Optimal. Leaf size=124 \[ \frac {(1-a x)^2}{3 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x}-\frac {2 (1-a x)^2 (1+a x) (5+2 a x)}{3 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3}-\frac {2 (1-a x)^{3/2} (1+a x)^{3/2} \text {ArcSin}(a x)}{a^4 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.25, antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6294, 6264,
100, 148, 41, 222} \begin {gather*} \frac {(1-a x)^2}{3 a^2 x \left (c-\frac {c}{a^2 x^2}\right )^{3/2}}-\frac {2 (a x+1)^{3/2} (1-a x)^{3/2} \text {ArcSin}(a x)}{a^4 x^3 \left (c-\frac {c}{a^2 x^2}\right )^{3/2}}-\frac {2 (a x+1) (2 a x+5) (1-a x)^2}{3 a^4 x^3 \left (c-\frac {c}{a^2 x^2}\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 41
Rule 100
Rule 148
Rule 222
Rule 6264
Rule 6294
Rubi steps
\begin {align*} \int \frac {e^{-2 \tanh ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{3/2}} \, dx &=\frac {\left ((1-a x)^{3/2} (1+a x)^{3/2}\right ) \int \frac {e^{-2 \tanh ^{-1}(a x)} x^3}{(1-a x)^{3/2} (1+a x)^{3/2}} \, dx}{\left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3}\\ &=\frac {\left ((1-a x)^{3/2} (1+a x)^{3/2}\right ) \int \frac {x^3}{\sqrt {1-a x} (1+a x)^{5/2}} \, dx}{\left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3}\\ &=\frac {(1-a x)^2}{3 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x}-\frac {\left ((1-a x)^{3/2} (1+a x)^{3/2}\right ) \int \frac {x (2-4 a x)}{\sqrt {1-a x} (1+a x)^{3/2}} \, dx}{3 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3}\\ &=\frac {(1-a x)^2}{3 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x}-\frac {2 (1-a x)^2 (1+a x) (5+2 a x)}{3 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3}-\frac {\left (2 (1-a x)^{3/2} (1+a x)^{3/2}\right ) \int \frac {1}{\sqrt {1-a x} \sqrt {1+a x}} \, dx}{a^3 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3}\\ &=\frac {(1-a x)^2}{3 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x}-\frac {2 (1-a x)^2 (1+a x) (5+2 a x)}{3 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3}-\frac {\left (2 (1-a x)^{3/2} (1+a x)^{3/2}\right ) \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{a^3 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3}\\ &=\frac {(1-a x)^2}{3 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x}-\frac {2 (1-a x)^2 (1+a x) (5+2 a x)}{3 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3}-\frac {2 (1-a x)^{3/2} (1+a x)^{3/2} \sin ^{-1}(a x)}{a^4 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.10, size = 95, normalized size = 0.77 \begin {gather*} \frac {10+4 a x-11 a^2 x^2-3 a^3 x^3+6 (1+a x) \sqrt {-1+a^2 x^2} \log \left (a x+\sqrt {-1+a^2 x^2}\right )}{3 a^2 c \sqrt {c-\frac {c}{a^2 x^2}} x (1+a x)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(325\) vs.
\(2(110)=220\).
time = 0.07, size = 326, normalized size = 2.63
method | result | size |
risch | \(-\frac {a^{2} x^{2}-1}{a^{2} c x \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}}-\frac {\left (-\frac {2 \ln \left (\frac {c \,a^{2} x}{\sqrt {c \,a^{2}}}+\sqrt {a^{2} c \,x^{2}-c}\right )}{a^{3} \sqrt {c \,a^{2}}}-\frac {\sqrt {c \,a^{2} \left (x +\frac {1}{a}\right )^{2}-2 a c \left (x +\frac {1}{a}\right )}}{3 a^{6} c \left (x +\frac {1}{a}\right )^{2}}+\frac {8 \sqrt {c \,a^{2} \left (x +\frac {1}{a}\right )^{2}-2 a c \left (x +\frac {1}{a}\right )}}{3 a^{5} c \left (x +\frac {1}{a}\right )}\right ) a^{2} \sqrt {c \left (a^{2} x^{2}-1\right )}}{c x \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}}\) | \(204\) |
default | \(\frac {\left (-3 c^{\frac {3}{2}} \sqrt {\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}}\, a^{3} x^{3}+4 \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, c^{\frac {3}{2}} a^{2} x^{2}-15 x^{2} a^{2} c^{\frac {3}{2}} \sqrt {\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}}+6 \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, \ln \left (\sqrt {c}\, x +\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\right ) \sqrt {\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}}\, a^{2} c x +4 \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, c^{\frac {3}{2}} a x +6 \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, \ln \left (\sqrt {c}\, x +\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\right ) \sqrt {\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}}\, a c -2 \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, c^{\frac {3}{2}}+12 c^{\frac {3}{2}} \sqrt {\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}}\right ) \left (a x -1\right )}{3 \sqrt {\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}}\, x^{3} \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}\right )^{\frac {3}{2}} a^{4} c^{\frac {3}{2}}}\) | \(326\) |
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 [A]
time = 0.43, size = 280, normalized size = 2.26 \begin {gather*} \left [\frac {3 \, {\left (a^{2} x^{2} + 2 \, a x + 1\right )} \sqrt {c} \log \left (2 \, a^{2} c x^{2} + 2 \, a^{2} \sqrt {c} x^{2} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - c\right ) - {\left (3 \, a^{3} x^{3} + 14 \, a^{2} x^{2} + 10 \, a x\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{3 \, {\left (a^{3} c^{2} x^{2} + 2 \, a^{2} c^{2} x + a c^{2}\right )}}, -\frac {6 \, {\left (a^{2} x^{2} + 2 \, a x + 1\right )} \sqrt {-c} \arctan \left (\frac {a^{2} \sqrt {-c} x^{2} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right ) + {\left (3 \, a^{3} x^{3} + 14 \, a^{2} x^{2} + 10 \, a x\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{3 \, {\left (a^{3} c^{2} x^{2} + 2 \, a^{2} c^{2} x + a c^{2}\right )}}\right ] \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 \frac {a x}{a c x \sqrt {c - \frac {c}{a^{2} x^{2}}} + c \sqrt {c - \frac {c}{a^{2} x^{2}}} - \frac {c \sqrt {c - \frac {c}{a^{2} x^{2}}}}{a x} - \frac {c \sqrt {c - \frac {c}{a^{2} x^{2}}}}{a^{2} x^{2}}}\, dx - \int \left (- \frac {1}{a c x \sqrt {c - \frac {c}{a^{2} x^{2}}} + c \sqrt {c - \frac {c}{a^{2} x^{2}}} - \frac {c \sqrt {c - \frac {c}{a^{2} x^{2}}}}{a x} - \frac {c \sqrt {c - \frac {c}{a^{2} x^{2}}}}{a^{2} x^{2}}}\right )\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} -\int \frac {a^2\,x^2-1}{{\left (c-\frac {c}{a^2\,x^2}\right )}^{3/2}\,{\left (a\,x+1\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________