3.84 \(\int \text {sech}^2(a+b x) \tanh ^2(a+b x) \, dx\)

Optimal. Leaf size=15 \[ \frac {\tanh ^3(a+b x)}{3 b} \]

[Out]

1/3*tanh(b*x+a)^3/b

________________________________________________________________________________________

Rubi [A]  time = 0.03, antiderivative size = 15, normalized size of antiderivative = 1.00, 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, 30} \[ \frac {\tanh ^3(a+b x)}{3 b} \]

Antiderivative was successfully verified.

[In]

Int[Sech[a + b*x]^2*Tanh[a + b*x]^2,x]

[Out]

Tanh[a + b*x]^3/(3*b)

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 2607

Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[1/f, Subst[Int[(b*x)
^n*(1 + x^2)^(m/2 - 1), x], x, Tan[e + f*x]], x] /; FreeQ[{b, e, f, n}, x] && IntegerQ[m/2] &&  !(IntegerQ[(n
- 1)/2] && LtQ[0, n, m - 1])

Rubi steps

\begin {align*} \int \text {sech}^2(a+b x) \tanh ^2(a+b x) \, dx &=\frac {i \operatorname {Subst}\left (\int x^2 \, dx,x,i \tanh (a+b x)\right )}{b}\\ &=\frac {\tanh ^3(a+b x)}{3 b}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 15, normalized size = 1.00 \[ \frac {\tanh ^3(a+b x)}{3 b} \]

Antiderivative was successfully verified.

[In]

Integrate[Sech[a + b*x]^2*Tanh[a + b*x]^2,x]

[Out]

Tanh[a + b*x]^3/(3*b)

________________________________________________________________________________________

fricas [B]  time = 0.42, size = 138, normalized size = 9.20 \[ -\frac {8 \, {\left (\cosh \left (b x + a\right )^{2} + \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2}\right )}}{3 \, {\left (b \cosh \left (b x + a\right )^{4} + 4 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + b \sinh \left (b x + a\right )^{4} + 4 \, b \cosh \left (b x + a\right )^{2} + 2 \, {\left (3 \, b \cosh \left (b x + a\right )^{2} + 2 \, b\right )} \sinh \left (b x + a\right )^{2} + 4 \, {\left (b \cosh \left (b x + a\right )^{3} + b \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + 3 \, b\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(b*x+a)^2*tanh(b*x+a)^2,x, algorithm="fricas")

[Out]

-8/3*(cosh(b*x + a)^2 + cosh(b*x + a)*sinh(b*x + a) + sinh(b*x + a)^2)/(b*cosh(b*x + a)^4 + 4*b*cosh(b*x + a)*
sinh(b*x + a)^3 + b*sinh(b*x + a)^4 + 4*b*cosh(b*x + a)^2 + 2*(3*b*cosh(b*x + a)^2 + 2*b)*sinh(b*x + a)^2 + 4*
(b*cosh(b*x + a)^3 + b*cosh(b*x + a))*sinh(b*x + a) + 3*b)

________________________________________________________________________________________

giac [B]  time = 0.13, size = 31, normalized size = 2.07 \[ -\frac {2 \, {\left (3 \, e^{\left (4 \, b x + 4 \, a\right )} + 1\right )}}{3 \, b {\left (e^{\left (2 \, b x + 2 \, a\right )} + 1\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(b*x+a)^2*tanh(b*x+a)^2,x, algorithm="giac")

[Out]

-2/3*(3*e^(4*b*x + 4*a) + 1)/(b*(e^(2*b*x + 2*a) + 1)^3)

________________________________________________________________________________________

maple [B]  time = 0.32, size = 42, normalized size = 2.80 \[ \frac {-\frac {\sinh \left (b x +a \right )}{2 \cosh \left (b x +a \right )^{3}}+\frac {\left (\frac {2}{3}+\frac {\mathrm {sech}\left (b x +a \right )^{2}}{3}\right ) \tanh \left (b x +a \right )}{2}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sech(b*x+a)^2*tanh(b*x+a)^2,x)

[Out]

1/b*(-1/2*sinh(b*x+a)/cosh(b*x+a)^3+1/2*(2/3+1/3*sech(b*x+a)^2)*tanh(b*x+a))

________________________________________________________________________________________

maxima [A]  time = 0.33, size = 13, normalized size = 0.87 \[ \frac {\tanh \left (b x + a\right )^{3}}{3 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(b*x+a)^2*tanh(b*x+a)^2,x, algorithm="maxima")

[Out]

1/3*tanh(b*x + a)^3/b

________________________________________________________________________________________

mupad [B]  time = 0.07, size = 31, normalized size = 2.07 \[ -\frac {2\,\left (3\,{\mathrm {e}}^{4\,a+4\,b\,x}+1\right )}{3\,b\,{\left ({\mathrm {e}}^{2\,a+2\,b\,x}+1\right )}^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tanh(a + b*x)^2/cosh(a + b*x)^2,x)

[Out]

-(2*(3*exp(4*a + 4*b*x) + 1))/(3*b*(exp(2*a + 2*b*x) + 1)^3)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \tanh ^{2}{\left (a + b x \right )} \operatorname {sech}^{2}{\left (a + b x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(b*x+a)**2*tanh(b*x+a)**2,x)

[Out]

Integral(tanh(a + b*x)**2*sech(a + b*x)**2, x)

________________________________________________________________________________________