Optimal. Leaf size=75 \[ -\frac {2 (1-a x)}{5 a \left (c-a^2 c x^2\right )^{5/2}}+\frac {x}{5 c \left (c-a^2 c x^2\right )^{3/2}}+\frac {2 x}{5 c^2 \sqrt {c-a^2 c x^2}} \]
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Rubi [A]
time = 0.05, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6277, 667, 198,
197} \begin {gather*} \frac {2 x}{5 c^2 \sqrt {c-a^2 c x^2}}+\frac {x}{5 c \left (c-a^2 c x^2\right )^{3/2}}-\frac {2 (1-a x)}{5 a \left (c-a^2 c x^2\right )^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 197
Rule 198
Rule 667
Rule 6277
Rubi steps
\begin {align*} \int \frac {e^{-2 \tanh ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{5/2}} \, dx &=c \int \frac {(1-a x)^2}{\left (c-a^2 c x^2\right )^{7/2}} \, dx\\ &=-\frac {2 (1-a x)}{5 a \left (c-a^2 c x^2\right )^{5/2}}+\frac {3}{5} \int \frac {1}{\left (c-a^2 c x^2\right )^{5/2}} \, dx\\ &=-\frac {2 (1-a x)}{5 a \left (c-a^2 c x^2\right )^{5/2}}+\frac {x}{5 c \left (c-a^2 c x^2\right )^{3/2}}+\frac {2 \int \frac {1}{\left (c-a^2 c x^2\right )^{3/2}} \, dx}{5 c}\\ &=-\frac {2 (1-a x)}{5 a \left (c-a^2 c x^2\right )^{5/2}}+\frac {x}{5 c \left (c-a^2 c x^2\right )^{3/2}}+\frac {2 x}{5 c^2 \sqrt {c-a^2 c x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 79, normalized size = 1.05 \begin {gather*} \frac {\sqrt {1-a^2 x^2} \left (-2+a x+4 a^2 x^2+2 a^3 x^3\right )}{5 a c^2 \sqrt {1-a x} (1+a x)^{5/2} \sqrt {c-a^2 c x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(187\) vs.
\(2(63)=126\).
time = 0.06, size = 188, normalized size = 2.51
method | result | size |
gosper | \(\frac {\left (a x -1\right )^{2} \left (2 a^{3} x^{3}+4 a^{2} x^{2}+a x -2\right )}{5 \left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}} a}\) | \(47\) |
trager | \(-\frac {\left (2 a^{3} x^{3}+4 a^{2} x^{2}+a x -2\right ) \sqrt {-a^{2} c \,x^{2}+c}}{5 c^{3} \left (a x +1\right )^{3} a \left (a x -1\right )}\) | \(57\) |
default | \(-\frac {x}{3 c \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}-\frac {2 x}{3 c^{2} \sqrt {-a^{2} c \,x^{2}+c}}+\frac {-\frac {2}{5 a c \left (x +\frac {1}{a}\right ) \left (-c \,a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a c \left (x +\frac {1}{a}\right )\right )^{\frac {3}{2}}}+\frac {8 a \left (-\frac {-2 a^{2} c \left (x +\frac {1}{a}\right )+2 a c}{6 a^{2} c^{2} \left (-c \,a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a c \left (x +\frac {1}{a}\right )\right )^{\frac {3}{2}}}-\frac {-2 a^{2} c \left (x +\frac {1}{a}\right )+2 a c}{3 a^{2} c^{3} \sqrt {-c \,a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a c \left (x +\frac {1}{a}\right )}}\right )}{5}}{a}\) | \(188\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 79, normalized size = 1.05 \begin {gather*} -\frac {2}{5 \, {\left ({\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} a^{2} c x + {\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} a c\right )}} + \frac {2 \, x}{5 \, \sqrt {-a^{2} c x^{2} + c} c^{2}} + \frac {x}{5 \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 75, normalized size = 1.00 \begin {gather*} -\frac {{\left (2 \, a^{3} x^{3} + 4 \, a^{2} x^{2} + a x - 2\right )} \sqrt {-a^{2} c x^{2} + c}}{5 \, {\left (a^{5} c^{3} x^{4} + 2 \, a^{4} c^{3} x^{3} - 2 \, a^{2} c^{3} x - a c^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \frac {a x}{a^{5} c^{2} x^{5} \sqrt {- a^{2} c x^{2} + c} + a^{4} c^{2} x^{4} \sqrt {- a^{2} c x^{2} + c} - 2 a^{3} c^{2} x^{3} \sqrt {- a^{2} c x^{2} + c} - 2 a^{2} c^{2} x^{2} \sqrt {- a^{2} c x^{2} + c} + a c^{2} x \sqrt {- a^{2} c x^{2} + c} + c^{2} \sqrt {- a^{2} c x^{2} + c}}\, dx - \int \left (- \frac {1}{a^{5} c^{2} x^{5} \sqrt {- a^{2} c x^{2} + c} + a^{4} c^{2} x^{4} \sqrt {- a^{2} c x^{2} + c} - 2 a^{3} c^{2} x^{3} \sqrt {- a^{2} c x^{2} + c} - 2 a^{2} c^{2} x^{2} \sqrt {- a^{2} c x^{2} + c} + a c^{2} x \sqrt {- a^{2} c x^{2} + c} + c^{2} \sqrt {- a^{2} c x^{2} + c}}\right )\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 220 vs.
\(2 (62) = 124\).
time = 0.43, size = 220, normalized size = 2.93 \begin {gather*} \frac {a^{3} {\left (\frac {5}{a^{3} c^{2} \sqrt {-c + \frac {2 \, c}{a x + 1}} \mathrm {sgn}\left (\frac {1}{a x + 1}\right ) \mathrm {sgn}\left (a\right )} - \frac {a^{12} {\left (c - \frac {2 \, c}{a x + 1}\right )}^{2} c^{20} \sqrt {-c + \frac {2 \, c}{a x + 1}} \mathrm {sgn}\left (\frac {1}{a x + 1}\right )^{4} \mathrm {sgn}\left (a\right )^{4} + 15 \, a^{12} c^{22} \sqrt {-c + \frac {2 \, c}{a x + 1}} \mathrm {sgn}\left (\frac {1}{a x + 1}\right )^{4} \mathrm {sgn}\left (a\right )^{4} + 5 \, a^{12} c^{21} {\left (-c + \frac {2 \, c}{a x + 1}\right )}^{\frac {3}{2}} \mathrm {sgn}\left (\frac {1}{a x + 1}\right )^{4} \mathrm {sgn}\left (a\right )^{4}}{a^{15} c^{25} \mathrm {sgn}\left (\frac {1}{a x + 1}\right )^{5} \mathrm {sgn}\left (a\right )^{5}}\right )} - \frac {16 \, \mathrm {sgn}\left (\frac {1}{a x + 1}\right ) \mathrm {sgn}\left (a\right )}{\sqrt {-c} c^{2}}}{40 \, {\left | a \right |}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.09, size = 56, normalized size = 0.75 \begin {gather*} -\frac {\sqrt {c-a^2\,c\,x^2}\,\left (2\,a^3\,x^3+4\,a^2\,x^2+a\,x-2\right )}{5\,a\,c^3\,\left (a\,x-1\right )\,{\left (a\,x+1\right )}^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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