Optimal. Leaf size=123 \[ \frac {7}{8} c^4 x \sqrt {1-a^2 x^2}+\frac {7 c^4 \left (1-a^2 x^2\right )^{3/2}}{12 a}+\frac {7 c^4 (1-a x) \left (1-a^2 x^2\right )^{3/2}}{20 a}+\frac {c^4 (1-a x)^2 \left (1-a^2 x^2\right )^{3/2}}{5 a}+\frac {7 c^4 \text {ArcSin}(a x)}{8 a} \]
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Rubi [A]
time = 0.06, antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {6262, 685, 655,
201, 222} \begin {gather*} \frac {c^4 (1-a x)^2 \left (1-a^2 x^2\right )^{3/2}}{5 a}+\frac {7 c^4 (1-a x) \left (1-a^2 x^2\right )^{3/2}}{20 a}+\frac {7 c^4 \left (1-a^2 x^2\right )^{3/2}}{12 a}+\frac {7}{8} c^4 x \sqrt {1-a^2 x^2}+\frac {7 c^4 \text {ArcSin}(a x)}{8 a} \end {gather*}
Antiderivative was successfully verified.
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Rule 201
Rule 222
Rule 655
Rule 685
Rule 6262
Rubi steps
\begin {align*} \int e^{\tanh ^{-1}(a x)} (c-a c x)^4 \, dx &=c \int (c-a c x)^3 \sqrt {1-a^2 x^2} \, dx\\ &=\frac {c^4 (1-a x)^2 \left (1-a^2 x^2\right )^{3/2}}{5 a}+\frac {1}{5} \left (7 c^2\right ) \int (c-a c x)^2 \sqrt {1-a^2 x^2} \, dx\\ &=\frac {7 c^4 (1-a x) \left (1-a^2 x^2\right )^{3/2}}{20 a}+\frac {c^4 (1-a x)^2 \left (1-a^2 x^2\right )^{3/2}}{5 a}+\frac {1}{4} \left (7 c^3\right ) \int (c-a c x) \sqrt {1-a^2 x^2} \, dx\\ &=\frac {7 c^4 \left (1-a^2 x^2\right )^{3/2}}{12 a}+\frac {7 c^4 (1-a x) \left (1-a^2 x^2\right )^{3/2}}{20 a}+\frac {c^4 (1-a x)^2 \left (1-a^2 x^2\right )^{3/2}}{5 a}+\frac {1}{4} \left (7 c^4\right ) \int \sqrt {1-a^2 x^2} \, dx\\ &=\frac {7}{8} c^4 x \sqrt {1-a^2 x^2}+\frac {7 c^4 \left (1-a^2 x^2\right )^{3/2}}{12 a}+\frac {7 c^4 (1-a x) \left (1-a^2 x^2\right )^{3/2}}{20 a}+\frac {c^4 (1-a x)^2 \left (1-a^2 x^2\right )^{3/2}}{5 a}+\frac {1}{8} \left (7 c^4\right ) \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx\\ &=\frac {7}{8} c^4 x \sqrt {1-a^2 x^2}+\frac {7 c^4 \left (1-a^2 x^2\right )^{3/2}}{12 a}+\frac {7 c^4 (1-a x) \left (1-a^2 x^2\right )^{3/2}}{20 a}+\frac {c^4 (1-a x)^2 \left (1-a^2 x^2\right )^{3/2}}{5 a}+\frac {7 c^4 \sin ^{-1}(a x)}{8 a}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 75, normalized size = 0.61 \begin {gather*} -\frac {c^4 \left (\sqrt {1-a^2 x^2} \left (-136-15 a x+112 a^2 x^2-90 a^3 x^3+24 a^4 x^4\right )+210 \text {ArcSin}\left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )\right )}{120 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. \(293\) vs.
\(2(105)=210\).
time = 0.00, size = 294, normalized size = 2.39
method | result | size |
risch | \(\frac {\left (24 a^{4} x^{4}-90 a^{3} x^{3}+112 a^{2} x^{2}-15 a x -136\right ) \left (a^{2} x^{2}-1\right ) c^{4}}{120 a \sqrt {-a^{2} x^{2}+1}}+\frac {7 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right ) c^{4}}{8 \sqrt {a^{2}}}\) | \(91\) |
meijerg | \(-\frac {c^{4} \left (-\frac {16 \sqrt {\pi }}{15}+\frac {\sqrt {\pi }\, \left (6 a^{4} x^{4}+8 a^{2} x^{2}+16\right ) \sqrt {-a^{2} x^{2}+1}}{15}\right )}{2 a \sqrt {\pi }}-\frac {3 c^{4} \left (-\frac {\sqrt {\pi }\, x \left (-a^{2}\right )^{\frac {5}{2}} \left (10 a^{2} x^{2}+15\right ) \sqrt {-a^{2} x^{2}+1}}{20 a^{4}}+\frac {3 \sqrt {\pi }\, \left (-a^{2}\right )^{\frac {5}{2}} \arcsin \left (a x \right )}{4 a^{5}}\right )}{2 \sqrt {\pi }\, \sqrt {-a^{2}}}+\frac {c^{4} \left (\frac {4 \sqrt {\pi }}{3}-\frac {\sqrt {\pi }\, \left (4 a^{2} x^{2}+8\right ) \sqrt {-a^{2} x^{2}+1}}{6}\right )}{a \sqrt {\pi }}-\frac {c^{4} \left (-\frac {\sqrt {\pi }\, x \left (-a^{2}\right )^{\frac {3}{2}} \sqrt {-a^{2} x^{2}+1}}{a^{2}}+\frac {\sqrt {\pi }\, \left (-a^{2}\right )^{\frac {3}{2}} \arcsin \left (a x \right )}{a^{3}}\right )}{\sqrt {\pi }\, \sqrt {-a^{2}}}+\frac {3 c^{4} \left (-2 \sqrt {\pi }+2 \sqrt {\pi }\, \sqrt {-a^{2} x^{2}+1}\right )}{2 a \sqrt {\pi }}+\frac {c^{4} \arcsin \left (a x \right )}{a}\) | \(277\) |
default | \(c^{4} \left (a^{5} \left (-\frac {x^{4} \sqrt {-a^{2} x^{2}+1}}{5 a^{2}}+\frac {-\frac {4 x^{2} \sqrt {-a^{2} x^{2}+1}}{15 a^{2}}-\frac {8 \sqrt {-a^{2} x^{2}+1}}{15 a^{4}}}{a^{2}}\right )-3 a^{4} \left (-\frac {x^{3} \sqrt {-a^{2} x^{2}+1}}{4 a^{2}}+\frac {-\frac {3 x \sqrt {-a^{2} x^{2}+1}}{8 a^{2}}+\frac {3 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{8 a^{2} \sqrt {a^{2}}}}{a^{2}}\right )+2 a^{3} \left (-\frac {x^{2} \sqrt {-a^{2} x^{2}+1}}{3 a^{2}}-\frac {2 \sqrt {-a^{2} x^{2}+1}}{3 a^{4}}\right )+2 a^{2} \left (-\frac {x \sqrt {-a^{2} x^{2}+1}}{2 a^{2}}+\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 a^{2} \sqrt {a^{2}}}\right )+\frac {3 \sqrt {-a^{2} x^{2}+1}}{a}+\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{\sqrt {a^{2}}}\right )\) | \(294\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.46, size = 118, normalized size = 0.96 \begin {gather*} -\frac {1}{5} \, \sqrt {-a^{2} x^{2} + 1} a^{3} c^{4} x^{4} + \frac {3}{4} \, \sqrt {-a^{2} x^{2} + 1} a^{2} c^{4} x^{3} - \frac {14}{15} \, \sqrt {-a^{2} x^{2} + 1} a c^{4} x^{2} + \frac {1}{8} \, \sqrt {-a^{2} x^{2} + 1} c^{4} x + \frac {7 \, c^{4} \arcsin \left (a x\right )}{8 \, a} + \frac {17 \, \sqrt {-a^{2} x^{2} + 1} c^{4}}{15 \, a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 92, normalized size = 0.75 \begin {gather*} -\frac {210 \, c^{4} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + {\left (24 \, a^{4} c^{4} x^{4} - 90 \, a^{3} c^{4} x^{3} + 112 \, a^{2} c^{4} x^{2} - 15 \, a c^{4} x - 136 \, c^{4}\right )} \sqrt {-a^{2} x^{2} + 1}}{120 \, 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. 226 vs.
\(2 (104) = 208\).
time = 5.23, size = 226, normalized size = 1.84 \begin {gather*} \begin {cases} \frac {3 c^{4} \sqrt {- a^{2} x^{2} + 1} + 2 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 ) + 2 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 ) - 3 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 ) + 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 ) + 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 = 0.46, size = 78, normalized size = 0.63 \begin {gather*} \frac {7 \, c^{4} \arcsin \left (a x\right ) \mathrm {sgn}\left (a\right )}{8 \, {\left | a \right |}} + \frac {1}{120} \, \sqrt {-a^{2} x^{2} + 1} {\left (\frac {136 \, c^{4}}{a} + {\left (15 \, c^{4} - 2 \, {\left (56 \, a c^{4} + 3 \, {\left (4 \, a^{3} c^{4} x - 15 \, a^{2} c^{4}\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 = 0.00, size = 128, normalized size = 1.04 \begin {gather*} \frac {c^4\,x\,\sqrt {1-a^2\,x^2}}{8}+\frac {7\,c^4\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{8\,\sqrt {-a^2}}+\frac {17\,c^4\,\sqrt {1-a^2\,x^2}}{15\,a}-\frac {14\,a\,c^4\,x^2\,\sqrt {1-a^2\,x^2}}{15}+\frac {3\,a^2\,c^4\,x^3\,\sqrt {1-a^2\,x^2}}{4}-\frac {a^3\,c^4\,x^4\,\sqrt {1-a^2\,x^2}}{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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