Optimal. Leaf size=27 \[ -\frac {4}{99} b x^{11/2}+\frac {2}{9} x^{9/2} \tanh ^{-1}(\tanh (a+b x)) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.01, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2199, 30}
\begin {gather*} \frac {2}{9} x^{9/2} \tanh ^{-1}(\tanh (a+b x))-\frac {4}{99} b x^{11/2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 30
Rule 2199
Rubi steps
\begin {align*} \int x^{7/2} \tanh ^{-1}(\tanh (a+b x)) \, dx &=\frac {2}{9} x^{9/2} \tanh ^{-1}(\tanh (a+b x))-\frac {1}{9} (2 b) \int x^{9/2} \, dx\\ &=-\frac {4}{99} b x^{11/2}+\frac {2}{9} x^{9/2} \tanh ^{-1}(\tanh (a+b x))\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.03, size = 23, normalized size = 0.85 \begin {gather*} \frac {2}{99} x^{9/2} \left (-2 b x+11 \tanh ^{-1}(\tanh (a+b x))\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.11, size = 20, normalized size = 0.74
method | result | size |
derivativedivides | \(-\frac {4 b \,x^{\frac {11}{2}}}{99}+\frac {2 x^{\frac {9}{2}} \arctanh \left (\tanh \left (b x +a \right )\right )}{9}\) | \(20\) |
default | \(-\frac {4 b \,x^{\frac {11}{2}}}{99}+\frac {2 x^{\frac {9}{2}} \arctanh \left (\tanh \left (b x +a \right )\right )}{9}\) | \(20\) |
risch | \(\frac {2 x^{\frac {9}{2}} \ln \left ({\mathrm e}^{b x +a}\right )}{9}-\frac {\left (-11 i \pi \,x^{4} \mathrm {csgn}\left (\frac {i}{{\mathrm e}^{2 b x +2 a}+1}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{2 b x +2 a}}{{\mathrm e}^{2 b x +2 a}+1}\right )^{2}-11 i \pi \,x^{4} \mathrm {csgn}\left (i {\mathrm e}^{2 b x +2 a}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{2 b x +2 a}}{{\mathrm e}^{2 b x +2 a}+1}\right )^{2}+11 i \pi \,x^{4} \mathrm {csgn}\left (\frac {i}{{\mathrm e}^{2 b x +2 a}+1}\right ) \mathrm {csgn}\left (i {\mathrm e}^{2 b x +2 a}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{2 b x +2 a}}{{\mathrm e}^{2 b x +2 a}+1}\right )+11 i \pi \,x^{4} \mathrm {csgn}\left (\frac {i {\mathrm e}^{2 b x +2 a}}{{\mathrm e}^{2 b x +2 a}+1}\right )^{3}+11 i \pi \,x^{4} \mathrm {csgn}\left (i {\mathrm e}^{b x +a}\right )^{2} \mathrm {csgn}\left (i {\mathrm e}^{2 b x +2 a}\right )+11 i \pi \,x^{4} \mathrm {csgn}\left (i {\mathrm e}^{2 b x +2 a}\right )^{3}-22 i \pi \,x^{4} \mathrm {csgn}\left (i {\mathrm e}^{b x +a}\right ) \mathrm {csgn}\left (i {\mathrm e}^{2 b x +2 a}\right )^{2}+8 b \,x^{5}\right ) \sqrt {x}}{198}\) | \(310\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.26, size = 19, normalized size = 0.70 \begin {gather*} -\frac {4}{99} \, b x^{\frac {11}{2}} + \frac {2}{9} \, x^{\frac {9}{2}} \operatorname {artanh}\left (\tanh \left (b x + a\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.33, size = 18, normalized size = 0.67 \begin {gather*} \frac {2}{99} \, {\left (9 \, b x^{5} + 11 \, a x^{4}\right )} \sqrt {x} \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 = 0.39, size = 13, normalized size = 0.48 \begin {gather*} \frac {2}{11} \, b x^{\frac {11}{2}} + \frac {2}{9} \, a x^{\frac {9}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 1.24, size = 57, normalized size = 2.11 \begin {gather*} \frac {x^{9/2}\,\ln \left (\frac {{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )}{9}-\frac {4\,b\,x^{11/2}}{99}-\frac {x^{9/2}\,\ln \left (\frac {1}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )}{9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________