Optimal. Leaf size=106 \[ -5 b^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {\tanh ^{-1}(\tanh (a+b x))}}\right ) \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )+5 b^2 \sqrt {x} \sqrt {\tanh ^{-1}(\tanh (a+b x))}-\frac {2 \tanh ^{-1}(\tanh (a+b x))^{5/2}}{3 x^{3/2}}-\frac {10 b \tanh ^{-1}(\tanh (a+b x))^{3/2}}{3 \sqrt {x}} \]
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Rubi [A] time = 0.06, antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {2168, 2169, 2165} \[ 5 b^2 \sqrt {x} \sqrt {\tanh ^{-1}(\tanh (a+b x))}-5 b^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {\tanh ^{-1}(\tanh (a+b x))}}\right ) \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )-\frac {2 \tanh ^{-1}(\tanh (a+b x))^{5/2}}{3 x^{3/2}}-\frac {10 b \tanh ^{-1}(\tanh (a+b x))^{3/2}}{3 \sqrt {x}} \]
Antiderivative was successfully verified.
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Rule 2165
Rule 2168
Rule 2169
Rubi steps
\begin {align*} \int \frac {\tanh ^{-1}(\tanh (a+b x))^{5/2}}{x^{5/2}} \, dx &=-\frac {2 \tanh ^{-1}(\tanh (a+b x))^{5/2}}{3 x^{3/2}}+\frac {1}{3} (5 b) \int \frac {\tanh ^{-1}(\tanh (a+b x))^{3/2}}{x^{3/2}} \, dx\\ &=-\frac {10 b \tanh ^{-1}(\tanh (a+b x))^{3/2}}{3 \sqrt {x}}-\frac {2 \tanh ^{-1}(\tanh (a+b x))^{5/2}}{3 x^{3/2}}+\left (5 b^2\right ) \int \frac {\sqrt {\tanh ^{-1}(\tanh (a+b x))}}{\sqrt {x}} \, dx\\ &=5 b^2 \sqrt {x} \sqrt {\tanh ^{-1}(\tanh (a+b x))}-\frac {10 b \tanh ^{-1}(\tanh (a+b x))^{3/2}}{3 \sqrt {x}}-\frac {2 \tanh ^{-1}(\tanh (a+b x))^{5/2}}{3 x^{3/2}}-\frac {1}{2} \left (5 b^2 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )\right ) \int \frac {1}{\sqrt {x} \sqrt {\tanh ^{-1}(\tanh (a+b x))}} \, dx\\ &=-5 b^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {\tanh ^{-1}(\tanh (a+b x))}}\right ) \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )+5 b^2 \sqrt {x} \sqrt {\tanh ^{-1}(\tanh (a+b x))}-\frac {10 b \tanh ^{-1}(\tanh (a+b x))^{3/2}}{3 \sqrt {x}}-\frac {2 \tanh ^{-1}(\tanh (a+b x))^{5/2}}{3 x^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 97, normalized size = 0.92 \[ 5 b^{3/2} \left (\tanh ^{-1}(\tanh (a+b x))-b x\right ) \log \left (\sqrt {b} \sqrt {\tanh ^{-1}(\tanh (a+b x))}+b \sqrt {x}\right )+\frac {\sqrt {\tanh ^{-1}(\tanh (a+b x))} \left (-10 b x \tanh ^{-1}(\tanh (a+b x))-2 \tanh ^{-1}(\tanh (a+b x))^2+15 b^2 x^2\right )}{3 x^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 138, normalized size = 1.30 \[ \left [\frac {15 \, a b^{\frac {3}{2}} x^{2} \log \left (2 \, b x + 2 \, \sqrt {b x + a} \sqrt {b} \sqrt {x} + a\right ) + 2 \, {\left (3 \, b^{2} x^{2} - 14 \, a b x - 2 \, a^{2}\right )} \sqrt {b x + a} \sqrt {x}}{6 \, x^{2}}, -\frac {15 \, a \sqrt {-b} b x^{2} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-b}}{b \sqrt {x}}\right ) - {\left (3 \, b^{2} x^{2} - 14 \, a b x - 2 \, a^{2}\right )} \sqrt {b x + a} \sqrt {x}}{3 \, x^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.25, size = 501, normalized size = 4.73 \[ -\frac {2 \arctanh \left (\tanh \left (b x +a \right )\right )^{\frac {7}{2}}}{3 \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x \right ) x^{\frac {3}{2}}}-\frac {8 b \arctanh \left (\tanh \left (b x +a \right )\right )^{\frac {7}{2}}}{3 \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x \right )^{2} \sqrt {x}}+\frac {8 b^{2} \sqrt {x}\, \arctanh \left (\tanh \left (b x +a \right )\right )^{\frac {5}{2}}}{3 \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x \right )^{2}}+\frac {10 b^{2} a \sqrt {x}\, \arctanh \left (\tanh \left (b x +a \right )\right )^{\frac {3}{2}}}{3 \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x \right )^{2}}+\frac {5 b^{2} a^{2} \sqrt {x}\, \sqrt {\arctanh \left (\tanh \left (b x +a \right )\right )}}{\left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x \right )^{2}}+\frac {5 b^{\frac {3}{2}} \ln \left (\sqrt {b}\, \sqrt {x}+\sqrt {\arctanh \left (\tanh \left (b x +a \right )\right )}\right ) a^{3}}{\left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x \right )^{2}}+\frac {15 b^{\frac {3}{2}} a^{2} \ln \left (\sqrt {b}\, \sqrt {x}+\sqrt {\arctanh \left (\tanh \left (b x +a \right )\right )}\right ) \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x -a \right )}{\left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x \right )^{2}}+\frac {10 b^{2} a \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x -a \right ) \sqrt {x}\, \sqrt {\arctanh \left (\tanh \left (b x +a \right )\right )}}{\left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x \right )^{2}}+\frac {15 b^{\frac {3}{2}} a \ln \left (\sqrt {b}\, \sqrt {x}+\sqrt {\arctanh \left (\tanh \left (b x +a \right )\right )}\right ) \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x -a \right )^{2}}{\left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x \right )^{2}}+\frac {10 b^{2} \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x -a \right ) \sqrt {x}\, \arctanh \left (\tanh \left (b x +a \right )\right )^{\frac {3}{2}}}{3 \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x \right )^{2}}+\frac {5 b^{2} \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x -a \right )^{2} \sqrt {x}\, \sqrt {\arctanh \left (\tanh \left (b x +a \right )\right )}}{\left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x \right )^{2}}+\frac {5 b^{\frac {3}{2}} \ln \left (\sqrt {b}\, \sqrt {x}+\sqrt {\arctanh \left (\tanh \left (b x +a \right )\right )}\right ) \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x -a \right )^{3}}{\left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x \right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {artanh}\left (\tanh \left (b x + a\right )\right )^{\frac {5}{2}}}{x^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\mathrm {atanh}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}^{5/2}}{x^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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