Optimal. Leaf size=135 \[ -\frac {7 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {b x-\tanh ^{-1}(\tanh (a+b x))}}\right ) \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^{5/2}}{b^{9/2}}+\frac {7 \sqrt {x} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2}{b^4}+\frac {7 x^{3/2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}{3 b^3}-\frac {x^{7/2}}{b \tanh ^{-1}(\tanh (a+b x))}+\frac {7 x^{5/2}}{5 b^2} \]
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Rubi [A] time = 0.10, antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2168, 2159, 2162} \[ \frac {7 x^{3/2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}{3 b^3}+\frac {7 \sqrt {x} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2}{b^4}-\frac {7 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {b x-\tanh ^{-1}(\tanh (a+b x))}}\right ) \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^{5/2}}{b^{9/2}}-\frac {x^{7/2}}{b \tanh ^{-1}(\tanh (a+b x))}+\frac {7 x^{5/2}}{5 b^2} \]
Antiderivative was successfully verified.
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Rule 2159
Rule 2162
Rule 2168
Rubi steps
\begin {align*} \int \frac {x^{7/2}}{\tanh ^{-1}(\tanh (a+b x))^2} \, dx &=-\frac {x^{7/2}}{b \tanh ^{-1}(\tanh (a+b x))}+\frac {7 \int \frac {x^{5/2}}{\tanh ^{-1}(\tanh (a+b x))} \, dx}{2 b}\\ &=\frac {7 x^{5/2}}{5 b^2}-\frac {x^{7/2}}{b \tanh ^{-1}(\tanh (a+b x))}-\frac {\left (7 \left (-b x+\tanh ^{-1}(\tanh (a+b x))\right )\right ) \int \frac {x^{3/2}}{\tanh ^{-1}(\tanh (a+b x))} \, dx}{2 b^2}\\ &=\frac {7 x^{5/2}}{5 b^2}+\frac {7 x^{3/2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}{3 b^3}-\frac {x^{7/2}}{b \tanh ^{-1}(\tanh (a+b x))}+\frac {\left (7 \left (-b x+\tanh ^{-1}(\tanh (a+b x))\right )^2\right ) \int \frac {\sqrt {x}}{\tanh ^{-1}(\tanh (a+b x))} \, dx}{2 b^3}\\ &=\frac {7 x^{5/2}}{5 b^2}+\frac {7 x^{3/2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}{3 b^3}+\frac {7 \sqrt {x} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2}{b^4}-\frac {x^{7/2}}{b \tanh ^{-1}(\tanh (a+b x))}-\frac {\left (7 \left (-b x+\tanh ^{-1}(\tanh (a+b x))\right )^3\right ) \int \frac {1}{\sqrt {x} \tanh ^{-1}(\tanh (a+b x))} \, dx}{2 b^4}\\ &=\frac {7 x^{5/2}}{5 b^2}+\frac {7 x^{3/2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}{3 b^3}+\frac {7 \sqrt {x} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2}{b^4}-\frac {7 \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 )^{5/2}}{b^{9/2}}-\frac {x^{7/2}}{b \tanh ^{-1}(\tanh (a+b x))}\\ \end {align*}
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Mathematica [A] time = 0.21, size = 144, normalized size = 1.07 \[ -\frac {7 \left (\tanh ^{-1}(\tanh (a+b x))-b x\right )^{5/2} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {\tanh ^{-1}(\tanh (a+b x))-b x}}\right )}{b^{9/2}}+\frac {\sqrt {x} \left (\tanh ^{-1}(\tanh (a+b x))-b x\right )^3}{b^4 \tanh ^{-1}(\tanh (a+b x))}+\frac {6 \sqrt {x} \left (\tanh ^{-1}(\tanh (a+b x))-b x\right )^2}{b^4}-\frac {4 x^{3/2} \left (\tanh ^{-1}(\tanh (a+b x))-b x\right )}{3 b^3}+\frac {2 x^{5/2}}{5 b^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 188, normalized size = 1.39 \[ \left [\frac {105 \, {\left (a^{2} b x + a^{3}\right )} \sqrt {-\frac {a}{b}} \log \left (\frac {b x - 2 \, b \sqrt {x} \sqrt {-\frac {a}{b}} - a}{b x + a}\right ) + 2 \, {\left (6 \, b^{3} x^{3} - 14 \, a b^{2} x^{2} + 70 \, a^{2} b x + 105 \, a^{3}\right )} \sqrt {x}}{30 \, {\left (b^{5} x + a b^{4}\right )}}, -\frac {105 \, {\left (a^{2} b x + a^{3}\right )} \sqrt {\frac {a}{b}} \arctan \left (\frac {b \sqrt {x} \sqrt {\frac {a}{b}}}{a}\right ) - {\left (6 \, b^{3} x^{3} - 14 \, a b^{2} x^{2} + 70 \, a^{2} b x + 105 \, a^{3}\right )} \sqrt {x}}{15 \, {\left (b^{5} x + a b^{4}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 76, normalized size = 0.56 \[ -\frac {7 \, a^{3} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} b^{4}} + \frac {a^{3} \sqrt {x}}{{\left (b x + a\right )} b^{4}} + \frac {2 \, {\left (3 \, b^{8} x^{\frac {5}{2}} - 10 \, a b^{7} x^{\frac {3}{2}} + 45 \, a^{2} b^{6} \sqrt {x}\right )}}{15 \, b^{10}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.27, size = 452, normalized size = 3.35 \[ \frac {2 x^{\frac {5}{2}}}{5 b^{2}}-\frac {4 x^{\frac {3}{2}} a}{3 b^{3}}-\frac {4 x^{\frac {3}{2}} \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x -a \right )}{3 b^{3}}+\frac {6 a^{2} \sqrt {x}}{b^{4}}+\frac {12 a \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x -a \right ) \sqrt {x}}{b^{4}}+\frac {6 \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x -a \right )^{2} \sqrt {x}}{b^{4}}+\frac {\sqrt {x}\, a^{3}}{b^{4} \arctanh \left (\tanh \left (b x +a \right )\right )}+\frac {3 \sqrt {x}\, a^{2} \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x -a \right )}{b^{4} \arctanh \left (\tanh \left (b x +a \right )\right )}+\frac {3 \sqrt {x}\, a \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x -a \right )^{2}}{b^{4} \arctanh \left (\tanh \left (b x +a \right )\right )}+\frac {\sqrt {x}\, \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x -a \right )^{3}}{b^{4} \arctanh \left (\tanh \left (b x +a \right )\right )}-\frac {7 \arctan \left (\frac {b \sqrt {x}}{\sqrt {\left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x \right ) b}}\right ) a^{3}}{b^{4} \sqrt {\left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x \right ) b}}-\frac {21 \arctan \left (\frac {b \sqrt {x}}{\sqrt {\left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x \right ) b}}\right ) a^{2} \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x -a \right )}{b^{4} \sqrt {\left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x \right ) b}}-\frac {21 \arctan \left (\frac {b \sqrt {x}}{\sqrt {\left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x \right ) b}}\right ) a \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x -a \right )^{2}}{b^{4} \sqrt {\left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x \right ) b}}-\frac {7 \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 -a \right )^{3}}{b^{4} \sqrt {\left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x \right ) b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 75, normalized size = 0.56 \[ \frac {6 \, b^{3} x^{\frac {7}{2}} - 14 \, a b^{2} x^{\frac {5}{2}} + 70 \, a^{2} b x^{\frac {3}{2}} + 105 \, a^{3} \sqrt {x}}{15 \, {\left (b^{5} x + a b^{4}\right )}} - \frac {7 \, a^{3} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.61, size = 523, normalized size = 3.87 \[ \frac {2\,x^{5/2}}{5\,b^2}+\frac {2\,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\,b^3}+\frac {3\,\sqrt {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 )}^2}{2\,b^4}+\frac {7\,\sqrt {2}\,\ln \left (\frac {64\,b^{19/2}\,\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 (\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 )\,\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}}\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 )}^{5/2}}{16\,b^{9/2}}-\frac {\sqrt {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 )}^3}{4\,b^4\,\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 )} \]
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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