3.7.47 \(\int (a \coth (x)+b \text {csch}(x))^2 \, dx\) [647]

Optimal. Leaf size=27 \[ a^2 x-(b+a \cosh (x)) (a+b \cosh (x)) \text {csch}(x)+a b \sinh (x) \]

[Out]

a^2*x-(b+a*cosh(x))*(a+b*cosh(x))*csch(x)+a*b*sinh(x)

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Rubi [A]
time = 0.05, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {4477, 2770, 2717} \begin {gather*} a^2 x+a b \sinh (x)-\text {csch}(x) (a \cosh (x)+b) (a+b \cosh (x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a*Coth[x] + b*Csch[x])^2,x]

[Out]

a^2*x - (b + a*Cosh[x])*(a + b*Cosh[x])*Csch[x] + a*b*Sinh[x]

Rule 2717

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 2770

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[(-(g*C
os[e + f*x])^(p + 1))*(a + b*Sin[e + f*x])^(m - 1)*((b + a*Sin[e + f*x])/(f*g*(p + 1))), x] + Dist[1/(g^2*(p +
1)), Int[(g*Cos[e + f*x])^(p + 2)*(a + b*Sin[e + f*x])^(m - 2)*(b^2*(m - 1) + a^2*(p + 2) + a*b*(m + p + 1)*S
in[e + f*x]), x], x] /; FreeQ[{a, b, e, f, g}, x] && NeQ[a^2 - b^2, 0] && GtQ[m, 1] && LtQ[p, -1] && (Integers
Q[2*m, 2*p] || IntegerQ[m])

Rule 4477

Int[(cot[(c_.) + (d_.)*(x_)]^(n_.)*(a_.) + csc[(c_.) + (d_.)*(x_)]^(n_.)*(b_.))^(p_)*(u_.), x_Symbol] :> Int[A
ctivateTrig[u]*Csc[c + d*x]^(n*p)*(b + a*Cos[c + d*x]^n)^p, x] /; FreeQ[{a, b, c, d}, x] && IntegersQ[n, p]

Rubi steps

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

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Mathematica [A]
time = 0.09, size = 23, normalized size = 0.85 \begin {gather*} -\left (\left (a^2+b^2\right ) \coth (x)\right )+a (a x-2 b \text {csch}(x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a*Coth[x] + b*Csch[x])^2,x]

[Out]

-((a^2 + b^2)*Coth[x]) + a*(a*x - 2*b*Csch[x])

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Maple [A]
time = 0.61, size = 30, normalized size = 1.11

method	result	size
risch	\(a^{2} x -\frac {2 \left (2 b \,{\mathrm e}^{x} a +a^{2}+b^{2}\right )}{{\mathrm e}^{2 x}-1}\)	\(30\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*coth(x)+b*csch(x))^2,x,method=_RETURNVERBOSE)

[Out]

a^2*x-2*(2*b*exp(x)*a+a^2+b^2)/(exp(2*x)-1)

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Maxima [A]
time = 0.25, size = 45, normalized size = 1.67 \begin {gather*} a^{2} {\left (x + \frac {2}{e^{\left (-2 \, x\right )} - 1}\right )} + \frac {4 \, a b}{e^{\left (-x\right )} - e^{x}} + \frac {2 \, b^{2}}{e^{\left (-2 \, x\right )} - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.35, size = 37, normalized size = 1.37 \begin {gather*} -\frac {2 \, a b + {\left (a^{2} + b^{2}\right )} \cosh \left (x\right ) - {\left (a^{2} x + a^{2} + b^{2}\right )} \sinh \left (x\right )}{\sinh \left (x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(2*a*b + (a^2 + b^2)*cosh(x) - (a^2*x + a^2 + b^2)*sinh(x))/sinh(x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*coth(x)+b*csch(x))**2,x)

[Out]

Integral((a*coth(x) + b*csch(x))**2, x)

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Giac [A]
time = 0.40, size = 29, normalized size = 1.07 \begin {gather*} a^{2} x - \frac {2 \, {\left (2 \, a b e^{x} + a^{2} + b^{2}\right )}}{e^{\left (2 \, x\right )} - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 1.61, size = 33, normalized size = 1.22 \begin {gather*} a^2\,x-\frac {2\,a^2+4\,{\mathrm {e}}^x\,a\,b+2\,b^2}{{\mathrm {e}}^{2\,x}-1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b/sinh(x) + a*coth(x))^2,x)

[Out]

a^2*x - (2*a^2 + 2*b^2 + 4*a*b*exp(x))/(exp(2*x) - 1)

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