Optimal. Leaf size=187 \[ -\frac {8 c^3 (1-a x)^5 \sqrt {c-a^2 c x^2}}{5 a \sqrt {1-a^2 x^2}}+\frac {2 c^3 (1-a x)^6 \sqrt {c-a^2 c x^2}}{a \sqrt {1-a^2 x^2}}-\frac {6 c^3 (1-a x)^7 \sqrt {c-a^2 c x^2}}{7 a \sqrt {1-a^2 x^2}}+\frac {c^3 (1-a x)^8 \sqrt {c-a^2 c x^2}}{8 a \sqrt {1-a^2 x^2}} \]
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Rubi [A]
time = 0.08, antiderivative size = 187, 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^3 (1-a x)^8 \sqrt {c-a^2 c x^2}}{8 a \sqrt {1-a^2 x^2}}-\frac {6 c^3 (1-a x)^7 \sqrt {c-a^2 c x^2}}{7 a \sqrt {1-a^2 x^2}}+\frac {2 c^3 (1-a x)^6 \sqrt {c-a^2 c x^2}}{a \sqrt {1-a^2 x^2}}-\frac {8 c^3 (1-a x)^5 \sqrt {c-a^2 c x^2}}{5 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^{-\tanh ^{-1}(a x)} \left (c-a^2 c x^2\right )^{7/2} \, dx &=\frac {\left (c^3 \sqrt {c-a^2 c x^2}\right ) \int e^{-\tanh ^{-1}(a x)} \left (1-a^2 x^2\right )^{7/2} \, dx}{\sqrt {1-a^2 x^2}}\\ &=\frac {\left (c^3 \sqrt {c-a^2 c x^2}\right ) \int (1-a x)^4 (1+a x)^3 \, dx}{\sqrt {1-a^2 x^2}}\\ &=\frac {\left (c^3 \sqrt {c-a^2 c x^2}\right ) \int \left (8 (1-a x)^4-12 (1-a x)^5+6 (1-a x)^6-(1-a x)^7\right ) \, dx}{\sqrt {1-a^2 x^2}}\\ &=-\frac {8 c^3 (1-a x)^5 \sqrt {c-a^2 c x^2}}{5 a \sqrt {1-a^2 x^2}}+\frac {2 c^3 (1-a x)^6 \sqrt {c-a^2 c x^2}}{a \sqrt {1-a^2 x^2}}-\frac {6 c^3 (1-a x)^7 \sqrt {c-a^2 c x^2}}{7 a \sqrt {1-a^2 x^2}}+\frac {c^3 (1-a x)^8 \sqrt {c-a^2 c x^2}}{8 a \sqrt {1-a^2 x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 68, normalized size = 0.36 \begin {gather*} \frac {c^3 (-1+a x)^5 \sqrt {c-a^2 c x^2} \left (93+185 a x+135 a^2 x^2+35 a^3 x^3\right )}{280 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 = 0.52
method | result | size |
gosper | \(\frac {x \left (35 a^{7} x^{7}-40 x^{6} a^{6}-140 x^{5} a^{5}+168 a^{4} x^{4}+210 a^{3} x^{3}-280 a^{2} x^{2}-140 a x +280\right ) \left (-a^{2} c \,x^{2}+c \right )^{\frac {7}{2}} \sqrt {-a^{2} x^{2}+1}}{280 \left (a x -1\right )^{4} \left (a x +1\right )^{4}}\) | \(97\) |
default | \(-\frac {\sqrt {-c \left (a^{2} x^{2}-1\right )}\, \sqrt {-a^{2} x^{2}+1}\, c^{3} x \left (35 a^{7} x^{7}-40 x^{6} a^{6}-140 x^{5} a^{5}+168 a^{4} x^{4}+210 a^{3} x^{3}-280 a^{2} x^{2}-140 a x +280\right )}{280 \left (a^{2} x^{2}-1\right )}\) | \(98\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 120, normalized size = 0.64 \begin {gather*} -\frac {{\left (35 \, a^{7} c^{3} x^{8} - 40 \, a^{6} c^{3} x^{7} - 140 \, a^{5} c^{3} x^{6} + 168 \, a^{4} c^{3} x^{5} + 210 \, a^{3} c^{3} x^{4} - 280 \, a^{2} c^{3} x^{3} - 140 \, a c^{3} x^{2} + 280 \, c^{3} x\right )} \sqrt {-a^{2} c x^{2} + c} \sqrt {-a^{2} x^{2} + 1}}{280 \, {\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 {\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )} \left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {7}{2}}}{a x + 1}\, 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 [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (c-a^2\,c\,x^2\right )}^{7/2}\,\sqrt {1-a^2\,x^2}}{a\,x+1} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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