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

Optimal. Leaf size=25 \[ \frac{b \cosh (c+d x)}{d}-\frac{a \tanh ^{-1}(\cosh (c+d x))}{d} \]

[Out]

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

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Rubi [A]  time = 0.0350184, antiderivative size = 25, 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 = {3014, 3770} \[ \frac{b \cosh (c+d x)}{d}-\frac{a \tanh ^{-1}(\cosh (c+d x))}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 3014

Int[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_) + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> -Simp[(C*Cos[
e + f*x]*(b*Sin[e + f*x])^(m + 1))/(b*f*(m + 2)), x] + Dist[(A*(m + 2) + C*(m + 1))/(m + 2), Int[(b*Sin[e + f*
x])^m, x], x] /; FreeQ[{b, e, f, A, C, m}, x] &&  !LtQ[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 \text{csch}(c+d x) \left (a+b \sinh ^2(c+d x)\right ) \, dx &=\frac{b \cosh (c+d x)}{d}+a \int \text{csch}(c+d x) \, dx\\ &=-\frac{a \tanh ^{-1}(\cosh (c+d x))}{d}+\frac{b \cosh (c+d x)}{d}\\ \end{align*}

Mathematica [B]  time = 0.0315701, size = 62, normalized size = 2.48 \[ \frac{a \log \left (\sinh \left (\frac{c}{2}+\frac{d x}{2}\right )\right )}{d}-\frac{a \log \left (\cosh \left (\frac{c}{2}+\frac{d x}{2}\right )\right )}{d}+\frac{b \sinh (c) \sinh (d x)}{d}+\frac{b \cosh (c) \cosh (d x)}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.029, size = 24, normalized size = 1. \begin{align*}{\frac{-2\,a{\it Artanh} \left ({{\rm e}^{dx+c}} \right ) +b\cosh \left ( dx+c \right ) }{d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.04059, size = 58, normalized size = 2.32 \begin{align*} \frac{1}{2} \, b{\left (\frac{e^{\left (d x + c\right )}}{d} + \frac{e^{\left (-d x - c\right )}}{d}\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*sinh(d*x+c)^2),x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 1.9737, size = 374, normalized size = 14.96 \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} - 2 \,{\left (a \cosh \left (d x + c\right ) + a \sinh \left (d x + c\right )\right )} \log \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) + 1\right ) + 2 \,{\left (a \cosh \left (d x + c\right ) + a \sinh \left (d x + c\right )\right )} \log \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) - 1\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(csch(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 - 2*(a*cosh(d*x + c) + a*sinh(d*x
+ c))*log(cosh(d*x + c) + sinh(d*x + c) + 1) + 2*(a*cosh(d*x + c) + a*sinh(d*x + c))*log(cosh(d*x + c) + sinh
(d*x + c) - 1) + b)/(d*cosh(d*x + c) + d*sinh(d*x + c))

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [B]  time = 1.2144, size = 78, normalized size = 3.12 \begin{align*} \frac{b e^{\left (d x + c\right )}}{2 \, d} + \frac{b e^{\left (-d x - c\right )}}{2 \, d} - \frac{a \log \left (e^{\left (d x + c\right )} + 1\right )}{d} + \frac{a \log \left ({\left | e^{\left (d x + c\right )} - 1 \right |}\right )}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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