Optimal. Leaf size=65 \[ -\frac {3 c \sqrt {1-a^2 x^2}}{2 a}-\frac {c \left (1-a^2 x^2\right )^{3/2}}{2 a (1-a x)}+\frac {3 c \text {ArcSin}(a x)}{2 a} \]
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Rubi [A]
time = 0.03, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {6262, 679, 222}
\begin {gather*} -\frac {c \left (1-a^2 x^2\right )^{3/2}}{2 a (1-a x)}-\frac {3 c \sqrt {1-a^2 x^2}}{2 a}+\frac {3 c \text {ArcSin}(a x)}{2 a} \end {gather*}
Antiderivative was successfully verified.
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Rule 222
Rule 679
Rule 6262
Rubi steps
\begin {align*} \int e^{3 \tanh ^{-1}(a x)} (c-a c x) \, dx &=c^3 \int \frac {\left (1-a^2 x^2\right )^{3/2}}{(c-a c x)^2} \, dx\\ &=-\frac {c \left (1-a^2 x^2\right )^{3/2}}{2 a (1-a x)}+\frac {1}{2} \left (3 c^2\right ) \int \frac {\sqrt {1-a^2 x^2}}{c-a c x} \, dx\\ &=-\frac {3 c \sqrt {1-a^2 x^2}}{2 a}-\frac {c \left (1-a^2 x^2\right )^{3/2}}{2 a (1-a x)}+\frac {1}{2} (3 c) \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {3 c \sqrt {1-a^2 x^2}}{2 a}-\frac {c \left (1-a^2 x^2\right )^{3/2}}{2 a (1-a x)}+\frac {3 c \sin ^{-1}(a x)}{2 a}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 48, normalized size = 0.74 \begin {gather*} -\frac {c \left ((4+a x) \sqrt {1-a^2 x^2}+6 \text {ArcSin}\left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )\right )}{2 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. \(157\) vs.
\(2(55)=110\).
time = 0.81, size = 158, normalized size = 2.43
method | result | size |
risch | \(\frac {\left (a x +4\right ) \left (a^{2} x^{2}-1\right ) c}{2 a \sqrt {-a^{2} x^{2}+1}}+\frac {3 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right ) c}{2 \sqrt {a^{2}}}\) | \(62\) |
default | \(-c \left (a^{4} \left (-\frac {x^{3}}{2 a^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {\frac {3 x}{2 a^{2} \sqrt {-a^{2} x^{2}+1}}-\frac {3 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 a^{2} \sqrt {a^{2}}}}{a^{2}}\right )+2 a^{3} \left (-\frac {x^{2}}{a^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {2}{a^{4} \sqrt {-a^{2} x^{2}+1}}\right )-\frac {2}{a \sqrt {-a^{2} x^{2}+1}}-\frac {x}{\sqrt {-a^{2} x^{2}+1}}\right )\) | \(158\) |
meijerg | \(-\frac {2 c \left (\sqrt {\pi }-\frac {\sqrt {\pi }}{\sqrt {-a^{2} x^{2}+1}}\right )}{a \sqrt {\pi }}+\frac {c x}{\sqrt {-a^{2} x^{2}+1}}-\frac {c \left (\frac {\sqrt {\pi }\, x \left (-a^{2}\right )^{\frac {5}{2}} \left (-5 a^{2} x^{2}+15\right )}{10 a^{4} \sqrt {-a^{2} x^{2}+1}}-\frac {3 \sqrt {\pi }\, \left (-a^{2}\right )^{\frac {5}{2}} \arcsin \left (a x \right )}{2 a^{5}}\right )}{\sqrt {\pi }\, \sqrt {-a^{2}}}-\frac {2 c \left (-2 \sqrt {\pi }+\frac {\sqrt {\pi }\, \left (-4 a^{2} x^{2}+8\right )}{4 \sqrt {-a^{2} x^{2}+1}}\right )}{a \sqrt {\pi }}\) | \(160\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, size = 85, normalized size = 1.31 \begin {gather*} \frac {a^{2} c x^{3}}{2 \, \sqrt {-a^{2} x^{2} + 1}} + \frac {2 \, a c x^{2}}{\sqrt {-a^{2} x^{2} + 1}} - \frac {c x}{2 \, \sqrt {-a^{2} x^{2} + 1}} + \frac {3 \, c \arcsin \left (a x\right )}{2 \, a} - \frac {2 \, c}{\sqrt {-a^{2} x^{2} + 1} a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 52, normalized size = 0.80 \begin {gather*} -\frac {6 \, c \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + \sqrt {-a^{2} x^{2} + 1} {\left (a c x + 4 \, c\right )}}{2 \, a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 6.02, size = 165, normalized size = 2.54 \begin {gather*} a^{2} c \left (\begin {cases} - \frac {i x \sqrt {a^{2} x^{2} - 1}}{2 a^{2}} - \frac {i \operatorname {acosh}{\left (a x \right )}}{2 a^{3}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\\frac {x^{3}}{2 \sqrt {- a^{2} x^{2} + 1}} - \frac {x}{2 a^{2} \sqrt {- a^{2} x^{2} + 1}} + \frac {\operatorname {asin}{\left (a x \right )}}{2 a^{3}} & \text {otherwise} \end {cases}\right ) + 2 a c \left (\begin {cases} \frac {x^{2}}{2} & \text {for}\: a^{2} = 0 \\- \frac {\sqrt {- a^{2} x^{2} + 1}}{a^{2}} & \text {otherwise} \end {cases}\right ) + c \left (\begin {cases} \sqrt {\frac {1}{a^{2}}} \operatorname {asin}{\left (x \sqrt {a^{2}} \right )} & \text {for}\: a^{2} > 0 \\\sqrt {- \frac {1}{a^{2}}} \operatorname {asinh}{\left (x \sqrt {- a^{2}} \right )} & \text {for}\: a^{2} < 0 \end {cases}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.43, size = 38, normalized size = 0.58 \begin {gather*} \frac {3 \, c \arcsin \left (a x\right ) \mathrm {sgn}\left (a\right )}{2 \, {\left | a \right |}} - \frac {1}{2} \, \sqrt {-a^{2} x^{2} + 1} {\left (c x + \frac {4 \, c}{a}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.03, size = 58, normalized size = 0.89 \begin {gather*} \frac {\frac {3\,c\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{2}+\sqrt {1-a^2\,x^2}\,\left (\frac {2\,a\,c}{\sqrt {-a^2}}-\frac {c\,x\,\sqrt {-a^2}}{2}\right )}{\sqrt {-a^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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