Optimal. Leaf size=145 \[ \frac {5 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {\tanh ^{-1}(\tanh (a+b x))}}\right ) \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^3}{8 b^{7/2}}+\frac {x^{5/2} \sqrt {\tanh ^{-1}(\tanh (a+b x))}}{3 b}+\frac {5 x^{3/2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right ) \sqrt {\tanh ^{-1}(\tanh (a+b x))}}{12 b^2}+\frac {5 \sqrt {x} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2 \sqrt {\tanh ^{-1}(\tanh (a+b x))}}{8 b^3} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.06, antiderivative size = 145, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {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 )^3}{8 b^{7/2}}+\frac {5 \sqrt {x} \sqrt {\tanh ^{-1}(\tanh (a+b x))} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2}{8 b^3}+\frac {5 x^{3/2} \sqrt {\tanh ^{-1}(\tanh (a+b x))} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}{12 b^2}+\frac {x^{5/2} \sqrt {\tanh ^{-1}(\tanh (a+b x))}}{3 b} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2196
Rule 2200
Rubi steps
\begin {align*} \int \frac {x^{5/2}}{\sqrt {\tanh ^{-1}(\tanh (a+b x))}} \, dx &=\frac {x^{5/2} \sqrt {\tanh ^{-1}(\tanh (a+b x))}}{3 b}-\frac {\left (5 \left (-b x+\tanh ^{-1}(\tanh (a+b x))\right )\right ) \int \frac {x^{3/2}}{\sqrt {\tanh ^{-1}(\tanh (a+b x))}} \, dx}{6 b}\\ &=\frac {x^{5/2} \sqrt {\tanh ^{-1}(\tanh (a+b x))}}{3 b}+\frac {5 x^{3/2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right ) \sqrt {\tanh ^{-1}(\tanh (a+b x))}}{12 b^2}+\frac {\left (5 \left (-b x+\tanh ^{-1}(\tanh (a+b x))\right )^2\right ) \int \frac {\sqrt {x}}{\sqrt {\tanh ^{-1}(\tanh (a+b x))}} \, dx}{8 b^2}\\ &=\frac {x^{5/2} \sqrt {\tanh ^{-1}(\tanh (a+b x))}}{3 b}+\frac {5 x^{3/2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right ) \sqrt {\tanh ^{-1}(\tanh (a+b x))}}{12 b^2}+\frac {5 \sqrt {x} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2 \sqrt {\tanh ^{-1}(\tanh (a+b x))}}{8 b^3}-\frac {\left (5 \left (-b x+\tanh ^{-1}(\tanh (a+b x))\right )^3\right ) \int \frac {1}{\sqrt {x} \sqrt {\tanh ^{-1}(\tanh (a+b x))}} \, dx}{16 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 )^3}{8 b^{7/2}}+\frac {x^{5/2} \sqrt {\tanh ^{-1}(\tanh (a+b x))}}{3 b}+\frac {5 x^{3/2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right ) \sqrt {\tanh ^{-1}(\tanh (a+b x))}}{12 b^2}+\frac {5 \sqrt {x} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2 \sqrt {\tanh ^{-1}(\tanh (a+b x))}}{8 b^3}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.06, size = 105, normalized size = 0.72 \begin {gather*} \frac {\sqrt {x} \sqrt {\tanh ^{-1}(\tanh (a+b x))} \left (33 b^2 x^2-40 b x \tanh ^{-1}(\tanh (a+b x))+15 \tanh ^{-1}(\tanh (a+b x))^2\right )}{24 b^3}+\frac {5 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^3 \log \left (b \sqrt {x}+\sqrt {b} \sqrt {\tanh ^{-1}(\tanh (a+b x))}\right )}{8 b^{7/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(303\) vs.
\(2(117)=234\).
time = 0.13, size = 304, normalized size = 2.10
method | result | size |
derivativedivides | \(\frac {x^{\frac {5}{2}} \sqrt {\arctanh \left (\tanh \left (b x +a \right )\right )}}{3 b}-\frac {5 a \,x^{\frac {3}{2}} \sqrt {\arctanh \left (\tanh \left (b x +a \right )\right )}}{12 b^{2}}+\frac {5 a^{2} \sqrt {x}\, \sqrt {\arctanh \left (\tanh \left (b x +a \right )\right )}}{8 b^{3}}-\frac {5 \ln \left (\sqrt {b}\, \sqrt {x}+\sqrt {\arctanh \left (\tanh \left (b x +a \right )\right )}\right ) a^{3}}{8 b^{\frac {7}{2}}}-\frac {15 a^{2} \ln \left (\sqrt {b}\, \sqrt {x}+\sqrt {\arctanh \left (\tanh \left (b x +a \right )\right )}\right ) \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x -a \right )}{8 b^{\frac {7}{2}}}+\frac {5 a \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x -a \right ) \sqrt {x}\, \sqrt {\arctanh \left (\tanh \left (b x +a \right )\right )}}{4 b^{3}}-\frac {15 a \ln \left (\sqrt {b}\, \sqrt {x}+\sqrt {\arctanh \left (\tanh \left (b x +a \right )\right )}\right ) \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x -a \right )^{2}}{8 b^{\frac {7}{2}}}-\frac {5 \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x -a \right ) x^{\frac {3}{2}} \sqrt {\arctanh \left (\tanh \left (b x +a \right )\right )}}{12 b^{2}}+\frac {5 \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x -a \right )^{2} \sqrt {x}\, \sqrt {\arctanh \left (\tanh \left (b x +a \right )\right )}}{8 b^{3}}-\frac {5 \ln \left (\sqrt {b}\, \sqrt {x}+\sqrt {\arctanh \left (\tanh \left (b x +a \right )\right )}\right ) \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x -a \right )^{3}}{8 b^{\frac {7}{2}}}\) | \(304\) |
default | \(\frac {x^{\frac {5}{2}} \sqrt {\arctanh \left (\tanh \left (b x +a \right )\right )}}{3 b}-\frac {5 a \,x^{\frac {3}{2}} \sqrt {\arctanh \left (\tanh \left (b x +a \right )\right )}}{12 b^{2}}+\frac {5 a^{2} \sqrt {x}\, \sqrt {\arctanh \left (\tanh \left (b x +a \right )\right )}}{8 b^{3}}-\frac {5 \ln \left (\sqrt {b}\, \sqrt {x}+\sqrt {\arctanh \left (\tanh \left (b x +a \right )\right )}\right ) a^{3}}{8 b^{\frac {7}{2}}}-\frac {15 a^{2} \ln \left (\sqrt {b}\, \sqrt {x}+\sqrt {\arctanh \left (\tanh \left (b x +a \right )\right )}\right ) \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x -a \right )}{8 b^{\frac {7}{2}}}+\frac {5 a \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x -a \right ) \sqrt {x}\, \sqrt {\arctanh \left (\tanh \left (b x +a \right )\right )}}{4 b^{3}}-\frac {15 a \ln \left (\sqrt {b}\, \sqrt {x}+\sqrt {\arctanh \left (\tanh \left (b x +a \right )\right )}\right ) \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x -a \right )^{2}}{8 b^{\frac {7}{2}}}-\frac {5 \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x -a \right ) x^{\frac {3}{2}} \sqrt {\arctanh \left (\tanh \left (b x +a \right )\right )}}{12 b^{2}}+\frac {5 \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x -a \right )^{2} \sqrt {x}\, \sqrt {\arctanh \left (\tanh \left (b x +a \right )\right )}}{8 b^{3}}-\frac {5 \ln \left (\sqrt {b}\, \sqrt {x}+\sqrt {\arctanh \left (\tanh \left (b x +a \right )\right )}\right ) \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x -a \right )^{3}}{8 b^{\frac {7}{2}}}\) | \(304\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.37, size = 140, normalized size = 0.97 \begin {gather*} \left [\frac {15 \, a^{3} \sqrt {b} \log \left (2 \, b x - 2 \, \sqrt {b x + a} \sqrt {b} \sqrt {x} + a\right ) + 2 \, {\left (8 \, b^{3} x^{2} - 10 \, a b^{2} x + 15 \, a^{2} b\right )} \sqrt {b x + a} \sqrt {x}}{48 \, b^{4}}, \frac {15 \, a^{3} \sqrt {-b} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-b}}{b \sqrt {x}}\right ) + {\left (8 \, b^{3} x^{2} - 10 \, a b^{2} x + 15 \, a^{2} b\right )} \sqrt {b x + a} \sqrt {x}}{24 \, b^{4}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.38, size = 64, normalized size = 0.44 \begin {gather*} \frac {1}{24} \, \sqrt {b x + a} {\left (2 \, x {\left (\frac {4 \, x}{b} - \frac {5 \, a}{b^{2}}\right )} + \frac {15 \, a^{2}}{b^{3}}\right )} \sqrt {x} + \frac {5 \, a^{3} \log \left ({\left | -\sqrt {b} \sqrt {x} + \sqrt {b x + a} \right |}\right )}{8 \, b^{\frac {7}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^{5/2}}{\sqrt {\mathrm {atanh}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________