Optimal. Leaf size=111 \[ \frac {5 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {\tanh ^{-1}(\tanh (a+b x))}}\right ) \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}{b^{7/2}}-\frac {2 x^{5/2}}{3 b \tanh ^{-1}(\tanh (a+b x))^{3/2}}-\frac {10 x^{3/2}}{3 b^2 \sqrt {\tanh ^{-1}(\tanh (a+b x))}}+\frac {5 \sqrt {x} \sqrt {\tanh ^{-1}(\tanh (a+b x))}}{b^3} \]
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Rubi [A]
time = 0.05, antiderivative size = 111, 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 = {2199, 2200,
2196} \begin {gather*} \frac {5 \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 )}{b^{7/2}}+\frac {5 \sqrt {x} \sqrt {\tanh ^{-1}(\tanh (a+b x))}}{b^3}-\frac {10 x^{3/2}}{3 b^2 \sqrt {\tanh ^{-1}(\tanh (a+b x))}}-\frac {2 x^{5/2}}{3 b \tanh ^{-1}(\tanh (a+b x))^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2196
Rule 2199
Rule 2200
Rubi steps
\begin {align*} \int \frac {x^{5/2}}{\tanh ^{-1}(\tanh (a+b x))^{5/2}} \, dx &=-\frac {2 x^{5/2}}{3 b \tanh ^{-1}(\tanh (a+b x))^{3/2}}+\frac {5 \int \frac {x^{3/2}}{\tanh ^{-1}(\tanh (a+b x))^{3/2}} \, dx}{3 b}\\ &=-\frac {2 x^{5/2}}{3 b \tanh ^{-1}(\tanh (a+b x))^{3/2}}-\frac {10 x^{3/2}}{3 b^2 \sqrt {\tanh ^{-1}(\tanh (a+b x))}}+\frac {5 \int \frac {\sqrt {x}}{\sqrt {\tanh ^{-1}(\tanh (a+b x))}} \, dx}{b^2}\\ &=-\frac {2 x^{5/2}}{3 b \tanh ^{-1}(\tanh (a+b x))^{3/2}}-\frac {10 x^{3/2}}{3 b^2 \sqrt {\tanh ^{-1}(\tanh (a+b x))}}+\frac {5 \sqrt {x} \sqrt {\tanh ^{-1}(\tanh (a+b x))}}{b^3}-\frac {\left (5 \left (-b x+\tanh ^{-1}(\tanh (a+b x))\right )\right ) \int \frac {1}{\sqrt {x} \sqrt {\tanh ^{-1}(\tanh (a+b x))}} \, dx}{2 b^3}\\ &=\frac {5 \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 )}{b^{7/2}}-\frac {2 x^{5/2}}{3 b \tanh ^{-1}(\tanh (a+b x))^{3/2}}-\frac {10 x^{3/2}}{3 b^2 \sqrt {\tanh ^{-1}(\tanh (a+b x))}}+\frac {5 \sqrt {x} \sqrt {\tanh ^{-1}(\tanh (a+b x))}}{b^3}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 101, normalized size = 0.91 \begin {gather*} -\frac {\sqrt {x} \left (2 b^2 x^2+10 b x \tanh ^{-1}(\tanh (a+b x))-15 \tanh ^{-1}(\tanh (a+b x))^2\right )}{3 b^3 \tanh ^{-1}(\tanh (a+b x))^{3/2}}+\frac {5 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right ) \log \left (b \sqrt {x}+\sqrt {b} \sqrt {\tanh ^{-1}(\tanh (a+b x))}\right )}{b^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(179\) vs.
\(2(87)=174\).
time = 0.13, size = 180, normalized size = 1.62
method | result | size |
derivativedivides | \(\frac {x^{\frac {5}{2}}}{b \arctanh \left (\tanh \left (b x +a \right )\right )^{\frac {3}{2}}}+\frac {5 a \,x^{\frac {3}{2}}}{3 b^{2} \arctanh \left (\tanh \left (b x +a \right )\right )^{\frac {3}{2}}}+\frac {5 a \sqrt {x}}{b^{3} \sqrt {\arctanh \left (\tanh \left (b x +a \right )\right )}}-\frac {5 a \ln \left (\sqrt {b}\, \sqrt {x}+\sqrt {\arctanh \left (\tanh \left (b x +a \right )\right )}\right )}{b^{\frac {7}{2}}}+\frac {5 \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x -a \right ) x^{\frac {3}{2}}}{3 b^{2} \arctanh \left (\tanh \left (b x +a \right )\right )^{\frac {3}{2}}}+\frac {5 \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x -a \right ) \sqrt {x}}{b^{3} \sqrt {\arctanh \left (\tanh \left (b x +a \right )\right )}}-\frac {5 \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x -a \right ) \ln \left (\sqrt {b}\, \sqrt {x}+\sqrt {\arctanh \left (\tanh \left (b x +a \right )\right )}\right )}{b^{\frac {7}{2}}}\) | \(180\) |
default | \(\frac {x^{\frac {5}{2}}}{b \arctanh \left (\tanh \left (b x +a \right )\right )^{\frac {3}{2}}}+\frac {5 a \,x^{\frac {3}{2}}}{3 b^{2} \arctanh \left (\tanh \left (b x +a \right )\right )^{\frac {3}{2}}}+\frac {5 a \sqrt {x}}{b^{3} \sqrt {\arctanh \left (\tanh \left (b x +a \right )\right )}}-\frac {5 a \ln \left (\sqrt {b}\, \sqrt {x}+\sqrt {\arctanh \left (\tanh \left (b x +a \right )\right )}\right )}{b^{\frac {7}{2}}}+\frac {5 \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x -a \right ) x^{\frac {3}{2}}}{3 b^{2} \arctanh \left (\tanh \left (b x +a \right )\right )^{\frac {3}{2}}}+\frac {5 \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x -a \right ) \sqrt {x}}{b^{3} \sqrt {\arctanh \left (\tanh \left (b x +a \right )\right )}}-\frac {5 \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x -a \right ) \ln \left (\sqrt {b}\, \sqrt {x}+\sqrt {\arctanh \left (\tanh \left (b x +a \right )\right )}\right )}{b^{\frac {7}{2}}}\) | \(180\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 214, normalized size = 1.93 \begin {gather*} \left [\frac {15 \, {\left (a b^{2} x^{2} + 2 \, a^{2} b x + a^{3}\right )} \sqrt {b} \log \left (2 \, b x - 2 \, \sqrt {b x + a} \sqrt {b} \sqrt {x} + a\right ) + 2 \, {\left (3 \, b^{3} x^{2} + 20 \, a b^{2} x + 15 \, a^{2} b\right )} \sqrt {b x + a} \sqrt {x}}{6 \, {\left (b^{6} x^{2} + 2 \, a b^{5} x + a^{2} b^{4}\right )}}, \frac {15 \, {\left (a b^{2} x^{2} + 2 \, a^{2} b x + a^{3}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-b}}{b \sqrt {x}}\right ) + {\left (3 \, b^{3} x^{2} + 20 \, a b^{2} x + 15 \, a^{2} b\right )} \sqrt {b x + a} \sqrt {x}}{3 \, {\left (b^{6} x^{2} + 2 \, a b^{5} x + a^{2} b^{4}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.39, size = 61, normalized size = 0.55 \begin {gather*} \frac {{\left (x {\left (\frac {3 \, x}{b} + \frac {20 \, a}{b^{2}}\right )} + \frac {15 \, a^{2}}{b^{3}}\right )} \sqrt {x}}{3 \, {\left (b x + a\right )}^{\frac {3}{2}}} + \frac {5 \, a \log \left ({\left | -\sqrt {b} \sqrt {x} + \sqrt {b x + a} \right |}\right )}{b^{\frac {7}{2}}} \end {gather*}
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 \begin {gather*} \int \frac {x^{5/2}}{{\mathrm {atanh}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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