3.287 \(\int \text{sech}(c+d x) (a+b \sinh ^2(c+d x)) \, dx\)

Optimal. Leaf size=28 \[ \frac{(a-b) \tan ^{-1}(\sinh (c+d x))}{d}+\frac{b \sinh (c+d x)}{d} \]

[Out]

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

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Rubi [A]  time = 0.0319274, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {3190, 388, 203} \[ \frac{(a-b) \tan ^{-1}(\sinh (c+d x))}{d}+\frac{b \sinh (c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

Int[Sech[c + d*x]*(a + b*Sinh[c + d*x]^2),x]

[Out]

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

Rule 3190

Int[cos[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_.), x_Symbol] :> With[{ff = Free
Factors[Sin[e + f*x], x]}, Dist[ff/f, Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b*ff^2*x^2)^p, x], x, Sin[e +
f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]

Rule 388

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

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

Mathematica [A]  time = 0.013777, size = 37, normalized size = 1.32 \[ \frac{a \tan ^{-1}(\sinh (c+d x))}{d}+\frac{b \sinh (c+d x)}{d}-\frac{b \tan ^{-1}(\sinh (c+d x))}{d} \]

Antiderivative was successfully verified.

[In]

Integrate[Sech[c + d*x]*(a + b*Sinh[c + d*x]^2),x]

[Out]

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

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Maple [A]  time = 0.028, size = 39, normalized size = 1.4 \begin{align*}{\frac{b\sinh \left ( dx+c \right ) }{d}}+2\,{\frac{a\arctan \left ({{\rm e}^{dx+c}} \right ) }{d}}-2\,{\frac{b\arctan \left ({{\rm e}^{dx+c}} \right ) }{d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.51922, size = 76, normalized size = 2.71 \begin{align*} \frac{1}{2} \, b{\left (\frac{4 \, \arctan \left (e^{\left (-d x - c\right )}\right )}{d} + \frac{e^{\left (d x + c\right )}}{d} - \frac{e^{\left (-d x - c\right )}}{d}\right )} + \frac{a \arctan \left (\sinh \left (d x + c\right )\right )}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*b*(4*arctan(e^(-d*x - c))/d + e^(d*x + c)/d - e^(-d*x - c)/d) + a*arctan(sinh(d*x + c))/d

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Fricas [B]  time = 1.51464, size = 282, normalized size = 10.07 \begin{align*} \frac{b \cosh \left (d x + c\right )^{2} + 2 \, b \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + b \sinh \left (d x + c\right )^{2} + 4 \,{\left ({\left (a - b\right )} \cosh \left (d x + c\right ) +{\left (a - b\right )} \sinh \left (d x + c\right )\right )} \arctan \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )\right ) - b}{2 \,{\left (d \cosh \left (d x + c\right ) + d \sinh \left (d x + c\right )\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.12814, size = 61, normalized size = 2.18 \begin{align*} \frac{2 \,{\left (a - b\right )} \arctan \left (e^{\left (d x + c\right )}\right )}{d} + \frac{b e^{\left (d x + c\right )}}{2 \, d} - \frac{b e^{\left (-d x - c\right )}}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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