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*ln(c*x^n))-b*n*x*coth(a+b*ln(c*x^n))*csch(a+b*ln(c*x^n))

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Rubi [C]  time = 0.14, 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 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])

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]

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*}

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Mathematica [A]  time = 0.42, 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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fricas [B]  time = 1.58, size = 187, normalized size = 4.45 \[ -\frac {2 \, {\left ({\left (b n + 1\right )} x \cosh \left (b n \log \relax (x) + b \log \relax (c) + a\right )^{2} + 2 \, {\left (b n + 1\right )} x \cosh \left (b n \log \relax (x) + b \log \relax (c) + a\right ) \sinh \left (b n \log \relax (x) + b \log \relax (c) + a\right ) + {\left (b n + 1\right )} x \sinh \left (b n \log \relax (x) + b \log \relax (c) + a\right )^{2} + {\left (b n - 1\right )} x\right )}}{\cosh \left (b n \log \relax (x) + b \log \relax (c) + a\right )^{3} + 3 \, \cosh \left (b n \log \relax (x) + b \log \relax (c) + a\right ) \sinh \left (b n \log \relax (x) + b \log \relax (c) + a\right )^{2} + \sinh \left (b n \log \relax (x) + b \log \relax (c) + a\right )^{3} + 3 \, {\left (\cosh \left (b n \log \relax (x) + b \log \relax (c) + a\right )^{2} - 1\right )} \sinh \left (b n \log \relax (x) + b \log \relax (c) + a\right ) - \cosh \left (b n \log \relax (x) + b \log \relax (c) + a\right )} \]

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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giac [B]  time = 0.55, size = 215, normalized size = 5.12 \[ -\frac {2 \, b c^{3 \, b} n x x^{3 \, b n} e^{\left (3 \, a\right )}}{c^{4 \, b} x^{4 \, b n} e^{\left (4 \, a\right )} - 2 \, c^{2 \, b} x^{2 \, b n} e^{\left (2 \, a\right )} + 1} - \frac {2 \, b c^{b} n x x^{b n} e^{a}}{c^{4 \, b} x^{4 \, b n} e^{\left (4 \, a\right )} - 2 \, c^{2 \, b} x^{2 \, b n} e^{\left (2 \, a\right )} + 1} - \frac {2 \, c^{3 \, b} x x^{3 \, b n} e^{\left (3 \, a\right )}}{c^{4 \, b} x^{4 \, b n} e^{\left (4 \, a\right )} - 2 \, c^{2 \, b} x^{2 \, b n} e^{\left (2 \, a\right )} + 1} + \frac {2 \, c^{b} x x^{b n} e^{a}}{c^{4 \, b} x^{4 \, b n} e^{\left (4 \, a\right )} - 2 \, c^{2 \, b} x^{2 \, b n} e^{\left (2 \, a\right )} + 1} \]

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]

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

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maple [C]  time = 1.11, size = 509, normalized size = 12.12 \[ -\frac {2 c^{b} \left (x^{n}\right )^{b} x \left (n b \left (x^{n}\right )^{2 b} c^{2 b} {\mathrm e}^{3 a} {\mathrm e}^{-\frac {3 i b \mathrm {csgn}\left (i c \,x^{n}\right )^{3} \pi }{2}} {\mathrm e}^{\frac {3 i b \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \mathrm {csgn}\left (i c \right ) \pi }{2}} {\mathrm e}^{\frac {3 i b \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \mathrm {csgn}\left (i x^{n}\right ) \pi }{2}} {\mathrm e}^{-\frac {3 i b \,\mathrm {csgn}\left (i c \,x^{n}\right ) \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \pi }{2}}+{\mathrm e}^{a} {\mathrm e}^{-\frac {i b \mathrm {csgn}\left (i c \,x^{n}\right )^{3} \pi }{2}} {\mathrm e}^{\frac {i b \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \mathrm {csgn}\left (i c \right ) \pi }{2}} {\mathrm e}^{\frac {i b \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \mathrm {csgn}\left (i x^{n}\right ) \pi }{2}} {\mathrm e}^{-\frac {i b \,\mathrm {csgn}\left (i c \,x^{n}\right ) \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \pi }{2}} b n +\left (x^{n}\right )^{2 b} c^{2 b} {\mathrm e}^{3 a} {\mathrm e}^{-\frac {3 i b \mathrm {csgn}\left (i c \,x^{n}\right )^{3} \pi }{2}} {\mathrm e}^{\frac {3 i b \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \mathrm {csgn}\left (i c \right ) \pi }{2}} {\mathrm e}^{\frac {3 i b \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \mathrm {csgn}\left (i x^{n}\right ) \pi }{2}} {\mathrm e}^{-\frac {3 i b \,\mathrm {csgn}\left (i c \,x^{n}\right ) \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \pi }{2}}-{\mathrm e}^{a} {\mathrm e}^{-\frac {i b \mathrm {csgn}\left (i c \,x^{n}\right )^{3} \pi }{2}} {\mathrm e}^{\frac {i b \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \mathrm {csgn}\left (i c \right ) \pi }{2}} {\mathrm e}^{\frac {i b \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \mathrm {csgn}\left (i x^{n}\right ) \pi }{2}} {\mathrm e}^{-\frac {i b \,\mathrm {csgn}\left (i c \,x^{n}\right ) \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \pi }{2}}\right )}{\left (\left (x^{n}\right )^{2 b} c^{2 b} {\mathrm e}^{2 a} {\mathrm e}^{-i b \mathrm {csgn}\left (i c \,x^{n}\right )^{3} \pi } {\mathrm e}^{i b \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \mathrm {csgn}\left (i c \right ) \pi } {\mathrm e}^{i b \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \mathrm {csgn}\left (i x^{n}\right ) \pi } {\mathrm e}^{-i b \,\mathrm {csgn}\left (i c \,x^{n}\right ) \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \pi }-1\right )^{2}} \]

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

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maxima [B]  time = 0.71, size = 95, normalized size = 2.26 \[ -\frac {2 \, {\left ({\left (b c^{3 \, b} n + c^{3 \, b}\right )} x e^{\left (3 \, b \log \left (x^{n}\right ) + 3 \, a\right )} + {\left (b c^{b} n - c^{b}\right )} x e^{\left (b \log \left (x^{n}\right ) + a\right )}\right )}}{c^{4 \, b} e^{\left (4 \, b \log \left (x^{n}\right ) + 4 \, a\right )} - 2 \, c^{2 \, b} e^{\left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )} + 1} \]

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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mupad [B]  time = 1.52, size = 65, normalized size = 1.55 \[ -\frac {2\,x\,{\mathrm {e}}^a\,{\left (c\,x^n\right )}^b\,\left (b\,n+{\mathrm {e}}^{2\,a}\,{\left (c\,x^n\right )}^{2\,b}+b\,n\,{\mathrm {e}}^{2\,a}\,{\left (c\,x^n\right )}^{2\,b}-1\right )}{{\left ({\mathrm {e}}^{2\,a}\,{\left (c\,x^n\right )}^{2\,b}-1\right )}^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (2 b^{2} n^{2} \operatorname {csch}^{2}{\left (a + b \log {\left (c x^{n} \right )} \right )} + b^{2} n^{2} - 1\right ) \operatorname {csch}{\left (a + b \log {\left (c x^{n} \right )} \right )}\, dx \]

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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