Optimal. Leaf size=167 \[ \frac {55}{128} c^4 x \sqrt {1-a^2 x^2}+\frac {55}{192} c^4 x \left (1-a^2 x^2\right )^{3/2}+\frac {11}{48} c^4 x \left (1-a^2 x^2\right )^{5/2}+\frac {11 c^4 \left (1-a^2 x^2\right )^{7/2}}{56 a}+\frac {11 c^4 (1-a x) \left (1-a^2 x^2\right )^{7/2}}{72 a}+\frac {c^4 (1-a x)^2 \left (1-a^2 x^2\right )^{7/2}}{9 a}+\frac {55 c^4 \text {ArcSin}(a x)}{128 a} \]
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Rubi [A]
time = 0.07, antiderivative size = 167, 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 = {6274, 685, 655,
201, 222} \begin {gather*} \frac {c^4 (1-a x)^2 \left (1-a^2 x^2\right )^{7/2}}{9 a}+\frac {11 c^4 (1-a x) \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 \text {ArcSin}(a x)}{128 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 )^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 = 0.10, size = 107, normalized size = 0.64 \begin {gather*} -\frac {c^4 \left (\sqrt {1-a^2 x^2} \left (-3712-4599 a x+10240 a^2 x^2-3066 a^3 x^3-8448 a^4 x^4+7224 a^5 x^5+1024 a^6 x^6-3024 a^7 x^7+896 a^8 x^8\right )+6930 \text {ArcSin}\left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )\right )}{8064 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. \(390\) vs.
\(2(141)=282\).
time = 0.06, size = 391, normalized size = 2.34
method | result | size |
risch | \(\frac {\left (896 a^{8} x^{8}-3024 a^{7} x^{7}+1024 x^{6} a^{6}+7224 x^{5} a^{5}-8448 a^{4} x^{4}-3066 a^{3} x^{3}+10240 a^{2} x^{2}-4599 a x -3712\right ) \left (a^{2} x^{2}-1\right ) c^{4}}{8064 a \sqrt {-a^{2} x^{2}+1}}+\frac {55 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right ) c^{4}}{128 \sqrt {a^{2}}}\) | \(123\) |
default | \(c^{4} \left (a^{5} \left (-\frac {x^{4} \left (-a^{2} x^{2}+1\right )^{\frac {5}{2}}}{9 a^{2}}+\frac {-\frac {4 x^{2} \left (-a^{2} x^{2}+1\right )^{\frac {5}{2}}}{63 a^{2}}-\frac {8 \left (-a^{2} x^{2}+1\right )^{\frac {5}{2}}}{315 a^{4}}}{a^{2}}\right )-3 a^{4} \left (-\frac {x^{3} \left (-a^{2} x^{2}+1\right )^{\frac {5}{2}}}{8 a^{2}}+\frac {-\frac {x \left (-a^{2} x^{2}+1\right )^{\frac {5}{2}}}{16 a^{2}}+\frac {\frac {x \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{4}+\frac {3 x \sqrt {-a^{2} x^{2}+1}}{8}+\frac {3 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{8 \sqrt {a^{2}}}}{16 a^{2}}}{a^{2}}\right )+2 a^{3} \left (-\frac {x^{2} \left (-a^{2} x^{2}+1\right )^{\frac {5}{2}}}{7 a^{2}}-\frac {2 \left (-a^{2} x^{2}+1\right )^{\frac {5}{2}}}{35 a^{4}}\right )+2 a^{2} \left (-\frac {x \left (-a^{2} x^{2}+1\right )^{\frac {5}{2}}}{6 a^{2}}+\frac {\frac {x \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{4}+\frac {3 x \sqrt {-a^{2} x^{2}+1}}{8}+\frac {3 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{8 \sqrt {a^{2}}}}{6 a^{2}}\right )+\frac {3 \left (-a^{2} x^{2}+1\right )^{\frac {5}{2}}}{5 a}+\frac {x \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{4}+\frac {3 x \sqrt {-a^{2} x^{2}+1}}{8}+\frac {3 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{8 \sqrt {a^{2}}}\right )\) | \(391\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.47, size = 154, normalized size = 0.92 \begin {gather*} -\frac {1}{9} \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {5}{2}} a^{3} c^{4} x^{4} + \frac {3}{8} \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {5}{2}} a^{2} c^{4} x^{3} - \frac {22}{63} \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {5}{2}} a c^{4} x^{2} - \frac {7}{48} \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {5}{2}} c^{4} x + \frac {55}{192} \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} c^{4} x + \frac {29 \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {5}{2}} c^{4}}{63 \, a} + \frac {55}{128} \, \sqrt {-a^{2} x^{2} + 1} c^{4} x + \frac {55 \, c^{4} \arcsin \left (a x\right )}{128 \, a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.32, size = 136, normalized size = 0.81 \begin {gather*} -\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} \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 = 32.58, size = 994, normalized size = 5.95 \begin {gather*} - a^{7} c^{4} \left (\begin {cases} \frac {x^{8} \sqrt {- a^{2} x^{2} + 1}}{9} - \frac {x^{6} \sqrt {- a^{2} x^{2} + 1}}{63 a^{2}} - \frac {2 x^{4} \sqrt {- a^{2} x^{2} + 1}}{105 a^{4}} - \frac {8 x^{2} \sqrt {- a^{2} x^{2} + 1}}{315 a^{6}} - \frac {16 \sqrt {- a^{2} x^{2} + 1}}{315 a^{8}} & \text {for}\: a \neq 0 \\\frac {x^{8}}{8} & \text {otherwise} \end {cases}\right ) + 3 a^{6} c^{4} \left (\begin {cases} \frac {i a^{2} x^{9}}{8 \sqrt {a^{2} x^{2} - 1}} - \frac {7 i x^{7}}{48 \sqrt {a^{2} x^{2} - 1}} - \frac {i x^{5}}{192 a^{2} \sqrt {a^{2} x^{2} - 1}} - \frac {5 i x^{3}}{384 a^{4} \sqrt {a^{2} x^{2} - 1}} + \frac {5 i x}{128 a^{6} \sqrt {a^{2} x^{2} - 1}} - \frac {5 i \operatorname {acosh}{\left (a x \right )}}{128 a^{7}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\- \frac {a^{2} x^{9}}{8 \sqrt {- a^{2} x^{2} + 1}} + \frac {7 x^{7}}{48 \sqrt {- a^{2} x^{2} + 1}} + \frac {x^{5}}{192 a^{2} \sqrt {- a^{2} x^{2} + 1}} + \frac {5 x^{3}}{384 a^{4} \sqrt {- a^{2} x^{2} + 1}} - \frac {5 x}{128 a^{6} \sqrt {- a^{2} x^{2} + 1}} + \frac {5 \operatorname {asin}{\left (a x \right )}}{128 a^{7}} & \text {otherwise} \end {cases}\right ) - a^{5} c^{4} \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 ) - 5 a^{4} c^{4} \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 ) + 5 a^{3} c^{4} \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 ) + a^{2} c^{4} \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^{4} \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^{4} \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 = 126, normalized size = 0.75 \begin {gather*} \frac {55 \, c^{4} \arcsin \left (a x\right ) \mathrm {sgn}\left (a\right )}{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 )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.07, size = 220, normalized size = 1.32 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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