Optimal. Leaf size=93 \[ 2 b^{5/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {\tanh ^{-1}(\tanh (a+b x))}}\right )-\frac {2 b^2 \sqrt {\tanh ^{-1}(\tanh (a+b x))}}{\sqrt {x}}-\frac {2 b \tanh ^{-1}(\tanh (a+b x))^{3/2}}{3 x^{3/2}}-\frac {2 \tanh ^{-1}(\tanh (a+b x))^{5/2}}{5 x^{5/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.06, antiderivative size = 93, 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 = {2168, 2165} \[ 2 b^{5/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {\tanh ^{-1}(\tanh (a+b x))}}\right )-\frac {2 b^2 \sqrt {\tanh ^{-1}(\tanh (a+b x))}}{\sqrt {x}}-\frac {2 b \tanh ^{-1}(\tanh (a+b x))^{3/2}}{3 x^{3/2}}-\frac {2 \tanh ^{-1}(\tanh (a+b x))^{5/2}}{5 x^{5/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2165
Rule 2168
Rubi steps
\begin {align*} \int \frac {\tanh ^{-1}(\tanh (a+b x))^{5/2}}{x^{7/2}} \, dx &=-\frac {2 \tanh ^{-1}(\tanh (a+b x))^{5/2}}{5 x^{5/2}}+b \int \frac {\tanh ^{-1}(\tanh (a+b x))^{3/2}}{x^{5/2}} \, dx\\ &=-\frac {2 b \tanh ^{-1}(\tanh (a+b x))^{3/2}}{3 x^{3/2}}-\frac {2 \tanh ^{-1}(\tanh (a+b x))^{5/2}}{5 x^{5/2}}+b^2 \int \frac {\sqrt {\tanh ^{-1}(\tanh (a+b x))}}{x^{3/2}} \, dx\\ &=-\frac {2 b^2 \sqrt {\tanh ^{-1}(\tanh (a+b x))}}{\sqrt {x}}-\frac {2 b \tanh ^{-1}(\tanh (a+b x))^{3/2}}{3 x^{3/2}}-\frac {2 \tanh ^{-1}(\tanh (a+b x))^{5/2}}{5 x^{5/2}}+b^3 \int \frac {1}{\sqrt {x} \sqrt {\tanh ^{-1}(\tanh (a+b x))}} \, dx\\ &=2 b^{5/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {\tanh ^{-1}(\tanh (a+b x))}}\right )-\frac {2 b^2 \sqrt {\tanh ^{-1}(\tanh (a+b x))}}{\sqrt {x}}-\frac {2 b \tanh ^{-1}(\tanh (a+b x))^{3/2}}{3 x^{3/2}}-\frac {2 \tanh ^{-1}(\tanh (a+b x))^{5/2}}{5 x^{5/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.06, size = 95, normalized size = 1.02 \[ -\frac {2 \left (-15 b^{5/2} x^{5/2} \log \left (\sqrt {b} \sqrt {\tanh ^{-1}(\tanh (a+b x))}+b \sqrt {x}\right )+15 b^2 x^2 \sqrt {\tanh ^{-1}(\tanh (a+b x))}+5 b x \tanh ^{-1}(\tanh (a+b x))^{3/2}+3 \tanh ^{-1}(\tanh (a+b x))^{5/2}\right )}{15 x^{5/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.90, size = 137, normalized size = 1.47 \[ \left [\frac {15 \, b^{\frac {5}{2}} x^{3} \log \left (2 \, b x + 2 \, \sqrt {b x + a} \sqrt {b} \sqrt {x} + a\right ) - 2 \, {\left (23 \, b^{2} x^{2} + 11 \, a b x + 3 \, a^{2}\right )} \sqrt {b x + a} \sqrt {x}}{15 \, x^{3}}, -\frac {2 \, {\left (15 \, \sqrt {-b} b^{2} x^{3} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-b}}{b \sqrt {x}}\right ) + {\left (23 \, b^{2} x^{2} + 11 \, a b x + 3 \, a^{2}\right )} \sqrt {b x + a} \sqrt {x}\right )}}{15 \, x^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.25, size = 532, normalized size = 5.72 \[ -\frac {2 \arctanh \left (\tanh \left (b x +a \right )\right )^{\frac {7}{2}}}{5 \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x \right ) x^{\frac {5}{2}}}-\frac {4 b \arctanh \left (\tanh \left (b x +a \right )\right )^{\frac {7}{2}}}{15 \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x \right )^{2} x^{\frac {3}{2}}}-\frac {16 b^{2} \arctanh \left (\tanh \left (b x +a \right )\right )^{\frac {7}{2}}}{15 \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x \right )^{3} \sqrt {x}}+\frac {16 b^{3} \sqrt {x}\, \arctanh \left (\tanh \left (b x +a \right )\right )^{\frac {5}{2}}}{15 \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x \right )^{3}}+\frac {4 b^{3} a \sqrt {x}\, \arctanh \left (\tanh \left (b x +a \right )\right )^{\frac {3}{2}}}{3 \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x \right )^{3}}+\frac {2 b^{3} a^{2} \sqrt {x}\, \sqrt {\arctanh \left (\tanh \left (b x +a \right )\right )}}{\left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x \right )^{3}}+\frac {2 b^{\frac {5}{2}} \ln \left (\sqrt {b}\, \sqrt {x}+\sqrt {\arctanh \left (\tanh \left (b x +a \right )\right )}\right ) a^{3}}{\left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x \right )^{3}}+\frac {6 b^{\frac {5}{2}} 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 )}{\left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x \right )^{3}}+\frac {4 b^{3} 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 )}}{\left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x \right )^{3}}+\frac {6 b^{\frac {5}{2}} 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}}{\left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x \right )^{3}}+\frac {4 b^{3} \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}}}{3 \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x \right )^{3}}+\frac {2 b^{3} \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 )}}{\left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x \right )^{3}}+\frac {2 b^{\frac {5}{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 )^{3}}{\left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x \right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {artanh}\left (\tanh \left (b x + a\right )\right )^{\frac {5}{2}}}{x^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\mathrm {atanh}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}^{5/2}}{x^{7/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________