Optimal. Leaf size=124 \[ -\frac {c^2 x \sqrt {1-a^2 x^2}}{16 a^3}-\frac {c^2 x^2 \left (1-a^2 x^2\right )^{3/2}}{5 a^2}+\frac {c^2 x^3 \left (1-a^2 x^2\right )^{3/2}}{6 a}-\frac {c^2 (16-15 a x) \left (1-a^2 x^2\right )^{3/2}}{120 a^4}-\frac {c^2 \text {ArcSin}(a x)}{16 a^4} \]
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Rubi [A]
time = 0.10, antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {6263, 847, 794,
201, 222} \begin {gather*} -\frac {c^2 \text {ArcSin}(a x)}{16 a^4}-\frac {c^2 x^2 \left (1-a^2 x^2\right )^{3/2}}{5 a^2}+\frac {c^2 x^3 \left (1-a^2 x^2\right )^{3/2}}{6 a}-\frac {c^2 (16-15 a x) \left (1-a^2 x^2\right )^{3/2}}{120 a^4}-\frac {c^2 x \sqrt {1-a^2 x^2}}{16 a^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 201
Rule 222
Rule 794
Rule 847
Rule 6263
Rubi steps
\begin {align*} \int e^{\tanh ^{-1}(a x)} x^3 (c-a c x)^2 \, dx &=c \int x^3 (c-a c x) \sqrt {1-a^2 x^2} \, dx\\ &=\frac {c^2 x^3 \left (1-a^2 x^2\right )^{3/2}}{6 a}-\frac {c \int x^2 \left (3 a c-6 a^2 c x\right ) \sqrt {1-a^2 x^2} \, dx}{6 a^2}\\ &=-\frac {c^2 x^2 \left (1-a^2 x^2\right )^{3/2}}{5 a^2}+\frac {c^2 x^3 \left (1-a^2 x^2\right )^{3/2}}{6 a}+\frac {c \int x \left (12 a^2 c-15 a^3 c x\right ) \sqrt {1-a^2 x^2} \, dx}{30 a^4}\\ &=-\frac {c^2 x^2 \left (1-a^2 x^2\right )^{3/2}}{5 a^2}+\frac {c^2 x^3 \left (1-a^2 x^2\right )^{3/2}}{6 a}-\frac {c^2 (16-15 a x) \left (1-a^2 x^2\right )^{3/2}}{120 a^4}-\frac {c^2 \int \sqrt {1-a^2 x^2} \, dx}{8 a^3}\\ &=-\frac {c^2 x \sqrt {1-a^2 x^2}}{16 a^3}-\frac {c^2 x^2 \left (1-a^2 x^2\right )^{3/2}}{5 a^2}+\frac {c^2 x^3 \left (1-a^2 x^2\right )^{3/2}}{6 a}-\frac {c^2 (16-15 a x) \left (1-a^2 x^2\right )^{3/2}}{120 a^4}-\frac {c^2 \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{16 a^3}\\ &=-\frac {c^2 x \sqrt {1-a^2 x^2}}{16 a^3}-\frac {c^2 x^2 \left (1-a^2 x^2\right )^{3/2}}{5 a^2}+\frac {c^2 x^3 \left (1-a^2 x^2\right )^{3/2}}{6 a}-\frac {c^2 (16-15 a x) \left (1-a^2 x^2\right )^{3/2}}{120 a^4}-\frac {c^2 \sin ^{-1}(a x)}{16 a^4}\\ \end {align*}
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Mathematica [A]
time = 0.09, size = 70, normalized size = 0.56 \begin {gather*} \frac {c^2 \left (\sqrt {1-a^2 x^2} \left (-32+15 a x-16 a^2 x^2+10 a^3 x^3+48 a^4 x^4-40 a^5 x^5\right )-15 \text {ArcSin}(a x)\right )}{240 a^4} \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. \(294\) vs.
\(2(106)=212\).
time = 0.92, size = 295, normalized size = 2.38
method | result | size |
risch | \(\frac {\left (40 a^{5} x^{5}-48 a^{4} x^{4}-10 a^{3} x^{3}+16 a^{2} x^{2}-15 a x +32\right ) \left (a^{2} x^{2}-1\right ) c^{2}}{240 a^{4} \sqrt {-a^{2} x^{2}+1}}-\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right ) c^{2}}{16 a^{3} \sqrt {a^{2}}}\) | \(102\) |
meijerg | \(-\frac {c^{2} \left (-\frac {\sqrt {\pi }\, x \left (-a^{2}\right )^{\frac {7}{2}} \left (56 a^{4} x^{4}+70 a^{2} x^{2}+105\right ) \sqrt {-a^{2} x^{2}+1}}{168 a^{6}}+\frac {5 \sqrt {\pi }\, \left (-a^{2}\right )^{\frac {7}{2}} \arcsin \left (a x \right )}{8 a^{7}}\right )}{2 a^{3} \sqrt {\pi }\, \sqrt {-a^{2}}}+\frac {c^{2} \left (-\frac {16 \sqrt {\pi }}{15}+\frac {\sqrt {\pi }\, \left (6 a^{4} x^{4}+8 a^{2} x^{2}+16\right ) \sqrt {-a^{2} x^{2}+1}}{15}\right )}{2 a^{4} \sqrt {\pi }}-\frac {c^{2} \left (-\frac {\sqrt {\pi }\, x \left (-a^{2}\right )^{\frac {5}{2}} \left (10 a^{2} x^{2}+15\right ) \sqrt {-a^{2} x^{2}+1}}{20 a^{4}}+\frac {3 \sqrt {\pi }\, \left (-a^{2}\right )^{\frac {5}{2}} \arcsin \left (a x \right )}{4 a^{5}}\right )}{2 a^{3} \sqrt {\pi }\, \sqrt {-a^{2}}}+\frac {c^{2} \left (\frac {4 \sqrt {\pi }}{3}-\frac {\sqrt {\pi }\, \left (4 a^{2} x^{2}+8\right ) \sqrt {-a^{2} x^{2}+1}}{6}\right )}{2 a^{4} \sqrt {\pi }}\) | \(258\) |
default | \(c^{2} \left (a^{3} \left (-\frac {x^{5} \sqrt {-a^{2} x^{2}+1}}{6 a^{2}}+\frac {-\frac {5 x^{3} \sqrt {-a^{2} x^{2}+1}}{24 a^{2}}+\frac {5 \left (-\frac {3 x \sqrt {-a^{2} x^{2}+1}}{8 a^{2}}+\frac {3 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{8 a^{2} \sqrt {a^{2}}}\right )}{6 a^{2}}}{a^{2}}\right )-a^{2} \left (-\frac {x^{4} \sqrt {-a^{2} x^{2}+1}}{5 a^{2}}+\frac {-\frac {4 x^{2} \sqrt {-a^{2} x^{2}+1}}{15 a^{2}}-\frac {8 \sqrt {-a^{2} x^{2}+1}}{15 a^{4}}}{a^{2}}\right )-a \left (-\frac {x^{3} \sqrt {-a^{2} x^{2}+1}}{4 a^{2}}+\frac {-\frac {3 x \sqrt {-a^{2} x^{2}+1}}{8 a^{2}}+\frac {3 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{8 a^{2} \sqrt {a^{2}}}}{a^{2}}\right )-\frac {x^{2} \sqrt {-a^{2} x^{2}+1}}{3 a^{2}}-\frac {2 \sqrt {-a^{2} x^{2}+1}}{3 a^{4}}\right )\) | \(295\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.46, size = 141, normalized size = 1.14 \begin {gather*} -\frac {1}{6} \, \sqrt {-a^{2} x^{2} + 1} a c^{2} x^{5} + \frac {1}{5} \, \sqrt {-a^{2} x^{2} + 1} c^{2} x^{4} + \frac {\sqrt {-a^{2} x^{2} + 1} c^{2} x^{3}}{24 \, a} - \frac {\sqrt {-a^{2} x^{2} + 1} c^{2} x^{2}}{15 \, a^{2}} + \frac {\sqrt {-a^{2} x^{2} + 1} c^{2} x}{16 \, a^{3}} - \frac {c^{2} \arcsin \left (a x\right )}{16 \, a^{4}} - \frac {2 \, \sqrt {-a^{2} x^{2} + 1} c^{2}}{15 \, a^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 104, normalized size = 0.84 \begin {gather*} \frac {30 \, c^{2} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) - {\left (40 \, a^{5} c^{2} x^{5} - 48 \, a^{4} c^{2} x^{4} - 10 \, a^{3} c^{2} x^{3} + 16 \, a^{2} c^{2} x^{2} - 15 \, a c^{2} x + 32 \, c^{2}\right )} \sqrt {-a^{2} x^{2} + 1}}{240 \, a^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 12.73, size = 486, normalized size = 3.92 \begin {gather*} a^{3} c^{2} \left (\begin {cases} - \frac {i x^{7}}{6 \sqrt {a^{2} x^{2} - 1}} - \frac {i x^{5}}{24 a^{2} \sqrt {a^{2} x^{2} - 1}} - \frac {5 i x^{3}}{48 a^{4} \sqrt {a^{2} x^{2} - 1}} + \frac {5 i x}{16 a^{6} \sqrt {a^{2} x^{2} - 1}} - \frac {5 i \operatorname {acosh}{\left (a x \right )}}{16 a^{7}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\\frac {x^{7}}{6 \sqrt {- a^{2} x^{2} + 1}} + \frac {x^{5}}{24 a^{2} \sqrt {- a^{2} x^{2} + 1}} + \frac {5 x^{3}}{48 a^{4} \sqrt {- a^{2} x^{2} + 1}} - \frac {5 x}{16 a^{6} \sqrt {- a^{2} x^{2} + 1}} + \frac {5 \operatorname {asin}{\left (a x \right )}}{16 a^{7}} & \text {otherwise} \end {cases}\right ) - a^{2} c^{2} \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 c^{2} \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 ) + c^{2} \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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 92, normalized size = 0.74 \begin {gather*} -\frac {1}{240} \, \sqrt {-a^{2} x^{2} + 1} {\left ({\left (2 \, {\left ({\left (4 \, {\left (5 \, a c^{2} x - 6 \, c^{2}\right )} x - \frac {5 \, c^{2}}{a}\right )} x + \frac {8 \, c^{2}}{a^{2}}\right )} x - \frac {15 \, c^{2}}{a^{3}}\right )} x + \frac {32 \, c^{2}}{a^{4}}\right )} - \frac {c^{2} \arcsin \left (a x\right ) \mathrm {sgn}\left (a\right )}{16 \, a^{3} {\left | a \right |}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.79, size = 154, normalized size = 1.24 \begin {gather*} \frac {c^2\,x^4\,\sqrt {1-a^2\,x^2}}{5}-\frac {2\,c^2\,\sqrt {1-a^2\,x^2}}{15\,a^4}+\frac {c^2\,x\,\sqrt {1-a^2\,x^2}}{16\,a^3}-\frac {a\,c^2\,x^5\,\sqrt {1-a^2\,x^2}}{6}-\frac {c^2\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{16\,a^3\,\sqrt {-a^2}}+\frac {c^2\,x^3\,\sqrt {1-a^2\,x^2}}{24\,a}-\frac {c^2\,x^2\,\sqrt {1-a^2\,x^2}}{15\,a^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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