Optimal. Leaf size=19 \[ \frac{\tanh ^{n+1}(a+b x)}{b (n+1)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0351163, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {2607, 32} \[ \frac{\tanh ^{n+1}(a+b x)}{b (n+1)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2607
Rule 32
Rubi steps
\begin{align*} \int \text{sech}^2(a+b x) \tanh ^n(a+b x) \, dx &=-\frac{i \operatorname{Subst}\left (\int (-i x)^n \, dx,x,i \tanh (a+b x)\right )}{b}\\ &=\frac{\tanh ^{1+n}(a+b x)}{b (1+n)}\\ \end{align*}
Mathematica [A] time = 0.0191072, size = 19, normalized size = 1. \[ \frac{\tanh ^{n+1}(a+b x)}{b (n+1)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.014, size = 20, normalized size = 1.1 \begin{align*}{\frac{ \left ( \tanh \left ( bx+a \right ) \right ) ^{n+1}}{b \left ( n+1 \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 2.0714, size = 190, normalized size = 10. \begin{align*} \frac{\cosh \left (n \log \left (\frac{\sinh \left (b x + a\right )}{\cosh \left (b x + a\right )}\right )\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right ) \sinh \left (n \log \left (\frac{\sinh \left (b x + a\right )}{\cosh \left (b x + a\right )}\right )\right )}{{\left (b n + b\right )} \cosh \left (b x + a\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \tanh ^{n}{\left (a + b x \right )} \operatorname{sech}^{2}{\left (a + b x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \tanh \left (b x + a\right )^{n} \operatorname{sech}\left (b x + a\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]