Optimal. Leaf size=145 \[ \frac {9}{16} c^3 x \sqrt {1-a^2 x^2}+\frac {3}{8} c^3 x \left (1-a^2 x^2\right )^{3/2}+\frac {3 c^3 \left (1-a^2 x^2\right )^{5/2}}{10 a}+\frac {3 c^3 (1-a x) \left (1-a^2 x^2\right )^{5/2}}{14 a}+\frac {c^3 (1-a x)^2 \left (1-a^2 x^2\right )^{5/2}}{7 a}+\frac {9 c^3 \text {ArcSin}(a x)}{16 a} \]
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Rubi [A]
time = 0.06, antiderivative size = 145, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {6274, 685, 655,
201, 222} \begin {gather*} \frac {c^3 (1-a x)^2 \left (1-a^2 x^2\right )^{5/2}}{7 a}+\frac {3 c^3 (1-a x) \left (1-a^2 x^2\right )^{5/2}}{14 a}+\frac {3 c^3 \left (1-a^2 x^2\right )^{5/2}}{10 a}+\frac {3}{8} c^3 x \left (1-a^2 x^2\right )^{3/2}+\frac {9}{16} c^3 x \sqrt {1-a^2 x^2}+\frac {9 c^3 \text {ArcSin}(a x)}{16 a} \end {gather*}
Antiderivative was successfully verified.
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Rule 201
Rule 222
Rule 655
Rule 685
Rule 6274
Rubi steps
\begin {align*} \int e^{-3 \tanh ^{-1}(a x)} \left (c-a^2 c x^2\right )^3 \, dx &=c^3 \int (1-a x)^3 \left (1-a^2 x^2\right )^{3/2} \, dx\\ &=\frac {c^3 (1-a x)^2 \left (1-a^2 x^2\right )^{5/2}}{7 a}+\frac {1}{7} \left (9 c^3\right ) \int (1-a x)^2 \left (1-a^2 x^2\right )^{3/2} \, dx\\ &=\frac {3 c^3 (1-a x) \left (1-a^2 x^2\right )^{5/2}}{14 a}+\frac {c^3 (1-a x)^2 \left (1-a^2 x^2\right )^{5/2}}{7 a}+\frac {1}{2} \left (3 c^3\right ) \int (1-a x) \left (1-a^2 x^2\right )^{3/2} \, dx\\ &=\frac {3 c^3 \left (1-a^2 x^2\right )^{5/2}}{10 a}+\frac {3 c^3 (1-a x) \left (1-a^2 x^2\right )^{5/2}}{14 a}+\frac {c^3 (1-a x)^2 \left (1-a^2 x^2\right )^{5/2}}{7 a}+\frac {1}{2} \left (3 c^3\right ) \int \left (1-a^2 x^2\right )^{3/2} \, dx\\ &=\frac {3}{8} c^3 x \left (1-a^2 x^2\right )^{3/2}+\frac {3 c^3 \left (1-a^2 x^2\right )^{5/2}}{10 a}+\frac {3 c^3 (1-a x) \left (1-a^2 x^2\right )^{5/2}}{14 a}+\frac {c^3 (1-a x)^2 \left (1-a^2 x^2\right )^{5/2}}{7 a}+\frac {1}{8} \left (9 c^3\right ) \int \sqrt {1-a^2 x^2} \, dx\\ &=\frac {9}{16} c^3 x \sqrt {1-a^2 x^2}+\frac {3}{8} c^3 x \left (1-a^2 x^2\right )^{3/2}+\frac {3 c^3 \left (1-a^2 x^2\right )^{5/2}}{10 a}+\frac {3 c^3 (1-a x) \left (1-a^2 x^2\right )^{5/2}}{14 a}+\frac {c^3 (1-a x)^2 \left (1-a^2 x^2\right )^{5/2}}{7 a}+\frac {1}{16} \left (9 c^3\right ) \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx\\ &=\frac {9}{16} c^3 x \sqrt {1-a^2 x^2}+\frac {3}{8} c^3 x \left (1-a^2 x^2\right )^{3/2}+\frac {3 c^3 \left (1-a^2 x^2\right )^{5/2}}{10 a}+\frac {3 c^3 (1-a x) \left (1-a^2 x^2\right )^{5/2}}{14 a}+\frac {c^3 (1-a x)^2 \left (1-a^2 x^2\right )^{5/2}}{7 a}+\frac {9 c^3 \sin ^{-1}(a x)}{16 a}\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 91, normalized size = 0.63 \begin {gather*} \frac {c^3 \left (\sqrt {1-a^2 x^2} \left (368+245 a x-656 a^2 x^2+350 a^3 x^3+208 a^4 x^4-280 a^5 x^5+80 a^6 x^6\right )-630 \text {ArcSin}\left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )\right )}{560 a} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.06, size = 210, normalized size = 1.45
method | result | size |
risch | \(-\frac {\left (80 x^{6} a^{6}-280 x^{5} a^{5}+208 a^{4} x^{4}+350 a^{3} x^{3}-656 a^{2} x^{2}+245 a x +368\right ) \left (a^{2} x^{2}-1\right ) c^{3}}{560 a \sqrt {-a^{2} x^{2}+1}}+\frac {9 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right ) c^{3}}{16 \sqrt {a^{2}}}\) | \(107\) |
default | \(-c^{3} \left (a^{3} \left (-\frac {x^{2} \left (-a^{2} x^{2}+1\right )^{\frac {5}{2}}}{7 a^{2}}-\frac {2 \left (-a^{2} x^{2}+1\right )^{\frac {5}{2}}}{35 a^{4}}\right )-3 a^{2} \left (-\frac {x \left (-a^{2} x^{2}+1\right )^{\frac {5}{2}}}{6 a^{2}}+\frac {\frac {x \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{4}+\frac {3 x \sqrt {-a^{2} x^{2}+1}}{8}+\frac {3 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{8 \sqrt {a^{2}}}}{6 a^{2}}\right )-\frac {3 \left (-a^{2} x^{2}+1\right )^{\frac {5}{2}}}{5 a}-\frac {x \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{4}-\frac {3 x \sqrt {-a^{2} x^{2}+1}}{8}-\frac {3 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{8 \sqrt {a^{2}}}\right )\) | \(210\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.47, size = 108, normalized size = 0.74 \begin {gather*} \frac {1}{7} \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {5}{2}} a c^{3} x^{2} - \frac {1}{2} \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {5}{2}} c^{3} x + \frac {3}{8} \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} c^{3} x + \frac {23 \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {5}{2}} c^{3}}{35 \, a} + \frac {9}{16} \, \sqrt {-a^{2} x^{2} + 1} c^{3} x + \frac {9 \, c^{3} \arcsin \left (a x\right )}{16 \, a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.33, size = 115, normalized size = 0.79 \begin {gather*} -\frac {630 \, c^{3} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) - {\left (80 \, a^{6} c^{3} x^{6} - 280 \, a^{5} c^{3} x^{5} + 208 \, a^{4} c^{3} x^{4} + 350 \, a^{3} c^{3} x^{3} - 656 \, a^{2} c^{3} x^{2} + 245 \, a c^{3} x + 368 \, c^{3}\right )} \sqrt {-a^{2} x^{2} + 1}}{560 \, a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 10.75, size = 631, normalized size = 4.35 \begin {gather*} a^{5} c^{3} \left (\begin {cases} \frac {x^{6} \sqrt {- a^{2} x^{2} + 1}}{7} - \frac {x^{4} \sqrt {- a^{2} x^{2} + 1}}{35 a^{2}} - \frac {4 x^{2} \sqrt {- a^{2} x^{2} + 1}}{105 a^{4}} - \frac {8 \sqrt {- a^{2} x^{2} + 1}}{105 a^{6}} & \text {for}\: a \neq 0 \\\frac {x^{6}}{6} & \text {otherwise} \end {cases}\right ) - 3 a^{4} c^{3} \left (\begin {cases} \frac {i a^{2} x^{7}}{6 \sqrt {a^{2} x^{2} - 1}} - \frac {5 i x^{5}}{24 \sqrt {a^{2} x^{2} - 1}} - \frac {i x^{3}}{48 a^{2} \sqrt {a^{2} x^{2} - 1}} + \frac {i x}{16 a^{4} \sqrt {a^{2} x^{2} - 1}} - \frac {i \operatorname {acosh}{\left (a x \right )}}{16 a^{5}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\- \frac {a^{2} x^{7}}{6 \sqrt {- a^{2} x^{2} + 1}} + \frac {5 x^{5}}{24 \sqrt {- a^{2} x^{2} + 1}} + \frac {x^{3}}{48 a^{2} \sqrt {- a^{2} x^{2} + 1}} - \frac {x}{16 a^{4} \sqrt {- a^{2} x^{2} + 1}} + \frac {\operatorname {asin}{\left (a x \right )}}{16 a^{5}} & \text {otherwise} \end {cases}\right ) + 2 a^{3} c^{3} \left (\begin {cases} \frac {x^{4} \sqrt {- a^{2} x^{2} + 1}}{5} - \frac {x^{2} \sqrt {- a^{2} x^{2} + 1}}{15 a^{2}} - \frac {2 \sqrt {- a^{2} x^{2} + 1}}{15 a^{4}} & \text {for}\: a \neq 0 \\\frac {x^{4}}{4} & \text {otherwise} \end {cases}\right ) + 2 a^{2} c^{3} \left (\begin {cases} \frac {i a^{2} x^{5}}{4 \sqrt {a^{2} x^{2} - 1}} - \frac {3 i x^{3}}{8 \sqrt {a^{2} x^{2} - 1}} + \frac {i x}{8 a^{2} \sqrt {a^{2} x^{2} - 1}} - \frac {i \operatorname {acosh}{\left (a x \right )}}{8 a^{3}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\- \frac {a^{2} x^{5}}{4 \sqrt {- a^{2} x^{2} + 1}} + \frac {3 x^{3}}{8 \sqrt {- a^{2} x^{2} + 1}} - \frac {x}{8 a^{2} \sqrt {- a^{2} x^{2} + 1}} + \frac {\operatorname {asin}{\left (a x \right )}}{8 a^{3}} & \text {otherwise} \end {cases}\right ) - 3 a c^{3} \left (\begin {cases} \frac {x^{2}}{2} & \text {for}\: a^{2} = 0 \\- \frac {\left (- a^{2} x^{2} + 1\right )^{\frac {3}{2}}}{3 a^{2}} & \text {otherwise} \end {cases}\right ) + c^{3} \left (\begin {cases} \frac {i x \sqrt {a^{2} x^{2} - 1}}{2} - \frac {i \operatorname {acosh}{\left (a x \right )}}{2 a} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\- \frac {a^{2} x^{3}}{2 \sqrt {- a^{2} x^{2} + 1}} + \frac {x}{2 \sqrt {- a^{2} x^{2} + 1}} + \frac {\operatorname {asin}{\left (a x \right )}}{2 a} & \text {otherwise} \end {cases}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.43, size = 102, normalized size = 0.70 \begin {gather*} \frac {9 \, c^{3} \arcsin \left (a x\right ) \mathrm {sgn}\left (a\right )}{16 \, {\left | a \right |}} + \frac {1}{560} \, \sqrt {-a^{2} x^{2} + 1} {\left (\frac {368 \, c^{3}}{a} + {\left (245 \, c^{3} - 2 \, {\left (328 \, a c^{3} - {\left (175 \, a^{2} c^{3} + 4 \, {\left (26 \, a^{3} c^{3} + 5 \, {\left (2 \, a^{5} c^{3} x - 7 \, a^{4} c^{3}\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 174, normalized size = 1.20 \begin {gather*} \frac {7\,c^3\,x\,\sqrt {1-a^2\,x^2}}{16}+\frac {9\,c^3\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{16\,\sqrt {-a^2}}+\frac {23\,c^3\,\sqrt {1-a^2\,x^2}}{35\,a}-\frac {41\,a\,c^3\,x^2\,\sqrt {1-a^2\,x^2}}{35}+\frac {5\,a^2\,c^3\,x^3\,\sqrt {1-a^2\,x^2}}{8}+\frac {13\,a^3\,c^3\,x^4\,\sqrt {1-a^2\,x^2}}{35}-\frac {a^4\,c^3\,x^5\,\sqrt {1-a^2\,x^2}}{2}+\frac {a^5\,c^3\,x^6\,\sqrt {1-a^2\,x^2}}{7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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