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

Optimal. Leaf size=27 \[ \frac{b \text{sech}(c+d x)}{d}-\frac{(a+b) \tanh ^{-1}(\cosh (c+d x))}{d} \]

[Out]

-(((a + b)*ArcTanh[Cosh[c + d*x]])/d) + (b*Sech[c + d*x])/d

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Rubi [A]  time = 0.044651, antiderivative size = 27, 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 = {4133, 453, 206} \[ \frac{b \text{sech}(c+d x)}{d}-\frac{(a+b) \tanh ^{-1}(\cosh (c+d x))}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(((a + b)*ArcTanh[Cosh[c + d*x]])/d) + (b*Sech[c + d*x])/d

Rule 4133

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

Rule 453

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(c*(e*x)^(m
+ 1)*(a + b*x^n)^(p + 1))/(a*e*(m + 1)), x] + Dist[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a*e^n*(m + 1)), In
t[(e*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && (IntegerQ[n] ||
GtQ[e, 0]) && ((GtQ[n, 0] && LtQ[m, -1]) || (LtQ[n, 0] && GtQ[m + n, -1])) &&  !ILtQ[p, -1]

Rule 206

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

Rubi steps

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

Mathematica [B]  time = 0.049223, size = 67, normalized size = 2.48 \[ \frac{a \log \left (\sinh \left (\frac{c}{2}+\frac{d x}{2}\right )\right )}{d}-\frac{a \log \left (\cosh \left (\frac{c}{2}+\frac{d x}{2}\right )\right )}{d}+\frac{b \text{sech}(c+d x)}{d}+\frac{b \log \left (\tanh \left (\frac{1}{2} (c+d x)\right )\right )}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((a*Log[Cosh[c/2 + (d*x)/2]])/d) + (a*Log[Sinh[c/2 + (d*x)/2]])/d + (b*Log[Tanh[(c + d*x)/2]])/d + (b*Sech[c
+ d*x])/d

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Maple [A]  time = 0.028, size = 36, normalized size = 1.3 \begin{align*}{\frac{-2\,a{\it Artanh} \left ({{\rm e}^{dx+c}} \right ) +b \left ( \left ( \cosh \left ( dx+c \right ) \right ) ^{-1}-2\,{\it Artanh} \left ({{\rm e}^{dx+c}} \right ) \right ) }{d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*(-2*a*arctanh(exp(d*x+c))+b*(1/cosh(d*x+c)-2*arctanh(exp(d*x+c))))

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Maxima [B]  time = 1.07553, size = 108, normalized size = 4. \begin{align*} -b{\left (\frac{\log \left (e^{\left (-d x - c\right )} + 1\right )}{d} - \frac{\log \left (e^{\left (-d x - c\right )} - 1\right )}{d} - \frac{2 \, e^{\left (-d x - c\right )}}{d{\left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}}\right )} + \frac{a \log \left (\tanh \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-b*(log(e^(-d*x - c) + 1)/d - log(e^(-d*x - c) - 1)/d - 2*e^(-d*x - c)/(d*(e^(-2*d*x - 2*c) + 1))) + a*log(tan
h(1/2*d*x + 1/2*c))/d

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Fricas [B]  time = 2.51786, size = 541, normalized size = 20.04 \begin{align*} \frac{2 \, b \cosh \left (d x + c\right ) -{\left ({\left (a + b\right )} \cosh \left (d x + c\right )^{2} + 2 \,{\left (a + b\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) +{\left (a + b\right )} \sinh \left (d x + c\right )^{2} + a + b\right )} \log \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) + 1\right ) +{\left ({\left (a + b\right )} \cosh \left (d x + c\right )^{2} + 2 \,{\left (a + b\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) +{\left (a + b\right )} \sinh \left (d x + c\right )^{2} + a + b\right )} \log \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) - 1\right ) + 2 \, b \sinh \left (d x + c\right )}{d \cosh \left (d x + c\right )^{2} + 2 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + d \sinh \left (d x + c\right )^{2} + d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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 ) \operatorname{csch}{\left (c + d x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.14546, size = 104, normalized size = 3.85 \begin{align*} -\frac{{\left (a + b\right )} \log \left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )} + 2\right )}{2 \, d} + \frac{{\left (a + b\right )} \log \left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )} - 2\right )}{2 \, d} + \frac{2 \, b}{d{\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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