Optimal. Leaf size=165 \[ -\frac {c^4 (a x+1)^2 \left (1-a^2 x^2\right )^{7/2}}{9 a}-\frac {11 c^4 (a x+1) \left (1-a^2 x^2\right )^{7/2}}{72 a}-\frac {11 c^4 \left (1-a^2 x^2\right )^{7/2}}{56 a}+\frac {11}{48} c^4 x \left (1-a^2 x^2\right )^{5/2}+\frac {55}{192} c^4 x \left (1-a^2 x^2\right )^{3/2}+\frac {55}{128} c^4 x \sqrt {1-a^2 x^2}+\frac {55 c^4 \sin ^{-1}(a x)}{128 a} \]
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Rubi [A] time = 0.09, antiderivative size = 165, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {6138, 671, 641, 195, 216} \[ -\frac {c^4 (a x+1)^2 \left (1-a^2 x^2\right )^{7/2}}{9 a}-\frac {11 c^4 (a x+1) \left (1-a^2 x^2\right )^{7/2}}{72 a}-\frac {11 c^4 \left (1-a^2 x^2\right )^{7/2}}{56 a}+\frac {11}{48} c^4 x \left (1-a^2 x^2\right )^{5/2}+\frac {55}{192} c^4 x \left (1-a^2 x^2\right )^{3/2}+\frac {55}{128} c^4 x \sqrt {1-a^2 x^2}+\frac {55 c^4 \sin ^{-1}(a x)}{128 a} \]
Antiderivative was successfully verified.
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Rule 195
Rule 216
Rule 641
Rule 671
Rule 6138
Rubi steps
\begin {align*} \int e^{3 \tanh ^{-1}(a x)} \left (c-a^2 c x^2\right )^4 \, dx &=c^4 \int (1+a x)^3 \left (1-a^2 x^2\right )^{5/2} \, dx\\ &=-\frac {c^4 (1+a x)^2 \left (1-a^2 x^2\right )^{7/2}}{9 a}+\frac {1}{9} \left (11 c^4\right ) \int (1+a x)^2 \left (1-a^2 x^2\right )^{5/2} \, dx\\ &=-\frac {11 c^4 (1+a x) \left (1-a^2 x^2\right )^{7/2}}{72 a}-\frac {c^4 (1+a x)^2 \left (1-a^2 x^2\right )^{7/2}}{9 a}+\frac {1}{8} \left (11 c^4\right ) \int (1+a x) \left (1-a^2 x^2\right )^{5/2} \, dx\\ &=-\frac {11 c^4 \left (1-a^2 x^2\right )^{7/2}}{56 a}-\frac {11 c^4 (1+a x) \left (1-a^2 x^2\right )^{7/2}}{72 a}-\frac {c^4 (1+a x)^2 \left (1-a^2 x^2\right )^{7/2}}{9 a}+\frac {1}{8} \left (11 c^4\right ) \int \left (1-a^2 x^2\right )^{5/2} \, dx\\ &=\frac {11}{48} c^4 x \left (1-a^2 x^2\right )^{5/2}-\frac {11 c^4 \left (1-a^2 x^2\right )^{7/2}}{56 a}-\frac {11 c^4 (1+a x) \left (1-a^2 x^2\right )^{7/2}}{72 a}-\frac {c^4 (1+a x)^2 \left (1-a^2 x^2\right )^{7/2}}{9 a}+\frac {1}{48} \left (55 c^4\right ) \int \left (1-a^2 x^2\right )^{3/2} \, dx\\ &=\frac {55}{192} c^4 x \left (1-a^2 x^2\right )^{3/2}+\frac {11}{48} c^4 x \left (1-a^2 x^2\right )^{5/2}-\frac {11 c^4 \left (1-a^2 x^2\right )^{7/2}}{56 a}-\frac {11 c^4 (1+a x) \left (1-a^2 x^2\right )^{7/2}}{72 a}-\frac {c^4 (1+a x)^2 \left (1-a^2 x^2\right )^{7/2}}{9 a}+\frac {1}{64} \left (55 c^4\right ) \int \sqrt {1-a^2 x^2} \, dx\\ &=\frac {55}{128} c^4 x \sqrt {1-a^2 x^2}+\frac {55}{192} c^4 x \left (1-a^2 x^2\right )^{3/2}+\frac {11}{48} c^4 x \left (1-a^2 x^2\right )^{5/2}-\frac {11 c^4 \left (1-a^2 x^2\right )^{7/2}}{56 a}-\frac {11 c^4 (1+a x) \left (1-a^2 x^2\right )^{7/2}}{72 a}-\frac {c^4 (1+a x)^2 \left (1-a^2 x^2\right )^{7/2}}{9 a}+\frac {1}{128} \left (55 c^4\right ) \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx\\ &=\frac {55}{128} c^4 x \sqrt {1-a^2 x^2}+\frac {55}{192} c^4 x \left (1-a^2 x^2\right )^{3/2}+\frac {11}{48} c^4 x \left (1-a^2 x^2\right )^{5/2}-\frac {11 c^4 \left (1-a^2 x^2\right )^{7/2}}{56 a}-\frac {11 c^4 (1+a x) \left (1-a^2 x^2\right )^{7/2}}{72 a}-\frac {c^4 (1+a x)^2 \left (1-a^2 x^2\right )^{7/2}}{9 a}+\frac {55 c^4 \sin ^{-1}(a x)}{128 a}\\ \end {align*}
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Mathematica [A] time = 0.15, size = 107, normalized size = 0.65 \[ \frac {c^4 \left (\sqrt {1-a^2 x^2} \left (896 a^8 x^8+3024 a^7 x^7+1024 a^6 x^6-7224 a^5 x^5-8448 a^4 x^4+3066 a^3 x^3+10240 a^2 x^2+4599 a x-3712\right )-6930 \sin ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )\right )}{8064 a} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.65, size = 137, normalized size = 0.83 \[ -\frac {6930 \, c^{4} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) - {\left (896 \, a^{8} c^{4} x^{8} + 3024 \, a^{7} c^{4} x^{7} + 1024 \, a^{6} c^{4} x^{6} - 7224 \, a^{5} c^{4} x^{5} - 8448 \, a^{4} c^{4} x^{4} + 3066 \, a^{3} c^{4} x^{3} + 10240 \, a^{2} c^{4} x^{2} + 4599 \, a c^{4} x - 3712 \, c^{4}\right )} \sqrt {-a^{2} x^{2} + 1}}{8064 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.01, size = 125, normalized size = 0.76 \[ \frac {55 \, c^{4} \arcsin \left (a x\right ) \mathrm {sgn}\relax (a)}{128 \, {\left | a \right |}} - \frac {1}{8064} \, \sqrt {-a^{2} x^{2} + 1} {\left (\frac {3712 \, c^{4}}{a} - {\left (4599 \, c^{4} + 2 \, {\left (5120 \, a c^{4} + {\left (1533 \, a^{2} c^{4} - 4 \, {\left (1056 \, a^{3} c^{4} + {\left (903 \, a^{4} c^{4} - 2 \, {\left (64 \, a^{5} c^{4} + 7 \, {\left (8 \, a^{7} c^{4} x + 27 \, a^{6} c^{4}\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.25, size = 275, normalized size = 1.67 \[ -\frac {c^{4} a^{9} x^{10}}{9 \sqrt {-a^{2} x^{2}+1}}-\frac {c^{4} a^{7} x^{8}}{63 \sqrt {-a^{2} x^{2}+1}}+\frac {74 c^{4} a^{5} x^{6}}{63 \sqrt {-a^{2} x^{2}+1}}-\frac {146 c^{4} a^{3} x^{4}}{63 \sqrt {-a^{2} x^{2}+1}}+\frac {109 c^{4} a \,x^{2}}{63 \sqrt {-a^{2} x^{2}+1}}-\frac {3 c^{4} a^{8} x^{9}}{8 \sqrt {-a^{2} x^{2}+1}}+\frac {61 c^{4} a^{6} x^{7}}{48 \sqrt {-a^{2} x^{2}+1}}-\frac {245 c^{4} a^{4} x^{5}}{192 \sqrt {-a^{2} x^{2}+1}}-\frac {73 c^{4} a^{2} x^{3}}{384 \sqrt {-a^{2} x^{2}+1}}+\frac {55 c^{4} \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{128 \sqrt {a^{2}}}+\frac {73 c^{4} x}{128 \sqrt {-a^{2} x^{2}+1}}-\frac {29 c^{4}}{63 a \sqrt {-a^{2} x^{2}+1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 256, normalized size = 1.55 \[ -\frac {a^{9} c^{4} x^{10}}{9 \, \sqrt {-a^{2} x^{2} + 1}} - \frac {3 \, a^{8} c^{4} x^{9}}{8 \, \sqrt {-a^{2} x^{2} + 1}} - \frac {a^{7} c^{4} x^{8}}{63 \, \sqrt {-a^{2} x^{2} + 1}} + \frac {61 \, a^{6} c^{4} x^{7}}{48 \, \sqrt {-a^{2} x^{2} + 1}} + \frac {74 \, a^{5} c^{4} x^{6}}{63 \, \sqrt {-a^{2} x^{2} + 1}} - \frac {245 \, a^{4} c^{4} x^{5}}{192 \, \sqrt {-a^{2} x^{2} + 1}} - \frac {146 \, a^{3} c^{4} x^{4}}{63 \, \sqrt {-a^{2} x^{2} + 1}} - \frac {73 \, a^{2} c^{4} x^{3}}{384 \, \sqrt {-a^{2} x^{2} + 1}} + \frac {109 \, a c^{4} x^{2}}{63 \, \sqrt {-a^{2} x^{2} + 1}} + \frac {73 \, c^{4} x}{128 \, \sqrt {-a^{2} x^{2} + 1}} + \frac {55 \, c^{4} \arcsin \left (a x\right )}{128 \, a} - \frac {29 \, c^{4}}{63 \, \sqrt {-a^{2} x^{2} + 1} a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.93, size = 220, normalized size = 1.33 \[ \frac {73\,c^4\,x\,\sqrt {1-a^2\,x^2}}{128}+\frac {55\,c^4\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{128\,\sqrt {-a^2}}-\frac {29\,c^4\,\sqrt {1-a^2\,x^2}}{63\,a}+\frac {80\,a\,c^4\,x^2\,\sqrt {1-a^2\,x^2}}{63}+\frac {73\,a^2\,c^4\,x^3\,\sqrt {1-a^2\,x^2}}{192}-\frac {22\,a^3\,c^4\,x^4\,\sqrt {1-a^2\,x^2}}{21}-\frac {43\,a^4\,c^4\,x^5\,\sqrt {1-a^2\,x^2}}{48}+\frac {8\,a^5\,c^4\,x^6\,\sqrt {1-a^2\,x^2}}{63}+\frac {3\,a^6\,c^4\,x^7\,\sqrt {1-a^2\,x^2}}{8}+\frac {a^7\,c^4\,x^8\,\sqrt {1-a^2\,x^2}}{9} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 46.24, size = 996, normalized size = 6.04 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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