Optimal. Leaf size=121 \[ -\frac {2 \tanh ^{-1}(\tanh (a+b x))^{5/2}}{\sqrt {x}}+\frac {5}{2} b \sqrt {x} \tanh ^{-1}(\tanh (a+b x))^{3/2}-\frac {15}{4} b \sqrt {x} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right ) \sqrt {\tanh ^{-1}(\tanh (a+b x))}+\frac {15}{4} \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {\tanh ^{-1}(\tanh (a+b x))}}\right ) \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2 \]
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Rubi [A] time = 0.07, antiderivative size = 121, 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} \[ -\frac {2 \tanh ^{-1}(\tanh (a+b x))^{5/2}}{\sqrt {x}}+\frac {5}{2} b \sqrt {x} \tanh ^{-1}(\tanh (a+b x))^{3/2}-\frac {15}{4} b \sqrt {x} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right ) \sqrt {\tanh ^{-1}(\tanh (a+b x))}+\frac {15}{4} \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {\tanh ^{-1}(\tanh (a+b x))}}\right ) \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2 \]
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^{3/2}} \, dx &=-\frac {2 \tanh ^{-1}(\tanh (a+b x))^{5/2}}{\sqrt {x}}+(5 b) \int \frac {\tanh ^{-1}(\tanh (a+b x))^{3/2}}{\sqrt {x}} \, dx\\ &=\frac {5}{2} b \sqrt {x} \tanh ^{-1}(\tanh (a+b x))^{3/2}-\frac {2 \tanh ^{-1}(\tanh (a+b x))^{5/2}}{\sqrt {x}}-\frac {1}{4} \left (15 b \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )\right ) \int \frac {\sqrt {\tanh ^{-1}(\tanh (a+b x))}}{\sqrt {x}} \, dx\\ &=-\frac {15}{4} b \sqrt {x} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right ) \sqrt {\tanh ^{-1}(\tanh (a+b x))}+\frac {5}{2} b \sqrt {x} \tanh ^{-1}(\tanh (a+b x))^{3/2}-\frac {2 \tanh ^{-1}(\tanh (a+b x))^{5/2}}{\sqrt {x}}+\frac {1}{8} \left (15 b \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2\right ) \int \frac {1}{\sqrt {x} \sqrt {\tanh ^{-1}(\tanh (a+b x))}} \, dx\\ &=\frac {15}{4} \sqrt {b} \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 )^2-\frac {15}{4} b \sqrt {x} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right ) \sqrt {\tanh ^{-1}(\tanh (a+b x))}+\frac {5}{2} b \sqrt {x} \tanh ^{-1}(\tanh (a+b x))^{3/2}-\frac {2 \tanh ^{-1}(\tanh (a+b x))^{5/2}}{\sqrt {x}}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 101, normalized size = 0.83 \[ \frac {15}{4} \sqrt {b} \left (\tanh ^{-1}(\tanh (a+b x))-b x\right )^2 \log \left (\sqrt {b} \sqrt {\tanh ^{-1}(\tanh (a+b x))}+b \sqrt {x}\right )-\frac {\sqrt {\tanh ^{-1}(\tanh (a+b x))} \left (-25 b x \tanh ^{-1}(\tanh (a+b x))+8 \tanh ^{-1}(\tanh (a+b x))^2+15 b^2 x^2\right )}{4 \sqrt {x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 137, normalized size = 1.13 \[ \left [\frac {15 \, a^{2} \sqrt {b} x \log \left (2 \, b x + 2 \, \sqrt {b x + a} \sqrt {b} \sqrt {x} + a\right ) + 2 \, {\left (2 \, b^{2} x^{2} + 9 \, a b x - 8 \, a^{2}\right )} \sqrt {b x + a} \sqrt {x}}{8 \, x}, -\frac {15 \, a^{2} \sqrt {-b} x \arctan \left (\frac {\sqrt {b x + a} \sqrt {-b}}{b \sqrt {x}}\right ) - {\left (2 \, b^{2} x^{2} + 9 \, a b x - 8 \, a^{2}\right )} \sqrt {b x + a} \sqrt {x}}{4 \, x}\right ] \]
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 = 460, normalized size = 3.80 \[ -\frac {2 \arctanh \left (\tanh \left (b x +a \right )\right )^{\frac {7}{2}}}{\left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x \right ) \sqrt {x}}+\frac {2 b \sqrt {x}\, \arctanh \left (\tanh \left (b x +a \right )\right )^{\frac {5}{2}}}{\arctanh \left (\tanh \left (b x +a \right )\right )-b x}+\frac {5 b a \sqrt {x}\, \arctanh \left (\tanh \left (b x +a \right )\right )^{\frac {3}{2}}}{2 \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x \right )}+\frac {15 b \,a^{2} \sqrt {x}\, \sqrt {\arctanh \left (\tanh \left (b x +a \right )\right )}}{4 \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x \right )}+\frac {15 \sqrt {b}\, \ln \left (\sqrt {b}\, \sqrt {x}+\sqrt {\arctanh \left (\tanh \left (b x +a \right )\right )}\right ) a^{3}}{4 \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x \right )}+\frac {45 \sqrt {b}\, 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 )}{4 \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x \right )}+\frac {15 b 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 )}}{2 \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x \right )}+\frac {45 \sqrt {b}\, 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}}{4 \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x \right )}+\frac {5 b \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}}}{2 \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x \right )}+\frac {15 b \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 )}}{4 \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x \right )}+\frac {15 \sqrt {b}\, \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}}{4 \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x \right )} \]
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 \[ \int \frac {\operatorname {artanh}\left (\tanh \left (b x + a\right )\right )^{\frac {5}{2}}}{x^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {{\mathrm {atanh}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}^{5/2}}{x^{3/2}} \,d x \]
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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