Optimal. Leaf size=172 \[ \frac {7 b^{5/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {b x-\tanh ^{-1}(\tanh (a+b x))}}\right )}{\left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^{9/2}}-\frac {7 b^2}{\sqrt {x} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^4}-\frac {7 b}{3 x^{3/2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^3}-\frac {7}{5 x^{5/2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2}-\frac {1}{b x^{7/2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}-\frac {1}{b x^{7/2} \tanh ^{-1}(\tanh (a+b x))} \]
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Rubi [A]
time = 0.10, antiderivative size = 172, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2199, 2194,
2193} \begin {gather*} \frac {7 b^{5/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {b x-\tanh ^{-1}(\tanh (a+b x))}}\right )}{\left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^{9/2}}-\frac {7 b^2}{\sqrt {x} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^4}-\frac {7 b}{3 x^{3/2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^3}-\frac {7}{5 x^{5/2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2}-\frac {1}{b x^{7/2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}-\frac {1}{b x^{7/2} \tanh ^{-1}(\tanh (a+b x))} \end {gather*}
Antiderivative was successfully verified.
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Rule 2193
Rule 2194
Rule 2199
Rubi steps
\begin {align*} \int \frac {1}{x^{7/2} \tanh ^{-1}(\tanh (a+b x))^2} \, dx &=-\frac {1}{b x^{7/2} \tanh ^{-1}(\tanh (a+b x))}-\frac {7 \int \frac {1}{x^{9/2} \tanh ^{-1}(\tanh (a+b x))} \, dx}{2 b}\\ &=-\frac {1}{b x^{7/2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}-\frac {1}{b x^{7/2} \tanh ^{-1}(\tanh (a+b x))}+\frac {7 \int \frac {1}{x^{7/2} \tanh ^{-1}(\tanh (a+b x))} \, dx}{2 \left (-b x+\tanh ^{-1}(\tanh (a+b x))\right )}\\ &=-\frac {7}{5 x^{5/2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2}-\frac {1}{b x^{7/2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}-\frac {1}{b x^{7/2} \tanh ^{-1}(\tanh (a+b x))}-\frac {(7 b) \int \frac {1}{x^{5/2} \tanh ^{-1}(\tanh (a+b x))} \, dx}{2 \left (-b x+\tanh ^{-1}(\tanh (a+b x))\right )^2}\\ &=-\frac {7 b}{3 x^{3/2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^3}-\frac {7}{5 x^{5/2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2}-\frac {1}{b x^{7/2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}-\frac {1}{b x^{7/2} \tanh ^{-1}(\tanh (a+b x))}+\frac {\left (7 b^2\right ) \int \frac {1}{x^{3/2} \tanh ^{-1}(\tanh (a+b x))} \, dx}{2 \left (-b x+\tanh ^{-1}(\tanh (a+b x))\right )^3}\\ &=-\frac {7 b^2}{\sqrt {x} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^4}-\frac {7 b}{3 x^{3/2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^3}-\frac {7}{5 x^{5/2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2}-\frac {1}{b x^{7/2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}-\frac {1}{b x^{7/2} \tanh ^{-1}(\tanh (a+b x))}+\frac {\left (7 b^3\right ) \int \frac {1}{\sqrt {x} \tanh ^{-1}(\tanh (a+b x))} \, dx}{2 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right ) \left (-b x+\tanh ^{-1}(\tanh (a+b x))\right )^3}\\ &=\frac {7 b^{5/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {b x-\tanh ^{-1}(\tanh (a+b x))}}\right )}{\left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^{9/2}}-\frac {7 b^2}{\sqrt {x} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^4}-\frac {7 b}{3 x^{3/2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^3}-\frac {7}{5 x^{5/2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2}-\frac {1}{b x^{7/2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}-\frac {1}{b x^{7/2} \tanh ^{-1}(\tanh (a+b x))}\\ \end {align*}
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Mathematica [A]
time = 0.17, size = 139, normalized size = 0.81 \begin {gather*} -\frac {7 b^{5/2} \text {ArcTan}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {-b x+\tanh ^{-1}(\tanh (a+b x))}}\right )}{\left (-b x+\tanh ^{-1}(\tanh (a+b x))\right )^{9/2}}-\frac {b^3 \sqrt {x}}{\tanh ^{-1}(\tanh (a+b x)) \left (-b x+\tanh ^{-1}(\tanh (a+b x))\right )^4}-\frac {2 \left (58 b^2 x^2-16 b x \tanh ^{-1}(\tanh (a+b x))+3 \tanh ^{-1}(\tanh (a+b x))^2\right )}{15 x^{5/2} \left (-b x+\tanh ^{-1}(\tanh (a+b x))\right )^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.94, size = 151, normalized size = 0.88
method | result | size |
derivativedivides | \(-\frac {2}{5 \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x \right )^{2} x^{\frac {5}{2}}}-\frac {6 b^{2}}{\left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x \right )^{4} \sqrt {x}}+\frac {4 b}{3 \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x \right )^{3} x^{\frac {3}{2}}}-\frac {b^{3} \sqrt {x}}{\left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x \right )^{4} \arctanh \left (\tanh \left (b x +a \right )\right )}-\frac {7 b^{3} \arctan \left (\frac {b \sqrt {x}}{\sqrt {\left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x \right ) b}}\right )}{\left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x \right )^{4} \sqrt {\left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x \right ) b}}\) | \(151\) |
default | \(-\frac {2}{5 \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x \right )^{2} x^{\frac {5}{2}}}-\frac {6 b^{2}}{\left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x \right )^{4} \sqrt {x}}+\frac {4 b}{3 \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x \right )^{3} x^{\frac {3}{2}}}-\frac {b^{3} \sqrt {x}}{\left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x \right )^{4} \arctanh \left (\tanh \left (b x +a \right )\right )}-\frac {7 b^{3} \arctan \left (\frac {b \sqrt {x}}{\sqrt {\left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x \right ) b}}\right )}{\left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x \right )^{4} \sqrt {\left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x \right ) b}}\) | \(151\) |
risch | \(\text {Expression too large to display}\) | \(46076\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, size = 75, normalized size = 0.44 \begin {gather*} -\frac {105 \, b^{3} x^{3} + 70 \, a b^{2} x^{2} - 14 \, a^{2} b x + 6 \, a^{3}}{15 \, {\left (a^{4} b x^{\frac {7}{2}} + a^{5} x^{\frac {5}{2}}\right )}} - \frac {7 \, b^{3} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} a^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 210, normalized size = 1.22 \begin {gather*} \left [\frac {105 \, {\left (b^{3} x^{4} + a b^{2} x^{3}\right )} \sqrt {-\frac {b}{a}} \log \left (\frac {b x - 2 \, a \sqrt {x} \sqrt {-\frac {b}{a}} - a}{b x + a}\right ) - 2 \, {\left (105 \, b^{3} x^{3} + 70 \, a b^{2} x^{2} - 14 \, a^{2} b x + 6 \, a^{3}\right )} \sqrt {x}}{30 \, {\left (a^{4} b x^{4} + a^{5} x^{3}\right )}}, \frac {105 \, {\left (b^{3} x^{4} + a b^{2} x^{3}\right )} \sqrt {\frac {b}{a}} \arctan \left (\frac {a \sqrt {\frac {b}{a}}}{b \sqrt {x}}\right ) - {\left (105 \, b^{3} x^{3} + 70 \, a b^{2} x^{2} - 14 \, a^{2} b x + 6 \, a^{3}\right )} \sqrt {x}}{15 \, {\left (a^{4} b x^{4} + a^{5} x^{3}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^{\frac {7}{2}} \operatorname {atanh}^{2}{\left (\tanh {\left (a + b x \right )} \right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 70, normalized size = 0.41 \begin {gather*} -\frac {7 \, b^{3} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} a^{4}} - \frac {b^{3} \sqrt {x}}{{\left (b x + a\right )} a^{4}} - \frac {2 \, {\left (45 \, b^{2} x^{2} - 10 \, a b x + 3 \, a^{2}\right )}}{15 \, a^{4} x^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.14, size = 1051, normalized size = 6.11 \begin {gather*} \frac {\frac {96\,b^2}{{\left (\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+2\,b\,x\right )}^3}-\frac {224\,b^3\,x}{{\left (\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+2\,b\,x\right )}^4}}{\sqrt {x}\,\left (\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )-\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )\right )}-\frac {32\,b}{3\,x^{3/2}\,{\left (\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+2\,b\,x\right )}^3}-\frac {8}{5\,x^{5/2}\,{\left (\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+2\,b\,x\right )}^2}+\frac {56\,\sqrt {2}\,b^{5/2}\,\ln \left (-\frac {\sqrt {b}\,\sqrt {\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+2\,b\,x}\,\left (\sqrt {2}\,\left (\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+2\,b\,x\right )+4\,\sqrt {b}\,\sqrt {x}\,\sqrt {\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+2\,b\,x}+2\,\sqrt {2}\,b\,x\right )\,\left ({\left (2\,a-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+2\,b\,x\right )}^8+112\,a^2\,{\left (2\,a-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+2\,b\,x\right )}^6-448\,a^3\,{\left (2\,a-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+2\,b\,x\right )}^5+1120\,a^4\,{\left (2\,a-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+2\,b\,x\right )}^4-1792\,a^5\,{\left (2\,a-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+2\,b\,x\right )}^3+1792\,a^6\,{\left (2\,a-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+2\,b\,x\right )}^2+256\,a^8-16\,a\,{\left (2\,a-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+2\,b\,x\right )}^7-1024\,a^7\,\left (2\,a-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+2\,b\,x\right )\right )}{2\,\left (\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )-\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )\right )}\right )}{{\left (\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+2\,b\,x\right )}^{9/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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