Optimal. Leaf size=158 \[ -\frac {7 c^4 x \sqrt {1-a^2 x^2}}{16 a}-\frac {7 c^4 \left (1-a^2 x^2\right )^{3/2}}{24 a^2}-\frac {7 c^4 (1-a x) \left (1-a^2 x^2\right )^{3/2}}{40 a^2}-\frac {c^4 (1-a x)^2 \left (1-a^2 x^2\right )^{3/2}}{10 a^2}-\frac {c^4 (1-a x)^3 \left (1-a^2 x^2\right )^{3/2}}{6 a^2}-\frac {7 c^4 \text {ArcSin}(a x)}{16 a^2} \]
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Rubi [A]
time = 0.10, antiderivative size = 158, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {6263, 809, 685,
655, 201, 222} \begin {gather*} -\frac {7 c^4 \text {ArcSin}(a x)}{16 a^2}-\frac {c^4 (1-a x)^3 \left (1-a^2 x^2\right )^{3/2}}{6 a^2}-\frac {c^4 (1-a x)^2 \left (1-a^2 x^2\right )^{3/2}}{10 a^2}-\frac {7 c^4 (1-a x) \left (1-a^2 x^2\right )^{3/2}}{40 a^2}-\frac {7 c^4 \left (1-a^2 x^2\right )^{3/2}}{24 a^2}-\frac {7 c^4 x \sqrt {1-a^2 x^2}}{16 a} \end {gather*}
Antiderivative was successfully verified.
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Rule 201
Rule 222
Rule 655
Rule 685
Rule 809
Rule 6263
Rubi steps
\begin {align*} \int e^{\tanh ^{-1}(a x)} x (c-a c x)^4 \, dx &=c \int x (c-a c x)^3 \sqrt {1-a^2 x^2} \, dx\\ &=-\frac {c^4 (1-a x)^3 \left (1-a^2 x^2\right )^{3/2}}{6 a^2}-\frac {c \int (c-a c x)^3 \sqrt {1-a^2 x^2} \, dx}{2 a}\\ &=-\frac {c^4 (1-a x)^2 \left (1-a^2 x^2\right )^{3/2}}{10 a^2}-\frac {c^4 (1-a x)^3 \left (1-a^2 x^2\right )^{3/2}}{6 a^2}-\frac {\left (7 c^2\right ) \int (c-a c x)^2 \sqrt {1-a^2 x^2} \, dx}{10 a}\\ &=-\frac {7 c^4 (1-a x) \left (1-a^2 x^2\right )^{3/2}}{40 a^2}-\frac {c^4 (1-a x)^2 \left (1-a^2 x^2\right )^{3/2}}{10 a^2}-\frac {c^4 (1-a x)^3 \left (1-a^2 x^2\right )^{3/2}}{6 a^2}-\frac {\left (7 c^3\right ) \int (c-a c x) \sqrt {1-a^2 x^2} \, dx}{8 a}\\ &=-\frac {7 c^4 \left (1-a^2 x^2\right )^{3/2}}{24 a^2}-\frac {7 c^4 (1-a x) \left (1-a^2 x^2\right )^{3/2}}{40 a^2}-\frac {c^4 (1-a x)^2 \left (1-a^2 x^2\right )^{3/2}}{10 a^2}-\frac {c^4 (1-a x)^3 \left (1-a^2 x^2\right )^{3/2}}{6 a^2}-\frac {\left (7 c^4\right ) \int \sqrt {1-a^2 x^2} \, dx}{8 a}\\ &=-\frac {7 c^4 x \sqrt {1-a^2 x^2}}{16 a}-\frac {7 c^4 \left (1-a^2 x^2\right )^{3/2}}{24 a^2}-\frac {7 c^4 (1-a x) \left (1-a^2 x^2\right )^{3/2}}{40 a^2}-\frac {c^4 (1-a x)^2 \left (1-a^2 x^2\right )^{3/2}}{10 a^2}-\frac {c^4 (1-a x)^3 \left (1-a^2 x^2\right )^{3/2}}{6 a^2}-\frac {\left (7 c^4\right ) \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{16 a}\\ &=-\frac {7 c^4 x \sqrt {1-a^2 x^2}}{16 a}-\frac {7 c^4 \left (1-a^2 x^2\right )^{3/2}}{24 a^2}-\frac {7 c^4 (1-a x) \left (1-a^2 x^2\right )^{3/2}}{40 a^2}-\frac {c^4 (1-a x)^2 \left (1-a^2 x^2\right )^{3/2}}{10 a^2}-\frac {c^4 (1-a x)^3 \left (1-a^2 x^2\right )^{3/2}}{6 a^2}-\frac {7 c^4 \sin ^{-1}(a x)}{16 a^2}\\ \end {align*}
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Mathematica [A]
time = 0.09, size = 83, normalized size = 0.53 \begin {gather*} -\frac {c^4 \left (\sqrt {1-a^2 x^2} \left (176-105 a x-32 a^2 x^2+170 a^3 x^3-144 a^4 x^4+40 a^5 x^5\right )-210 \text {ArcSin}\left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )\right )}{240 a^2} \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. \(371\) vs.
\(2(136)=272\).
time = 0.78, size = 372, normalized size = 2.35
method | result | size |
risch | \(\frac {\left (40 a^{5} x^{5}-144 a^{4} x^{4}+170 a^{3} x^{3}-32 a^{2} x^{2}-105 a x +176\right ) \left (a^{2} x^{2}-1\right ) c^{4}}{240 a^{2} \sqrt {-a^{2} x^{2}+1}}-\frac {7 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right ) c^{4}}{16 a \sqrt {a^{2}}}\) | \(102\) |
meijerg | \(-\frac {c^{4} \left (-\frac {\sqrt {\pi }\, x \left (-a^{2}\right )^{\frac {7}{2}} \left (56 a^{4} x^{4}+70 a^{2} x^{2}+105\right ) \sqrt {-a^{2} x^{2}+1}}{168 a^{6}}+\frac {5 \sqrt {\pi }\, \left (-a^{2}\right )^{\frac {7}{2}} \arcsin \left (a x \right )}{8 a^{7}}\right )}{2 a \sqrt {\pi }\, \sqrt {-a^{2}}}+\frac {3 c^{4} \left (-\frac {16 \sqrt {\pi }}{15}+\frac {\sqrt {\pi }\, \left (6 a^{4} x^{4}+8 a^{2} x^{2}+16\right ) \sqrt {-a^{2} x^{2}+1}}{15}\right )}{2 a^{2} \sqrt {\pi }}+\frac {c^{4} \left (-\frac {\sqrt {\pi }\, x \left (-a^{2}\right )^{\frac {5}{2}} \left (10 a^{2} x^{2}+15\right ) \sqrt {-a^{2} x^{2}+1}}{20 a^{4}}+\frac {3 \sqrt {\pi }\, \left (-a^{2}\right )^{\frac {5}{2}} \arcsin \left (a x \right )}{4 a^{5}}\right )}{a \sqrt {\pi }\, \sqrt {-a^{2}}}+\frac {c^{4} \left (\frac {4 \sqrt {\pi }}{3}-\frac {\sqrt {\pi }\, \left (4 a^{2} x^{2}+8\right ) \sqrt {-a^{2} x^{2}+1}}{6}\right )}{a^{2} \sqrt {\pi }}+\frac {3 c^{4} \left (-\frac {\sqrt {\pi }\, x \left (-a^{2}\right )^{\frac {3}{2}} \sqrt {-a^{2} x^{2}+1}}{a^{2}}+\frac {\sqrt {\pi }\, \left (-a^{2}\right )^{\frac {3}{2}} \arcsin \left (a x \right )}{a^{3}}\right )}{2 a \sqrt {\pi }\, \sqrt {-a^{2}}}-\frac {c^{4} \left (-2 \sqrt {\pi }+2 \sqrt {\pi }\, \sqrt {-a^{2} x^{2}+1}\right )}{2 a^{2} \sqrt {\pi }}\) | \(355\) |
default | \(c^{4} \left (a^{5} \left (-\frac {x^{5} \sqrt {-a^{2} x^{2}+1}}{6 a^{2}}+\frac {-\frac {5 x^{3} \sqrt {-a^{2} x^{2}+1}}{24 a^{2}}+\frac {5 \left (-\frac {3 x \sqrt {-a^{2} x^{2}+1}}{8 a^{2}}+\frac {3 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{8 a^{2} \sqrt {a^{2}}}\right )}{6 a^{2}}}{a^{2}}\right )-3 a^{4} \left (-\frac {x^{4} \sqrt {-a^{2} x^{2}+1}}{5 a^{2}}+\frac {-\frac {4 x^{2} \sqrt {-a^{2} x^{2}+1}}{15 a^{2}}-\frac {8 \sqrt {-a^{2} x^{2}+1}}{15 a^{4}}}{a^{2}}\right )+2 a^{3} \left (-\frac {x^{3} \sqrt {-a^{2} x^{2}+1}}{4 a^{2}}+\frac {-\frac {3 x \sqrt {-a^{2} x^{2}+1}}{8 a^{2}}+\frac {3 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{8 a^{2} \sqrt {a^{2}}}}{a^{2}}\right )+2 a^{2} \left (-\frac {x^{2} \sqrt {-a^{2} x^{2}+1}}{3 a^{2}}-\frac {2 \sqrt {-a^{2} x^{2}+1}}{3 a^{4}}\right )-3 a \left (-\frac {x \sqrt {-a^{2} x^{2}+1}}{2 a^{2}}+\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 a^{2} \sqrt {a^{2}}}\right )-\frac {\sqrt {-a^{2} x^{2}+1}}{a^{2}}\right )\) | \(372\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.47, size = 141, normalized size = 0.89 \begin {gather*} -\frac {1}{6} \, \sqrt {-a^{2} x^{2} + 1} a^{3} c^{4} x^{5} + \frac {3}{5} \, \sqrt {-a^{2} x^{2} + 1} a^{2} c^{4} x^{4} - \frac {17}{24} \, \sqrt {-a^{2} x^{2} + 1} a c^{4} x^{3} + \frac {2}{15} \, \sqrt {-a^{2} x^{2} + 1} c^{4} x^{2} + \frac {7 \, \sqrt {-a^{2} x^{2} + 1} c^{4} x}{16 \, a} - \frac {7 \, c^{4} \arcsin \left (a x\right )}{16 \, a^{2}} - \frac {11 \, \sqrt {-a^{2} x^{2} + 1} c^{4}}{15 \, a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 104, normalized size = 0.66 \begin {gather*} \frac {210 \, c^{4} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) - {\left (40 \, a^{5} c^{4} x^{5} - 144 \, a^{4} c^{4} x^{4} + 170 \, a^{3} c^{4} x^{3} - 32 \, a^{2} c^{4} x^{2} - 105 \, a c^{4} x + 176 \, c^{4}\right )} \sqrt {-a^{2} x^{2} + 1}}{240 \, a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 14.11, size = 617, normalized size = 3.91 \begin {gather*} a^{5} c^{4} \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 ) - 3 a^{4} c^{4} \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 ) + 2 a^{3} c^{4} \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 ) + 2 a^{2} c^{4} \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 ) - 3 a c^{4} \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 ) + c^{4} \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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.47, size = 94, normalized size = 0.59 \begin {gather*} -\frac {7 \, c^{4} \arcsin \left (a x\right ) \mathrm {sgn}\left (a\right )}{16 \, a {\left | a \right |}} - \frac {1}{240} \, \sqrt {-a^{2} x^{2} + 1} {\left (\frac {176 \, c^{4}}{a^{2}} - {\left (\frac {105 \, c^{4}}{a} + 2 \, {\left (16 \, c^{4} - {\left (85 \, a c^{4} + 4 \, {\left (5 \, a^{3} c^{4} x - 18 \, a^{2} c^{4}\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 = 154, normalized size = 0.97 \begin {gather*} \frac {2\,c^4\,x^2\,\sqrt {1-a^2\,x^2}}{15}-\frac {11\,c^4\,\sqrt {1-a^2\,x^2}}{15\,a^2}+\frac {7\,c^4\,x\,\sqrt {1-a^2\,x^2}}{16\,a}-\frac {17\,a\,c^4\,x^3\,\sqrt {1-a^2\,x^2}}{24}-\frac {7\,c^4\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{16\,a\,\sqrt {-a^2}}+\frac {3\,a^2\,c^4\,x^4\,\sqrt {1-a^2\,x^2}}{5}-\frac {a^3\,c^4\,x^5\,\sqrt {1-a^2\,x^2}}{6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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