Integrand size = 10, antiderivative size = 445 \[ \int e^{\frac {1}{4} \text {arctanh}(a x)} \, dx=-\frac {(1-a x)^{7/8} \sqrt [8]{1+a x}}{a}-\frac {\sqrt {2+\sqrt {2}} \arctan \left (\frac {\sqrt {2-\sqrt {2}}-\frac {2 \sqrt [8]{1+a x}}{\sqrt [8]{1-a x}}}{\sqrt {2+\sqrt {2}}}\right )}{4 a}-\frac {\sqrt {2-\sqrt {2}} \arctan \left (\frac {\sqrt {2+\sqrt {2}}-\frac {2 \sqrt [8]{1+a x}}{\sqrt [8]{1-a x}}}{\sqrt {2-\sqrt {2}}}\right )}{4 a}+\frac {\sqrt {2+\sqrt {2}} \arctan \left (\frac {\sqrt {2-\sqrt {2}}+\frac {2 \sqrt [8]{1+a x}}{\sqrt [8]{1-a x}}}{\sqrt {2+\sqrt {2}}}\right )}{4 a}+\frac {\sqrt {2-\sqrt {2}} \arctan \left (\frac {\sqrt {2+\sqrt {2}}+\frac {2 \sqrt [8]{1+a x}}{\sqrt [8]{1-a x}}}{\sqrt {2-\sqrt {2}}}\right )}{4 a}+\frac {\sqrt {2-\sqrt {2}} \text {arctanh}\left (\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1+a x}}{\sqrt [8]{1-a x} \left (1+\frac {\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )}\right )}{4 a}+\frac {\sqrt {2+\sqrt {2}} \text {arctanh}\left (\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1+a x}}{\sqrt [8]{1-a x} \left (1+\frac {\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )}\right )}{4 a} \] Output:
-(-a*x+1)^(7/8)*(a*x+1)^(1/8)/a-1/4*(2+2^(1/2))^(1/2)*arctan(((2-2^(1/2))^ (1/2)-2*(a*x+1)^(1/8)/(-a*x+1)^(1/8))/(2+2^(1/2))^(1/2))/a-1/4*(2-2^(1/2)) ^(1/2)*arctan(((2+2^(1/2))^(1/2)-2*(a*x+1)^(1/8)/(-a*x+1)^(1/8))/(2-2^(1/2 ))^(1/2))/a+1/4*(2+2^(1/2))^(1/2)*arctan(((2-2^(1/2))^(1/2)+2*(a*x+1)^(1/8 )/(-a*x+1)^(1/8))/(2+2^(1/2))^(1/2))/a+1/4*(2-2^(1/2))^(1/2)*arctan(((2+2^ (1/2))^(1/2)+2*(a*x+1)^(1/8)/(-a*x+1)^(1/8))/(2-2^(1/2))^(1/2))/a+1/4*(2-2 ^(1/2))^(1/2)*arctanh((2-2^(1/2))^(1/2)*(a*x+1)^(1/8)/(-a*x+1)^(1/8)/(1+(a *x+1)^(1/4)/(-a*x+1)^(1/4)))/a+1/4*(2+2^(1/2))^(1/2)*arctanh((2+2^(1/2))^( 1/2)*(a*x+1)^(1/8)/(-a*x+1)^(1/8)/(1+(a*x+1)^(1/4)/(-a*x+1)^(1/4)))/a
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.07 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.11 \[ \int e^{\frac {1}{4} \text {arctanh}(a x)} \, dx=\frac {2 e^{\frac {1}{4} \text {arctanh}(a x)} \left (-\frac {1}{1+e^{2 \text {arctanh}(a x)}}+\operatorname {Hypergeometric2F1}\left (\frac {1}{8},1,\frac {9}{8},-e^{2 \text {arctanh}(a x)}\right )\right )}{a} \] Input:
Integrate[E^(ArcTanh[a*x]/4),x]
Output:
(2*E^(ArcTanh[a*x]/4)*(-(1 + E^(2*ArcTanh[a*x]))^(-1) + Hypergeometric2F1[ 1/8, 1, 9/8, -E^(2*ArcTanh[a*x])]))/a
Time = 1.26 (sec) , antiderivative size = 589, normalized size of antiderivative = 1.32, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.200, Rules used = {6675, 60, 73, 854, 828, 1442, 1483, 1142, 25, 1083, 217, 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.
| \(\displaystyle \int e^{\frac {1}{4} \text {arctanh}(a x)} \, dx\) |
| \(\Big \downarrow \) 6675 |
| \(\displaystyle \int \frac {\sqrt [8]{a x+1}}{\sqrt [8]{1-a x}}dx\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {1}{4} \int \frac {1}{\sqrt [8]{1-a x} (a x+1)^{7/8}}dx-\frac {(1-a x)^{7/8} \sqrt [8]{a x+1}}{a}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -\frac {2 \int \frac {(1-a x)^{3/4}}{(a x+1)^{7/8}}d\sqrt [8]{1-a x}}{a}-\frac {(1-a x)^{7/8} \sqrt [8]{a x+1}}{a}\) |
| \(\Big \downarrow \) 854 |
| \(\displaystyle -\frac {2 \int \frac {(1-a x)^{3/4}}{2-a x}d\frac {\sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}}{a}-\frac {(1-a x)^{7/8} \sqrt [8]{a x+1}}{a}\) |
| \(\Big \downarrow \) 828 |
| \(\displaystyle -\frac {2 \left (\frac {\int \frac {\sqrt {1-a x}}{\sqrt {1-a x}-\sqrt {2} \sqrt [4]{1-a x}+1}d\frac {\sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}}{2 \sqrt {2}}-\frac {\int \frac {\sqrt {1-a x}}{\sqrt {1-a x}+\sqrt {2} \sqrt [4]{1-a x}+1}d\frac {\sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}}{2 \sqrt {2}}\right )}{a}-\frac {(1-a x)^{7/8} \sqrt [8]{a x+1}}{a}\) |
| \(\Big \downarrow \) 1442 |
| \(\displaystyle -\frac {2 \left (\frac {\frac {\sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}-\int \frac {1-\sqrt {2} \sqrt [4]{1-a x}}{\sqrt {1-a x}-\sqrt {2} \sqrt [4]{1-a x}+1}d\frac {\sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}}{2 \sqrt {2}}-\frac {\frac {\sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}-\int \frac {\sqrt {2} \sqrt [4]{1-a x}+1}{\sqrt {1-a x}+\sqrt {2} \sqrt [4]{1-a x}+1}d\frac {\sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}}{2 \sqrt {2}}\right )}{a}-\frac {(1-a x)^{7/8} \sqrt [8]{a x+1}}{a}\) |
| \(\Big \downarrow \) 1483 |
| \(\displaystyle -\frac {2 \left (\frac {-\frac {\int \frac {\sqrt {2+\sqrt {2}}-\frac {\left (1+\sqrt {2}\right ) \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}}{\sqrt [4]{1-a x}-\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+1}d\frac {\sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}}{2 \sqrt {2+\sqrt {2}}}-\frac {\int \frac {\frac {\left (1+\sqrt {2}\right ) \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+\sqrt {2+\sqrt {2}}}{\sqrt [4]{1-a x}+\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+1}d\frac {\sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}}{2 \sqrt {2+\sqrt {2}}}+\frac {\sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}}{2 \sqrt {2}}-\frac {-\frac {\int \frac {\sqrt {2-\sqrt {2}}-\frac {\left (1-\sqrt {2}\right ) \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}}{\sqrt [4]{1-a x}-\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+1}d\frac {\sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}}{2 \sqrt {2-\sqrt {2}}}-\frac {\int \frac {\frac {\left (1-\sqrt {2}\right ) \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+\sqrt {2-\sqrt {2}}}{\sqrt [4]{1-a x}+\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+1}d\frac {\sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}}{2 \sqrt {2-\sqrt {2}}}+\frac {\sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}}{2 \sqrt {2}}\right )}{a}-\frac {(1-a x)^{7/8} \sqrt [8]{a x+1}}{a}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle -\frac {2 \left (\frac {-\frac {-\frac {1}{2} \sqrt {2-\sqrt {2}} \int \frac {1}{\sqrt [4]{1-a x}-\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+1}d\frac {\sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}-\frac {1}{2} \left (1+\sqrt {2}\right ) \int -\frac {\sqrt {2+\sqrt {2}}-\frac {2 \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}}{\sqrt [4]{1-a x}-\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+1}d\frac {\sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}}{2 \sqrt {2+\sqrt {2}}}-\frac {\frac {1}{2} \left (1+\sqrt {2}\right ) \int \frac {\frac {2 \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+\sqrt {2+\sqrt {2}}}{\sqrt [4]{1-a x}+\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+1}d\frac {\sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}-\frac {1}{2} \sqrt {2-\sqrt {2}} \int \frac {1}{\sqrt [4]{1-a x}+\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+1}d\frac {\sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}}{2 \sqrt {2+\sqrt {2}}}+\frac {\sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}}{2 \sqrt {2}}-\frac {-\frac {\frac {1}{2} \sqrt {2+\sqrt {2}} \int \frac {1}{\sqrt [4]{1-a x}-\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+1}d\frac {\sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}-\frac {1}{2} \left (1-\sqrt {2}\right ) \int -\frac {\sqrt {2-\sqrt {2}}-\frac {2 \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}}{\sqrt [4]{1-a x}-\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+1}d\frac {\sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}}{2 \sqrt {2-\sqrt {2}}}-\frac {\frac {1}{2} \sqrt {2+\sqrt {2}} \int \frac {1}{\sqrt [4]{1-a x}+\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+1}d\frac {\sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+\frac {1}{2} \left (1-\sqrt {2}\right ) \int \frac {\frac {2 \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+\sqrt {2-\sqrt {2}}}{\sqrt [4]{1-a x}+\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+1}d\frac {\sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}}{2 \sqrt {2-\sqrt {2}}}+\frac {\sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}}{2 \sqrt {2}}\right )}{a}-\frac {(1-a x)^{7/8} \sqrt [8]{a x+1}}{a}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {2 \left (\frac {-\frac {\frac {1}{2} \left (1+\sqrt {2}\right ) \int \frac {\sqrt {2+\sqrt {2}}-\frac {2 \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}}{\sqrt [4]{1-a x}-\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+1}d\frac {\sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}-\frac {1}{2} \sqrt {2-\sqrt {2}} \int \frac {1}{\sqrt [4]{1-a x}-\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+1}d\frac {\sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}}{2 \sqrt {2+\sqrt {2}}}-\frac {\frac {1}{2} \left (1+\sqrt {2}\right ) \int \frac {\frac {2 \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+\sqrt {2+\sqrt {2}}}{\sqrt [4]{1-a x}+\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+1}d\frac {\sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}-\frac {1}{2} \sqrt {2-\sqrt {2}} \int \frac {1}{\sqrt [4]{1-a x}+\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+1}d\frac {\sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}}{2 \sqrt {2+\sqrt {2}}}+\frac {\sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}}{2 \sqrt {2}}-\frac {-\frac {\frac {1}{2} \sqrt {2+\sqrt {2}} \int \frac {1}{\sqrt [4]{1-a x}-\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+1}d\frac {\sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+\frac {1}{2} \left (1-\sqrt {2}\right ) \int \frac {\sqrt {2-\sqrt {2}}-\frac {2 \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}}{\sqrt [4]{1-a x}-\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+1}d\frac {\sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}}{2 \sqrt {2-\sqrt {2}}}-\frac {\frac {1}{2} \sqrt {2+\sqrt {2}} \int \frac {1}{\sqrt [4]{1-a x}+\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+1}d\frac {\sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+\frac {1}{2} \left (1-\sqrt {2}\right ) \int \frac {\frac {2 \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+\sqrt {2-\sqrt {2}}}{\sqrt [4]{1-a x}+\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+1}d\frac {\sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}}{2 \sqrt {2-\sqrt {2}}}+\frac {\sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}}{2 \sqrt {2}}\right )}{a}-\frac {(1-a x)^{7/8} \sqrt [8]{a x+1}}{a}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle -\frac {2 \left (\frac {-\frac {\sqrt {2-\sqrt {2}} \int \frac {1}{-\sqrt [4]{1-a x}+\sqrt {2}-2}d\left (\frac {2 \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}-\sqrt {2+\sqrt {2}}\right )+\frac {1}{2} \left (1+\sqrt {2}\right ) \int \frac {\sqrt {2+\sqrt {2}}-\frac {2 \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}}{\sqrt [4]{1-a x}-\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+1}d\frac {\sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}}{2 \sqrt {2+\sqrt {2}}}-\frac {\sqrt {2-\sqrt {2}} \int \frac {1}{-\sqrt [4]{1-a x}+\sqrt {2}-2}d\left (\frac {2 \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+\sqrt {2+\sqrt {2}}\right )+\frac {1}{2} \left (1+\sqrt {2}\right ) \int \frac {\frac {2 \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+\sqrt {2+\sqrt {2}}}{\sqrt [4]{1-a x}+\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+1}d\frac {\sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}}{2 \sqrt {2+\sqrt {2}}}+\frac {\sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}}{2 \sqrt {2}}-\frac {-\frac {\frac {1}{2} \left (1-\sqrt {2}\right ) \int \frac {\sqrt {2-\sqrt {2}}-\frac {2 \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}}{\sqrt [4]{1-a x}-\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+1}d\frac {\sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}-\sqrt {2+\sqrt {2}} \int \frac {1}{-\sqrt [4]{1-a x}-\sqrt {2}-2}d\left (\frac {2 \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}-\sqrt {2-\sqrt {2}}\right )}{2 \sqrt {2-\sqrt {2}}}-\frac {\frac {1}{2} \left (1-\sqrt {2}\right ) \int \frac {\frac {2 \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+\sqrt {2-\sqrt {2}}}{\sqrt [4]{1-a x}+\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+1}d\frac {\sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}-\sqrt {2+\sqrt {2}} \int \frac {1}{-\sqrt [4]{1-a x}-\sqrt {2}-2}d\left (\frac {2 \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+\sqrt {2-\sqrt {2}}\right )}{2 \sqrt {2-\sqrt {2}}}+\frac {\sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}}{2 \sqrt {2}}\right )}{a}-\frac {(1-a x)^{7/8} \sqrt [8]{a x+1}}{a}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle -\frac {2 \left (\frac {-\frac {\frac {1}{2} \left (1+\sqrt {2}\right ) \int \frac {\sqrt {2+\sqrt {2}}-\frac {2 \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}}{\sqrt [4]{1-a x}-\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+1}d\frac {\sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}-\arctan \left (\frac {\frac {2 \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}-\sqrt {2+\sqrt {2}}}{\sqrt {2-\sqrt {2}}}\right )}{2 \sqrt {2+\sqrt {2}}}-\frac {\frac {1}{2} \left (1+\sqrt {2}\right ) \int \frac {\frac {2 \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+\sqrt {2+\sqrt {2}}}{\sqrt [4]{1-a x}+\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+1}d\frac {\sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}-\arctan \left (\frac {\frac {2 \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+\sqrt {2+\sqrt {2}}}{\sqrt {2-\sqrt {2}}}\right )}{2 \sqrt {2+\sqrt {2}}}+\frac {\sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}}{2 \sqrt {2}}-\frac {-\frac {\arctan \left (\frac {\frac {2 \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}-\sqrt {2-\sqrt {2}}}{\sqrt {2+\sqrt {2}}}\right )+\frac {1}{2} \left (1-\sqrt {2}\right ) \int \frac {\sqrt {2-\sqrt {2}}-\frac {2 \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}}{\sqrt [4]{1-a x}-\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+1}d\frac {\sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}}{2 \sqrt {2-\sqrt {2}}}-\frac {\arctan \left (\frac {\frac {2 \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+\sqrt {2-\sqrt {2}}}{\sqrt {2+\sqrt {2}}}\right )+\frac {1}{2} \left (1-\sqrt {2}\right ) \int \frac {\frac {2 \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+\sqrt {2-\sqrt {2}}}{\sqrt [4]{1-a x}+\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+1}d\frac {\sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}}{2 \sqrt {2-\sqrt {2}}}+\frac {\sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}}{2 \sqrt {2}}\right )}{a}-\frac {(1-a x)^{7/8} \sqrt [8]{a x+1}}{a}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle -\frac {2 \left (\frac {-\frac {-\arctan \left (\frac {\frac {2 \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}-\sqrt {2+\sqrt {2}}}{\sqrt {2-\sqrt {2}}}\right )-\frac {1}{2} \left (1+\sqrt {2}\right ) \log \left (\sqrt [4]{1-a x}-\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+1\right )}{2 \sqrt {2+\sqrt {2}}}-\frac {\frac {1}{2} \left (1+\sqrt {2}\right ) \log \left (\sqrt [4]{1-a x}+\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+1\right )-\arctan \left (\frac {\frac {2 \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+\sqrt {2+\sqrt {2}}}{\sqrt {2-\sqrt {2}}}\right )}{2 \sqrt {2+\sqrt {2}}}+\frac {\sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}}{2 \sqrt {2}}-\frac {-\frac {\arctan \left (\frac {\frac {2 \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}-\sqrt {2-\sqrt {2}}}{\sqrt {2+\sqrt {2}}}\right )-\frac {1}{2} \left (1-\sqrt {2}\right ) \log \left (\sqrt [4]{1-a x}-\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+1\right )}{2 \sqrt {2-\sqrt {2}}}-\frac {\arctan \left (\frac {\frac {2 \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+\sqrt {2-\sqrt {2}}}{\sqrt {2+\sqrt {2}}}\right )+\frac {1}{2} \left (1-\sqrt {2}\right ) \log \left (\sqrt [4]{1-a x}+\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+1\right )}{2 \sqrt {2-\sqrt {2}}}+\frac {\sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}}{2 \sqrt {2}}\right )}{a}-\frac {(1-a x)^{7/8} \sqrt [8]{a x+1}}{a}\) |
Input:
Int[E^(ArcTanh[a*x]/4),x]
Output:
-(((1 - a*x)^(7/8)*(1 + a*x)^(1/8))/a) - (2*(-1/2*((1 - a*x)^(1/8)/(1 + a* x)^(1/8) - (ArcTan[(-Sqrt[2 - Sqrt[2]] + (2*(1 - a*x)^(1/8))/(1 + a*x)^(1/ 8))/Sqrt[2 + Sqrt[2]]] - ((1 - Sqrt[2])*Log[1 + (1 - a*x)^(1/4) - (Sqrt[2 - Sqrt[2]]*(1 - a*x)^(1/8))/(1 + a*x)^(1/8)])/2)/(2*Sqrt[2 - Sqrt[2]]) - ( ArcTan[(Sqrt[2 - Sqrt[2]] + (2*(1 - a*x)^(1/8))/(1 + a*x)^(1/8))/Sqrt[2 + Sqrt[2]]] + ((1 - Sqrt[2])*Log[1 + (1 - a*x)^(1/4) + (Sqrt[2 - Sqrt[2]]*(1 - a*x)^(1/8))/(1 + a*x)^(1/8)])/2)/(2*Sqrt[2 - Sqrt[2]]))/Sqrt[2] + ((1 - a*x)^(1/8)/(1 + a*x)^(1/8) - (-ArcTan[(-Sqrt[2 + Sqrt[2]] + (2*(1 - a*x)^ (1/8))/(1 + a*x)^(1/8))/Sqrt[2 - Sqrt[2]]] - ((1 + Sqrt[2])*Log[1 + (1 - a *x)^(1/4) - (Sqrt[2 + Sqrt[2]]*(1 - a*x)^(1/8))/(1 + a*x)^(1/8)])/2)/(2*Sq rt[2 + Sqrt[2]]) - (-ArcTan[(Sqrt[2 + Sqrt[2]] + (2*(1 - a*x)^(1/8))/(1 + a*x)^(1/8))/Sqrt[2 - Sqrt[2]]] + ((1 + Sqrt[2])*Log[1 + (1 - a*x)^(1/4) + (Sqrt[2 + Sqrt[2]]*(1 - a*x)^(1/8))/(1 + a*x)^(1/8)])/2)/(2*Sqrt[2 + Sqrt[ 2]]))/(2*Sqrt[2])))/a
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[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{r = Numerator[R t[a/b, 4]], s = Denominator[Rt[a/b, 4]]}, Simp[s^3/(2*Sqrt[2]*b*r) Int[x^ (m - n/4)/(r^2 - Sqrt[2]*r*s*x^(n/4) + s^2*x^(n/2)), x], x] - Simp[s^3/(2*S qrt[2]*b*r) Int[x^(m - n/4)/(r^2 + Sqrt[2]*r*s*x^(n/4) + s^2*x^(n/2)), x] , x]] /; FreeQ[{a, b}, x] && IGtQ[n/4, 0] && IGtQ[m, 0] && LtQ[m, n - 1] && GtQ[a/b, 0]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^(p + (m + 1)/n) Subst[Int[x^m/(1 - b*x^n)^(p + (m + 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)] && IntegersQ[m, p + (m + 1)/n]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{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_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[(2*c*d - b*e)/(2*c) Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
Int[((d_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[d^3*(d*x)^(m - 3)*((a + b*x^2 + c*x^4)^(p + 1)/(c*(m + 4*p + 1))), x] - Simp[d^4/(c*(m + 4*p + 1)) Int[(d*x)^(m - 4)*Simp[a*(m - 3) + b*(m + 2*p - 1)*x^2, x]*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, p}, x ] && NeQ[b^2 - 4*a*c, 0] && GtQ[m, 3] && NeQ[m + 4*p + 1, 0] && IntegerQ[2* p] && (IntegerQ[p] || IntegerQ[m])
Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : > With[{q = Rt[a/c, 2]}, With[{r = Rt[2*q - b/c, 2]}, Simp[1/(2*c*q*r) In t[(d*r - (d - e*q)*x)/(q - r*x + x^2), x], x] + Simp[1/(2*c*q*r) Int[(d*r + (d - e*q)*x)/(q + r*x + x^2), x], x]]] /; FreeQ[{a, b, c, d, e}, x] && N eQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NegQ[b^2 - 4*a*c]
Int[E^(ArcTanh[(a_.)*(x_)]*(n_)), x_Symbol] :> Int[(1 + a*x)^(n/2)/(1 - a*x )^(n/2), x] /; FreeQ[{a, n}, x] && !IntegerQ[(n - 1)/2]
\[\int {\left (\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}^{\frac {1}{4}}d x\]
Input:
int(((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/4),x)
Output:
int(((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/4),x)
Result contains complex when optimal does not.
Time = 0.13 (sec) , antiderivative size = 419, normalized size of antiderivative = 0.94 \[ \int e^{\frac {1}{4} \text {arctanh}(a x)} \, dx=-\frac {-\left (i + 1\right ) \, \sqrt {2} a \left (-\frac {1}{a^{8}}\right )^{\frac {1}{8}} \log \left (\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} a \left (-\frac {1}{a^{8}}\right )^{\frac {1}{8}} + \left (-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}\right )^{\frac {1}{4}}\right ) + \left (i - 1\right ) \, \sqrt {2} a \left (-\frac {1}{a^{8}}\right )^{\frac {1}{8}} \log \left (-\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} a \left (-\frac {1}{a^{8}}\right )^{\frac {1}{8}} + \left (-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}\right )^{\frac {1}{4}}\right ) - \left (i - 1\right ) \, \sqrt {2} a \left (-\frac {1}{a^{8}}\right )^{\frac {1}{8}} \log \left (\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} a \left (-\frac {1}{a^{8}}\right )^{\frac {1}{8}} + \left (-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}\right )^{\frac {1}{4}}\right ) + \left (i + 1\right ) \, \sqrt {2} a \left (-\frac {1}{a^{8}}\right )^{\frac {1}{8}} \log \left (-\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} a \left (-\frac {1}{a^{8}}\right )^{\frac {1}{8}} + \left (-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}\right )^{\frac {1}{4}}\right ) - 2 \, a \left (-\frac {1}{a^{8}}\right )^{\frac {1}{8}} \log \left (a \left (-\frac {1}{a^{8}}\right )^{\frac {1}{8}} + \left (-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}\right )^{\frac {1}{4}}\right ) - 2 i \, a \left (-\frac {1}{a^{8}}\right )^{\frac {1}{8}} \log \left (i \, a \left (-\frac {1}{a^{8}}\right )^{\frac {1}{8}} + \left (-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}\right )^{\frac {1}{4}}\right ) + 2 i \, a \left (-\frac {1}{a^{8}}\right )^{\frac {1}{8}} \log \left (-i \, a \left (-\frac {1}{a^{8}}\right )^{\frac {1}{8}} + \left (-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}\right )^{\frac {1}{4}}\right ) + 2 \, a \left (-\frac {1}{a^{8}}\right )^{\frac {1}{8}} \log \left (-a \left (-\frac {1}{a^{8}}\right )^{\frac {1}{8}} + \left (-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}\right )^{\frac {1}{4}}\right ) - 8 \, {\left (a x - 1\right )} \left (-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}\right )^{\frac {1}{4}}}{8 \, a} \] Input:
integrate(((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/4),x, algorithm="fricas")
Output:
-1/8*(-(I + 1)*sqrt(2)*a*(-1/a^8)^(1/8)*log((1/2*I + 1/2)*sqrt(2)*a*(-1/a^ 8)^(1/8) + (-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4)) + (I - 1)*sqrt(2)*a*(-1/ a^8)^(1/8)*log(-(1/2*I - 1/2)*sqrt(2)*a*(-1/a^8)^(1/8) + (-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4)) - (I - 1)*sqrt(2)*a*(-1/a^8)^(1/8)*log((1/2*I - 1/2) *sqrt(2)*a*(-1/a^8)^(1/8) + (-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4)) + (I + 1)*sqrt(2)*a*(-1/a^8)^(1/8)*log(-(1/2*I + 1/2)*sqrt(2)*a*(-1/a^8)^(1/8) + (-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4)) - 2*a*(-1/a^8)^(1/8)*log(a*(-1/a^8) ^(1/8) + (-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4)) - 2*I*a*(-1/a^8)^(1/8)*log (I*a*(-1/a^8)^(1/8) + (-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4)) + 2*I*a*(-1/a ^8)^(1/8)*log(-I*a*(-1/a^8)^(1/8) + (-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4)) + 2*a*(-1/a^8)^(1/8)*log(-a*(-1/a^8)^(1/8) + (-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4)) - 8*(a*x - 1)*(-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4))/a
\[ \int e^{\frac {1}{4} \text {arctanh}(a x)} \, dx=\int \sqrt [4]{\frac {a x + 1}{\sqrt {- a^{2} x^{2} + 1}}}\, dx \] Input:
integrate(((a*x+1)/(-a**2*x**2+1)**(1/2))**(1/4),x)
Output:
Integral(((a*x + 1)/sqrt(-a**2*x**2 + 1))**(1/4), x)
\[ \int e^{\frac {1}{4} \text {arctanh}(a x)} \, dx=\int { \left (\frac {a x + 1}{\sqrt {-a^{2} x^{2} + 1}}\right )^{\frac {1}{4}} \,d x } \] Input:
integrate(((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/4),x, algorithm="maxima")
Output:
integrate(((a*x + 1)/sqrt(-a^2*x^2 + 1))^(1/4), x)
\[ \int e^{\frac {1}{4} \text {arctanh}(a x)} \, dx=\int { \left (\frac {a x + 1}{\sqrt {-a^{2} x^{2} + 1}}\right )^{\frac {1}{4}} \,d x } \] Input:
integrate(((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/4),x, algorithm="giac")
Output:
integrate(((a*x + 1)/sqrt(-a^2*x^2 + 1))^(1/4), x)
Timed out. \[ \int e^{\frac {1}{4} \text {arctanh}(a x)} \, dx=\int {\left (\frac {a\,x+1}{\sqrt {1-a^2\,x^2}}\right )}^{1/4} \,d x \] Input:
int(((a*x + 1)/(1 - a^2*x^2)^(1/2))^(1/4),x)
Output:
int(((a*x + 1)/(1 - a^2*x^2)^(1/2))^(1/4), x)
\[ \int e^{\frac {1}{4} \text {arctanh}(a x)} \, dx=\int \frac {\left (a x +1\right )^{\frac {1}{4}}}{\left (-a^{2} x^{2}+1\right )^{\frac {1}{8}}}d x \] Input:
int(((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/4),x)
Output:
int((a*x + 1)**(1/4)/( - a**2*x**2 + 1)**(1/8),x)