3.11.21 \(\int \text {csch}^2(x) (-1+\sinh ^2(x)) \, dx\) [1021]

Optimal. Leaf size=4 \[ x+\coth (x) \]

[Out]

x+coth(x)

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Rubi [A]
time = 0.01, antiderivative size = 4, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {3091, 8} \begin {gather*} x+\coth (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Csch[x]^2*(-1 + Sinh[x]^2),x]

[Out]

x + Coth[x]

Rule 8

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

Rule 3091

Int[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_) + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[A*Cos[e +
f*x]*((b*Sin[e + f*x])^(m + 1)/(b*f*(m + 1))), x] + Dist[(A*(m + 2) + C*(m + 1))/(b^2*(m + 1)), Int[(b*Sin[e
+ f*x])^(m + 2), x], x] /; FreeQ[{b, e, f, A, C}, x] && LtQ[m, -1]

Rubi steps

\begin {align*} \int \text {csch}^2(x) \left (-1+\sinh ^2(x)\right ) \, dx &=\coth (x)+\int 1 \, dx\\ &=x+\coth (x)\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 4, normalized size = 1.00 \begin {gather*} x+\coth (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Csch[x]^2*(-1 + Sinh[x]^2),x]

[Out]

x + Coth[x]

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(12\) vs. \(2(4)=8\).
time = 0.69, size = 13, normalized size = 3.25

method	result	size
risch	\(x +\frac {2}{{\mathrm e}^{2 x}-1}\)	\(13\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csch(x)^2*(-1+sinh(x)^2),x,method=_RETURNVERBOSE)

[Out]

x+2/(exp(2*x)-1)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 12 vs. \(2 (4) = 8\).
time = 0.26, size = 12, normalized size = 3.00 \begin {gather*} x - \frac {2}{e^{\left (-2 \, x\right )} - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(x)^2*(-1+sinh(x)^2),x, algorithm="maxima")

[Out]

x - 2/(e^(-2*x) - 1)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 14 vs. \(2 (4) = 8\).
time = 0.34, size = 14, normalized size = 3.50 \begin {gather*} \frac {{\left (x - 1\right )} \sinh \left (x\right ) + \cosh \left (x\right )}{\sinh \left (x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(x)^2*(-1+sinh(x)^2),x, algorithm="fricas")

[Out]

((x - 1)*sinh(x) + cosh(x))/sinh(x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (\sinh {\left (x \right )} - 1\right ) \left (\sinh {\left (x \right )} + 1\right ) \operatorname {csch}^{2}{\left (x \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(x)**2*(-1+sinh(x)**2),x)

[Out]

Integral((sinh(x) - 1)*(sinh(x) + 1)*csch(x)**2, x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 12 vs. \(2 (4) = 8\).
time = 0.41, size = 12, normalized size = 3.00 \begin {gather*} x + \frac {2}{e^{\left (2 \, x\right )} - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(x)^2*(-1+sinh(x)^2),x, algorithm="giac")

[Out]

x + 2/(e^(2*x) - 1)

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Mupad [B]
time = 1.72, size = 12, normalized size = 3.00 \begin {gather*} x+\frac {2}{{\mathrm {e}}^{2\,x}-1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((sinh(x)^2 - 1)/sinh(x)^2,x)

[Out]

x + 2/(exp(2*x) - 1)

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