Optimal. Leaf size=185 \[ \frac {8 c^4 (1+a x)^7 \sqrt {c-a^2 c x^2}}{7 a \sqrt {1-a^2 x^2}}-\frac {3 c^4 (1+a x)^8 \sqrt {c-a^2 c x^2}}{2 a \sqrt {1-a^2 x^2}}+\frac {2 c^4 (1+a x)^9 \sqrt {c-a^2 c x^2}}{3 a \sqrt {1-a^2 x^2}}-\frac {c^4 (1+a x)^{10} \sqrt {c-a^2 c x^2}}{10 a \sqrt {1-a^2 x^2}} \]
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Rubi [A]
time = 0.08, antiderivative size = 185, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {6278, 6275, 45}
\begin {gather*} -\frac {c^4 (a x+1)^{10} \sqrt {c-a^2 c x^2}}{10 a \sqrt {1-a^2 x^2}}+\frac {2 c^4 (a x+1)^9 \sqrt {c-a^2 c x^2}}{3 a \sqrt {1-a^2 x^2}}-\frac {3 c^4 (a x+1)^8 \sqrt {c-a^2 c x^2}}{2 a \sqrt {1-a^2 x^2}}+\frac {8 c^4 (a x+1)^7 \sqrt {c-a^2 c x^2}}{7 a \sqrt {1-a^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 6275
Rule 6278
Rubi steps
\begin {align*} \int e^{3 \tanh ^{-1}(a x)} \left (c-a^2 c x^2\right )^{9/2} \, dx &=\frac {\left (c^4 \sqrt {c-a^2 c x^2}\right ) \int e^{3 \tanh ^{-1}(a x)} \left (1-a^2 x^2\right )^{9/2} \, dx}{\sqrt {1-a^2 x^2}}\\ &=\frac {\left (c^4 \sqrt {c-a^2 c x^2}\right ) \int (1-a x)^3 (1+a x)^6 \, dx}{\sqrt {1-a^2 x^2}}\\ &=\frac {\left (c^4 \sqrt {c-a^2 c x^2}\right ) \int \left (8 (1+a x)^6-12 (1+a x)^7+6 (1+a x)^8-(1+a x)^9\right ) \, dx}{\sqrt {1-a^2 x^2}}\\ &=\frac {8 c^4 (1+a x)^7 \sqrt {c-a^2 c x^2}}{7 a \sqrt {1-a^2 x^2}}-\frac {3 c^4 (1+a x)^8 \sqrt {c-a^2 c x^2}}{2 a \sqrt {1-a^2 x^2}}+\frac {2 c^4 (1+a x)^9 \sqrt {c-a^2 c x^2}}{3 a \sqrt {1-a^2 x^2}}-\frac {c^4 (1+a x)^{10} \sqrt {c-a^2 c x^2}}{10 a \sqrt {1-a^2 x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 68, normalized size = 0.37 \begin {gather*} -\frac {c^4 (1+a x)^7 \sqrt {c-a^2 c x^2} \left (-44+98 a x-77 a^2 x^2+21 a^3 x^3\right )}{210 a \sqrt {1-a^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 98, normalized size = 0.53
method | result | size |
gosper | \(\frac {x \left (21 a^{9} x^{9}+70 a^{8} x^{8}-240 x^{6} a^{6}-210 x^{5} a^{5}+252 a^{4} x^{4}+420 a^{3} x^{3}-315 a x -210\right ) \left (-a^{2} c \,x^{2}+c \right )^{\frac {9}{2}}}{210 \left (a x -1\right )^{3} \left (a x +1\right )^{3} \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}\) | \(97\) |
default | \(\frac {\sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a^{2} x^{2}-1\right )}\, c^{4} x \left (21 a^{9} x^{9}+70 a^{8} x^{8}-240 x^{6} a^{6}-210 x^{5} a^{5}+252 a^{4} x^{4}+420 a^{3} x^{3}-315 a x -210\right )}{210 a^{2} x^{2}-210}\) | \(98\) |
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. 409 vs.
\(2 (161) = 322\).
time = 0.29, size = 409, normalized size = 2.21 \begin {gather*} -\frac {1}{7} \, a^{6} c^{\frac {9}{2}} x^{7} + \frac {3}{5} \, a^{4} c^{\frac {9}{2}} x^{5} - a^{2} c^{\frac {9}{2}} x^{3} + c^{\frac {9}{2}} x + \frac {1}{40} \, {\left (\frac {4 \, a^{8} c^{5} x^{12}}{\sqrt {a^{4} c x^{4} - 2 \, a^{2} c x^{2} + c}} - \frac {19 \, a^{6} c^{5} x^{10}}{\sqrt {a^{4} c x^{4} - 2 \, a^{2} c x^{2} + c}} + \frac {35 \, a^{4} c^{5} x^{8}}{\sqrt {a^{4} c x^{4} - 2 \, a^{2} c x^{2} + c}} - \frac {30 \, a^{2} c^{5} x^{6}}{\sqrt {a^{4} c x^{4} - 2 \, a^{2} c x^{2} + c}} + \frac {10 \, c^{5} x^{4}}{\sqrt {a^{4} c x^{4} - 2 \, a^{2} c x^{2} + c}}\right )} a^{3} - \frac {1}{105} \, {\left (35 \, a^{6} c^{\frac {9}{2}} x^{9} - 135 \, a^{4} c^{\frac {9}{2}} x^{7} + 189 \, a^{2} c^{\frac {9}{2}} x^{5} - 105 \, c^{\frac {9}{2}} x^{3}\right )} a^{2} + \frac {3}{8} \, {\left (\frac {a^{8} c^{5} x^{10}}{\sqrt {a^{4} c x^{4} - 2 \, a^{2} c x^{2} + c}} - \frac {5 \, a^{6} c^{5} x^{8}}{\sqrt {a^{4} c x^{4} - 2 \, a^{2} c x^{2} + c}} + \frac {10 \, a^{4} c^{5} x^{6}}{\sqrt {a^{4} c x^{4} - 2 \, a^{2} c x^{2} + c}} - \frac {10 \, a^{2} c^{5} x^{4}}{\sqrt {a^{4} c x^{4} - 2 \, a^{2} c x^{2} + c}} + \frac {4 \, c^{5}}{\sqrt {a^{4} c x^{4} - 2 \, a^{2} c x^{2} + c} a^{2}}\right )} a \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.33, size = 120, normalized size = 0.65 \begin {gather*} \frac {{\left (21 \, a^{9} c^{4} x^{10} + 70 \, a^{8} c^{4} x^{9} - 240 \, a^{6} c^{4} x^{7} - 210 \, a^{5} c^{4} x^{6} + 252 \, a^{4} c^{4} x^{5} + 420 \, a^{3} c^{4} x^{4} - 315 \, a c^{4} x^{2} - 210 \, c^{4} x\right )} \sqrt {-a^{2} c x^{2} + c} \sqrt {-a^{2} x^{2} + 1}}{210 \, {\left (a^{2} x^{2} - 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {9}{2}} \left (a x + 1\right )^{3}}{\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.09, size = 106, normalized size = 0.57 \begin {gather*} \frac {\sqrt {c-a^2\,c\,x^2}\,\left (-\frac {a^9\,c^4\,x^{10}}{10}-\frac {a^8\,c^4\,x^9}{3}+\frac {8\,a^6\,c^4\,x^7}{7}+a^5\,c^4\,x^6-\frac {6\,a^4\,c^4\,x^5}{5}-2\,a^3\,c^4\,x^4+\frac {3\,a\,c^4\,x^2}{2}+c^4\,x\right )}{\sqrt {1-a^2\,x^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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