3.146 \(\int \text{csch}(c+d x) (a+b \sinh ^3(c+d x)) \, dx\)

Optimal. Leaf size=40 \[ -\frac{a \tanh ^{-1}(\cosh (c+d x))}{d}+\frac{b \sinh (c+d x) \cosh (c+d x)}{2 d}-\frac{b x}{2} \]

[Out]

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

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Rubi [A]  time = 0.0559113, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {3220, 3770, 2635, 8} \[ -\frac{a \tanh ^{-1}(\cosh (c+d x))}{d}+\frac{b \sinh (c+d x) \cosh (c+d x)}{2 d}-\frac{b x}{2} \]

Antiderivative was successfully verified.

[In]

Int[Csch[c + d*x]*(a + b*Sinh[c + d*x]^3),x]

[Out]

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

Rule 3220

Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^(n_))^(p_.), x_Symbol] :> Int[ExpandTr
ig[sin[e + f*x]^m*(a + b*sin[e + f*x]^n)^p, x], x] /; FreeQ[{a, b, e, f}, x] && IntegersQ[m, p] && (EqQ[n, 4]
|| GtQ[p, 0] || (EqQ[p, -1] && IntegerQ[n]))

Rule 3770

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

Rule 2635

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Sin[c + d*x])^(n - 1))/(d*n),
x] + Dist[(b^2*(n - 1))/n, Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integer
Q[2*n]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

\begin{align*} \int \text{csch}(c+d x) \left (a+b \sinh ^3(c+d x)\right ) \, dx &=i \int \left (-i a \text{csch}(c+d x)-i b \sinh ^2(c+d x)\right ) \, dx\\ &=a \int \text{csch}(c+d x) \, dx+b \int \sinh ^2(c+d x) \, dx\\ &=-\frac{a \tanh ^{-1}(\cosh (c+d x))}{d}+\frac{b \cosh (c+d x) \sinh (c+d x)}{2 d}-\frac{1}{2} b \int 1 \, dx\\ &=-\frac{b x}{2}-\frac{a \tanh ^{-1}(\cosh (c+d x))}{d}+\frac{b \cosh (c+d x) \sinh (c+d x)}{2 d}\\ \end{align*}

Mathematica [A]  time = 0.0542062, size = 72, normalized size = 1.8 \[ \frac{a \log \left (\sinh \left (\frac{c}{2}+\frac{d x}{2}\right )\right )}{d}-\frac{a \log \left (\cosh \left (\frac{c}{2}+\frac{d x}{2}\right )\right )}{d}+\frac{b (-c-d x)}{2 d}+\frac{b \sinh (2 (c+d x))}{4 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Csch[c + d*x]*(a + b*Sinh[c + d*x]^3),x]

[Out]

(b*(-c - d*x))/(2*d) - (a*Log[Cosh[c/2 + (d*x)/2]])/d + (a*Log[Sinh[c/2 + (d*x)/2]])/d + (b*Sinh[2*(c + d*x)])
/(4*d)

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Maple [A]  time = 0.036, size = 40, normalized size = 1. \begin{align*}{\frac{1}{d} \left ( -2\,a{\it Artanh} \left ({{\rm e}^{dx+c}} \right ) +b \left ({\frac{\cosh \left ( dx+c \right ) \sinh \left ( dx+c \right ) }{2}}-{\frac{dx}{2}}-{\frac{c}{2}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csch(d*x+c)*(a+b*sinh(d*x+c)^3),x)

[Out]

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

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Maxima [A]  time = 1.1445, size = 68, normalized size = 1.7 \begin{align*} -\frac{1}{8} \, b{\left (4 \, x - \frac{e^{\left (2 \, d x + 2 \, c\right )}}{d} + \frac{e^{\left (-2 \, d x - 2 \, 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)^3),x, algorithm="maxima")

[Out]

-1/8*b*(4*x - e^(2*d*x + 2*c)/d + e^(-2*d*x - 2*c)/d) + a*log(tanh(1/2*d*x + 1/2*c))/d

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Fricas [B]  time = 1.99806, size = 721, normalized size = 18.02 \begin{align*} -\frac{4 \, b d x \cosh \left (d x + c\right )^{2} - b \cosh \left (d x + c\right )^{4} - 4 \, b \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} - b \sinh \left (d x + c\right )^{4} + 2 \,{\left (2 \, b d x - 3 \, b \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )^{2} + 8 \,{\left (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}\right )} \log \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) + 1\right ) - 8 \,{\left (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}\right )} \log \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) - 1\right ) + 4 \,{\left (2 \, b d x \cosh \left (d x + c\right ) - b \cosh \left (d x + c\right )^{3}\right )} \sinh \left (d x + c\right ) + b}{8 \,{\left (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}\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)^3),x, algorithm="fricas")

[Out]

-1/8*(4*b*d*x*cosh(d*x + c)^2 - b*cosh(d*x + c)^4 - 4*b*cosh(d*x + c)*sinh(d*x + c)^3 - b*sinh(d*x + c)^4 + 2*
(2*b*d*x - 3*b*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 8*(a*cosh(d*x + c)^2 + 2*a*cosh(d*x + c)*sinh(d*x + c) + a*s
inh(d*x + c)^2)*log(cosh(d*x + c) + sinh(d*x + c) + 1) - 8*(a*cosh(d*x + c)^2 + 2*a*cosh(d*x + c)*sinh(d*x + c
) + a*sinh(d*x + c)^2)*log(cosh(d*x + c) + sinh(d*x + c) - 1) + 4*(2*b*d*x*cosh(d*x + c) - b*cosh(d*x + c)^3)*
sinh(d*x + c) + b)/(d*cosh(d*x + c)^2 + 2*d*cosh(d*x + c)*sinh(d*x + c) + d*sinh(d*x + c)^2)

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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)**3),x)

[Out]

Timed out

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Giac [A]  time = 1.13918, size = 97, normalized size = 2.42 \begin{align*} -\frac{{\left (d x + c\right )} b}{2 \, d} + \frac{b e^{\left (2 \, d x + 2 \, c\right )}}{8 \, d} - \frac{b e^{\left (-2 \, d x - 2 \, c\right )}}{8 \, 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)^3),x, algorithm="giac")

[Out]

-1/2*(d*x + c)*b/d + 1/8*b*e^(2*d*x + 2*c)/d - 1/8*b*e^(-2*d*x - 2*c)/d - a*log(e^(d*x + c) + 1)/d + a*log(abs
(e^(d*x + c) - 1))/d