Optimal. Leaf size=67 \[ -\frac {a^4 c^2 x^{m+5}}{m+5}-\frac {2 a^3 c^2 x^{m+4}}{m+4}+\frac {2 a c^2 x^{m+2}}{m+2}+\frac {c^2 x^{m+1}}{m+1} \]
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Rubi [A] time = 0.09, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {6150, 75} \[ -\frac {2 a^3 c^2 x^{m+4}}{m+4}-\frac {a^4 c^2 x^{m+5}}{m+5}+\frac {2 a c^2 x^{m+2}}{m+2}+\frac {c^2 x^{m+1}}{m+1} \]
Antiderivative was successfully verified.
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Rule 75
Rule 6150
Rubi steps
\begin {align*} \int e^{2 \tanh ^{-1}(a x)} x^m \left (c-a^2 c x^2\right )^2 \, dx &=c^2 \int x^m (1-a x) (1+a x)^3 \, dx\\ &=c^2 \int \left (x^m+2 a x^{1+m}-2 a^3 x^{3+m}-a^4 x^{4+m}\right ) \, dx\\ &=\frac {c^2 x^{1+m}}{1+m}+\frac {2 a c^2 x^{2+m}}{2+m}-\frac {2 a^3 c^2 x^{4+m}}{4+m}-\frac {a^4 c^2 x^{5+m}}{5+m}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 69, normalized size = 1.03 \[ \frac {c^2 x^{m+1} \left (2 (m+3) \left (\frac {a^3 x^3}{m+4}+\frac {3 a^2 x^2}{m+3}+\frac {3 a x}{m+2}+\frac {1}{m+1}\right )-(a x+1)^4\right )}{m+5} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.68, size = 179, normalized size = 2.67 \[ -\frac {{\left ({\left (a^{4} c^{2} m^{3} + 7 \, a^{4} c^{2} m^{2} + 14 \, a^{4} c^{2} m + 8 \, a^{4} c^{2}\right )} x^{5} + 2 \, {\left (a^{3} c^{2} m^{3} + 8 \, a^{3} c^{2} m^{2} + 17 \, a^{3} c^{2} m + 10 \, a^{3} c^{2}\right )} x^{4} - 2 \, {\left (a c^{2} m^{3} + 10 \, a c^{2} m^{2} + 29 \, a c^{2} m + 20 \, a c^{2}\right )} x^{2} - {\left (c^{2} m^{3} + 11 \, c^{2} m^{2} + 38 \, c^{2} m + 40 \, c^{2}\right )} x\right )} x^{m}}{m^{4} + 12 \, m^{3} + 49 \, m^{2} + 78 \, m + 40} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int -\frac {{\left (a^{2} c x^{2} - c\right )}^{2} {\left (a x + 1\right )}^{2} x^{m}}{a^{2} x^{2} - 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.03, size = 146, normalized size = 2.18 \[ -\frac {c^{2} x^{1+m} \left (a^{4} m^{3} x^{4}+7 a^{4} m^{2} x^{4}+14 a^{4} m \,x^{4}+2 a^{3} m^{3} x^{3}+8 x^{4} a^{4}+16 a^{3} m^{2} x^{3}+34 a^{3} m \,x^{3}+20 x^{3} a^{3}-2 a \,m^{3} x -20 a \,m^{2} x -58 a m x -m^{3}-40 a x -11 m^{2}-38 m -40\right )}{\left (5+m \right ) \left (4+m \right ) \left (2+m \right ) \left (1+m \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.37, size = 114, normalized size = 1.70 \[ -\frac {{\left ({\left (m^{3} + 7 \, m^{2} + 14 \, m + 8\right )} a^{4} c^{2} x^{5} + 2 \, {\left (m^{3} + 8 \, m^{2} + 17 \, m + 10\right )} a^{3} c^{2} x^{4} - 2 \, {\left (m^{3} + 10 \, m^{2} + 29 \, m + 20\right )} a c^{2} x^{2} - {\left (m^{3} + 11 \, m^{2} + 38 \, m + 40\right )} c^{2} x\right )} x^{m}}{m^{4} + 12 \, m^{3} + 49 \, m^{2} + 78 \, m + 40} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.02, size = 173, normalized size = 2.58 \[ x^m\,\left (\frac {c^2\,x\,\left (m^3+11\,m^2+38\,m+40\right )}{m^4+12\,m^3+49\,m^2+78\,m+40}+\frac {2\,a\,c^2\,x^2\,\left (m^3+10\,m^2+29\,m+20\right )}{m^4+12\,m^3+49\,m^2+78\,m+40}-\frac {a^4\,c^2\,x^5\,\left (m^3+7\,m^2+14\,m+8\right )}{m^4+12\,m^3+49\,m^2+78\,m+40}-\frac {2\,a^3\,c^2\,x^4\,\left (m^3+8\,m^2+17\,m+10\right )}{m^4+12\,m^3+49\,m^2+78\,m+40}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.85, size = 706, normalized size = 10.54 \[ \begin {cases} - a^{4} c^{2} \log {\relax (x )} + \frac {2 a^{3} c^{2}}{x} - \frac {2 a c^{2}}{3 x^{3}} - \frac {c^{2}}{4 x^{4}} & \text {for}\: m = -5 \\- a^{4} c^{2} x - 2 a^{3} c^{2} \log {\relax (x )} - \frac {a c^{2}}{x^{2}} - \frac {c^{2}}{3 x^{3}} & \text {for}\: m = -4 \\- \frac {a^{4} c^{2} x^{3}}{3} - a^{3} c^{2} x^{2} + 2 a c^{2} \log {\relax (x )} - \frac {c^{2}}{x} & \text {for}\: m = -2 \\- \frac {a^{4} c^{2} x^{4}}{4} - \frac {2 a^{3} c^{2} x^{3}}{3} + 2 a c^{2} x + c^{2} \log {\relax (x )} & \text {for}\: m = -1 \\- \frac {a^{4} c^{2} m^{3} x^{5} x^{m}}{m^{4} + 12 m^{3} + 49 m^{2} + 78 m + 40} - \frac {7 a^{4} c^{2} m^{2} x^{5} x^{m}}{m^{4} + 12 m^{3} + 49 m^{2} + 78 m + 40} - \frac {14 a^{4} c^{2} m x^{5} x^{m}}{m^{4} + 12 m^{3} + 49 m^{2} + 78 m + 40} - \frac {8 a^{4} c^{2} x^{5} x^{m}}{m^{4} + 12 m^{3} + 49 m^{2} + 78 m + 40} - \frac {2 a^{3} c^{2} m^{3} x^{4} x^{m}}{m^{4} + 12 m^{3} + 49 m^{2} + 78 m + 40} - \frac {16 a^{3} c^{2} m^{2} x^{4} x^{m}}{m^{4} + 12 m^{3} + 49 m^{2} + 78 m + 40} - \frac {34 a^{3} c^{2} m x^{4} x^{m}}{m^{4} + 12 m^{3} + 49 m^{2} + 78 m + 40} - \frac {20 a^{3} c^{2} x^{4} x^{m}}{m^{4} + 12 m^{3} + 49 m^{2} + 78 m + 40} + \frac {2 a c^{2} m^{3} x^{2} x^{m}}{m^{4} + 12 m^{3} + 49 m^{2} + 78 m + 40} + \frac {20 a c^{2} m^{2} x^{2} x^{m}}{m^{4} + 12 m^{3} + 49 m^{2} + 78 m + 40} + \frac {58 a c^{2} m x^{2} x^{m}}{m^{4} + 12 m^{3} + 49 m^{2} + 78 m + 40} + \frac {40 a c^{2} x^{2} x^{m}}{m^{4} + 12 m^{3} + 49 m^{2} + 78 m + 40} + \frac {c^{2} m^{3} x x^{m}}{m^{4} + 12 m^{3} + 49 m^{2} + 78 m + 40} + \frac {11 c^{2} m^{2} x x^{m}}{m^{4} + 12 m^{3} + 49 m^{2} + 78 m + 40} + \frac {38 c^{2} m x x^{m}}{m^{4} + 12 m^{3} + 49 m^{2} + 78 m + 40} + \frac {40 c^{2} x x^{m}}{m^{4} + 12 m^{3} + 49 m^{2} + 78 m + 40} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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