\(\int \sinh ^2(a+b x) \tanh (a+b x) \, dx\) [380]

Optimal result

Integrand size = 15, antiderivative size = 28 \[ \int \sinh ^2(a+b x) \tanh (a+b x) \, dx=\frac {\cosh ^2(a+b x)}{2 b}-\frac {\log (\cosh (a+b x))}{b} \]

[Out]

1/2*cosh(b*x+a)^2/b-ln(cosh(b*x+a))/b

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {2670, 14} \[ \int \sinh ^2(a+b x) \tanh (a+b x) \, dx=\frac {\cosh ^2(a+b x)}{2 b}-\frac {\log (\cosh (a+b x))}{b} \]

[In]

Int[Sinh[a + b*x]^2*Tanh[a + b*x],x]

[Out]

Cosh[a + b*x]^2/(2*b) - Log[Cosh[a + b*x]]/b

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 2670

Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*tan[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> Dist[-f^(-1), Subst[Int[(1 - x^2
)^((m + n - 1)/2)/x^n, x], x, Cos[e + f*x]], x] /; FreeQ[{e, f}, x] && IntegersQ[m, n, (m + n - 1)/2]

Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \frac {1-x^2}{x} \, dx,x,\cosh (a+b x)\right )}{b} \\ & = -\frac {\text {Subst}\left (\int \left (\frac {1}{x}-x\right ) \, dx,x,\cosh (a+b x)\right )}{b} \\ & = \frac {\cosh ^2(a+b x)}{2 b}-\frac {\log (\cosh (a+b x))}{b} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.89 \[ \int \sinh ^2(a+b x) \tanh (a+b x) \, dx=-\frac {-\frac {1}{2} \cosh ^2(a+b x)+\log (\cosh (a+b x))}{b} \]

[In]

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

[Out]

-((-1/2*Cosh[a + b*x]^2 + Log[Cosh[a + b*x]])/b)

Maple [A] (verified)

Time = 0.73 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.89

method	result	size
derivativedivides	\(\frac {\frac {\sinh \left (b x +a \right )^{2}}{2}-\ln \left (\cosh \left (b x +a \right )\right )}{b}\)	\(25\)
default	\(\frac {\frac {\sinh \left (b x +a \right )^{2}}{2}-\ln \left (\cosh \left (b x +a \right )\right )}{b}\)	\(25\)
risch	\(x +\frac {{\mathrm e}^{2 b x +2 a}}{8 b}+\frac {{\mathrm e}^{-2 b x -2 a}}{8 b}+\frac {2 a}{b}-\frac {\ln \left (1+{\mathrm e}^{2 b x +2 a}\right )}{b}\)	\(54\)

[In]

int(sech(b*x+a)*sinh(b*x+a)^3,x,method=_RETURNVERBOSE)

[Out]

1/b*(1/2*sinh(b*x+a)^2-ln(cosh(b*x+a)))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 197 vs. \(2 (26) = 52\).

Time = 0.26 (sec) , antiderivative size = 197, normalized size of antiderivative = 7.04 \[ \int \sinh ^2(a+b x) \tanh (a+b x) \, dx=\frac {8 \, b x \cosh \left (b x + a\right )^{2} + \cosh \left (b x + a\right )^{4} + 4 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + \sinh \left (b x + a\right )^{4} + 2 \, {\left (4 \, b x + 3 \, \cosh \left (b x + a\right )^{2}\right )} \sinh \left (b x + a\right )^{2} - 8 \, {\left (\cosh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2}\right )} \log \left (\frac {2 \, \cosh \left (b x + a\right )}{\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )}\right ) + 4 \, {\left (4 \, b x \cosh \left (b x + a\right ) + \cosh \left (b x + a\right )^{3}\right )} \sinh \left (b x + a\right ) + 1}{8 \, {\left (b \cosh \left (b x + a\right )^{2} + 2 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + b \sinh \left (b x + a\right )^{2}\right )}} \]

[In]

integrate(sech(b*x+a)*sinh(b*x+a)^3,x, algorithm="fricas")

[Out]

1/8*(8*b*x*cosh(b*x + a)^2 + cosh(b*x + a)^4 + 4*cosh(b*x + a)*sinh(b*x + a)^3 + sinh(b*x + a)^4 + 2*(4*b*x +
3*cosh(b*x + a)^2)*sinh(b*x + a)^2 - 8*(cosh(b*x + a)^2 + 2*cosh(b*x + a)*sinh(b*x + a) + sinh(b*x + a)^2)*log
(2*cosh(b*x + a)/(cosh(b*x + a) - sinh(b*x + a))) + 4*(4*b*x*cosh(b*x + a) + cosh(b*x + a)^3)*sinh(b*x + a) +
1)/(b*cosh(b*x + a)^2 + 2*b*cosh(b*x + a)*sinh(b*x + a) + b*sinh(b*x + a)^2)

Sympy [F]

\[ \int \sinh ^2(a+b x) \tanh (a+b x) \, dx=\int \sinh ^{3}{\left (a + b x \right )} \operatorname {sech}{\left (a + b x \right )}\, dx \]

[In]

integrate(sech(b*x+a)*sinh(b*x+a)**3,x)

[Out]

Integral(sinh(a + b*x)**3*sech(a + b*x), x)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 56 vs. \(2 (26) = 52\).

Time = 0.31 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.00 \[ \int \sinh ^2(a+b x) \tanh (a+b x) \, dx=-\frac {b x + a}{b} + \frac {e^{\left (2 \, b x + 2 \, a\right )}}{8 \, b} + \frac {e^{\left (-2 \, b x - 2 \, a\right )}}{8 \, b} - \frac {\log \left (e^{\left (-2 \, b x - 2 \, a\right )} + 1\right )}{b} \]

[In]

integrate(sech(b*x+a)*sinh(b*x+a)^3,x, algorithm="maxima")

[Out]

-(b*x + a)/b + 1/8*e^(2*b*x + 2*a)/b + 1/8*e^(-2*b*x - 2*a)/b - log(e^(-2*b*x - 2*a) + 1)/b

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (26) = 52\).

Time = 0.27 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.14 \[ \int \sinh ^2(a+b x) \tanh (a+b x) \, dx=\frac {8 \, b x - {\left (4 \, e^{\left (2 \, b x + 2 \, a\right )} - 1\right )} e^{\left (-2 \, b x - 2 \, a\right )} + 8 \, a + e^{\left (2 \, b x + 2 \, a\right )} - 8 \, \log \left (e^{\left (2 \, b x + 2 \, a\right )} + 1\right )}{8 \, b} \]

[In]

integrate(sech(b*x+a)*sinh(b*x+a)^3,x, algorithm="giac")

[Out]

1/8*(8*b*x - (4*e^(2*b*x + 2*a) - 1)*e^(-2*b*x - 2*a) + 8*a + e^(2*b*x + 2*a) - 8*log(e^(2*b*x + 2*a) + 1))/b

Mupad [B] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.71 \[ \int \sinh ^2(a+b x) \tanh (a+b x) \, dx=x-\frac {\ln \left ({\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1\right )}{b}+\frac {{\mathrm {e}}^{-2\,a-2\,b\,x}}{8\,b}+\frac {{\mathrm {e}}^{2\,a+2\,b\,x}}{8\,b} \]

[In]

int(sinh(a + b*x)^3/cosh(a + b*x),x)

[Out]

x - log(exp(2*a)*exp(2*b*x) + 1)/b + exp(- 2*a - 2*b*x)/(8*b) + exp(2*a + 2*b*x)/(8*b)