Optimal. Leaf size=83 \[ \frac {3}{8} c^4 x \sqrt {1-a^2 x^2}+\frac {1}{4} c^4 x \left (1-a^2 x^2\right )^{3/2}+\frac {c^4 \left (1-a^2 x^2\right )^{5/2}}{5 a}+\frac {3 c^4 \text {ArcSin}(a x)}{8 a} \]
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Rubi [A]
time = 0.04, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {6262, 655, 201,
222} \begin {gather*} \frac {c^4 \left (1-a^2 x^2\right )^{5/2}}{5 a}+\frac {1}{4} c^4 x \left (1-a^2 x^2\right )^{3/2}+\frac {3}{8} c^4 x \sqrt {1-a^2 x^2}+\frac {3 c^4 \text {ArcSin}(a x)}{8 a} \end {gather*}
Antiderivative was successfully verified.
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Rule 201
Rule 222
Rule 655
Rule 6262
Rubi steps
\begin {align*} \int e^{3 \tanh ^{-1}(a x)} (c-a c x)^4 \, dx &=c^3 \int (c-a c x) \left (1-a^2 x^2\right )^{3/2} \, dx\\ &=\frac {c^4 \left (1-a^2 x^2\right )^{5/2}}{5 a}+c^4 \int \left (1-a^2 x^2\right )^{3/2} \, dx\\ &=\frac {1}{4} c^4 x \left (1-a^2 x^2\right )^{3/2}+\frac {c^4 \left (1-a^2 x^2\right )^{5/2}}{5 a}+\frac {1}{4} \left (3 c^4\right ) \int \sqrt {1-a^2 x^2} \, dx\\ &=\frac {3}{8} c^4 x \sqrt {1-a^2 x^2}+\frac {1}{4} c^4 x \left (1-a^2 x^2\right )^{3/2}+\frac {c^4 \left (1-a^2 x^2\right )^{5/2}}{5 a}+\frac {1}{8} \left (3 c^4\right ) \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx\\ &=\frac {3}{8} c^4 x \sqrt {1-a^2 x^2}+\frac {1}{4} c^4 x \left (1-a^2 x^2\right )^{3/2}+\frac {c^4 \left (1-a^2 x^2\right )^{5/2}}{5 a}+\frac {3 c^4 \sin ^{-1}(a x)}{8 a}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 75, normalized size = 0.90 \begin {gather*} \frac {c^4 \left (\sqrt {1-a^2 x^2} \left (8+25 a x-16 a^2 x^2-10 a^3 x^3+8 a^4 x^4\right )-30 \text {ArcSin}\left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )\right )}{40 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. \(479\) vs.
\(2(69)=138\).
time = 1.20, size = 480, normalized size = 5.78
method | result | size |
risch | \(-\frac {\left (8 a^{4} x^{4}-10 a^{3} x^{3}-16 a^{2} x^{2}+25 a x +8\right ) \left (a^{2} x^{2}-1\right ) c^{4}}{40 a \sqrt {-a^{2} x^{2}+1}}+\frac {3 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right ) c^{4}}{8 \sqrt {a^{2}}}\) | \(91\) |
meijerg | \(\frac {3 c^{4} \left (\frac {\sqrt {\pi }\, x \left (-a^{2}\right )^{\frac {5}{2}} \left (-5 a^{2} x^{2}+15\right )}{10 a^{4} \sqrt {-a^{2} x^{2}+1}}-\frac {3 \sqrt {\pi }\, \left (-a^{2}\right )^{\frac {5}{2}} \arcsin \left (a x \right )}{2 a^{5}}\right )}{\sqrt {\pi }\, \sqrt {-a^{2}}}+\frac {3 c^{4} \left (-2 \sqrt {\pi }+\frac {\sqrt {\pi }\, \left (-4 a^{2} x^{2}+8\right )}{4 \sqrt {-a^{2} x^{2}+1}}\right )}{a \sqrt {\pi }}+\frac {3 c^{4} \left (\frac {\sqrt {\pi }\, x \left (-a^{2}\right )^{\frac {3}{2}}}{a^{2} \sqrt {-a^{2} x^{2}+1}}-\frac {\sqrt {\pi }\, \left (-a^{2}\right )^{\frac {3}{2}} \arcsin \left (a x \right )}{a^{3}}\right )}{\sqrt {\pi }\, \sqrt {-a^{2}}}+\frac {c^{4} \left (\sqrt {\pi }-\frac {\sqrt {\pi }}{\sqrt {-a^{2} x^{2}+1}}\right )}{a \sqrt {\pi }}+\frac {c^{4} x}{\sqrt {-a^{2} x^{2}+1}}+\frac {c^{4} \left (-\frac {16 \sqrt {\pi }}{5}+\frac {\sqrt {\pi }\, \left (-8 a^{6} x^{6}-16 a^{4} x^{4}-64 a^{2} x^{2}+128\right )}{40 \sqrt {-a^{2} x^{2}+1}}\right )}{a \sqrt {\pi }}+\frac {c^{4} \left (\frac {\sqrt {\pi }\, x \left (-a^{2}\right )^{\frac {7}{2}} \left (-14 a^{4} x^{4}-35 a^{2} x^{2}+105\right )}{56 a^{6} \sqrt {-a^{2} x^{2}+1}}-\frac {15 \sqrt {\pi }\, \left (-a^{2}\right )^{\frac {7}{2}} \arcsin \left (a x \right )}{8 a^{7}}\right )}{\sqrt {\pi }\, \sqrt {-a^{2}}}+\frac {3 c^{4} \left (\frac {8 \sqrt {\pi }}{3}-\frac {\sqrt {\pi }\, \left (-2 a^{4} x^{4}-8 a^{2} x^{2}+16\right )}{6 \sqrt {-a^{2} x^{2}+1}}\right )}{a \sqrt {\pi }}\) | \(420\) |
default | \(c^{4} \left (a^{7} \left (-\frac {x^{6}}{5 a^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {-\frac {2 x^{4}}{5 a^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {6 \left (-\frac {4 x^{2}}{3 a^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {8}{3 a^{4} \sqrt {-a^{2} x^{2}+1}}\right )}{5 a^{2}}}{a^{2}}\right )-a^{6} \left (-\frac {x^{5}}{4 a^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {-\frac {5 x^{3}}{8 a^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {5 \left (\frac {3 x}{2 a^{2} \sqrt {-a^{2} x^{2}+1}}-\frac {3 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 a^{2} \sqrt {a^{2}}}\right )}{4 a^{2}}}{a^{2}}\right )-3 a^{5} \left (-\frac {x^{4}}{3 a^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {-\frac {4 x^{2}}{3 a^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {8}{3 a^{4} \sqrt {-a^{2} x^{2}+1}}}{a^{2}}\right )+3 a^{4} \left (-\frac {x^{3}}{2 a^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {\frac {3 x}{2 a^{2} \sqrt {-a^{2} x^{2}+1}}-\frac {3 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 a^{2} \sqrt {a^{2}}}}{a^{2}}\right )+3 a^{3} \left (-\frac {x^{2}}{a^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {2}{a^{4} \sqrt {-a^{2} x^{2}+1}}\right )-3 a^{2} \left (\frac {x}{a^{2} \sqrt {-a^{2} x^{2}+1}}-\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{a^{2} \sqrt {a^{2}}}\right )-\frac {1}{a \sqrt {-a^{2} x^{2}+1}}+\frac {x}{\sqrt {-a^{2} x^{2}+1}}\right )\) | \(480\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 164 vs.
\(2 (69) = 138\).
time = 0.48, size = 164, normalized size = 1.98 \begin {gather*} -\frac {a^{5} c^{4} x^{6}}{5 \, \sqrt {-a^{2} x^{2} + 1}} + \frac {a^{4} c^{4} x^{5}}{4 \, \sqrt {-a^{2} x^{2} + 1}} + \frac {3 \, a^{3} c^{4} x^{4}}{5 \, \sqrt {-a^{2} x^{2} + 1}} - \frac {7 \, a^{2} c^{4} x^{3}}{8 \, \sqrt {-a^{2} x^{2} + 1}} - \frac {3 \, a c^{4} x^{2}}{5 \, \sqrt {-a^{2} x^{2} + 1}} + \frac {5 \, c^{4} x}{8 \, \sqrt {-a^{2} x^{2} + 1}} + \frac {3 \, c^{4} \arcsin \left (a x\right )}{8 \, a} + \frac {c^{4}}{5 \, \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 = 93, normalized size = 1.12 \begin {gather*} -\frac {30 \, c^{4} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) - {\left (8 \, a^{4} c^{4} x^{4} - 10 \, a^{3} c^{4} x^{3} - 16 \, a^{2} c^{4} x^{2} + 25 \, a c^{4} x + 8 \, c^{4}\right )} \sqrt {-a^{2} x^{2} + 1}}{40 \, a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 14.50, size = 459, normalized size = 5.53 \begin {gather*} - a^{5} c^{4} \left (\begin {cases} - \frac {x^{4} \sqrt {- a^{2} x^{2} + 1}}{5 a^{2}} - \frac {4 x^{2} \sqrt {- a^{2} x^{2} + 1}}{15 a^{4}} - \frac {8 \sqrt {- a^{2} x^{2} + 1}}{15 a^{6}} & \text {for}\: a \neq 0 \\\frac {x^{6}}{6} & \text {otherwise} \end {cases}\right ) + a^{4} c^{4} \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^{3} c^{4} \left (\begin {cases} - \frac {x^{2} \sqrt {- a^{2} x^{2} + 1}}{3 a^{2}} - \frac {2 \sqrt {- a^{2} x^{2} + 1}}{3 a^{4}} & \text {for}\: a \neq 0 \\\frac {x^{4}}{4} & \text {otherwise} \end {cases}\right ) - 2 a^{2} c^{4} \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 ) - a c^{4} \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^{4} \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 = 0.45, size = 78, normalized size = 0.94 \begin {gather*} \frac {3 \, c^{4} \arcsin \left (a x\right ) \mathrm {sgn}\left (a\right )}{8 \, {\left | a \right |}} + \frac {1}{40} \, \sqrt {-a^{2} x^{2} + 1} {\left (\frac {8 \, c^{4}}{a} + {\left (25 \, c^{4} - 2 \, {\left (8 \, a c^{4} - {\left (4 \, a^{3} c^{4} x - 5 \, a^{2} c^{4}\right )} x\right )} x\right )} x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.03, size = 128, normalized size = 1.54 \begin {gather*} \frac {5\,c^4\,x\,\sqrt {1-a^2\,x^2}}{8}+\frac {3\,c^4\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{8\,\sqrt {-a^2}}+\frac {c^4\,\sqrt {1-a^2\,x^2}}{5\,a}-\frac {2\,a\,c^4\,x^2\,\sqrt {1-a^2\,x^2}}{5}-\frac {a^2\,c^4\,x^3\,\sqrt {1-a^2\,x^2}}{4}+\frac {a^3\,c^4\,x^4\,\sqrt {1-a^2\,x^2}}{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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