Optimal. Leaf size=76 \[ c \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2} x-\frac {3 c \csc ^{-1}(a x)}{a}+\frac {3 c \tanh ^{-1}\left (\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}\right )}{a} \]
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Rubi [A]
time = 0.04, antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 8, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {6329, 100, 12,
132, 41, 222, 94, 214} \begin {gather*} c x \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/2}-\frac {3 c \csc ^{-1}(a x)}{a}+\frac {3 c \tanh ^{-1}\left (\sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )}{a} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 41
Rule 94
Rule 100
Rule 132
Rule 214
Rule 222
Rule 6329
Rubi steps
\begin {align*} \int e^{3 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right ) \, dx &=-\left (c \text {Subst}\left (\int \frac {\left (1+\frac {x}{a}\right )^{5/2}}{x^2 \sqrt {1-\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )\right )\\ &=c \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2} x+c \text {Subst}\left (\int -\frac {3 \sqrt {1+\frac {x}{a}}}{a x \sqrt {1-\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )\\ &=c \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2} x-\frac {(3 c) \text {Subst}\left (\int \frac {\sqrt {1+\frac {x}{a}}}{x \sqrt {1-\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{a}\\ &=c \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2} x-\frac {(3 c) \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x}{a}} \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{a^2}-\frac {(3 c) \text {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x}{a}} \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{a}\\ &=c \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2} x-\frac {(3 c) \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{a^2}+\frac {(3 c) \text {Subst}\left (\int \frac {1}{\frac {1}{a}-\frac {x^2}{a}} \, dx,x,\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}\right )}{a^2}\\ &=c \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2} x-\frac {3 c \csc ^{-1}(a x)}{a}+\frac {3 c \tanh ^{-1}\left (\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}\right )}{a}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 57, normalized size = 0.75 \begin {gather*} \frac {c \left (\sqrt {1-\frac {1}{a^2 x^2}} (1+a x)-3 \text {ArcSin}\left (\frac {1}{a x}\right )+3 \log \left (\left (1+\sqrt {1-\frac {1}{a^2 x^2}}\right ) x\right )\right )}{a} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(234\) vs.
\(2(68)=136\).
time = 0.10, size = 235, normalized size = 3.09
method | result | size |
risch | \(\frac {\left (a x -1\right ) c}{x \,a^{2} \sqrt {\frac {a x -1}{a x +1}}}+\frac {\left (a \sqrt {\left (a x +1\right ) \left (a x -1\right )}+\frac {3 a^{2} \ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}-1}\right )}{\sqrt {a^{2}}}-3 a \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )\right ) c \sqrt {\left (a x +1\right ) \left (a x -1\right )}}{a^{2} \left (a x +1\right ) \sqrt {\frac {a x -1}{a x +1}}}\) | \(134\) |
default | \(-\frac {\left (a x -1\right )^{2} c \left (-\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a^{2} x^{2}+\left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}+3 \sqrt {a^{2}}\, \sqrt {a^{2} x^{2}-1}\, a x +\ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{2} x +3 \sqrt {a^{2}}\, \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right ) a x -4 \sqrt {\left (a x +1\right ) \left (a x -1\right )}\, \sqrt {a^{2}}\, a x -4 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}}{\sqrt {a^{2}}}\right ) a^{2} x \right )}{\left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (a x +1\right ) \sqrt {\left (a x +1\right ) \left (a x -1\right )}\, a^{2} x \sqrt {a^{2}}}\) | \(235\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.47, size = 118, normalized size = 1.55 \begin {gather*} -a {\left (\frac {4 \, c \sqrt {\frac {a x - 1}{a x + 1}}}{\frac {{\left (a x - 1\right )}^{2} a^{2}}{{\left (a x + 1\right )}^{2}} - a^{2}} - \frac {6 \, c \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right )}{a^{2}} - \frac {3 \, c \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2}} + \frac {3 \, c \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 106, normalized size = 1.39 \begin {gather*} \frac {6 \, a c x \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right ) + 3 \, a c x \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 3 \, a c x \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) + {\left (a^{2} c x^{2} + 2 \, a c x + c\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{a^{2} x} \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*} \frac {c \left (\int \frac {a^{2}}{\frac {a x \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} - \frac {\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}}\, dx + \int \left (- \frac {1}{\frac {a x^{3} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} - \frac {x^{2} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}}\right )\, dx\right )}{a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.43, size = 130, normalized size = 1.71 \begin {gather*} \frac {6 \, c \arctan \left (-x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1}\right )}{a \mathrm {sgn}\left (a x + 1\right )} - \frac {3 \, c \log \left ({\left | -x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1} \right |}\right )}{{\left | a \right |} \mathrm {sgn}\left (a x + 1\right )} + \frac {\sqrt {a^{2} x^{2} - 1} c}{a \mathrm {sgn}\left (a x + 1\right )} + \frac {2 \, c}{{\left ({\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{2} + 1\right )} {\left | a \right |} \mathrm {sgn}\left (a x + 1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.09, size = 84, normalized size = 1.11 \begin {gather*} \frac {6\,c\,\mathrm {atan}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a}+\frac {6\,c\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a}+\frac {4\,c\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{a-\frac {a\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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