### 3.359 $$\int \sinh (a+b x) \tanh (a+b x) \, dx$$

Optimal. Leaf size=23 $\frac{\sinh (a+b x)}{b}-\frac{\tan ^{-1}(\sinh (a+b x))}{b}$

[Out]

-(ArcTan[Sinh[a + b*x]]/b) + Sinh[a + b*x]/b

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Rubi [A]  time = 0.0154926, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.231, Rules used = {2592, 321, 203} $\frac{\sinh (a+b x)}{b}-\frac{\tan ^{-1}(\sinh (a+b x))}{b}$

Antiderivative was successfully veriﬁed.

[In]

Int[Sinh[a + b*x]*Tanh[a + b*x],x]

[Out]

-(ArcTan[Sinh[a + b*x]]/b) + Sinh[a + b*x]/b

Rule 2592

Int[((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*tan[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> With[{ff = FreeFactors[S
in[e + f*x], x]}, Dist[ff/f, Subst[Int[(ff*x)^(m + n)/(a^2 - ff^2*x^2)^((n + 1)/2), x], x, (a*Sin[e + f*x])/ff
], x]] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n + 1)/2]

Rule 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rubi steps

\begin{align*} \int \sinh (a+b x) \tanh (a+b x) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^2}{1+x^2} \, dx,x,\sinh (a+b x)\right )}{b}\\ &=\frac{\sinh (a+b x)}{b}-\frac{\operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\sinh (a+b x)\right )}{b}\\ &=-\frac{\tan ^{-1}(\sinh (a+b x))}{b}+\frac{\sinh (a+b x)}{b}\\ \end{align*}

Mathematica [A]  time = 0.0116213, size = 23, normalized size = 1. $\frac{\sinh (a+b x)}{b}-\frac{\tan ^{-1}(\sinh (a+b x))}{b}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-(ArcTan[Sinh[a + b*x]]/b) + Sinh[a + b*x]/b

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Maple [A]  time = 0.013, size = 24, normalized size = 1. \begin{align*}{\frac{\sinh \left ( bx+a \right ) }{b}}-2\,{\frac{\arctan \left ({{\rm e}^{bx+a}} \right ) }{b}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sinh(b*x+a)/b-2*arctan(exp(b*x+a))/b

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Maxima [A]  time = 1.6059, size = 55, normalized size = 2.39 \begin{align*} \frac{2 \, \arctan \left (e^{\left (-b x - a\right )}\right )}{b} + \frac{e^{\left (b x + a\right )}}{2 \, b} - \frac{e^{\left (-b x - a\right )}}{2 \, b} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 2.08583, size = 254, normalized size = 11.04 \begin{align*} -\frac{4 \,{\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )} \arctan \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right ) - \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} + 1}{2 \,{\left (b \cosh \left (b x + a\right ) + b \sinh \left (b x + a\right )\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(4*(cosh(b*x + a) + sinh(b*x + a))*arctan(cosh(b*x + a) + sinh(b*x + a)) - cosh(b*x + a)^2 - 2*cosh(b*x +
a)*sinh(b*x + a) - sinh(b*x + a)^2 + 1)/(b*cosh(b*x + a) + b*sinh(b*x + a))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sinh ^{2}{\left (a + b x \right )} \operatorname{sech}{\left (a + b x \right )}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.16673, size = 51, normalized size = 2.22 \begin{align*} -\frac{2 \, \arctan \left (e^{\left (b x + a\right )}\right )}{b} + \frac{e^{\left (b x + a\right )}}{2 \, b} - \frac{e^{\left (-b x - a\right )}}{2 \, b} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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