Optimal. Leaf size=143 \[ \frac {9}{16} c^3 x \sqrt {1-a^2 x^2}+\frac {3}{8} c^3 x \left (1-a^2 x^2\right )^{3/2}-\frac {3 c^3 \left (1-a^2 x^2\right )^{5/2}}{10 a}-\frac {3 c^3 (1+a x) \left (1-a^2 x^2\right )^{5/2}}{14 a}-\frac {c^3 (1+a x)^2 \left (1-a^2 x^2\right )^{5/2}}{7 a}+\frac {9 c^3 \text {ArcSin}(a x)}{16 a} \]
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Rubi [A]
time = 0.06, antiderivative size = 143, 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 = {6273, 685, 655,
201, 222} \begin {gather*} -\frac {c^3 (a x+1)^2 \left (1-a^2 x^2\right )^{5/2}}{7 a}-\frac {3 c^3 (a x+1) \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 6273
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 = 0.09, size = 91, normalized size = 0.64 \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 [B] Leaf count of result is larger than twice the leaf count of optimal. \(585\) vs.
\(2(121)=242\).
time = 0.08, size = 586, normalized size = 4.10
method | result | size |
risch | \(\frac {\left (80 x^{6} a^{6}+280 x^{5} a^{5}+208 a^{4} x^{4}-350 a^{3} x^{3}-656 a^{2} x^{2}-245 a x +368\right ) \left (a^{2} x^{2}-1\right ) c^{3}}{560 a \sqrt {-a^{2} x^{2}+1}}+\frac {9 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right ) c^{3}}{16 \sqrt {a^{2}}}\) | \(107\) |
meijerg | \(-\frac {8 c^{3} \left (\frac {\sqrt {\pi }\, x \left (-a^{2}\right )^{\frac {7}{2}} \left (-14 a^{4} x^{4}-35 a^{2} x^{2}+105\right )}{56 a^{6} \sqrt {-a^{2} x^{2}+1}}-\frac {15 \sqrt {\pi }\, \left (-a^{2}\right )^{\frac {7}{2}} \arcsin \left (a x \right )}{8 a^{7}}\right )}{\sqrt {\pi }\, \sqrt {-a^{2}}}-\frac {6 c^{3} \left (\frac {\sqrt {\pi }\, x \left (-a^{2}\right )^{\frac {5}{2}} \left (-5 a^{2} x^{2}+15\right )}{10 a^{4} \sqrt {-a^{2} x^{2}+1}}-\frac {3 \sqrt {\pi }\, \left (-a^{2}\right )^{\frac {5}{2}} \arcsin \left (a x \right )}{2 a^{5}}\right )}{\sqrt {\pi }\, \sqrt {-a^{2}}}+\frac {c^{3} x}{\sqrt {-a^{2} x^{2}+1}}+\frac {c^{3} \left (\frac {128 \sqrt {\pi }}{35}-\frac {\sqrt {\pi }\, \left (-10 a^{8} x^{8}-16 x^{6} a^{6}-32 a^{4} x^{4}-128 a^{2} x^{2}+256\right )}{70 \sqrt {-a^{2} x^{2}+1}}\right )}{a \sqrt {\pi }}-\frac {6 c^{3} \left (\frac {8 \sqrt {\pi }}{3}-\frac {\sqrt {\pi }\, \left (-2 a^{4} x^{4}-8 a^{2} x^{2}+16\right )}{6 \sqrt {-a^{2} x^{2}+1}}\right )}{a \sqrt {\pi }}-\frac {8 c^{3} \left (-2 \sqrt {\pi }+\frac {\sqrt {\pi }\, \left (-4 a^{2} x^{2}+8\right )}{4 \sqrt {-a^{2} x^{2}+1}}\right )}{a \sqrt {\pi }}-\frac {3 c^{3} \left (\frac {\sqrt {\pi }\, x \left (-a^{2}\right )^{\frac {9}{2}} \left (-24 x^{6} a^{6}-42 a^{4} x^{4}-105 a^{2} x^{2}+315\right )}{144 a^{8} \sqrt {-a^{2} x^{2}+1}}-\frac {35 \sqrt {\pi }\, \left (-a^{2}\right )^{\frac {9}{2}} \arcsin \left (a x \right )}{16 a^{9}}\right )}{\sqrt {\pi }\, \sqrt {-a^{2}}}-\frac {3 c^{3} \left (\sqrt {\pi }-\frac {\sqrt {\pi }}{\sqrt {-a^{2} x^{2}+1}}\right )}{a \sqrt {\pi }}\) | \(457\) |
default | \(-c^{3} \left (a^{9} \left (-\frac {x^{8}}{7 a^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {-\frac {8 x^{6}}{35 a^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {8 \left (-\frac {2 x^{4}}{5 a^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {6 \left (-\frac {4 x^{2}}{3 a^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {8}{3 a^{4} \sqrt {-a^{2} x^{2}+1}}\right )}{5 a^{2}}\right )}{7 a^{2}}}{a^{2}}\right )+3 a^{8} \left (-\frac {x^{7}}{6 a^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {-\frac {7 x^{5}}{24 a^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {7 \left (-\frac {5 x^{3}}{8 a^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {5 \left (\frac {3 x}{2 a^{2} \sqrt {-a^{2} x^{2}+1}}-\frac {3 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 a^{2} \sqrt {a^{2}}}\right )}{4 a^{2}}\right )}{6 a^{2}}}{a^{2}}\right )-8 a^{6} \left (-\frac {x^{5}}{4 a^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {-\frac {5 x^{3}}{8 a^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {5 \left (\frac {3 x}{2 a^{2} \sqrt {-a^{2} x^{2}+1}}-\frac {3 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 a^{2} \sqrt {a^{2}}}\right )}{4 a^{2}}}{a^{2}}\right )-6 a^{5} \left (-\frac {x^{4}}{3 a^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {-\frac {4 x^{2}}{3 a^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {8}{3 a^{4} \sqrt {-a^{2} x^{2}+1}}}{a^{2}}\right )+6 a^{4} \left (-\frac {x^{3}}{2 a^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {\frac {3 x}{2 a^{2} \sqrt {-a^{2} x^{2}+1}}-\frac {3 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 a^{2} \sqrt {a^{2}}}}{a^{2}}\right )+8 a^{3} \left (-\frac {x^{2}}{a^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {2}{a^{4} \sqrt {-a^{2} x^{2}+1}}\right )-\frac {3}{a \sqrt {-a^{2} x^{2}+1}}-\frac {x}{\sqrt {-a^{2} x^{2}+1}}\right )\) | \(586\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.47, size = 210, normalized size = 1.47 \begin {gather*} \frac {a^{7} c^{3} x^{8}}{7 \, \sqrt {-a^{2} x^{2} + 1}} + \frac {a^{6} c^{3} x^{7}}{2 \, \sqrt {-a^{2} x^{2} + 1}} + \frac {8 \, a^{5} c^{3} x^{6}}{35 \, \sqrt {-a^{2} x^{2} + 1}} - \frac {9 \, a^{4} c^{3} x^{5}}{8 \, \sqrt {-a^{2} x^{2} + 1}} - \frac {54 \, a^{3} c^{3} x^{4}}{35 \, \sqrt {-a^{2} x^{2} + 1}} + \frac {3 \, a^{2} c^{3} x^{3}}{16 \, \sqrt {-a^{2} x^{2} + 1}} + \frac {64 \, a c^{3} x^{2}}{35 \, \sqrt {-a^{2} x^{2} + 1}} + \frac {7 \, c^{3} x}{16 \, \sqrt {-a^{2} x^{2} + 1}} + \frac {9 \, c^{3} \arcsin \left (a x\right )}{16 \, a} - \frac {23 \, c^{3}}{35 \, \sqrt {-a^{2} x^{2} + 1} a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.51, size = 114, normalized size = 0.80 \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 [A]
time = 23.81, size = 765, normalized size = 5.35 \begin {gather*} a^{7} c^{3} \left (\begin {cases} - \frac {x^{6} \sqrt {- a^{2} x^{2} + 1}}{7 a^{2}} - \frac {6 x^{4} \sqrt {- a^{2} x^{2} + 1}}{35 a^{4}} - \frac {8 x^{2} \sqrt {- a^{2} x^{2} + 1}}{35 a^{6}} - \frac {16 \sqrt {- a^{2} x^{2} + 1}}{35 a^{8}} & \text {for}\: a \neq 0 \\\frac {x^{8}}{8} & \text {otherwise} \end {cases}\right ) + 3 a^{6} c^{3} \left (\begin {cases} - \frac {i x^{7}}{6 \sqrt {a^{2} x^{2} - 1}} - \frac {i x^{5}}{24 a^{2} \sqrt {a^{2} x^{2} - 1}} - \frac {5 i x^{3}}{48 a^{4} \sqrt {a^{2} x^{2} - 1}} + \frac {5 i x}{16 a^{6} \sqrt {a^{2} x^{2} - 1}} - \frac {5 i \operatorname {acosh}{\left (a x \right )}}{16 a^{7}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\\frac {x^{7}}{6 \sqrt {- a^{2} x^{2} + 1}} + \frac {x^{5}}{24 a^{2} \sqrt {- a^{2} x^{2} + 1}} + \frac {5 x^{3}}{48 a^{4} \sqrt {- a^{2} x^{2} + 1}} - \frac {5 x}{16 a^{6} \sqrt {- a^{2} x^{2} + 1}} + \frac {5 \operatorname {asin}{\left (a x \right )}}{16 a^{7}} & \text {otherwise} \end {cases}\right ) + a^{5} c^{3} \left (\begin {cases} - \frac {x^{4} \sqrt {- a^{2} x^{2} + 1}}{5 a^{2}} - \frac {4 x^{2} \sqrt {- a^{2} x^{2} + 1}}{15 a^{4}} - \frac {8 \sqrt {- a^{2} x^{2} + 1}}{15 a^{6}} & \text {for}\: a \neq 0 \\\frac {x^{6}}{6} & \text {otherwise} \end {cases}\right ) - 5 a^{4} c^{3} \left (\begin {cases} - \frac {i x^{5}}{4 \sqrt {a^{2} x^{2} - 1}} - \frac {i x^{3}}{8 a^{2} \sqrt {a^{2} x^{2} - 1}} + \frac {3 i x}{8 a^{4} \sqrt {a^{2} x^{2} - 1}} - \frac {3 i \operatorname {acosh}{\left (a x \right )}}{8 a^{5}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\\frac {x^{5}}{4 \sqrt {- a^{2} x^{2} + 1}} + \frac {x^{3}}{8 a^{2} \sqrt {- a^{2} x^{2} + 1}} - \frac {3 x}{8 a^{4} \sqrt {- a^{2} x^{2} + 1}} + \frac {3 \operatorname {asin}{\left (a x \right )}}{8 a^{5}} & \text {otherwise} \end {cases}\right ) - 5 a^{3} c^{3} \left (\begin {cases} - \frac {x^{2} \sqrt {- a^{2} x^{2} + 1}}{3 a^{2}} - \frac {2 \sqrt {- a^{2} x^{2} + 1}}{3 a^{4}} & \text {for}\: a \neq 0 \\\frac {x^{4}}{4} & \text {otherwise} \end {cases}\right ) + a^{2} c^{3} \left (\begin {cases} - \frac {i x \sqrt {a^{2} x^{2} - 1}}{2 a^{2}} - \frac {i \operatorname {acosh}{\left (a x \right )}}{2 a^{3}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\\frac {x^{3}}{2 \sqrt {- a^{2} x^{2} + 1}} - \frac {x}{2 a^{2} \sqrt {- a^{2} x^{2} + 1}} + \frac {\operatorname {asin}{\left (a x \right )}}{2 a^{3}} & \text {otherwise} \end {cases}\right ) + 3 a c^{3} \left (\begin {cases} \frac {x^{2}}{2} & \text {for}\: a^{2} = 0 \\- \frac {\sqrt {- a^{2} x^{2} + 1}}{a^{2}} & \text {otherwise} \end {cases}\right ) + c^{3} \left (\begin {cases} \sqrt {\frac {1}{a^{2}}} \operatorname {asin}{\left (x \sqrt {a^{2}} \right )} & \text {for}\: a^{2} > 0 \\\sqrt {- \frac {1}{a^{2}}} \operatorname {asinh}{\left (x \sqrt {- a^{2}} \right )} & \text {for}\: a^{2} < 0 \end {cases}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.44, size = 102, normalized size = 0.71 \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 = 0.05, size = 174, normalized size = 1.22 \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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