Optimal. Leaf size=59 \[ \frac {3}{8} c^3 x \sqrt {1-a^2 x^2}+\frac {1}{4} c^3 x \left (1-a^2 x^2\right )^{3/2}+\frac {3 c^3 \text {ArcSin}(a x)}{8 a} \]
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Rubi [A]
time = 0.03, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6262, 201, 222}
\begin {gather*} \frac {1}{4} c^3 x \left (1-a^2 x^2\right )^{3/2}+\frac {3}{8} c^3 x \sqrt {1-a^2 x^2}+\frac {3 c^3 \text {ArcSin}(a x)}{8 a} \end {gather*}
Antiderivative was successfully verified.
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Rule 201
Rule 222
Rule 6262
Rubi steps
\begin {align*} \int e^{3 \tanh ^{-1}(a x)} (c-a c x)^3 \, dx &=c^3 \int \left (1-a^2 x^2\right )^{3/2} \, dx\\ &=\frac {1}{4} c^3 x \left (1-a^2 x^2\right )^{3/2}+\frac {1}{4} \left (3 c^3\right ) \int \sqrt {1-a^2 x^2} \, dx\\ &=\frac {3}{8} c^3 x \sqrt {1-a^2 x^2}+\frac {1}{4} c^3 x \left (1-a^2 x^2\right )^{3/2}+\frac {1}{8} \left (3 c^3\right ) \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx\\ &=\frac {3}{8} c^3 x \sqrt {1-a^2 x^2}+\frac {1}{4} c^3 x \left (1-a^2 x^2\right )^{3/2}+\frac {3 c^3 \sin ^{-1}(a x)}{8 a}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 44, normalized size = 0.75 \begin {gather*} \frac {c^3 \left (a x \left (5-2 a^2 x^2\right ) \sqrt {1-a^2 x^2}+3 \text {ArcSin}(a x)\right )}{8 a} \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. \(257\) vs.
\(2(49)=98\).
time = 1.13, size = 258, normalized size = 4.37
method | result | size |
risch | \(\frac {x \left (2 a^{2} x^{2}-5\right ) \left (a^{2} x^{2}-1\right ) c^{3}}{8 \sqrt {-a^{2} x^{2}+1}}+\frac {3 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right ) c^{3}}{8 \sqrt {a^{2}}}\) | \(69\) |
meijerg | \(\frac {3 c^{3} \left (\frac {\sqrt {\pi }\, x \left (-a^{2}\right )^{\frac {3}{2}}}{a^{2} \sqrt {-a^{2} x^{2}+1}}-\frac {\sqrt {\pi }\, \left (-a^{2}\right )^{\frac {3}{2}} \arcsin \left (a x \right )}{a^{3}}\right )}{\sqrt {\pi }\, \sqrt {-a^{2}}}+\frac {c^{3} x}{\sqrt {-a^{2} x^{2}+1}}+\frac {c^{3} \left (\frac {\sqrt {\pi }\, x \left (-a^{2}\right )^{\frac {7}{2}} \left (-14 a^{4} x^{4}-35 a^{2} x^{2}+105\right )}{56 a^{6} \sqrt {-a^{2} x^{2}+1}}-\frac {15 \sqrt {\pi }\, \left (-a^{2}\right )^{\frac {7}{2}} \arcsin \left (a x \right )}{8 a^{7}}\right )}{\sqrt {\pi }\, \sqrt {-a^{2}}}+\frac {3 c^{3} \left (\frac {\sqrt {\pi }\, x \left (-a^{2}\right )^{\frac {5}{2}} \left (-5 a^{2} x^{2}+15\right )}{10 a^{4} \sqrt {-a^{2} x^{2}+1}}-\frac {3 \sqrt {\pi }\, \left (-a^{2}\right )^{\frac {5}{2}} \arcsin \left (a x \right )}{2 a^{5}}\right )}{\sqrt {\pi }\, \sqrt {-a^{2}}}\) | \(234\) |
default | \(-c^{3} \left (a^{6} \left (-\frac {x^{5}}{4 a^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {-\frac {5 x^{3}}{8 a^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {5 \left (\frac {3 x}{2 a^{2} \sqrt {-a^{2} x^{2}+1}}-\frac {3 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 a^{2} \sqrt {a^{2}}}\right )}{4 a^{2}}}{a^{2}}\right )-3 a^{4} \left (-\frac {x^{3}}{2 a^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {\frac {3 x}{2 a^{2} \sqrt {-a^{2} x^{2}+1}}-\frac {3 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 a^{2} \sqrt {a^{2}}}}{a^{2}}\right )+3 a^{2} \left (\frac {x}{a^{2} \sqrt {-a^{2} x^{2}+1}}-\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{a^{2} \sqrt {a^{2}}}\right )-\frac {x}{\sqrt {-a^{2} x^{2}+1}}\right )\) | \(258\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.47, size = 77, normalized size = 1.31 \begin {gather*} \frac {a^{4} c^{3} x^{5}}{4 \, \sqrt {-a^{2} x^{2} + 1}} - \frac {7 \, a^{2} c^{3} x^{3}}{8 \, \sqrt {-a^{2} x^{2} + 1}} + \frac {5 \, c^{3} x}{8 \, \sqrt {-a^{2} x^{2} + 1}} + \frac {3 \, c^{3} \arcsin \left (a x\right )}{8 \, a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 65, normalized size = 1.10 \begin {gather*} -\frac {6 \, c^{3} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + {\left (2 \, a^{3} c^{3} x^{3} - 5 \, a c^{3} x\right )} \sqrt {-a^{2} x^{2} + 1}}{8 \, a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 5.83, size = 301, normalized size = 5.10 \begin {gather*} a^{4} c^{3} \left (\begin {cases} - \frac {i x^{5}}{4 \sqrt {a^{2} x^{2} - 1}} - \frac {i x^{3}}{8 a^{2} \sqrt {a^{2} x^{2} - 1}} + \frac {3 i x}{8 a^{4} \sqrt {a^{2} x^{2} - 1}} - \frac {3 i \operatorname {acosh}{\left (a x \right )}}{8 a^{5}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\\frac {x^{5}}{4 \sqrt {- a^{2} x^{2} + 1}} + \frac {x^{3}}{8 a^{2} \sqrt {- a^{2} x^{2} + 1}} - \frac {3 x}{8 a^{4} \sqrt {- a^{2} x^{2} + 1}} + \frac {3 \operatorname {asin}{\left (a x \right )}}{8 a^{5}} & \text {otherwise} \end {cases}\right ) - 2 a^{2} c^{3} \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 ) + c^{3} \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 = 48, normalized size = 0.81 \begin {gather*} \frac {3 \, c^{3} \arcsin \left (a x\right ) \mathrm {sgn}\left (a\right )}{8 \, {\left | a \right |}} - \frac {1}{8} \, {\left (2 \, a^{2} c^{3} x^{2} - 5 \, c^{3}\right )} \sqrt {-a^{2} x^{2} + 1} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.05, size = 64, normalized size = 1.08 \begin {gather*} \frac {5\,c^3\,x\,\sqrt {1-a^2\,x^2}}{8}+\frac {3\,c^3\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{8\,\sqrt {-a^2}}-\frac {a^2\,c^3\,x^3\,\sqrt {1-a^2\,x^2}}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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