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

Optimal. Leaf size=24 \[ \frac{a \sinh (c+d x)}{d}+\frac{b \tan ^{-1}(\sinh (c+d x))}{d} \]

[Out]

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

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Rubi [A]  time = 0.0298312, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {4045, 3770} \[ \frac{a \sinh (c+d x)}{d}+\frac{b \tan ^{-1}(\sinh (c+d x))}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 4045

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(m_.)*(csc[(e_.) + (f_.)*(x_)]^2*(C_.) + (A_)), x_Symbol] :> Simp[(A*Cot[e
+ f*x]*(b*Csc[e + f*x])^m)/(f*m), x] + Dist[(C*m + A*(m + 1))/(b^2*m), Int[(b*Csc[e + f*x])^(m + 2), x], x] /
; FreeQ[{b, e, f, A, C}, x] && NeQ[C*m + A*(m + 1), 0] && LeQ[m, -1]

Rule 3770

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

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

Mathematica [A]  time = 0.0171176, size = 35, normalized size = 1.46 \[ \frac{a \sinh (c) \cosh (d x)}{d}+\frac{a \cosh (c) \sinh (d x)}{d}+\frac{b \tan ^{-1}(\sinh (c+d x))}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.031, size = 26, normalized size = 1.1 \begin{align*}{\frac{a\sinh \left ( 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(cosh(d*x+c)*(a+b*sech(d*x+c)^2),x)

[Out]

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

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Maxima [A]  time = 1.60006, size = 38, normalized size = 1.58 \begin{align*} -\frac{2 \, b \arctan \left (e^{\left (-d x - c\right )}\right )}{d} + \frac{a \sinh \left (d x + c\right )}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 2.15015, size = 266, normalized size = 11.08 \begin{align*} \frac{a \cosh \left (d x + c\right )^{2} + 2 \, a \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + a \sinh \left (d x + c\right )^{2} + 4 \,{\left (b \cosh \left (d x + c\right ) + b \sinh \left (d x + c\right )\right )} \arctan \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )\right ) - a}{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(cosh(d*x+c)*(a+b*sech(d*x+c)^2),x, algorithm="fricas")

[Out]

1/2*(a*cosh(d*x + c)^2 + 2*a*cosh(d*x + c)*sinh(d*x + c) + a*sinh(d*x + c)^2 + 4*(b*cosh(d*x + c) + b*sinh(d*x
+ c))*arctan(cosh(d*x + c) + sinh(d*x + c)) - a)/(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 \operatorname{sech}^{2}{\left (c + d x \right )}\right ) \cosh{\left (c + d x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.19392, size = 55, normalized size = 2.29 \begin{align*} \frac{2 \, b \arctan \left (e^{\left (d x + c\right )}\right )}{d} + \frac{a e^{\left (d x + c\right )}}{2 \, d} - \frac{a e^{\left (-d x - c\right )}}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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