec (c+d x)) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \left (a \csc \left (c+d x+\frac {\pi }{2}\right )+a\right )^{4/3} \left (A+B \csc \left (c+d x+\frac {\pi }{2}\right )\right )dx\) |
| \(\Big \downarrow \) 4412 |
| \(\displaystyle A \int (\sec (c+d x) a+a)^{4/3}dx+B \int \sec (c+d x) (\sec (c+d x) a+a)^{4/3}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle A \int \left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^{4/3}dx+B \int \csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^{4/3}dx\) |
| \(\Big \downarrow \) 4266 |
| \(\displaystyle \frac {a A \sqrt [3]{a \sec (c+d x)+a} \int (\sec (c+d x)+1)^{4/3}dx}{\sqrt [3]{\sec (c+d x)+1}}+B \int \csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^{4/3}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a A \sqrt [3]{a \sec (c+d x)+a} \int \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )^{4/3}dx}{\sqrt [3]{\sec (c+d x)+1}}+B \int \csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^{4/3}dx\) |
| \(\Big \downarrow \) 4265 |
| \(\displaystyle B \int \csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^{4/3}dx-\frac {a A \tan (c+d x) \sqrt [3]{a \sec (c+d x)+a} \int \frac {\cos (c+d x) (\sec (c+d x)+1)^{5/6}}{\sqrt {1-\sec (c+d x)}}d\sec (c+d x)}{d \sqrt {1-\sec (c+d x)} (\sec (c+d x)+1)^{5/6}}\) |
| \(\Big \downarrow \) 149 |
| \(\displaystyle B \int \csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^{4/3}dx-\frac {6 a A \tan (c+d x) \sqrt [3]{a \sec (c+d x)+a} \int \frac {\cos (c+d x) (\sec (c+d x)+1)^{5/3}}{\sqrt {1-\sec (c+d x)}}d\sqrt [6]{\sec (c+d x)+1}}{d \sqrt {1-\sec (c+d x)} (\sec (c+d x)+1)^{5/6}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {6 a A \tan (c+d x) \sqrt [3]{a \sec (c+d x)+a} \int -\frac {\cos (c+d x) (\sec (c+d x)+1)^{5/3}}{\sqrt {1-\sec (c+d x)}}d\sqrt [6]{\sec (c+d x)+1}}{d \sqrt {1-\sec (c+d x)} (\sec (c+d x)+1)^{5/6}}+B \int \csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^{4/3}dx\) |
| \(\Big \downarrow \) 1012 |
| \(\displaystyle B \int \csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^{4/3}dx+\frac {3 \sqrt {2} a A \tan (c+d x) (\sec (c+d x)+1) \sqrt [3]{a \sec (c+d x)+a} \operatorname {AppellF1}\left (\frac {11}{6},1,\frac {1}{2},\frac {17}{6},\sec (c+d x)+1,\frac {1}{2} (\sec (c+d x)+1)\right )}{11 d \sqrt {1-\sec (c+d x)}}\) |
| \(\Big \downarrow \) 4315 |
| \(\displaystyle \frac {a B \sqrt [3]{a \sec (c+d x)+a} \int \sec (c+d x) (\sec (c+d x)+1)^{4/3}dx}{\sqrt [3]{\sec (c+d x)+1}}+\frac {3 \sqrt {2} a A \tan (c+d x) (\sec (c+d x)+1) \sqrt [3]{a \sec (c+d x)+a} \operatorname {AppellF1}\left (\frac {11}{6},1,\frac {1}{2},\frac {17}{6},\sec (c+d x)+1,\frac {1}{2} (\sec (c+d x)+1)\right )}{11 d \sqrt {1-\sec (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a B \sqrt [3]{a \sec (c+d x)+a} \int \csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )^{4/3}dx}{\sqrt [3]{\sec (c+d x)+1}}+\frac {3 \sqrt {2} a A \tan (c+d x) (\sec (c+d x)+1) \sqrt [3]{a \sec (c+d x)+a} \operatorname {AppellF1}\left (\frac {11}{6},1,\frac {1}{2},\frac {17}{6},\sec (c+d x)+1,\frac {1}{2} (\sec (c+d x)+1)\right )}{11 d \sqrt {1-\sec (c+d x)}}\) |
| \(\Big \downarrow \) 4314 |
| \(\displaystyle \frac {3 \sqrt {2} a A \tan (c+d x) (\sec (c+d x)+1) \sqrt [3]{a \sec (c+d x)+a} \operatorname {AppellF1}\left (\frac {11}{6},1,\frac {1}{2},\frac {17}{6},\sec (c+d x)+1,\frac {1}{2} (\sec (c+d x)+1)\right )}{11 d \sqrt {1-\sec (c+d x)}}-\frac {a B \tan (c+d x) \sqrt [3]{a \sec (c+d x)+a} \int \frac {(\sec (c+d x)+1)^{5/6}}{\sqrt {1-\sec (c+d x)}}d\sec (c+d x)}{d \sqrt {1-\sec (c+d x)} (\sec (c+d x)+1)^{5/6}}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {3 \sqrt {2} a A \tan (c+d x) (\sec (c+d x)+1) \sqrt [3]{a \sec (c+d x)+a} \operatorname {AppellF1}\left (\frac {11}{6},1,\frac {1}{2},\frac {17}{6},\sec (c+d x)+1,\frac {1}{2} (\sec (c+d x)+1)\right )}{11 d \sqrt {1-\sec (c+d x)}}-\frac {a B \tan (c+d x) \sqrt [3]{a \sec (c+d x)+a} \left (\frac {5}{4} \int \frac {1}{\sqrt {1-\sec (c+d x)} \sqrt [6]{\sec (c+d x)+1}}d\sec (c+d x)-\frac {3}{4} \sqrt {1-\sec (c+d x)} (\sec (c+d x)+1)^{5/6}\right )}{d \sqrt {1-\sec (c+d x)} (\sec (c+d x)+1)^{5/6}}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {3 \sqrt {2} a A \tan (c+d x) (\sec (c+d x)+1) \sqrt [3]{a \sec (c+d x)+a} \operatorname {AppellF1}\left (\frac {11}{6},1,\frac {1}{2},\frac {17}{6},\sec (c+d x)+1,\frac {1}{2} (\sec (c+d x)+1)\right )}{11 d \sqrt {1-\sec (c+d x)}}-\frac {a B \tan (c+d x) \sqrt [3]{a \sec (c+d x)+a} \left (\frac {15}{2} \int \frac {(\sec (c+d x)+1)^{2/3}}{\sqrt {1-\sec (c+d x)}}d\sqrt [6]{\sec (c+d x)+1}-\frac {3}{4} \sqrt {1-\sec (c+d x)} (\sec (c+d x)+1)^{5/6}\right )}{d \sqrt {1-\sec (c+d x)} (\sec (c+d x)+1)^{5/6}}\) |
| \(\Big \downarrow \) 837 |
| \(\displaystyle \frac {3 \sqrt {2} a A \tan (c+d x) (\sec (c+d x)+1) \sqrt [3]{a \sec (c+d x)+a} \operatorname {AppellF1}\left (\frac {11}{6},1,\frac {1}{2},\frac {17}{6},\sec (c+d x)+1,\frac {1}{2} (\sec (c+d x)+1)\right )}{11 d \sqrt {1-\sec (c+d x)}}-\frac {a B \tan (c+d x) \sqrt [3]{a \sec (c+d x)+a} \left (\frac {15}{2} \left (-\frac {\left (1-\sqrt {3}\right ) \int \frac {1}{\sqrt {1-\sec (c+d x)}}d\sqrt [6]{\sec (c+d x)+1}}{\sqrt [3]{2}}-\frac {1}{2} \int -\frac {2 (\sec (c+d x)+1)^{2/3}+2^{2/3} \left (1-\sqrt {3}\right )}{\sqrt {1-\sec (c+d x)}}d\sqrt [6]{\sec (c+d x)+1}\right )-\frac {3}{4} \sqrt {1-\sec (c+d x)} (\sec (c+d x)+1)^{5/6}\right )}{d \sqrt {1-\sec (c+d x)} (\sec (c+d x)+1)^{5/6}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {3 \sqrt {2} a A \tan (c+d x) (\sec (c+d x)+1) \sqrt [3]{a \sec (c+d x)+a} \operatorname {AppellF1}\left (\frac {11}{6},1,\frac {1}{2},\frac {17}{6},\sec (c+d x)+1,\frac {1}{2} (\sec (c+d x)+1)\right )}{11 d \sqrt {1-\sec (c+d x)}}-\frac {a B \tan (c+d x) \sqrt [3]{a \sec (c+d x)+a} \left (\frac {15}{2} \left (\frac {1}{2} \int \frac {2^{2/3} \left (\sqrt [3]{2} (\sec (c+d x)+1)^{2/3}-\sqrt {3}+1\right )}{\sqrt {1-\sec (c+d x)}}d\sqrt [6]{\sec (c+d x)+1}-\frac {\left (1-\sqrt {3}\right ) \int \frac {1}{\sqrt {1-\sec (c+d x)}}d\sqrt [6]{\sec (c+d x)+1}}{\sqrt [3]{2}}\right )-\frac {3}{4} \sqrt {1-\sec (c+d x)} (\sec (c+d x)+1)^{5/6}\right )}{d \sqrt {1-\sec (c+d x)} (\sec (c+d x)+1)^{5/6}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3 \sqrt {2} a A \tan (c+d x) (\sec (c+d x)+1) \sqrt [3]{a \sec (c+d x)+a} \operatorname {AppellF1}\left (\frac {11}{6},1,\frac {1}{2},\frac {17}{6},\sec (c+d x)+1,\frac {1}{2} (\sec (c+d x)+1)\right )}{11 d \sqrt {1-\sec (c+d x)}}-\frac {a B \tan (c+d x) \sqrt [3]{a \sec (c+d x)+a} \left (\frac {15}{2} \left (\frac {\int \frac {\sqrt [3]{2} (\sec (c+d x)+1)^{2/3}-\sqrt {3}+1}{\sqrt {1-\sec (c+d x)}}d\sqrt [6]{\sec (c+d x)+1}}{\sqrt [3]{2}}-\frac {\left (1-\sqrt {3}\right ) \int \frac {1}{\sqrt {1-\sec (c+d x)}}d\sqrt [6]{\sec (c+d x)+1}}{\sqrt [3]{2}}\right )-\frac {3}{4} \sqrt {1-\sec (c+d x)} (\sec (c+d x)+1)^{5/6}\right )}{d \sqrt {1-\sec (c+d x)} (\sec (c+d x)+1)^{5/6}}\) |
| \(\Big \downarrow \) 766 |
| \(\displaystyle \frac {3 \sqrt {2} a A \tan (c+d x) (\sec (c+d x)+1) \sqrt [3]{a \sec (c+d x)+a} \operatorname {AppellF1}\left (\frac {11}{6},1,\frac {1}{2},\frac {17}{6},\sec (c+d x)+1,\frac {1}{2} (\sec (c+d x)+1)\right )}{11 d \sqrt {1-\sec (c+d x)}}-\frac {a B \tan (c+d x) \sqrt [3]{a \sec (c+d x)+a} \left (\frac {15}{2} \left (\frac {\int \frac {\sqrt [3]{2} (\sec (c+d x)+1)^{2/3}-\sqrt {3}+1}{\sqrt {1-\sec (c+d x)}}d\sqrt [6]{\sec (c+d x)+1}}{\sqrt [3]{2}}-\frac {\left (1-\sqrt {3}\right ) \sqrt [6]{\sec (c+d x)+1} \left (\sqrt [3]{2}-\sqrt [3]{\sec (c+d x)+1}\right ) \sqrt {\frac {(\sec (c+d x)+1)^{2/3}+\sqrt [3]{2} \sqrt [3]{\sec (c+d x)+1}+2^{2/3}}{\left (\sqrt [3]{2}-\left (1+\sqrt {3}\right ) \sqrt [3]{\sec (c+d x)+1}\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {\sqrt [3]{2}-\left (1-\sqrt {3}\right ) \sqrt [3]{\sec (c+d x)+1}}{\sqrt [3]{2}-\left (1+\sqrt {3}\right ) \sqrt [3]{\sec (c+d x)+1}}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{2\ 2^{2/3} \sqrt [4]{3} \sqrt {1-\sec (c+d x)} \sqrt {-\frac {\sqrt [3]{\sec (c+d x)+1} \left (\sqrt [3]{2}-\sqrt [3]{\sec (c+d x)+1}\right )}{\left (\sqrt [3]{2}-\left (1+\sqrt {3}\right ) \sqrt [3]{\sec (c+d x)+1}\right )^2}}}\right )-\frac {3}{4} \sqrt {1-\sec (c+d x)} (\sec (c+d x)+1)^{5/6}\right )}{d \sqrt {1-\sec (c+d x)} (\sec (c+d x)+1)^{5/6}}\) |
| \(\Big \downarrow \) 2420 |
| \(\displaystyle \frac {3 \sqrt {2} a A \operatorname {AppellF1}\left (\frac {11}{6},1,\frac {1}{2},\frac {17}{6},\sec (c+d x)+1,\frac {1}{2} (\sec (c+d x)+1)\right ) (\sec (c+d x)+1) \sqrt [3]{\sec (c+d x) a+a} \tan (c+d x)}{11 d \sqrt {1-\sec (c+d x)}}-\frac {a B \sqrt [3]{\sec (c+d x) a+a} \left (\frac {15}{2} \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} 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 [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}}}{\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 ) \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 ) \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}}}{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}{4} \sqrt {1-\sec (c+d x)} (\sec (c+d x)+1)^{5/6}\right ) \tan (c+d x)}{d \sqrt {1-\sec (c+d x)} (\sec (c+d x)+1)^{5/6}}\) |
Input:
Int[(a + a*Sec[c + d*x])^(4/3)*(A + B*Sec[c + d*x]),x]
Output:
(3*Sqrt[2]*a*A*AppellF1[11/6, 1, 1/2, 17/6, 1 + Sec[c + d*x], (1 + Sec[c + d*x])/2]*(1 + Sec[c + d*x])*(a + a*Sec[c + d*x])^(1/3)*Tan[c + d*x])/(11* d*Sqrt[1 - Sec[c + d*x]]) - (a*B*(a + a*Sec[c + d*x])^(1/3)*((-3*Sqrt[1 - Sec[c + d*x]]*(1 + Sec[c + d*x])^(5/6))/4 + (15*(-1/2*((1 - Sqrt[3])*Ellip ticF[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]]*Sqr t[-(((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 + Se c[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)))/2)*Tan[c + d*x])/(d*Sqrt[1 - Sec[c + d*x]]*(1 ...
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/(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[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_) )^(p_.), x_] :> With[{k = Denominator[m]}, Simp[k/b Subst[Int[x^(k*(m + 1 ) - 1)*(c - a*(d/b) + d*(x^k/b))^n*(e - a*(f/b) + f*(x^k/b))^p, x], x, (a + b*x)^(1/k)], x]] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && FractionQ[m] && IntegerQ[2*n] && IntegerQ[p]
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[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_ ))^(q_), x_Symbol] :> Simp[a^p*c^q*((e*x)^(m + 1)/(e*(m + 1)))*AppellF1[(m + 1)/n, -p, -q, 1 + (m + 1)/n, (-b)*(x^n/a), (-d)*(x^n/c)], x] /; FreeQ[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n - 1] && (IntegerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])
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[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_), x_Symbol] :> Simp[a^n*(Cot [c + d*x]/(d*Sqrt[1 + Csc[c + d*x]]*Sqrt[1 - Csc[c + d*x]])) Subst[Int[(1 + b*(x/a))^(n - 1/2)/(x*Sqrt[1 - b*(x/a)]), x], x, Csc[c + d*x]], x] /; Fr eeQ[{a, b, c, d, n}, x] && EqQ[a^2 - b^2, 0] && !IntegerQ[2*n] && GtQ[a, 0 ]
Int[(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_), x_Symbol] :> Simp[a^IntPar t[n]*((a + b*Csc[c + d*x])^FracPart[n]/(1 + (b/a)*Csc[c + d*x])^FracPart[n] ) Int[(1 + (b/a)*Csc[c + d*x])^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[a^2 - b^2, 0] && !IntegerQ[2*n] && !GtQ[a, 0]
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_)]*(b_.) + (a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*(d _.) + (c_)), x_Symbol] :> Simp[c Int[(a + b*Csc[e + f*x])^m, x], x] + Sim p[d Int[(a + b*Csc[e + f*x])^m*Csc[e + f*x], x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && !IntegerQ[2*m]
\[\int \left (a +a \sec \left (d x +c \right )\right )^{\frac {4}{3}} \left (A +B \sec \left (d x +c \right )\right )d x\]
Input:
int((a+a*sec(d*x+c))^(4/3)*(A+B*sec(d*x+c)),x)
Output:
int((a+a*sec(d*x+c))^(4/3)*(A+B*sec(d*x+c)),x)
Timed out. \[ \int (a+a \sec (c+d x))^{4/3} (A+B \sec (c+d x)) \, dx=\text {Timed out} \] Input:
integrate((a+a*sec(d*x+c))^(4/3)*(A+B*sec(d*x+c)),x, algorithm="fricas")
Output:
Timed out
\[ \int (a+a \sec (c+d x))^{4/3} (A+B \sec (c+d x)) \, dx=\int \left (a \left (\sec {\left (c + d x \right )} + 1\right )\right )^{\frac {4}{3}} \left (A + B \sec {\left (c + d x \right )}\right )\, dx \] Input:
integrate((a+a*sec(d*x+c))**(4/3)*(A+B*sec(d*x+c)),x)
Output:
Integral((a*(sec(c + d*x) + 1))**(4/3)*(A + B*sec(c + d*x)), x)
\[ \int (a+a \sec (c+d x))^{4/3} (A+B \sec (c+d x)) \, dx=\int { {\left (B \sec \left (d x + c\right ) + A\right )} {\left (a \sec \left (d x + c\right ) + a\right )}^{\frac {4}{3}} \,d x } \] Input:
integrate((a+a*sec(d*x+c))^(4/3)*(A+B*sec(d*x+c)),x, algorithm="maxima")
Output:
integrate((B*sec(d*x + c) + A)*(a*sec(d*x + c) + a)^(4/3), x)
\[ \int (a+a \sec (c+d x))^{4/3} (A+B \sec (c+d x)) \, dx=\int { {\left (B \sec \left (d x + c\right ) + A\right )} {\left (a \sec \left (d x + c\right ) + a\right )}^{\frac {4}{3}} \,d x } \] Input:
integrate((a+a*sec(d*x+c))^(4/3)*(A+B*sec(d*x+c)),x, algorithm="giac")
Output:
integrate((B*sec(d*x + c) + A)*(a*sec(d*x + c) + a)^(4/3), x)
Timed out. \[ \int (a+a \sec (c+d x))^{4/3} (A+B \sec (c+d x)) \, dx=\int \left (A+\frac {B}{\cos \left (c+d\,x\right )}\right )\,{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^{4/3} \,d x \] Input:
int((A + B/cos(c + d*x))*(a + a/cos(c + d*x))^(4/3),x)
Output:
int((A + B/cos(c + d*x))*(a + a/cos(c + d*x))^(4/3), x)
\[ \int (a+a \sec (c+d x))^{4/3} (A+B \sec (c+d x)) \, dx=a^{\frac {4}{3}} \left (\left (\int \left (\sec \left (d x +c \right )+1\right )^{\frac {1}{3}}d x \right ) a +\left (\int \left (\sec \left (d x +c \right )+1\right )^{\frac {1}{3}} \sec \left (d x +c \right )^{2}d x \right ) b +\left (\int \left (\sec \left (d x +c \right )+1\right )^{\frac {1}{3}} \sec \left (d x +c \right )d x \right ) a +\left (\int \left (\sec \left (d x +c \right )+1\right )^{\frac {1}{3}} \sec \left (d x +c \right )d x \right ) b \right ) \] Input:
int((a+a*sec(d*x+c))^(4/3)*(A+B*sec(d*x+c)),x)
Output:
a**(1/3)*a*(int((sec(c + d*x) + 1)**(1/3),x)*a + int((sec(c + d*x) + 1)**( 1/3)*sec(c + d*x)**2,x)*b + int((sec(c + d*x) + 1)**(1/3)*sec(c + d*x),x)* a + int((sec(c + d*x) + 1)**(1/3)*sec(c + d*x),x)*b)