Optimal. Leaf size=47 \[ \frac {\left (a+b x^n\right ) \coth ^{-1}\left (a+b x^n\right )}{b n}+\frac {\log \left (1-\left (a+b x^n\right )^2\right )}{2 b n} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.04, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {6847, 6239,
6022, 266} \begin {gather*} \frac {\log \left (1-\left (a+b x^n\right )^2\right )}{2 b n}+\frac {\left (a+b x^n\right ) \coth ^{-1}\left (a+b x^n\right )}{b n} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 266
Rule 6022
Rule 6239
Rule 6847
Rubi steps
\begin {align*} \int x^{-1+n} \coth ^{-1}\left (a+b x^n\right ) \, dx &=\frac {\text {Subst}\left (\int \coth ^{-1}(a+b x) \, dx,x,x^n\right )}{n}\\ &=\frac {\text {Subst}\left (\int \coth ^{-1}(x) \, dx,x,a+b x^n\right )}{b n}\\ &=\frac {\left (a+b x^n\right ) \coth ^{-1}\left (a+b x^n\right )}{b n}-\frac {\text {Subst}\left (\int \frac {x}{1-x^2} \, dx,x,a+b x^n\right )}{b n}\\ &=\frac {\left (a+b x^n\right ) \coth ^{-1}\left (a+b x^n\right )}{b n}+\frac {\log \left (1-\left (a+b x^n\right )^2\right )}{2 b n}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.03, size = 42, normalized size = 0.89 \begin {gather*} \frac {2 \left (a+b x^n\right ) \coth ^{-1}\left (a+b x^n\right )+\log \left (1-\left (a+b x^n\right )^2\right )}{2 b n} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(117\) vs.
\(2(45)=90\).
time = 0.05, size = 118, normalized size = 2.51
method | result | size |
risch | \(\frac {x^{n} \ln \left (a +b \,x^{n}+1\right )}{2 n}-\frac {x^{n} \ln \left (a +b \,x^{n}-1\right )}{2 n}+\frac {\ln \left (x^{n}+\frac {1+a}{b}\right ) a}{2 n b}-\frac {\ln \left (x^{n}+\frac {-1+a}{b}\right ) a}{2 n b}+\frac {\ln \left (x^{n}+\frac {1+a}{b}\right )}{2 n b}+\frac {\ln \left (x^{n}+\frac {-1+a}{b}\right )}{2 n b}\) | \(118\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.26, size = 40, normalized size = 0.85 \begin {gather*} \frac {2 \, {\left (b x^{n} + a\right )} \operatorname {arcoth}\left (b x^{n} + a\right ) + \log \left (-{\left (b x^{n} + a\right )}^{2} + 1\right )}{2 \, b n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 108 vs.
\(2 (45) = 90\).
time = 0.36, size = 108, normalized size = 2.30 \begin {gather*} \frac {{\left (a + 1\right )} \log \left (b \cosh \left (n \log \left (x\right )\right ) + b \sinh \left (n \log \left (x\right )\right ) + a + 1\right ) - {\left (a - 1\right )} \log \left (b \cosh \left (n \log \left (x\right )\right ) + b \sinh \left (n \log \left (x\right )\right ) + a - 1\right ) + {\left (b \cosh \left (n \log \left (x\right )\right ) + b \sinh \left (n \log \left (x\right )\right )\right )} \log \left (\frac {b \cosh \left (n \log \left (x\right )\right ) + b \sinh \left (n \log \left (x\right )\right ) + a + 1}{b \cosh \left (n \log \left (x\right )\right ) + b \sinh \left (n \log \left (x\right )\right ) + a - 1}\right )}{2 \, b n} \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: HeuristicGCDFailed} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 119 vs.
\(2 (45) = 90\).
time = 0.42, size = 119, normalized size = 2.53 \begin {gather*} \frac {{\left ({\left (a + 1\right )} b - {\left (a - 1\right )} b\right )} {\left (\frac {\log \left (\frac {{\left | b x^{n} + a + 1 \right |}}{{\left | b x^{n} + a - 1 \right |}}\right )}{b^{2}} - \frac {\log \left ({\left | \frac {b x^{n} + a + 1}{b x^{n} + a - 1} - 1 \right |}\right )}{b^{2}} + \frac {\log \left (\frac {b x^{n} + a + 1}{b x^{n} + a - 1}\right )}{b^{2} {\left (\frac {b x^{n} + a + 1}{b x^{n} + a - 1} - 1\right )}}\right )}}{2 \, n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 2.51, size = 58, normalized size = 1.23 \begin {gather*} \frac {\frac {\ln \left (a^2+b^2\,x^{2\,n}+2\,a\,b\,x^n-1\right )}{2}+a\,\mathrm {acoth}\left (a+b\,x^n\right )}{b\,n}+\frac {x^n\,\mathrm {acoth}\left (a+b\,x^n\right )}{n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________