Optimal. Leaf size=135 \[ -\frac {35 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {b x-\tanh ^{-1}(\tanh (a+b x))}}\right ) \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^{3/2}}{4 b^{9/2}}+\frac {35 \sqrt {x} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}{4 b^4}-\frac {7 x^{5/2}}{4 b^2 \tanh ^{-1}(\tanh (a+b x))}-\frac {x^{7/2}}{2 b \tanh ^{-1}(\tanh (a+b x))^2}+\frac {35 x^{3/2}}{12 b^3} \]
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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} \[ -\frac {7 x^{5/2}}{4 b^2 \tanh ^{-1}(\tanh (a+b x))}+\frac {35 \sqrt {x} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}{4 b^4}-\frac {35 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {b x-\tanh ^{-1}(\tanh (a+b x))}}\right ) \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^{3/2}}{4 b^{9/2}}-\frac {x^{7/2}}{2 b \tanh ^{-1}(\tanh (a+b x))^2}+\frac {35 x^{3/2}}{12 b^3} \]
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))^3} \, dx &=-\frac {x^{7/2}}{2 b \tanh ^{-1}(\tanh (a+b x))^2}+\frac {7 \int \frac {x^{5/2}}{\tanh ^{-1}(\tanh (a+b x))^2} \, dx}{4 b}\\ &=-\frac {x^{7/2}}{2 b \tanh ^{-1}(\tanh (a+b x))^2}-\frac {7 x^{5/2}}{4 b^2 \tanh ^{-1}(\tanh (a+b x))}+\frac {35 \int \frac {x^{3/2}}{\tanh ^{-1}(\tanh (a+b x))} \, dx}{8 b^2}\\ &=\frac {35 x^{3/2}}{12 b^3}-\frac {x^{7/2}}{2 b \tanh ^{-1}(\tanh (a+b x))^2}-\frac {7 x^{5/2}}{4 b^2 \tanh ^{-1}(\tanh (a+b x))}-\frac {\left (35 \left (-b x+\tanh ^{-1}(\tanh (a+b x))\right )\right ) \int \frac {\sqrt {x}}{\tanh ^{-1}(\tanh (a+b x))} \, dx}{8 b^3}\\ &=\frac {35 x^{3/2}}{12 b^3}+\frac {35 \sqrt {x} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}{4 b^4}-\frac {x^{7/2}}{2 b \tanh ^{-1}(\tanh (a+b x))^2}-\frac {7 x^{5/2}}{4 b^2 \tanh ^{-1}(\tanh (a+b x))}+\frac {\left (35 \left (-b x+\tanh ^{-1}(\tanh (a+b x))\right )^2\right ) \int \frac {1}{\sqrt {x} \tanh ^{-1}(\tanh (a+b x))} \, dx}{8 b^4}\\ &=\frac {35 x^{3/2}}{12 b^3}+\frac {35 \sqrt {x} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}{4 b^4}-\frac {35 \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 )^{3/2}}{4 b^{9/2}}-\frac {x^{7/2}}{2 b \tanh ^{-1}(\tanh (a+b x))^2}-\frac {7 x^{5/2}}{4 b^2 \tanh ^{-1}(\tanh (a+b x))}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 147, normalized size = 1.09 \[ -\frac {21 b^{5/2} x^{5/2} \tanh ^{-1}(\tanh (a+b x))-140 b^{3/2} x^{3/2} \tanh ^{-1}(\tanh (a+b x))^2+105 \sqrt {b} \sqrt {x} \tanh ^{-1}(\tanh (a+b x))^3-105 \tanh ^{-1}(\tanh (a+b x))^2 \left (\tanh ^{-1}(\tanh (a+b x))-b x\right )^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {\tanh ^{-1}(\tanh (a+b x))-b x}}\right )+6 b^{7/2} x^{7/2}}{12 b^{9/2} \tanh ^{-1}(\tanh (a+b x))^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.57, size = 227, normalized size = 1.68 \[ \left [\frac {105 \, {\left (a b^{2} x^{2} + 2 \, a^{2} b x + a^{3}\right )} \sqrt {-\frac {a}{b}} \log \left (\frac {b x + 2 \, b \sqrt {x} \sqrt {-\frac {a}{b}} - a}{b x + a}\right ) + 2 \, {\left (8 \, b^{3} x^{3} - 56 \, a b^{2} x^{2} - 175 \, a^{2} b x - 105 \, a^{3}\right )} \sqrt {x}}{24 \, {\left (b^{6} x^{2} + 2 \, a b^{5} x + a^{2} b^{4}\right )}}, \frac {105 \, {\left (a b^{2} x^{2} + 2 \, a^{2} b x + a^{3}\right )} \sqrt {\frac {a}{b}} \arctan \left (\frac {b \sqrt {x} \sqrt {\frac {a}{b}}}{a}\right ) + {\left (8 \, b^{3} x^{3} - 56 \, a b^{2} x^{2} - 175 \, a^{2} b x - 105 \, a^{3}\right )} \sqrt {x}}{12 \, {\left (b^{6} x^{2} + 2 \, a b^{5} x + a^{2} b^{4}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 77, normalized size = 0.57 \[ \frac {35 \, a^{2} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{4 \, \sqrt {a b} b^{4}} - \frac {13 \, a^{2} b x^{\frac {3}{2}} + 11 \, a^{3} \sqrt {x}}{4 \, {\left (b x + a\right )}^{2} b^{4}} + \frac {2 \, {\left (b^{6} x^{\frac {3}{2}} - 9 \, a b^{5} \sqrt {x}\right )}}{3 \, b^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.27, size = 418, normalized size = 3.10 \[ \frac {2 x^{\frac {3}{2}}}{3 b^{3}}-\frac {6 a \sqrt {x}}{b^{4}}-\frac {6 \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x -a \right ) \sqrt {x}}{b^{4}}-\frac {13 a^{2} x^{\frac {3}{2}}}{4 b^{3} \arctanh \left (\tanh \left (b x +a \right )\right )^{2}}-\frac {13 x^{\frac {3}{2}} a \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x -a \right )}{2 b^{3} \arctanh \left (\tanh \left (b x +a \right )\right )^{2}}-\frac {13 x^{\frac {3}{2}} \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x -a \right )^{2}}{4 b^{3} \arctanh \left (\tanh \left (b x +a \right )\right )^{2}}-\frac {11 \sqrt {x}\, a^{3}}{4 b^{4} \arctanh \left (\tanh \left (b x +a \right )\right )^{2}}-\frac {33 \sqrt {x}\, a^{2} \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x -a \right )}{4 b^{4} \arctanh \left (\tanh \left (b x +a \right )\right )^{2}}-\frac {33 \sqrt {x}\, a \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x -a \right )^{2}}{4 b^{4} \arctanh \left (\tanh \left (b x +a \right )\right )^{2}}-\frac {11 \sqrt {x}\, \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x -a \right )^{3}}{4 b^{4} \arctanh \left (\tanh \left (b x +a \right )\right )^{2}}+\frac {35 \arctan \left (\frac {b \sqrt {x}}{\sqrt {\left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x \right ) b}}\right ) a^{2}}{4 b^{4} \sqrt {\left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x \right ) b}}+\frac {35 \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 {35 \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 )^{2}}{4 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 = 0.44, size = 86, normalized size = 0.64 \[ \frac {8 \, b^{3} x^{\frac {7}{2}} - 56 \, a b^{2} x^{\frac {5}{2}} - 175 \, a^{2} b x^{\frac {3}{2}} - 105 \, a^{3} \sqrt {x}}{12 \, {\left (b^{6} x^{2} + 2 \, a b^{5} x + a^{2} b^{4}\right )}} + \frac {35 \, a^{2} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{4 \, \sqrt {a b} b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.98, size = 571, normalized size = 4.23 \[ \frac {2\,x^{3/2}}{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 )}{b^4}+\frac {35\,\sqrt {2}\,\ln \left (\frac {256\,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 )}^{3/2}}{32\,b^{9/2}}-\frac {13\,\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}{8\,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 )}-\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 )}^2} \]
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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