Optimal. Leaf size=65 \[ 16 b^2 \sqrt {x} \tanh ^{-1}(\tanh (a+b x))-\frac {2 \tanh ^{-1}(\tanh (a+b x))^3}{3 x^{3/2}}-\frac {4 b \tanh ^{-1}(\tanh (a+b x))^2}{\sqrt {x}}-\frac {32}{3} b^3 x^{3/2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {2168, 30} \[ 16 b^2 \sqrt {x} \tanh ^{-1}(\tanh (a+b x))-\frac {2 \tanh ^{-1}(\tanh (a+b x))^3}{3 x^{3/2}}-\frac {4 b \tanh ^{-1}(\tanh (a+b x))^2}{\sqrt {x}}-\frac {32}{3} b^3 x^{3/2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 30
Rule 2168
Rubi steps
\begin {align*} \int \frac {\tanh ^{-1}(\tanh (a+b x))^3}{x^{5/2}} \, dx &=-\frac {2 \tanh ^{-1}(\tanh (a+b x))^3}{3 x^{3/2}}+(2 b) \int \frac {\tanh ^{-1}(\tanh (a+b x))^2}{x^{3/2}} \, dx\\ &=-\frac {4 b \tanh ^{-1}(\tanh (a+b x))^2}{\sqrt {x}}-\frac {2 \tanh ^{-1}(\tanh (a+b x))^3}{3 x^{3/2}}+\left (8 b^2\right ) \int \frac {\tanh ^{-1}(\tanh (a+b x))}{\sqrt {x}} \, dx\\ &=16 b^2 \sqrt {x} \tanh ^{-1}(\tanh (a+b x))-\frac {4 b \tanh ^{-1}(\tanh (a+b x))^2}{\sqrt {x}}-\frac {2 \tanh ^{-1}(\tanh (a+b x))^3}{3 x^{3/2}}-\left (16 b^3\right ) \int \sqrt {x} \, dx\\ &=-\frac {32}{3} b^3 x^{3/2}+16 b^2 \sqrt {x} \tanh ^{-1}(\tanh (a+b x))-\frac {4 b \tanh ^{-1}(\tanh (a+b x))^2}{\sqrt {x}}-\frac {2 \tanh ^{-1}(\tanh (a+b x))^3}{3 x^{3/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.03, size = 55, normalized size = 0.85 \[ -\frac {2 \left (-24 b^2 x^2 \tanh ^{-1}(\tanh (a+b x))+6 b x \tanh ^{-1}(\tanh (a+b x))^2+\tanh ^{-1}(\tanh (a+b x))^3+16 b^3 x^3\right )}{3 x^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.52, size = 34, normalized size = 0.52 \[ \frac {2 \, {\left (b^{3} x^{3} + 9 \, a b^{2} x^{2} - 9 \, a^{2} b x - a^{3}\right )}}{3 \, x^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.16, size = 34, normalized size = 0.52 \[ \frac {2}{3} \, b^{3} x^{\frac {3}{2}} + 6 \, a b^{2} \sqrt {x} - \frac {2 \, {\left (9 \, a^{2} b x + a^{3}\right )}}{3 \, x^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.26, size = 55, normalized size = 0.85 \[ -\frac {2 \arctanh \left (\tanh \left (b x +a \right )\right )^{3}}{3 x^{\frac {3}{2}}}+4 b \left (-\frac {\arctanh \left (\tanh \left (b x +a \right )\right )^{2}}{\sqrt {x}}+4 b \left (\arctanh \left (\tanh \left (b x +a \right )\right ) \sqrt {x}-\frac {2 b \,x^{\frac {3}{2}}}{3}\right )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.35, size = 55, normalized size = 0.85 \[ -\frac {4 \, b \operatorname {artanh}\left (\tanh \left (b x + a\right )\right )^{2}}{\sqrt {x}} - \frac {16}{3} \, {\left (2 \, b^{2} x^{\frac {3}{2}} - 3 \, b \sqrt {x} \operatorname {artanh}\left (\tanh \left (b x + a\right )\right )\right )} b - \frac {2 \, \operatorname {artanh}\left (\tanh \left (b x + a\right )\right )^{3}}{3 \, x^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.22, size = 182, normalized size = 2.80 \[ \frac {{\left (\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+2\,b\,x\right )}^3}{12\,x^{3/2}}+\frac {2\,b^3\,x^{3/2}}{3}-\frac {3\,b\,{\left (\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+2\,b\,x\right )}^2}{2\,\sqrt {x}}-3\,b^2\,\sqrt {x}\,\left (\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+2\,b\,x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 10.58, size = 66, normalized size = 1.02 \[ - \frac {32 b^{3} x^{\frac {3}{2}}}{3} + 16 b^{2} \sqrt {x} \operatorname {atanh}{\left (\tanh {\left (a + b x \right )} \right )} - \frac {4 b \operatorname {atanh}^{2}{\left (\tanh {\left (a + b x \right )} \right )}}{\sqrt {x}} - \frac {2 \operatorname {atanh}^{3}{\left (\tanh {\left (a + b x \right )} \right )}}{3 x^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________