Optimal. Leaf size=127 \[ -\frac {c^4 \left (1-a^2 x^2\right )^{9/2}}{9 a}+\frac {1}{8} c^4 x \left (1-a^2 x^2\right )^{7/2}+\frac {7}{48} c^4 x \left (1-a^2 x^2\right )^{5/2}+\frac {35}{192} c^4 x \left (1-a^2 x^2\right )^{3/2}+\frac {35}{128} c^4 x \sqrt {1-a^2 x^2}+\frac {35 c^4 \sin ^{-1}(a x)}{128 a} \]
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Rubi [A] time = 0.06, antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6138, 641, 195, 216} \[ -\frac {c^4 \left (1-a^2 x^2\right )^{9/2}}{9 a}+\frac {1}{8} c^4 x \left (1-a^2 x^2\right )^{7/2}+\frac {7}{48} c^4 x \left (1-a^2 x^2\right )^{5/2}+\frac {35}{192} c^4 x \left (1-a^2 x^2\right )^{3/2}+\frac {35}{128} c^4 x \sqrt {1-a^2 x^2}+\frac {35 c^4 \sin ^{-1}(a x)}{128 a} \]
Antiderivative was successfully verified.
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Rule 195
Rule 216
Rule 641
Rule 6138
Rubi steps
\begin {align*} \int e^{\tanh ^{-1}(a x)} \left (c-a^2 c x^2\right )^4 \, dx &=c^4 \int (1+a x) \left (1-a^2 x^2\right )^{7/2} \, dx\\ &=-\frac {c^4 \left (1-a^2 x^2\right )^{9/2}}{9 a}+c^4 \int \left (1-a^2 x^2\right )^{7/2} \, dx\\ &=\frac {1}{8} c^4 x \left (1-a^2 x^2\right )^{7/2}-\frac {c^4 \left (1-a^2 x^2\right )^{9/2}}{9 a}+\frac {1}{8} \left (7 c^4\right ) \int \left (1-a^2 x^2\right )^{5/2} \, dx\\ &=\frac {7}{48} c^4 x \left (1-a^2 x^2\right )^{5/2}+\frac {1}{8} c^4 x \left (1-a^2 x^2\right )^{7/2}-\frac {c^4 \left (1-a^2 x^2\right )^{9/2}}{9 a}+\frac {1}{48} \left (35 c^4\right ) \int \left (1-a^2 x^2\right )^{3/2} \, dx\\ &=\frac {35}{192} c^4 x \left (1-a^2 x^2\right )^{3/2}+\frac {7}{48} c^4 x \left (1-a^2 x^2\right )^{5/2}+\frac {1}{8} c^4 x \left (1-a^2 x^2\right )^{7/2}-\frac {c^4 \left (1-a^2 x^2\right )^{9/2}}{9 a}+\frac {1}{64} \left (35 c^4\right ) \int \sqrt {1-a^2 x^2} \, dx\\ &=\frac {35}{128} c^4 x \sqrt {1-a^2 x^2}+\frac {35}{192} c^4 x \left (1-a^2 x^2\right )^{3/2}+\frac {7}{48} c^4 x \left (1-a^2 x^2\right )^{5/2}+\frac {1}{8} c^4 x \left (1-a^2 x^2\right )^{7/2}-\frac {c^4 \left (1-a^2 x^2\right )^{9/2}}{9 a}+\frac {1}{128} \left (35 c^4\right ) \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx\\ &=\frac {35}{128} c^4 x \sqrt {1-a^2 x^2}+\frac {35}{192} c^4 x \left (1-a^2 x^2\right )^{3/2}+\frac {7}{48} c^4 x \left (1-a^2 x^2\right )^{5/2}+\frac {1}{8} c^4 x \left (1-a^2 x^2\right )^{7/2}-\frac {c^4 \left (1-a^2 x^2\right )^{9/2}}{9 a}+\frac {35 c^4 \sin ^{-1}(a x)}{128 a}\\ \end {align*}
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Mathematica [A] time = 0.16, size = 107, normalized size = 0.84 \[ -\frac {c^4 \left (\sqrt {1-a^2 x^2} \left (128 a^8 x^8+144 a^7 x^7-512 a^6 x^6-600 a^5 x^5+768 a^4 x^4+978 a^3 x^3-512 a^2 x^2-837 a x+128\right )+630 \sin ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )\right )}{1152 a} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.67, size = 136, normalized size = 1.07 \[ -\frac {630 \, c^{4} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + {\left (128 \, a^{8} c^{4} x^{8} + 144 \, a^{7} c^{4} x^{7} - 512 \, a^{6} c^{4} x^{6} - 600 \, a^{5} c^{4} x^{5} + 768 \, a^{4} c^{4} x^{4} + 978 \, a^{3} c^{4} x^{3} - 512 \, a^{2} c^{4} x^{2} - 837 \, a c^{4} x + 128 \, c^{4}\right )} \sqrt {-a^{2} x^{2} + 1}}{1152 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 127, normalized size = 1.00 \[ \frac {35 \, c^{4} \arcsin \left (a x\right ) \mathrm {sgn}\relax (a)}{128 \, {\left | a \right |}} - \frac {1}{1152} \, \sqrt {-a^{2} x^{2} + 1} {\left (\frac {128 \, c^{4}}{a} - {\left (837 \, c^{4} + 2 \, {\left (256 \, a c^{4} - {\left (489 \, a^{2} c^{4} + 4 \, {\left (96 \, a^{3} c^{4} - {\left (75 \, a^{4} c^{4} + 2 \, {\left (32 \, a^{5} c^{4} - {\left (8 \, a^{7} c^{4} x + 9 \, a^{6} c^{4}\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.09, size = 229, normalized size = 1.80 \[ \frac {93 c^{4} x \sqrt {-a^{2} x^{2}+1}}{128}-\frac {c^{4} \sqrt {-a^{2} x^{2}+1}}{9 a}-\frac {c^{4} a^{7} x^{8} \sqrt {-a^{2} x^{2}+1}}{9}+\frac {4 c^{4} a^{5} x^{6} \sqrt {-a^{2} x^{2}+1}}{9}-\frac {2 c^{4} a^{3} x^{4} \sqrt {-a^{2} x^{2}+1}}{3}+\frac {4 c^{4} a \,x^{2} \sqrt {-a^{2} x^{2}+1}}{9}-\frac {c^{4} a^{6} x^{7} \sqrt {-a^{2} x^{2}+1}}{8}+\frac {25 c^{4} a^{4} x^{5} \sqrt {-a^{2} x^{2}+1}}{48}-\frac {163 c^{4} a^{2} x^{3} \sqrt {-a^{2} x^{2}+1}}{192}+\frac {35 c^{4} \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{128 \sqrt {a^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 210, normalized size = 1.65 \[ -\frac {1}{9} \, \sqrt {-a^{2} x^{2} + 1} a^{7} c^{4} x^{8} - \frac {1}{8} \, \sqrt {-a^{2} x^{2} + 1} a^{6} c^{4} x^{7} + \frac {4}{9} \, \sqrt {-a^{2} x^{2} + 1} a^{5} c^{4} x^{6} + \frac {25}{48} \, \sqrt {-a^{2} x^{2} + 1} a^{4} c^{4} x^{5} - \frac {2}{3} \, \sqrt {-a^{2} x^{2} + 1} a^{3} c^{4} x^{4} - \frac {163}{192} \, \sqrt {-a^{2} x^{2} + 1} a^{2} c^{4} x^{3} + \frac {4}{9} \, \sqrt {-a^{2} x^{2} + 1} a c^{4} x^{2} + \frac {93}{128} \, \sqrt {-a^{2} x^{2} + 1} c^{4} x + \frac {35 \, c^{4} \arcsin \left (a x\right )}{128 \, a} - \frac {\sqrt {-a^{2} x^{2} + 1} c^{4}}{9 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.09, size = 118, normalized size = 0.93 \[ \frac {35\,c^4\,x\,\sqrt {1-a^2\,x^2}}{128}+\frac {35\,c^4\,x\,{\left (1-a^2\,x^2\right )}^{3/2}}{192}+\frac {7\,c^4\,x\,{\left (1-a^2\,x^2\right )}^{5/2}}{48}+\frac {c^4\,x\,{\left (1-a^2\,x^2\right )}^{7/2}}{8}-\frac {c^4\,{\left (1-a^2\,x^2\right )}^{9/2}}{9\,a}-\frac {35\,c^4\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )\,\sqrt {-a^2}}{128\,a^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 40.90, size = 452, normalized size = 3.56 \[ \begin {cases} \frac {- \frac {c^{4} \left (- a^{2} x^{2} + 1\right )^{\frac {3}{2}}}{3} + c^{4} \left (\begin {cases} \frac {a x \sqrt {- a^{2} x^{2} + 1}}{2} + \frac {\operatorname {asin}{\left (a x \right )}}{2} & \text {for}\: a x > -1 \wedge a x < 1 \end {cases}\right ) - 3 c^{4} \left (\begin {cases} - \frac {a x \left (- 2 a^{2} x^{2} + 1\right ) \sqrt {- a^{2} x^{2} + 1}}{8} + \frac {\operatorname {asin}{\left (a x \right )}}{8} & \text {for}\: a x > -1 \wedge a x < 1 \end {cases}\right ) - 3 c^{4} \left (\begin {cases} \frac {\left (- a^{2} x^{2} + 1\right )^{\frac {5}{2}}}{5} - \frac {\left (- a^{2} x^{2} + 1\right )^{\frac {3}{2}}}{3} & \text {for}\: a x > -1 \wedge a x < 1 \end {cases}\right ) + 3 c^{4} \left (\begin {cases} - \frac {a^{3} x^{3} \left (- a^{2} x^{2} + 1\right )^{\frac {3}{2}}}{6} - \frac {a x \left (- 2 a^{2} x^{2} + 1\right ) \sqrt {- a^{2} x^{2} + 1}}{16} + \frac {\operatorname {asin}{\left (a x \right )}}{16} & \text {for}\: a x > -1 \wedge a x < 1 \end {cases}\right ) + 3 c^{4} \left (\begin {cases} - \frac {\left (- a^{2} x^{2} + 1\right )^{\frac {7}{2}}}{7} + \frac {2 \left (- a^{2} x^{2} + 1\right )^{\frac {5}{2}}}{5} - \frac {\left (- a^{2} x^{2} + 1\right )^{\frac {3}{2}}}{3} & \text {for}\: a x > -1 \wedge a x < 1 \end {cases}\right ) - c^{4} \left (\begin {cases} - \frac {a^{3} x^{3} \left (- a^{2} x^{2} + 1\right )^{\frac {3}{2}}}{6} - \frac {a x \left (- 2 a^{2} x^{2} + 1\right ) \sqrt {- a^{2} x^{2} + 1}}{32} - \frac {a x \sqrt {- a^{2} x^{2} + 1} \left (- 16 a^{6} x^{6} + 24 a^{4} x^{4} - 10 a^{2} x^{2} + 1\right )}{128} + \frac {5 \operatorname {asin}{\left (a x \right )}}{128} & \text {for}\: a x > -1 \wedge a x < 1 \end {cases}\right ) - c^{4} \left (\begin {cases} \frac {\left (- a^{2} x^{2} + 1\right )^{\frac {9}{2}}}{9} - \frac {3 \left (- a^{2} x^{2} + 1\right )^{\frac {7}{2}}}{7} + \frac {3 \left (- a^{2} x^{2} + 1\right )^{\frac {5}{2}}}{5} - \frac {\left (- a^{2} x^{2} + 1\right )^{\frac {3}{2}}}{3} & \text {for}\: a x > -1 \wedge a x < 1 \end {cases}\right )}{a} & \text {for}\: a \neq 0 \\c^{4} x & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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