Optimal. Leaf size=255 \[ \frac {3 \sqrt [4]{1-a x} (1+a x)^{3/4}}{4 a}-\frac {(1-a x)^{5/4} (1+a x)^{3/4}}{2 a}+\frac {3 \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{4 \sqrt {2} a}-\frac {3 \text {ArcTan}\left (1+\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{4 \sqrt {2} a}+\frac {3 \log \left (1+\frac {\sqrt {1-a x}}{\sqrt {1+a x}}-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{8 \sqrt {2} a}-\frac {3 \log \left (1+\frac {\sqrt {1-a x}}{\sqrt {1+a x}}+\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{8 \sqrt {2} a} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.13, antiderivative size = 255, normalized size of antiderivative = 1.00, number of steps
used = 14, number of rules used = 10, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {6275, 52, 65,
246, 217, 1179, 642, 1176, 631, 210} \begin {gather*} \frac {3 \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )}{4 \sqrt {2} a}-\frac {3 \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{4 \sqrt {2} a}-\frac {(a x+1)^{3/4} (1-a x)^{5/4}}{2 a}+\frac {3 (a x+1)^{3/4} \sqrt [4]{1-a x}}{4 a}+\frac {3 \log \left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{8 \sqrt {2} a}-\frac {3 \log \left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}+\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{8 \sqrt {2} a} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 52
Rule 65
Rule 210
Rule 217
Rule 246
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 6275
Rubi steps
\begin {align*} \int e^{\frac {1}{2} \tanh ^{-1}(a x)} \sqrt {1-a^2 x^2} \, dx &=\int \sqrt [4]{1-a x} (1+a x)^{3/4} \, dx\\ &=-\frac {(1-a x)^{5/4} (1+a x)^{3/4}}{2 a}+\frac {3}{4} \int \frac {\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}} \, dx\\ &=\frac {3 \sqrt [4]{1-a x} (1+a x)^{3/4}}{4 a}-\frac {(1-a x)^{5/4} (1+a x)^{3/4}}{2 a}+\frac {3}{8} \int \frac {1}{(1-a x)^{3/4} \sqrt [4]{1+a x}} \, dx\\ &=\frac {3 \sqrt [4]{1-a x} (1+a x)^{3/4}}{4 a}-\frac {(1-a x)^{5/4} (1+a x)^{3/4}}{2 a}-\frac {3 \text {Subst}\left (\int \frac {1}{\sqrt [4]{2-x^4}} \, dx,x,\sqrt [4]{1-a x}\right )}{2 a}\\ &=\frac {3 \sqrt [4]{1-a x} (1+a x)^{3/4}}{4 a}-\frac {(1-a x)^{5/4} (1+a x)^{3/4}}{2 a}-\frac {3 \text {Subst}\left (\int \frac {1}{1+x^4} \, dx,x,\frac {\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{2 a}\\ &=\frac {3 \sqrt [4]{1-a x} (1+a x)^{3/4}}{4 a}-\frac {(1-a x)^{5/4} (1+a x)^{3/4}}{2 a}-\frac {3 \text {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\frac {\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{4 a}-\frac {3 \text {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\frac {\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{4 a}\\ &=\frac {3 \sqrt [4]{1-a x} (1+a x)^{3/4}}{4 a}-\frac {(1-a x)^{5/4} (1+a x)^{3/4}}{2 a}-\frac {3 \text {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\frac {\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{8 a}-\frac {3 \text {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\frac {\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{8 a}+\frac {3 \text {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\frac {\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{8 \sqrt {2} a}+\frac {3 \text {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\frac {\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{8 \sqrt {2} a}\\ &=\frac {3 \sqrt [4]{1-a x} (1+a x)^{3/4}}{4 a}-\frac {(1-a x)^{5/4} (1+a x)^{3/4}}{2 a}+\frac {3 \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 )}{8 \sqrt {2} a}-\frac {3 \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 )}{8 \sqrt {2} a}-\frac {3 \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{4 \sqrt {2} a}+\frac {3 \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{4 \sqrt {2} a}\\ &=\frac {3 \sqrt [4]{1-a x} (1+a x)^{3/4}}{4 a}-\frac {(1-a x)^{5/4} (1+a x)^{3/4}}{2 a}+\frac {3 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{4 \sqrt {2} a}-\frac {3 \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{4 \sqrt {2} a}+\frac {3 \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 )}{8 \sqrt {2} a}-\frac {3 \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 )}{8 \sqrt {2} a}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 0.01, size = 42, normalized size = 0.16 \begin {gather*} -\frac {4\ 2^{3/4} (1-a x)^{5/4} \, _2F_1\left (-\frac {3}{4},\frac {5}{4};\frac {9}{4};\frac {1}{2} (1-a x)\right )}{5 a} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \sqrt {\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}}\, \sqrt {-a^{2} x^{2}+1}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 525 vs.
\(2 (194) = 388\).
time = 0.38, size = 525, normalized size = 2.06 \begin {gather*} -\frac {12 \, \sqrt {2} a \frac {1}{a^{4}}^{\frac {1}{4}} \arctan \left (\sqrt {2} a \sqrt {\frac {\sqrt {2} {\left (a^{4} x - a^{3}\right )} \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} \frac {1}{a^{4}}^{\frac {3}{4}} + {\left (a^{3} x - a^{2}\right )} \sqrt {\frac {1}{a^{4}}} - \sqrt {-a^{2} x^{2} + 1}}{a x - 1}} \frac {1}{a^{4}}^{\frac {1}{4}} - \sqrt {2} a \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} \frac {1}{a^{4}}^{\frac {1}{4}} - 1\right ) + 12 \, \sqrt {2} a \frac {1}{a^{4}}^{\frac {1}{4}} \arctan \left (\sqrt {2} a \sqrt {-\frac {\sqrt {2} {\left (a^{4} x - a^{3}\right )} \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} \frac {1}{a^{4}}^{\frac {3}{4}} - {\left (a^{3} x - a^{2}\right )} \sqrt {\frac {1}{a^{4}}} + \sqrt {-a^{2} x^{2} + 1}}{a x - 1}} \frac {1}{a^{4}}^{\frac {1}{4}} - \sqrt {2} a \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} \frac {1}{a^{4}}^{\frac {1}{4}} + 1\right ) + 3 \, \sqrt {2} a \frac {1}{a^{4}}^{\frac {1}{4}} \log \left (\frac {\sqrt {2} {\left (a^{4} x - a^{3}\right )} \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} \frac {1}{a^{4}}^{\frac {3}{4}} + {\left (a^{3} x - a^{2}\right )} \sqrt {\frac {1}{a^{4}}} - \sqrt {-a^{2} x^{2} + 1}}{a x - 1}\right ) - 3 \, \sqrt {2} a \frac {1}{a^{4}}^{\frac {1}{4}} \log \left (-\frac {\sqrt {2} {\left (a^{4} x - a^{3}\right )} \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} \frac {1}{a^{4}}^{\frac {3}{4}} - {\left (a^{3} x - a^{2}\right )} \sqrt {\frac {1}{a^{4}}} + \sqrt {-a^{2} x^{2} + 1}}{a x - 1}\right ) - 4 \, \sqrt {-a^{2} x^{2} + 1} {\left (2 \, a x + 1\right )} \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}}}{16 \, a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {\frac {a x + 1}{\sqrt {- a^{2} x^{2} + 1}}} \sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \sqrt {1-a^2\,x^2}\,\sqrt {\frac {a\,x+1}{\sqrt {1-a^2\,x^2}}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________