Integrand size = 25, antiderivative size = 359 \[ \int e^{\frac {1}{2} \text {arctanh}(a x)} \left (1-a^2 x^2\right )^{5/2} \, dx=\frac {231 \sqrt [4]{1-a x} (1+a x)^{3/4}}{512 a}+\frac {231 (1-a x)^{5/4} (1+a x)^{3/4}}{1280 a}+\frac {77 (1-a x)^{9/4} (1+a x)^{3/4}}{960 a}-\frac {77 (1-a x)^{13/4} (1+a x)^{3/4}}{480 a}-\frac {11 (1-a x)^{13/4} (1+a x)^{7/4}}{60 a}-\frac {(1-a x)^{13/4} (1+a x)^{11/4}}{6 a}+\frac {231 \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{512 \sqrt {2} a}-\frac {231 \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{512 \sqrt {2} a}+\frac {231 \log \left (1+\frac {\sqrt {1-a x}}{\sqrt {1+a x}}-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{1024 \sqrt {2} a}-\frac {231 \log \left (1+\frac {\sqrt {1-a x}}{\sqrt {1+a x}}+\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{1024 \sqrt {2} a} \]
231/512*(-a*x+1)^(1/4)*(a*x+1)^(3/4)/a+231/1280*(-a*x+1)^(5/4)*(a*x+1)^(3/ 4)/a+77/960*(-a*x+1)^(9/4)*(a*x+1)^(3/4)/a-77/480*(-a*x+1)^(13/4)*(a*x+1)^ (3/4)/a-11/60*(-a*x+1)^(13/4)*(a*x+1)^(7/4)/a-1/6*(-a*x+1)^(13/4)*(a*x+1)^ (11/4)/a-231/1024*arctan(-1+(-a*x+1)^(1/4)*2^(1/2)/(a*x+1)^(1/4))/a*2^(1/2 )-231/1024*arctan(1+(-a*x+1)^(1/4)*2^(1/2)/(a*x+1)^(1/4))/a*2^(1/2)+231/20 48*ln(1-(-a*x+1)^(1/4)*2^(1/2)/(a*x+1)^(1/4)+(-a*x+1)^(1/2)/(a*x+1)^(1/2)) /a*2^(1/2)-231/2048*ln(1+(-a*x+1)^(1/4)*2^(1/2)/(a*x+1)^(1/4)+(-a*x+1)^(1/ 2)/(a*x+1)^(1/2))/a*2^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.02 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.12 \[ \int e^{\frac {1}{2} \text {arctanh}(a x)} \left (1-a^2 x^2\right )^{5/2} \, dx=-\frac {16\ 2^{3/4} (1-a x)^{13/4} \operatorname {Hypergeometric2F1}\left (-\frac {11}{4},\frac {13}{4},\frac {17}{4},\frac {1}{2} (1-a x)\right )}{13 a} \]
Time = 0.58 (sec) , antiderivative size = 360, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.680, Rules used = {6690, 60, 60, 60, 60, 60, 60, 73, 770, 755, 1476, 1082, 217, 1479, 25, 27, 1103}
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.
| \(\displaystyle \int \left (1-a^2 x^2\right )^{5/2} e^{\frac {1}{2} \text {arctanh}(a x)} \, dx\) |
| \(\Big \downarrow \) 6690 |
| \(\displaystyle \int (1-a x)^{9/4} (a x+1)^{11/4}dx\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {11}{12} \int (1-a x)^{9/4} (a x+1)^{7/4}dx-\frac {(1-a x)^{13/4} (a x+1)^{11/4}}{6 a}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {11}{12} \left (\frac {7}{10} \int (1-a x)^{9/4} (a x+1)^{3/4}dx-\frac {(1-a x)^{13/4} (a x+1)^{7/4}}{5 a}\right )-\frac {(1-a x)^{13/4} (a x+1)^{11/4}}{6 a}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {11}{12} \left (\frac {7}{10} \left (\frac {3}{8} \int \frac {(1-a x)^{9/4}}{\sqrt [4]{a x+1}}dx-\frac {(1-a x)^{13/4} (a x+1)^{3/4}}{4 a}\right )-\frac {(1-a x)^{13/4} (a x+1)^{7/4}}{5 a}\right )-\frac {(1-a x)^{13/4} (a x+1)^{11/4}}{6 a}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {11}{12} \left (\frac {7}{10} \left (\frac {3}{8} \left (\frac {3}{2} \int \frac {(1-a x)^{5/4}}{\sqrt [4]{a x+1}}dx+\frac {(a x+1)^{3/4} (1-a x)^{9/4}}{3 a}\right )-\frac {(1-a x)^{13/4} (a x+1)^{3/4}}{4 a}\right )-\frac {(1-a x)^{13/4} (a x+1)^{7/4}}{5 a}\right )-\frac {(1-a x)^{13/4} (a x+1)^{11/4}}{6 a}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {11}{12} \left (\frac {7}{10} \left (\frac {3}{8} \left (\frac {3}{2} \left (\frac {5}{4} \int \frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}dx+\frac {(a x+1)^{3/4} (1-a x)^{5/4}}{2 a}\right )+\frac {(a x+1)^{3/4} (1-a x)^{9/4}}{3 a}\right )-\frac {(1-a x)^{13/4} (a x+1)^{3/4}}{4 a}\right )-\frac {(1-a x)^{13/4} (a x+1)^{7/4}}{5 a}\right )-\frac {(1-a x)^{13/4} (a x+1)^{11/4}}{6 a}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {11}{12} \left (\frac {7}{10} \left (\frac {3}{8} \left (\frac {3}{2} \left (\frac {5}{4} \left (\frac {1}{2} \int \frac {1}{(1-a x)^{3/4} \sqrt [4]{a x+1}}dx+\frac {\sqrt [4]{1-a x} (a x+1)^{3/4}}{a}\right )+\frac {(a x+1)^{3/4} (1-a x)^{5/4}}{2 a}\right )+\frac {(a x+1)^{3/4} (1-a x)^{9/4}}{3 a}\right )-\frac {(1-a x)^{13/4} (a x+1)^{3/4}}{4 a}\right )-\frac {(1-a x)^{13/4} (a x+1)^{7/4}}{5 a}\right )-\frac {(1-a x)^{13/4} (a x+1)^{11/4}}{6 a}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {11}{12} \left (\frac {7}{10} \left (\frac {3}{8} \left (\frac {3}{2} \left (\frac {5}{4} \left (\frac {\sqrt [4]{1-a x} (a x+1)^{3/4}}{a}-\frac {2 \int \frac {1}{\sqrt [4]{a x+1}}d\sqrt [4]{1-a x}}{a}\right )+\frac {(a x+1)^{3/4} (1-a x)^{5/4}}{2 a}\right )+\frac {(a x+1)^{3/4} (1-a x)^{9/4}}{3 a}\right )-\frac {(1-a x)^{13/4} (a x+1)^{3/4}}{4 a}\right )-\frac {(1-a x)^{13/4} (a x+1)^{7/4}}{5 a}\right )-\frac {(1-a x)^{13/4} (a x+1)^{11/4}}{6 a}\) |
| \(\Big \downarrow \) 770 |
| \(\displaystyle \frac {11}{12} \left (\frac {7}{10} \left (\frac {3}{8} \left (\frac {3}{2} \left (\frac {5}{4} \left (\frac {\sqrt [4]{1-a x} (a x+1)^{3/4}}{a}-\frac {2 \int \frac {1}{2-a x}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}}{a}\right )+\frac {(a x+1)^{3/4} (1-a x)^{5/4}}{2 a}\right )+\frac {(a x+1)^{3/4} (1-a x)^{9/4}}{3 a}\right )-\frac {(1-a x)^{13/4} (a x+1)^{3/4}}{4 a}\right )-\frac {(1-a x)^{13/4} (a x+1)^{7/4}}{5 a}\right )-\frac {(1-a x)^{13/4} (a x+1)^{11/4}}{6 a}\) |
| \(\Big \downarrow \) 755 |
| \(\displaystyle \frac {11}{12} \left (\frac {7}{10} \left (\frac {3}{8} \left (\frac {3}{2} \left (\frac {5}{4} \left (\frac {\sqrt [4]{1-a x} (a x+1)^{3/4}}{a}-\frac {2 \left (\frac {1}{2} \int \frac {1-\sqrt {1-a x}}{2-a x}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+\frac {1}{2} \int \frac {\sqrt {1-a x}+1}{2-a x}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )}{a}\right )+\frac {(a x+1)^{3/4} (1-a x)^{5/4}}{2 a}\right )+\frac {(a x+1)^{3/4} (1-a x)^{9/4}}{3 a}\right )-\frac {(1-a x)^{13/4} (a x+1)^{3/4}}{4 a}\right )-\frac {(1-a x)^{13/4} (a x+1)^{7/4}}{5 a}\right )-\frac {(1-a x)^{13/4} (a x+1)^{11/4}}{6 a}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {11}{12} \left (\frac {7}{10} \left (\frac {3}{8} \left (\frac {3}{2} \left (\frac {5}{4} \left (\frac {\sqrt [4]{1-a x} (a x+1)^{3/4}}{a}-\frac {2 \left (\frac {1}{2} \int \frac {1-\sqrt {1-a x}}{2-a x}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+\frac {1}{2} \left (\frac {1}{2} \int \frac {1}{\sqrt {1-a x}-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+\frac {1}{2} \int \frac {1}{\sqrt {1-a x}+\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )\right )}{a}\right )+\frac {(a x+1)^{3/4} (1-a x)^{5/4}}{2 a}\right )+\frac {(a x+1)^{3/4} (1-a x)^{9/4}}{3 a}\right )-\frac {(1-a x)^{13/4} (a x+1)^{3/4}}{4 a}\right )-\frac {(1-a x)^{13/4} (a x+1)^{7/4}}{5 a}\right )-\frac {(1-a x)^{13/4} (a x+1)^{11/4}}{6 a}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {11}{12} \left (\frac {7}{10} \left (\frac {3}{8} \left (\frac {3}{2} \left (\frac {5}{4} \left (\frac {\sqrt [4]{1-a x} (a x+1)^{3/4}}{a}-\frac {2 \left (\frac {1}{2} \left (\frac {\int \frac {1}{-\sqrt {1-a x}-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )}{\sqrt {2}}-\frac {\int \frac {1}{-\sqrt {1-a x}-1}d\left (\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{\sqrt {2}}\right )+\frac {1}{2} \int \frac {1-\sqrt {1-a x}}{2-a x}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )}{a}\right )+\frac {(a x+1)^{3/4} (1-a x)^{5/4}}{2 a}\right )+\frac {(a x+1)^{3/4} (1-a x)^{9/4}}{3 a}\right )-\frac {(1-a x)^{13/4} (a x+1)^{3/4}}{4 a}\right )-\frac {(1-a x)^{13/4} (a x+1)^{7/4}}{5 a}\right )-\frac {(1-a x)^{13/4} (a x+1)^{11/4}}{6 a}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {11}{12} \left (\frac {7}{10} \left (\frac {3}{8} \left (\frac {3}{2} \left (\frac {5}{4} \left (\frac {\sqrt [4]{1-a x} (a x+1)^{3/4}}{a}-\frac {2 \left (\frac {1}{2} \int \frac {1-\sqrt {1-a x}}{2-a x}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )}{\sqrt {2}}\right )\right )}{a}\right )+\frac {(a x+1)^{3/4} (1-a x)^{5/4}}{2 a}\right )+\frac {(a x+1)^{3/4} (1-a x)^{9/4}}{3 a}\right )-\frac {(1-a x)^{13/4} (a x+1)^{3/4}}{4 a}\right )-\frac {(1-a x)^{13/4} (a x+1)^{7/4}}{5 a}\right )-\frac {(1-a x)^{13/4} (a x+1)^{11/4}}{6 a}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {11}{12} \left (\frac {7}{10} \left (\frac {3}{8} \left (\frac {3}{2} \left (\frac {5}{4} \left (\frac {\sqrt [4]{1-a x} (a x+1)^{3/4}}{a}-\frac {2 \left (\frac {1}{2} \left (-\frac {\int -\frac {\sqrt {2}-\frac {2 \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}}{\sqrt {1-a x}-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}}{2 \sqrt {2}}-\frac {\int -\frac {\sqrt {2} \left (\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{\sqrt {1-a x}+\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )}{\sqrt {2}}\right )\right )}{a}\right )+\frac {(a x+1)^{3/4} (1-a x)^{5/4}}{2 a}\right )+\frac {(a x+1)^{3/4} (1-a x)^{9/4}}{3 a}\right )-\frac {(1-a x)^{13/4} (a x+1)^{3/4}}{4 a}\right )-\frac {(1-a x)^{13/4} (a x+1)^{7/4}}{5 a}\right )-\frac {(1-a x)^{13/4} (a x+1)^{11/4}}{6 a}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {11}{12} \left (\frac {7}{10} \left (\frac {3}{8} \left (\frac {3}{2} \left (\frac {5}{4} \left (\frac {\sqrt [4]{1-a x} (a x+1)^{3/4}}{a}-\frac {2 \left (\frac {1}{2} \left (\frac {\int \frac {\sqrt {2}-\frac {2 \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}}{\sqrt {1-a x}-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}}{2 \sqrt {2}}+\frac {\int \frac {\sqrt {2} \left (\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{\sqrt {1-a x}+\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )}{\sqrt {2}}\right )\right )}{a}\right )+\frac {(a x+1)^{3/4} (1-a x)^{5/4}}{2 a}\right )+\frac {(a x+1)^{3/4} (1-a x)^{9/4}}{3 a}\right )-\frac {(1-a x)^{13/4} (a x+1)^{3/4}}{4 a}\right )-\frac {(1-a x)^{13/4} (a x+1)^{7/4}}{5 a}\right )-\frac {(1-a x)^{13/4} (a x+1)^{11/4}}{6 a}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {11}{12} \left (\frac {7}{10} \left (\frac {3}{8} \left (\frac {3}{2} \left (\frac {5}{4} \left (\frac {\sqrt [4]{1-a x} (a x+1)^{3/4}}{a}-\frac {2 \left (\frac {1}{2} \left (\frac {\int \frac {\sqrt {2}-\frac {2 \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}}{\sqrt {1-a x}-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}}{2 \sqrt {2}}+\frac {1}{2} \int \frac {\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1}{\sqrt {1-a x}+\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )}{\sqrt {2}}\right )\right )}{a}\right )+\frac {(a x+1)^{3/4} (1-a x)^{5/4}}{2 a}\right )+\frac {(a x+1)^{3/4} (1-a x)^{9/4}}{3 a}\right )-\frac {(1-a x)^{13/4} (a x+1)^{3/4}}{4 a}\right )-\frac {(1-a x)^{13/4} (a x+1)^{7/4}}{5 a}\right )-\frac {(1-a x)^{13/4} (a x+1)^{11/4}}{6 a}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {11}{12} \left (\frac {7}{10} \left (\frac {3}{8} \left (\frac {3}{2} \left (\frac {5}{4} \left (\frac {\sqrt [4]{1-a x} (a x+1)^{3/4}}{a}-\frac {2 \left (\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )}{\sqrt {2}}\right )+\frac {1}{2} \left (\frac {\log \left (\sqrt {1-a x}+\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\sqrt {1-a x}-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{2 \sqrt {2}}\right )\right )}{a}\right )+\frac {(a x+1)^{3/4} (1-a x)^{5/4}}{2 a}\right )+\frac {(a x+1)^{3/4} (1-a x)^{9/4}}{3 a}\right )-\frac {(1-a x)^{13/4} (a x+1)^{3/4}}{4 a}\right )-\frac {(1-a x)^{13/4} (a x+1)^{7/4}}{5 a}\right )-\frac {(1-a x)^{13/4} (a x+1)^{11/4}}{6 a}\) |
-1/6*((1 - a*x)^(13/4)*(1 + a*x)^(11/4))/a + (11*(-1/5*((1 - a*x)^(13/4)*( 1 + a*x)^(7/4))/a + (7*(-1/4*((1 - a*x)^(13/4)*(1 + a*x)^(3/4))/a + (3*((( 1 - a*x)^(9/4)*(1 + a*x)^(3/4))/(3*a) + (3*(((1 - a*x)^(5/4)*(1 + a*x)^(3/ 4))/(2*a) + (5*(((1 - a*x)^(1/4)*(1 + a*x)^(3/4))/a - (2*((-(ArcTan[1 - (S qrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)]/Sqrt[2]) + ArcTan[1 + (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)]/Sqrt[2])/2 + (-1/2*Log[1 + Sqrt[1 - a*x] - (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)]/Sqrt[2] + Log[1 + Sqrt[1 - a*x ] + (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)]/(2*Sqrt[2]))/2))/a))/4))/2) )/8))/10))/12
3.13.81.3.1 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/(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_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2] ], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*r) Int[(r - s*x^2)/(a + b*x^4) , x], x] + Simp[1/(2*r) Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] & & AtomQ[SplitProduct[SumBaseQ, b]]))
Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^(p + 1/n) Subst[In t[1/(1 - b*x^n)^(p + 1/n + 1), x], x, x/(a + b*x^n)^(1/n)], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[-1, p, 0] && NeQ[p, -2^(-1)] && IntegerQ[p + 1 /n]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b )], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /; Fre eQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 2*(d/e), 2]}, Simp[e/(2*c) Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ e/(2*c) Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ -2*(d/e), 2]}, Simp[e/(2*c*q) Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Simp[e/(2*c*q) Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*((c_) + (d_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[c^p Int[(1 - a*x)^(p - n/2)*(1 + a*x)^(p + n/2), x], x] /; FreeQ[{a , c, d, n, p}, x] && EqQ[a^2*c + d, 0] && (IntegerQ[p] || GtQ[c, 0])
\[\int \sqrt {\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}}\, \left (-a^{2} x^{2}+1\right )^{\frac {5}{2}}d x\]
Result contains complex when optimal does not.
Time = 0.27 (sec) , antiderivative size = 268, normalized size of antiderivative = 0.75 \[ \int e^{\frac {1}{2} \text {arctanh}(a x)} \left (1-a^2 x^2\right )^{5/2} \, dx=\frac {3465 \, a \left (-\frac {1}{a^{4}}\right )^{\frac {1}{4}} \log \left (a^{3} \left (-\frac {1}{a^{4}}\right )^{\frac {3}{4}} + \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}}\right ) - 3465 i \, a \left (-\frac {1}{a^{4}}\right )^{\frac {1}{4}} \log \left (i \, a^{3} \left (-\frac {1}{a^{4}}\right )^{\frac {3}{4}} + \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}}\right ) + 3465 i \, a \left (-\frac {1}{a^{4}}\right )^{\frac {1}{4}} \log \left (-i \, a^{3} \left (-\frac {1}{a^{4}}\right )^{\frac {3}{4}} + \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}}\right ) - 3465 \, a \left (-\frac {1}{a^{4}}\right )^{\frac {1}{4}} \log \left (-a^{3} \left (-\frac {1}{a^{4}}\right )^{\frac {3}{4}} + \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}}\right ) + 2 \, {\left (1280 \, a^{5} x^{5} + 128 \, a^{4} x^{4} - 4144 \, a^{3} x^{3} - 520 \, a^{2} x^{2} + 5174 \, a x + 1547\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}}}{15360 \, a} \]
1/15360*(3465*a*(-1/a^4)^(1/4)*log(a^3*(-1/a^4)^(3/4) + sqrt(-sqrt(-a^2*x^ 2 + 1)/(a*x - 1))) - 3465*I*a*(-1/a^4)^(1/4)*log(I*a^3*(-1/a^4)^(3/4) + sq rt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))) + 3465*I*a*(-1/a^4)^(1/4)*log(-I*a^3*(- 1/a^4)^(3/4) + sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))) - 3465*a*(-1/a^4)^(1/4 )*log(-a^3*(-1/a^4)^(3/4) + sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))) + 2*(1280 *a^5*x^5 + 128*a^4*x^4 - 4144*a^3*x^3 - 520*a^2*x^2 + 5174*a*x + 1547)*sqr t(-a^2*x^2 + 1)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)))/a
Timed out. \[ \int e^{\frac {1}{2} \text {arctanh}(a x)} \left (1-a^2 x^2\right )^{5/2} \, dx=\text {Timed out} \]
\[ \int e^{\frac {1}{2} \text {arctanh}(a x)} \left (1-a^2 x^2\right )^{5/2} \, dx=\int { {\left (-a^{2} x^{2} + 1\right )}^{\frac {5}{2}} \sqrt {\frac {a x + 1}{\sqrt {-a^{2} x^{2} + 1}}} \,d x } \]
\[ \int e^{\frac {1}{2} \text {arctanh}(a x)} \left (1-a^2 x^2\right )^{5/2} \, dx=\int { {\left (-a^{2} x^{2} + 1\right )}^{\frac {5}{2}} \sqrt {\frac {a x + 1}{\sqrt {-a^{2} x^{2} + 1}}} \,d x } \]
Timed out. \[ \int e^{\frac {1}{2} \text {arctanh}(a x)} \left (1-a^2 x^2\right )^{5/2} \, dx=\int {\left (1-a^2\,x^2\right )}^{5/2}\,\sqrt {\frac {a\,x+1}{\sqrt {1-a^2\,x^2}}} \,d x \]