Optimal. Leaf size=127 \[ \frac {35}{128} c^4 x \sqrt {1-a^2 x^2}+\frac {35}{192} c^4 x \left (1-a^2 x^2\right )^{3/2}+\frac {7}{48} c^4 x \left (1-a^2 x^2\right )^{5/2}+\frac {1}{8} c^4 x \left (1-a^2 x^2\right )^{7/2}-\frac {c^4 \left (1-a^2 x^2\right )^{9/2}}{9 a}+\frac {35 c^4 \text {ArcSin}(a x)}{128 a} \]
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Rubi [A]
time = 0.05, 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 = {6273, 655, 201,
222} \begin {gather*} -\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 \text {ArcSin}(a x)}{128 a} \end {gather*}
Antiderivative was successfully verified.
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Rule 201
Rule 222
Rule 655
Rule 6273
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 = 0.10, size = 107, normalized size = 0.84 \begin {gather*} -\frac {c^4 \left (\sqrt {1-a^2 x^2} \left (128-837 a x-512 a^2 x^2+978 a^3 x^3+768 a^4 x^4-600 a^5 x^5-512 a^6 x^6+144 a^7 x^7+128 a^8 x^8\right )+630 \text {ArcSin}\left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )\right )}{1152 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. \(746\) vs.
\(2(105)=210\).
time = 0.14, size = 747, normalized size = 5.88
method | result | size |
risch | \(\frac {\left (128 a^{8} x^{8}+144 a^{7} x^{7}-512 x^{6} a^{6}-600 x^{5} a^{5}+768 a^{4} x^{4}+978 a^{3} x^{3}-512 a^{2} x^{2}-837 a x +128\right ) \left (a^{2} x^{2}-1\right ) c^{4}}{1152 a \sqrt {-a^{2} x^{2}+1}}+\frac {35 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right ) c^{4}}{128 \sqrt {a^{2}}}\) | \(123\) |
meijerg | \(-\frac {c^{4} \left (-\frac {256 \sqrt {\pi }}{315}+\frac {\sqrt {\pi }\, \left (70 a^{8} x^{8}+80 x^{6} a^{6}+96 a^{4} x^{4}+128 a^{2} x^{2}+256\right ) \sqrt {-a^{2} x^{2}+1}}{315}\right )}{2 a \sqrt {\pi }}-\frac {2 c^{4} \left (\frac {32 \sqrt {\pi }}{35}-\frac {\sqrt {\pi }\, \left (40 x^{6} a^{6}+48 a^{4} x^{4}+64 a^{2} x^{2}+128\right ) \sqrt {-a^{2} x^{2}+1}}{140}\right )}{a \sqrt {\pi }}-\frac {3 c^{4} \left (-\frac {16 \sqrt {\pi }}{15}+\frac {\sqrt {\pi }\, \left (6 a^{4} x^{4}+8 a^{2} x^{2}+16\right ) \sqrt {-a^{2} x^{2}+1}}{15}\right )}{a \sqrt {\pi }}-\frac {2 c^{4} \left (\frac {4 \sqrt {\pi }}{3}-\frac {\sqrt {\pi }\, \left (4 a^{2} x^{2}+8\right ) \sqrt {-a^{2} x^{2}+1}}{6}\right )}{a \sqrt {\pi }}-\frac {c^{4} \left (-2 \sqrt {\pi }+2 \sqrt {\pi }\, \sqrt {-a^{2} x^{2}+1}\right )}{2 a \sqrt {\pi }}+\frac {c^{4} \left (-\frac {\sqrt {\pi }\, x \left (-a^{2}\right )^{\frac {9}{2}} \left (144 x^{6} a^{6}+168 a^{4} x^{4}+210 a^{2} x^{2}+315\right ) \sqrt {-a^{2} x^{2}+1}}{576 a^{8}}+\frac {35 \sqrt {\pi }\, \left (-a^{2}\right )^{\frac {9}{2}} \arcsin \left (a x \right )}{64 a^{9}}\right )}{2 \sqrt {\pi }\, \sqrt {-a^{2}}}+\frac {2 c^{4} \left (-\frac {\sqrt {\pi }\, x \left (-a^{2}\right )^{\frac {7}{2}} \left (56 a^{4} x^{4}+70 a^{2} x^{2}+105\right ) \sqrt {-a^{2} x^{2}+1}}{168 a^{6}}+\frac {5 \sqrt {\pi }\, \left (-a^{2}\right )^{\frac {7}{2}} \arcsin \left (a x \right )}{8 a^{7}}\right )}{\sqrt {\pi }\, \sqrt {-a^{2}}}+\frac {3 c^{4} \left (-\frac {\sqrt {\pi }\, x \left (-a^{2}\right )^{\frac {5}{2}} \left (10 a^{2} x^{2}+15\right ) \sqrt {-a^{2} x^{2}+1}}{20 a^{4}}+\frac {3 \sqrt {\pi }\, \left (-a^{2}\right )^{\frac {5}{2}} \arcsin \left (a x \right )}{4 a^{5}}\right )}{\sqrt {\pi }\, \sqrt {-a^{2}}}+\frac {2 c^{4} \left (-\frac {\sqrt {\pi }\, x \left (-a^{2}\right )^{\frac {3}{2}} \sqrt {-a^{2} x^{2}+1}}{a^{2}}+\frac {\sqrt {\pi }\, \left (-a^{2}\right )^{\frac {3}{2}} \arcsin \left (a x \right )}{a^{3}}\right )}{\sqrt {\pi }\, \sqrt {-a^{2}}}+\frac {c^{4} \arcsin \left (a x \right )}{a}\) | \(576\) |
default | \(c^{4} \left (a^{9} \left (-\frac {x^{8} \sqrt {-a^{2} x^{2}+1}}{9 a^{2}}+\frac {-\frac {8 x^{6} \sqrt {-a^{2} x^{2}+1}}{63 a^{2}}+\frac {8 \left (-\frac {6 x^{4} \sqrt {-a^{2} x^{2}+1}}{35 a^{2}}+\frac {6 \left (-\frac {4 x^{2} \sqrt {-a^{2} x^{2}+1}}{15 a^{2}}-\frac {8 \sqrt {-a^{2} x^{2}+1}}{15 a^{4}}\right )}{7 a^{2}}\right )}{9 a^{2}}}{a^{2}}\right )+a^{8} \left (-\frac {x^{7} \sqrt {-a^{2} x^{2}+1}}{8 a^{2}}+\frac {-\frac {7 x^{5} \sqrt {-a^{2} x^{2}+1}}{48 a^{2}}+\frac {7 \left (-\frac {5 x^{3} \sqrt {-a^{2} x^{2}+1}}{24 a^{2}}+\frac {5 \left (-\frac {3 x \sqrt {-a^{2} x^{2}+1}}{8 a^{2}}+\frac {3 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{8 a^{2} \sqrt {a^{2}}}\right )}{6 a^{2}}\right )}{8 a^{2}}}{a^{2}}\right )-4 a^{7} \left (-\frac {x^{6} \sqrt {-a^{2} x^{2}+1}}{7 a^{2}}+\frac {-\frac {6 x^{4} \sqrt {-a^{2} x^{2}+1}}{35 a^{2}}+\frac {6 \left (-\frac {4 x^{2} \sqrt {-a^{2} x^{2}+1}}{15 a^{2}}-\frac {8 \sqrt {-a^{2} x^{2}+1}}{15 a^{4}}\right )}{7 a^{2}}}{a^{2}}\right )-4 a^{6} \left (-\frac {x^{5} \sqrt {-a^{2} x^{2}+1}}{6 a^{2}}+\frac {-\frac {5 x^{3} \sqrt {-a^{2} x^{2}+1}}{24 a^{2}}+\frac {5 \left (-\frac {3 x \sqrt {-a^{2} x^{2}+1}}{8 a^{2}}+\frac {3 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{8 a^{2} \sqrt {a^{2}}}\right )}{6 a^{2}}}{a^{2}}\right )+6 a^{5} \left (-\frac {x^{4} \sqrt {-a^{2} x^{2}+1}}{5 a^{2}}+\frac {-\frac {4 x^{2} \sqrt {-a^{2} x^{2}+1}}{15 a^{2}}-\frac {8 \sqrt {-a^{2} x^{2}+1}}{15 a^{4}}}{a^{2}}\right )+6 a^{4} \left (-\frac {x^{3} \sqrt {-a^{2} x^{2}+1}}{4 a^{2}}+\frac {-\frac {3 x \sqrt {-a^{2} x^{2}+1}}{8 a^{2}}+\frac {3 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{8 a^{2} \sqrt {a^{2}}}}{a^{2}}\right )-4 a^{3} \left (-\frac {x^{2} \sqrt {-a^{2} x^{2}+1}}{3 a^{2}}-\frac {2 \sqrt {-a^{2} x^{2}+1}}{3 a^{4}}\right )-4 a^{2} \left (-\frac {x \sqrt {-a^{2} x^{2}+1}}{2 a^{2}}+\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 a^{2} \sqrt {a^{2}}}\right )-\frac {\sqrt {-a^{2} x^{2}+1}}{a}+\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{\sqrt {a^{2}}}\right )\) | \(747\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.47, size = 210, normalized size = 1.65 \begin {gather*} -\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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 136, normalized size = 1.07 \begin {gather*} -\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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 595 vs.
\(2 (110) = 220\).
time = 16.09, size = 595, normalized size = 4.69 \begin {gather*} \begin {cases} \frac {- c^{4} \sqrt {- a^{2} x^{2} + 1} - 4 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 ) - 4 c^{4} \left (\begin {cases} \frac {\left (- a^{2} x^{2} + 1\right )^{\frac {3}{2}}}{3} - \sqrt {- a^{2} x^{2} + 1} & \text {for}\: a x > -1 \wedge a x < 1 \end {cases}\right ) + 6 c^{4} \left (\begin {cases} \frac {a x \left (- 2 a^{2} x^{2} + 1\right ) \sqrt {- a^{2} x^{2} + 1}}{8} - \frac {a x \sqrt {- a^{2} x^{2} + 1}}{2} + \frac {3 \operatorname {asin}{\left (a x \right )}}{8} & \text {for}\: a x > -1 \wedge a x < 1 \end {cases}\right ) + 6 c^{4} \left (\begin {cases} - \frac {\left (- a^{2} x^{2} + 1\right )^{\frac {5}{2}}}{5} + \frac {2 \left (- a^{2} x^{2} + 1\right )^{\frac {3}{2}}}{3} - \sqrt {- a^{2} x^{2} + 1} & \text {for}\: a x > -1 \wedge a x < 1 \end {cases}\right ) - 4 c^{4} \left (\begin {cases} \frac {a^{3} x^{3} \left (- a^{2} x^{2} + 1\right )^{\frac {3}{2}}}{6} + \frac {3 a x \left (- 2 a^{2} x^{2} + 1\right ) \sqrt {- a^{2} x^{2} + 1}}{16} - \frac {a x \sqrt {- a^{2} x^{2} + 1}}{2} + \frac {5 \operatorname {asin}{\left (a x \right )}}{16} & \text {for}\: a x > -1 \wedge a x < 1 \end {cases}\right ) - 4 c^{4} \left (\begin {cases} \frac {\left (- a^{2} x^{2} + 1\right )^{\frac {7}{2}}}{7} - \frac {3 \left (- a^{2} x^{2} + 1\right )^{\frac {5}{2}}}{5} + \left (- a^{2} x^{2} + 1\right )^{\frac {3}{2}} - \sqrt {- a^{2} x^{2} + 1} & \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}}}{3} + \frac {7 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 {a x \sqrt {- a^{2} x^{2} + 1}}{2} + \frac {35 \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 {4 \left (- a^{2} x^{2} + 1\right )^{\frac {7}{2}}}{7} - \frac {6 \left (- a^{2} x^{2} + 1\right )^{\frac {5}{2}}}{5} + \frac {4 \left (- a^{2} x^{2} + 1\right )^{\frac {3}{2}}}{3} - \sqrt {- a^{2} x^{2} + 1} & \text {for}\: a x > -1 \wedge a x < 1 \end {cases}\right ) + c^{4} \operatorname {asin}{\left (a x \right )}}{a} & \text {for}\: a \neq 0 \\c^{4} x & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, size = 127, normalized size = 1.00 \begin {gather*} \frac {35 \, c^{4} \arcsin \left (a x\right ) \mathrm {sgn}\left (a\right )}{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 )} \end {gather*}
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 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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