3.160 \(\int (-(1-b^2 n^2) \text{csch}(a+b \log (c x^n))+2 b^2 n^2 \text{csch}^3(a+b \log (c x^n))) \, dx\)

Optimal. Leaf size=42 \[ -x \text{csch}\left (a+b \log \left (c x^n\right )\right )-b n x \coth \left (a+b \log \left (c x^n\right )\right ) \text{csch}\left (a+b \log \left (c x^n\right )\right ) \]

[Out]

-(x*Csch[a + b*Log[c*x^n]]) - b*n*x*Coth[a + b*Log[c*x^n]]*Csch[a + b*Log[c*x^n]]

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Rubi [C]  time = 0.135159, antiderivative size = 137, normalized size of antiderivative = 3.26, number of steps used = 9, number of rules used = 4, integrand size = 45, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.089, Rules used = {5546, 5548, 263, 364} \[ 2 e^a x (1-b n) \left (c x^n\right )^b \, _2F_1\left (1,\frac{b+\frac{1}{n}}{2 b};\frac{1}{2} \left (3+\frac{1}{b n}\right );e^{2 a} \left (c x^n\right )^{2 b}\right )-\frac{16 e^{3 a} b^2 n^2 x \left (c x^n\right )^{3 b} \, _2F_1\left (3,\frac{3 b+\frac{1}{n}}{2 b};\frac{1}{2} \left (5+\frac{1}{b n}\right );e^{2 a} \left (c x^n\right )^{2 b}\right )}{3 b n+1} \]

Warning: Unable to verify antiderivative.

[In]

Int[-((1 - b^2*n^2)*Csch[a + b*Log[c*x^n]]) + 2*b^2*n^2*Csch[a + b*Log[c*x^n]]^3,x]

[Out]

2*E^a*(1 - b*n)*x*(c*x^n)^b*Hypergeometric2F1[1, (b + n^(-1))/(2*b), (3 + 1/(b*n))/2, E^(2*a)*(c*x^n)^(2*b)] -
(16*b^2*E^(3*a)*n^2*x*(c*x^n)^(3*b)*Hypergeometric2F1[3, (3*b + n^(-1))/(2*b), (5 + 1/(b*n))/2, E^(2*a)*(c*x^
n)^(2*b)])/(1 + 3*b*n)

Rule 5546

Int[Csch[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)]^(p_.), x_Symbol] :> Dist[x/(n*(c*x^n)^(1/n)), Subst[Int[
x^(1/n - 1)*Csch[d*(a + b*Log[x])]^p, x], x, c*x^n], x] /; FreeQ[{a, b, c, d, n, p}, x] && (NeQ[c, 1] || NeQ[n
, 1])

Rule 5548

Int[Csch[((a_.) + Log[x_]*(b_.))*(d_.)]^(p_.)*((e_.)*(x_))^(m_.), x_Symbol] :> Dist[2^p/E^(a*d*p), Int[(e*x)^m
/(x^(b*d*p)*(1 - 1/(E^(2*a*d)*x^(2*b*d)))^p), x], x] /; FreeQ[{a, b, d, e, m}, x] && IntegerQ[p]

Rule 263

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[x^(m + n*p)*(b + a/x^n)^p, x] /; FreeQ[{a, b, m
, n}, x] && IntegerQ[p] && NegQ[n]

Rule 364

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a^p*(c*x)^(m + 1)*Hypergeometric2F1[-
p, (m + 1)/n, (m + 1)/n + 1, -((b*x^n)/a)])/(c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rubi steps

\begin{align*} \int \left (-\left (1-b^2 n^2\right ) \text{csch}\left (a+b \log \left (c x^n\right )\right )+2 b^2 n^2 \text{csch}^3\left (a+b \log \left (c x^n\right )\right )\right ) \, dx &=\left (2 b^2 n^2\right ) \int \text{csch}^3\left (a+b \log \left (c x^n\right )\right ) \, dx+\left (-1+b^2 n^2\right ) \int \text{csch}\left (a+b \log \left (c x^n\right )\right ) \, dx\\ &=\left (2 b^2 n x \left (c x^n\right )^{-1/n}\right ) \operatorname{Subst}\left (\int x^{-1+\frac{1}{n}} \text{csch}^3(a+b \log (x)) \, dx,x,c x^n\right )+\frac{\left (\left (-1+b^2 n^2\right ) x \left (c x^n\right )^{-1/n}\right ) \operatorname{Subst}\left (\int x^{-1+\frac{1}{n}} \text{csch}(a+b \log (x)) \, dx,x,c x^n\right )}{n}\\ &=\left (16 b^2 e^{-3 a} n x \left (c x^n\right )^{-1/n}\right ) \operatorname{Subst}\left (\int \frac{x^{-1-3 b+\frac{1}{n}}}{\left (1-e^{-2 a} x^{-2 b}\right )^3} \, dx,x,c x^n\right )+\frac{\left (2 e^{-a} \left (-1+b^2 n^2\right ) x \left (c x^n\right )^{-1/n}\right ) \operatorname{Subst}\left (\int \frac{x^{-1-b+\frac{1}{n}}}{1-e^{-2 a} x^{-2 b}} \, dx,x,c x^n\right )}{n}\\ &=\left (16 b^2 e^{-3 a} n x \left (c x^n\right )^{-1/n}\right ) \operatorname{Subst}\left (\int \frac{x^{-1+3 b+\frac{1}{n}}}{\left (-e^{-2 a}+x^{2 b}\right )^3} \, dx,x,c x^n\right )+\frac{\left (2 e^{-a} \left (-1+b^2 n^2\right ) x \left (c x^n\right )^{-1/n}\right ) \operatorname{Subst}\left (\int \frac{x^{-1+b+\frac{1}{n}}}{-e^{-2 a}+x^{2 b}} \, dx,x,c x^n\right )}{n}\\ &=2 e^a (1-b n) x \left (c x^n\right )^b \, _2F_1\left (1,\frac{b+\frac{1}{n}}{2 b};\frac{1}{2} \left (3+\frac{1}{b n}\right );e^{2 a} \left (c x^n\right )^{2 b}\right )-\frac{16 b^2 e^{3 a} n^2 x \left (c x^n\right )^{3 b} \, _2F_1\left (3,\frac{3 b+\frac{1}{n}}{2 b};\frac{1}{2} \left (5+\frac{1}{b n}\right );e^{2 a} \left (c x^n\right )^{2 b}\right )}{1+3 b n}\\ \end{align*}

Mathematica [A]  time = 0.387069, size = 30, normalized size = 0.71 \[ -x \left (b n \coth \left (a+b \log \left (c x^n\right )\right )+1\right ) \text{csch}\left (a+b \log \left (c x^n\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[-((1 - b^2*n^2)*Csch[a + b*Log[c*x^n]]) + 2*b^2*n^2*Csch[a + b*Log[c*x^n]]^3,x]

[Out]

-(x*(1 + b*n*Coth[a + b*Log[c*x^n]])*Csch[a + b*Log[c*x^n]])

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Maple [C]  time = 0.237, size = 509, normalized size = 12.1 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(-b^2*n^2+1)*csch(a+b*ln(c*x^n))+2*b^2*n^2*csch(a+b*ln(c*x^n))^3,x)

[Out]

-2*c^b*(x^n)^b*x/((c^b)^2*((x^n)^b)^2*exp(2*a)*exp(-I*b*Pi*csgn(I*c*x^n)^3)*exp(I*b*Pi*csgn(I*c*x^n)^2*csgn(I*
c))*exp(I*b*Pi*csgn(I*c*x^n)^2*csgn(I*x^n))*exp(-I*b*Pi*csgn(I*c*x^n)*csgn(I*c)*csgn(I*x^n))-1)^2*((c^b)^2*((x
^n)^b)^2*b*n*exp(3*a)*exp(-3/2*I*b*Pi*csgn(I*c*x^n)^3)*exp(3/2*I*b*Pi*csgn(I*c*x^n)^2*csgn(I*c))*exp(3/2*I*b*P
i*csgn(I*c*x^n)^2*csgn(I*x^n))*exp(-3/2*I*b*Pi*csgn(I*c*x^n)*csgn(I*c)*csgn(I*x^n))+exp(a)*exp(-1/2*I*b*Pi*csg
n(I*c*x^n)^3)*exp(1/2*I*b*Pi*csgn(I*c*x^n)^2*csgn(I*c))*exp(1/2*I*b*Pi*csgn(I*c*x^n)^2*csgn(I*x^n))*exp(-1/2*I
*b*Pi*csgn(I*c*x^n)*csgn(I*c)*csgn(I*x^n))*b*n+(c^b)^2*((x^n)^b)^2*exp(3*a)*exp(-3/2*I*b*Pi*csgn(I*c*x^n)^3)*e
xp(3/2*I*b*Pi*csgn(I*c*x^n)^2*csgn(I*c))*exp(3/2*I*b*Pi*csgn(I*c*x^n)^2*csgn(I*x^n))*exp(-3/2*I*b*Pi*csgn(I*c*
x^n)*csgn(I*c)*csgn(I*x^n))-exp(a)*exp(-1/2*I*b*Pi*csgn(I*c*x^n)^3)*exp(1/2*I*b*Pi*csgn(I*c*x^n)^2*csgn(I*c))*
exp(1/2*I*b*Pi*csgn(I*c*x^n)^2*csgn(I*x^n))*exp(-1/2*I*b*Pi*csgn(I*c*x^n)*csgn(I*c)*csgn(I*x^n)))

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Maxima [B]  time = 2.18901, size = 128, normalized size = 3.05 \begin{align*} -\frac{2 \,{\left ({\left (b c^{3 \, b} n + c^{3 \, b}\right )} x e^{\left (3 \, b \log \left (x^{n}\right ) + 3 \, a\right )} +{\left (b c^{b} n - c^{b}\right )} x e^{\left (b \log \left (x^{n}\right ) + a\right )}\right )}}{c^{4 \, b} e^{\left (4 \, b \log \left (x^{n}\right ) + 4 \, a\right )} - 2 \, c^{2 \, b} e^{\left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )} + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-(-b^2*n^2+1)*csch(a+b*log(c*x^n))+2*b^2*n^2*csch(a+b*log(c*x^n))^3,x, algorithm="maxima")

[Out]

-2*((b*c^(3*b)*n + c^(3*b))*x*e^(3*b*log(x^n) + 3*a) + (b*c^b*n - c^b)*x*e^(b*log(x^n) + a))/(c^(4*b)*e^(4*b*l
og(x^n) + 4*a) - 2*c^(2*b)*e^(2*b*log(x^n) + 2*a) + 1)

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Fricas [B]  time = 1.62483, size = 602, normalized size = 14.33 \begin{align*} -\frac{2 \,{\left ({\left (b n + 1\right )} x \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + 2 \,{\left (b n + 1\right )} x \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) +{\left (b n + 1\right )} x \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} +{\left (b n - 1\right )} x\right )}}{\cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{3} + 3 \, \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{3} + 3 \,{\left (\cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} - 1\right )} \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) - \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-(-b^2*n^2+1)*csch(a+b*log(c*x^n))+2*b^2*n^2*csch(a+b*log(c*x^n))^3,x, algorithm="fricas")

[Out]

-2*((b*n + 1)*x*cosh(b*n*log(x) + b*log(c) + a)^2 + 2*(b*n + 1)*x*cosh(b*n*log(x) + b*log(c) + a)*sinh(b*n*log
(x) + b*log(c) + a) + (b*n + 1)*x*sinh(b*n*log(x) + b*log(c) + a)^2 + (b*n - 1)*x)/(cosh(b*n*log(x) + b*log(c)
+ a)^3 + 3*cosh(b*n*log(x) + b*log(c) + a)*sinh(b*n*log(x) + b*log(c) + a)^2 + sinh(b*n*log(x) + b*log(c) + a
)^3 + 3*(cosh(b*n*log(x) + b*log(c) + a)^2 - 1)*sinh(b*n*log(x) + b*log(c) + a) - cosh(b*n*log(x) + b*log(c) +
a))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (2 b^{2} n^{2} \operatorname{csch}^{2}{\left (a + b \log{\left (c x^{n} \right )} \right )} + b^{2} n^{2} - 1\right ) \operatorname{csch}{\left (a + b \log{\left (c x^{n} \right )} \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-(-b**2*n**2+1)*csch(a+b*ln(c*x**n))+2*b**2*n**2*csch(a+b*ln(c*x**n))**3,x)

[Out]

Integral((2*b**2*n**2*csch(a + b*log(c*x**n))**2 + b**2*n**2 - 1)*csch(a + b*log(c*x**n)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int 2 \, b^{2} n^{2} \operatorname{csch}\left (b \log \left (c x^{n}\right ) + a\right )^{3} +{\left (b^{2} n^{2} - 1\right )} \operatorname{csch}\left (b \log \left (c x^{n}\right ) + a\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-(-b^2*n^2+1)*csch(a+b*log(c*x^n))+2*b^2*n^2*csch(a+b*log(c*x^n))^3,x, algorithm="giac")

[Out]

integrate(2*b^2*n^2*csch(b*log(c*x^n) + a)^3 + (b^2*n^2 - 1)*csch(b*log(c*x^n) + a), x)