Optimal. Leaf size=111 \[ -\frac {19 c x^2 \sqrt {1-a^2 x^2}}{15 a}-\frac {3}{4} c x^3 \sqrt {1-a^2 x^2}-\frac {1}{5} a c x^4 \sqrt {1-a^2 x^2}-\frac {c (304+195 a x) \sqrt {1-a^2 x^2}}{120 a^3}+\frac {13 c \text {ArcSin}(a x)}{8 a^3} \]
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Rubi [A]
time = 0.17, antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {6283, 1823,
847, 794, 222} \begin {gather*} \frac {13 c \text {ArcSin}(a x)}{8 a^3}-\frac {19 c x^2 \sqrt {1-a^2 x^2}}{15 a}-\frac {1}{5} a c x^4 \sqrt {1-a^2 x^2}-\frac {3}{4} c x^3 \sqrt {1-a^2 x^2}-\frac {c (195 a x+304) \sqrt {1-a^2 x^2}}{120 a^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 222
Rule 794
Rule 847
Rule 1823
Rule 6283
Rubi steps
\begin {align*} \int e^{3 \tanh ^{-1}(a x)} x^2 \left (c-a^2 c x^2\right ) \, dx &=c \int \frac {x^2 (1+a x)^3}{\sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {1}{5} a c x^4 \sqrt {1-a^2 x^2}-\frac {c \int \frac {x^2 \left (-5 a^2-19 a^3 x-15 a^4 x^2\right )}{\sqrt {1-a^2 x^2}} \, dx}{5 a^2}\\ &=-\frac {3}{4} c x^3 \sqrt {1-a^2 x^2}-\frac {1}{5} a c x^4 \sqrt {1-a^2 x^2}+\frac {c \int \frac {x^2 \left (65 a^4+76 a^5 x\right )}{\sqrt {1-a^2 x^2}} \, dx}{20 a^4}\\ &=-\frac {19 c x^2 \sqrt {1-a^2 x^2}}{15 a}-\frac {3}{4} c x^3 \sqrt {1-a^2 x^2}-\frac {1}{5} a c x^4 \sqrt {1-a^2 x^2}-\frac {c \int \frac {x \left (-152 a^5-195 a^6 x\right )}{\sqrt {1-a^2 x^2}} \, dx}{60 a^6}\\ &=-\frac {19 c x^2 \sqrt {1-a^2 x^2}}{15 a}-\frac {3}{4} c x^3 \sqrt {1-a^2 x^2}-\frac {1}{5} a c x^4 \sqrt {1-a^2 x^2}-\frac {c (304+195 a x) \sqrt {1-a^2 x^2}}{120 a^3}+\frac {(13 c) \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{8 a^2}\\ &=-\frac {19 c x^2 \sqrt {1-a^2 x^2}}{15 a}-\frac {3}{4} c x^3 \sqrt {1-a^2 x^2}-\frac {1}{5} a c x^4 \sqrt {1-a^2 x^2}-\frac {c (304+195 a x) \sqrt {1-a^2 x^2}}{120 a^3}+\frac {13 c \sin ^{-1}(a x)}{8 a^3}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 62, normalized size = 0.56 \begin {gather*} \frac {-c \sqrt {1-a^2 x^2} \left (304+195 a x+152 a^2 x^2+90 a^3 x^3+24 a^4 x^4\right )+195 c \text {ArcSin}(a x)}{120 a^3} \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. \(439\) vs.
\(2(93)=186\).
time = 0.06, size = 440, normalized size = 3.96
method | result | size |
risch | \(\frac {\left (24 a^{4} x^{4}+90 a^{3} x^{3}+152 a^{2} x^{2}+195 a x +304\right ) \left (a^{2} x^{2}-1\right ) c}{120 a^{3} \sqrt {-a^{2} x^{2}+1}}+\frac {13 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right ) c}{8 a^{2} \sqrt {a^{2}}}\) | \(90\) |
meijerg | \(\frac {2 c \left (\frac {\sqrt {\pi }\, x \left (-a^{2}\right )^{\frac {5}{2}} \left (-5 a^{2} x^{2}+15\right )}{10 a^{4} \sqrt {-a^{2} x^{2}+1}}-\frac {3 \sqrt {\pi }\, \left (-a^{2}\right )^{\frac {5}{2}} \arcsin \left (a x \right )}{2 a^{5}}\right )}{a^{2} \sqrt {\pi }\, \sqrt {-a^{2}}}-\frac {c \left (\frac {\sqrt {\pi }\, x \left (-a^{2}\right )^{\frac {3}{2}}}{a^{2} \sqrt {-a^{2} x^{2}+1}}-\frac {\sqrt {\pi }\, \left (-a^{2}\right )^{\frac {3}{2}} \arcsin \left (a x \right )}{a^{3}}\right )}{a^{2} \sqrt {\pi }\, \sqrt {-a^{2}}}-\frac {c \left (-\frac {16 \sqrt {\pi }}{5}+\frac {\sqrt {\pi }\, \left (-8 x^{6} a^{6}-16 a^{4} x^{4}-64 a^{2} x^{2}+128\right )}{40 \sqrt {-a^{2} x^{2}+1}}\right )}{a^{3} \sqrt {\pi }}+\frac {2 c \left (\frac {8 \sqrt {\pi }}{3}-\frac {\sqrt {\pi }\, \left (-2 a^{4} x^{4}-8 a^{2} x^{2}+16\right )}{6 \sqrt {-a^{2} x^{2}+1}}\right )}{a^{3} \sqrt {\pi }}+\frac {3 c \left (\frac {\sqrt {\pi }\, x \left (-a^{2}\right )^{\frac {7}{2}} \left (-14 a^{4} x^{4}-35 a^{2} x^{2}+105\right )}{56 a^{6} \sqrt {-a^{2} x^{2}+1}}-\frac {15 \sqrt {\pi }\, \left (-a^{2}\right )^{\frac {7}{2}} \arcsin \left (a x \right )}{8 a^{7}}\right )}{a^{2} \sqrt {\pi }\, \sqrt {-a^{2}}}+\frac {3 c \left (-2 \sqrt {\pi }+\frac {\sqrt {\pi }\, \left (-4 a^{2} x^{2}+8\right )}{4 \sqrt {-a^{2} x^{2}+1}}\right )}{a^{3} \sqrt {\pi }}\) | \(371\) |
default | \(-c \left (a^{5} \left (-\frac {x^{6}}{5 a^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {-\frac {2 x^{4}}{5 a^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {6 \left (-\frac {4 x^{2}}{3 a^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {8}{3 a^{4} \sqrt {-a^{2} x^{2}+1}}\right )}{5 a^{2}}}{a^{2}}\right )+3 a^{4} \left (-\frac {x^{5}}{4 a^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {-\frac {5 x^{3}}{8 a^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {5 \left (\frac {3 x}{2 a^{2} \sqrt {-a^{2} x^{2}+1}}-\frac {3 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 a^{2} \sqrt {a^{2}}}\right )}{4 a^{2}}}{a^{2}}\right )+2 a^{3} \left (-\frac {x^{4}}{3 a^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {-\frac {4 x^{2}}{3 a^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {8}{3 a^{4} \sqrt {-a^{2} x^{2}+1}}}{a^{2}}\right )-2 a^{2} \left (-\frac {x^{3}}{2 a^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {\frac {3 x}{2 a^{2} \sqrt {-a^{2} x^{2}+1}}-\frac {3 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 a^{2} \sqrt {a^{2}}}}{a^{2}}\right )-3 a \left (-\frac {x^{2}}{a^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {2}{a^{4} \sqrt {-a^{2} x^{2}+1}}\right )-\frac {x}{a^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{a^{2} \sqrt {a^{2}}}\right )\) | \(440\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, size = 148, normalized size = 1.33 \begin {gather*} \frac {a^{3} c x^{6}}{5 \, \sqrt {-a^{2} x^{2} + 1}} + \frac {3 \, a^{2} c x^{5}}{4 \, \sqrt {-a^{2} x^{2} + 1}} + \frac {16 \, a c x^{4}}{15 \, \sqrt {-a^{2} x^{2} + 1}} + \frac {7 \, c x^{3}}{8 \, \sqrt {-a^{2} x^{2} + 1}} + \frac {19 \, c x^{2}}{15 \, \sqrt {-a^{2} x^{2} + 1} a} - \frac {13 \, c x}{8 \, \sqrt {-a^{2} x^{2} + 1} a^{2}} + \frac {13 \, c \arcsin \left (a x\right )}{8 \, a^{3}} - \frac {38 \, c}{15 \, \sqrt {-a^{2} x^{2} + 1} a^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.33, size = 80, normalized size = 0.72 \begin {gather*} -\frac {390 \, c \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + {\left (24 \, a^{4} c x^{4} + 90 \, a^{3} c x^{3} + 152 \, a^{2} c x^{2} + 195 \, a c x + 304 \, c\right )} \sqrt {-a^{2} x^{2} + 1}}{120 \, a^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 12.03, size = 371, normalized size = 3.34 \begin {gather*} a^{3} c \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 ) + 3 a^{2} c \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 ) + 3 a c \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 ) + 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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.44, size = 69, normalized size = 0.62 \begin {gather*} -\frac {1}{120} \, \sqrt {-a^{2} x^{2} + 1} {\left ({\left (2 \, {\left (3 \, {\left (4 \, a c x + 15 \, c\right )} x + \frac {76 \, c}{a}\right )} x + \frac {195 \, c}{a^{2}}\right )} x + \frac {304 \, c}{a^{3}}\right )} + \frac {13 \, c \arcsin \left (a x\right ) \mathrm {sgn}\left (a\right )}{8 \, a^{2} {\left | a \right |}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.03, size = 119, normalized size = 1.07 \begin {gather*} \frac {13\,c\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{8\,a^2\,\sqrt {-a^2}}-\frac {3\,c\,x^3\,\sqrt {1-a^2\,x^2}}{4}-\frac {19\,c\,x^2\,\sqrt {1-a^2\,x^2}}{15\,a}-\frac {13\,c\,x\,\sqrt {1-a^2\,x^2}}{8\,a^2}-\frac {a\,c\,x^4\,\sqrt {1-a^2\,x^2}}{5}-\frac {38\,c\,\sqrt {1-a^2\,x^2}}{15\,a^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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