Optimal. Leaf size=31 \[ \frac{\tanh ^3(a+b x)}{3 b}-\frac{\tanh ^5(a+b x)}{5 b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.032713, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {2607, 14} \[ \frac{\tanh ^3(a+b x)}{3 b}-\frac{\tanh ^5(a+b x)}{5 b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2607
Rule 14
Rubi steps
\begin{align*} \int \text{sech}^4(a+b x) \tanh ^2(a+b x) \, dx &=\frac{i \operatorname{Subst}\left (\int x^2 \left (1+x^2\right ) \, dx,x,i \tanh (a+b x)\right )}{b}\\ &=\frac{i \operatorname{Subst}\left (\int \left (x^2+x^4\right ) \, dx,x,i \tanh (a+b x)\right )}{b}\\ &=\frac{\tanh ^3(a+b x)}{3 b}-\frac{\tanh ^5(a+b x)}{5 b}\\ \end{align*}
Mathematica [A] time = 0.0407975, size = 56, normalized size = 1.81 \[ \frac{2 \tanh (a+b x)}{15 b}-\frac{\tanh (a+b x) \text{sech}^4(a+b x)}{5 b}+\frac{\tanh (a+b x) \text{sech}^2(a+b x)}{15 b} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.019, size = 52, normalized size = 1.7 \begin{align*}{\frac{1}{b} \left ( -{\frac{\sinh \left ( bx+a \right ) }{4\, \left ( \cosh \left ( bx+a \right ) \right ) ^{5}}}+{\frac{\tanh \left ( bx+a \right ) }{4} \left ({\frac{8}{15}}+{\frac{ \left ({\rm sech} \left (bx+a\right ) \right ) ^{4}}{5}}+{\frac{4\, \left ({\rm sech} \left (bx+a\right ) \right ) ^{2}}{15}} \right ) } \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.12916, size = 373, normalized size = 12.03 \begin{align*} \frac{4 \, e^{\left (-2 \, b x - 2 \, a\right )}}{3 \, b{\left (5 \, e^{\left (-2 \, b x - 2 \, a\right )} + 10 \, e^{\left (-4 \, b x - 4 \, a\right )} + 10 \, e^{\left (-6 \, b x - 6 \, a\right )} + 5 \, e^{\left (-8 \, b x - 8 \, a\right )} + e^{\left (-10 \, b x - 10 \, a\right )} + 1\right )}} - \frac{4 \, e^{\left (-4 \, b x - 4 \, a\right )}}{3 \, b{\left (5 \, e^{\left (-2 \, b x - 2 \, a\right )} + 10 \, e^{\left (-4 \, b x - 4 \, a\right )} + 10 \, e^{\left (-6 \, b x - 6 \, a\right )} + 5 \, e^{\left (-8 \, b x - 8 \, a\right )} + e^{\left (-10 \, b x - 10 \, a\right )} + 1\right )}} + \frac{4 \, e^{\left (-6 \, b x - 6 \, a\right )}}{b{\left (5 \, e^{\left (-2 \, b x - 2 \, a\right )} + 10 \, e^{\left (-4 \, b x - 4 \, a\right )} + 10 \, e^{\left (-6 \, b x - 6 \, a\right )} + 5 \, e^{\left (-8 \, b x - 8 \, a\right )} + e^{\left (-10 \, b x - 10 \, a\right )} + 1\right )}} + \frac{4}{15 \, b{\left (5 \, e^{\left (-2 \, b x - 2 \, a\right )} + 10 \, e^{\left (-4 \, b x - 4 \, a\right )} + 10 \, e^{\left (-6 \, b x - 6 \, a\right )} + 5 \, e^{\left (-8 \, b x - 8 \, a\right )} + e^{\left (-10 \, b x - 10 \, a\right )} + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.85278, size = 836, normalized size = 26.97 \begin{align*} -\frac{8 \,{\left (8 \, \cosh \left (b x + a\right )^{3} + 24 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + 7 \, \sinh \left (b x + a\right )^{3} +{\left (21 \, \cosh \left (b x + a\right )^{2} - 5\right )} \sinh \left (b x + a\right )\right )}}{15 \,{\left (b \cosh \left (b x + a\right )^{7} + 7 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{6} + b \sinh \left (b x + a\right )^{7} + 5 \, b \cosh \left (b x + a\right )^{5} +{\left (21 \, b \cosh \left (b x + a\right )^{2} + 5 \, b\right )} \sinh \left (b x + a\right )^{5} + 5 \,{\left (7 \, b \cosh \left (b x + a\right )^{3} + 5 \, b \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{4} + 11 \, b \cosh \left (b x + a\right )^{3} +{\left (35 \, b \cosh \left (b x + a\right )^{4} + 50 \, b \cosh \left (b x + a\right )^{2} + 9 \, b\right )} \sinh \left (b x + a\right )^{3} +{\left (21 \, b \cosh \left (b x + a\right )^{5} + 50 \, b \cosh \left (b x + a\right )^{3} + 33 \, b \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{2} + 15 \, b \cosh \left (b x + a\right ) +{\left (7 \, b \cosh \left (b x + a\right )^{6} + 25 \, b \cosh \left (b x + a\right )^{4} + 27 \, b \cosh \left (b x + a\right )^{2} + 5 \, b\right )} \sinh \left (b x + a\right )\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 ^{2}{\left (a + b x \right )} \operatorname{sech}^{4}{\left (a + b x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.25431, size = 72, normalized size = 2.32 \begin{align*} -\frac{4 \,{\left (15 \, e^{\left (6 \, b x + 6 \, a\right )} - 5 \, e^{\left (4 \, b x + 4 \, a\right )} + 5 \, e^{\left (2 \, b x + 2 \, a\right )} + 1\right )}}{15 \, b{\left (e^{\left (2 \, b x + 2 \, a\right )} + 1\right )}^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]