3.147 \(\int \text{csch}^2(c+d x) (a+b \sinh ^3(c+d x)) \, dx\)

Optimal. Leaf size=24 \[ \frac{b \cosh (c+d x)}{d}-\frac{a \coth (c+d x)}{d} \]

[Out]

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

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Rubi [A]  time = 0.0500271, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {3220, 3767, 8, 2638} \[ \frac{b \cosh (c+d x)}{d}-\frac{a \coth (c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

Int[Csch[c + d*x]^2*(a + b*Sinh[c + d*x]^3),x]

[Out]

(b*Cosh[c + d*x])/d - (a*Coth[c + d*x])/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 3767

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rule 8

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

Rule 2638

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

Rubi steps

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

Mathematica [A]  time = 0.0291625, size = 35, normalized size = 1.46 \[ -\frac{a \coth (c+d x)}{d}+\frac{b \sinh (c) \sinh (d x)}{d}+\frac{b \cosh (c) \cosh (d x)}{d} \]

Antiderivative was successfully verified.

[In]

Integrate[Csch[c + d*x]^2*(a + b*Sinh[c + d*x]^3),x]

[Out]

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

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Maple [A]  time = 0.036, size = 23, normalized size = 1. \begin{align*}{\frac{-{\rm coth} \left (dx+c\right )a+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)^2*(a+b*sinh(d*x+c)^3),x)

[Out]

1/d*(-coth(d*x+c)*a+b*cosh(d*x+c))

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Maxima [A]  time = 1.13484, size = 63, normalized size = 2.62 \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{2 \, a}{d{\left (e^{\left (-2 \, d x - 2 \, c\right )} - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(d*x+c)^2*(a+b*sinh(d*x+c)^3),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.95052, size = 103, normalized size = 4.29 \begin{align*} -\frac{a \cosh \left (d x + c\right ) -{\left (b \cosh \left (d x + c\right ) + a\right )} \sinh \left (d x + c\right )}{d \sinh \left (d x + c\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(d*x+c)^2*(a+b*sinh(d*x+c)^3),x, algorithm="fricas")

[Out]

-(a*cosh(d*x + c) - (b*cosh(d*x + c) + a)*sinh(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)**2*(a+b*sinh(d*x+c)**3),x)

[Out]

Timed out

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Giac [B]  time = 1.20409, size = 84, normalized size = 3.5 \begin{align*} \frac{b e^{\left (d x + c\right )}}{2 \, d} + \frac{b e^{\left (2 \, d x + 2 \, c\right )} - 4 \, a e^{\left (d x + c\right )} - b}{2 \, d{\left (e^{\left (3 \, d x + 3 \, 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)^2*(a+b*sinh(d*x+c)^3),x, algorithm="giac")

[Out]

1/2*b*e^(d*x + c)/d + 1/2*(b*e^(2*d*x + 2*c) - 4*a*e^(d*x + c) - b)/(d*(e^(3*d*x + 3*c) - e^(d*x + c)))