3.145 \(\int \frac {x^6}{\text {csch}^{\frac {3}{2}}(2 \log (c x))} \, dx\)

Optimal. Leaf size=30 \[ \frac {x^7 \left (c^4-\frac {1}{x^4}\right )}{10 c^4 \text {csch}^{\frac {3}{2}}(2 \log (c x))} \]

[Out]

1/10*(c^4-1/x^4)*x^7/c^4/csch(2*ln(c*x))^(3/2)

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Rubi [A]  time = 0.04, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {5552, 5550, 264} \[ \frac {x^7 \left (c^4-\frac {1}{x^4}\right )}{10 c^4 \text {csch}^{\frac {3}{2}}(2 \log (c x))} \]

Antiderivative was successfully verified.

[In]

Int[x^6/Csch[2*Log[c*x]]^(3/2),x]

[Out]

((c^4 - x^(-4))*x^7)/(10*c^4*Csch[2*Log[c*x]]^(3/2))

Rule 264

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

Rule 5550

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

Rule 5552

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

Rubi steps

\begin {align*} \int \frac {x^6}{\text {csch}^{\frac {3}{2}}(2 \log (c x))} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x^6}{\text {csch}^{\frac {3}{2}}(2 \log (x))} \, dx,x,c x\right )}{c^7}\\ &=\frac {\operatorname {Subst}\left (\int \left (1-\frac {1}{x^4}\right )^{3/2} x^9 \, dx,x,c x\right )}{c^{10} \left (1-\frac {1}{c^4 x^4}\right )^{3/2} x^3 \text {csch}^{\frac {3}{2}}(2 \log (c x))}\\ &=\frac {\left (c^4-\frac {1}{x^4}\right ) x^7}{10 c^4 \text {csch}^{\frac {3}{2}}(2 \log (c x))}\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 44, normalized size = 1.47 \[ \frac {\left (c^4 x^4-1\right )^3 \sqrt {\frac {c^2 x^2}{2 c^4 x^4-2}}}{20 c^8 x} \]

Antiderivative was successfully verified.

[In]

Integrate[x^6/Csch[2*Log[c*x]]^(3/2),x]

[Out]

((-1 + c^4*x^4)^3*Sqrt[(c^2*x^2)/(-2 + 2*c^4*x^4)])/(20*c^8*x)

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fricas [B]  time = 1.03, size = 56, normalized size = 1.87 \[ \frac {\sqrt {2} {\left (c^{12} x^{12} - 3 \, c^{8} x^{8} + 3 \, c^{4} x^{4} - 1\right )} \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4} - 1}}}{40 \, c^{8} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6/csch(2*log(c*x))^(3/2),x, algorithm="fricas")

[Out]

1/40*sqrt(2)*(c^12*x^12 - 3*c^8*x^8 + 3*c^4*x^4 - 1)*sqrt(c^2*x^2/(c^4*x^4 - 1))/(c^8*x)

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6/csch(2*log(c*x))^(3/2),x, algorithm="giac")

[Out]

Exception raised: RuntimeError >> An error occurred running a Giac command:INPUT:sage2OUTPUT:Warning, integrat
ion of abs or sign assumes constant sign by intervals (correct if the argument is real):Check [abs(t_nostep)]W
arning, integration of abs or sign assumes constant sign by intervals (correct if the argument is real):Check
[abs(t_nostep)]Unable to divide, perhaps due to rounding error%%%{1,[8,4,1,0]%%%}+%%%{-1,[4,0,1,0]%%%} / %%%{1
,[0,2,0,1]%%%} Error: Bad Argument Value

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maple [A]  time = 0.14, size = 47, normalized size = 1.57 \[ \frac {\sqrt {2}\, x \left (c^{8} x^{8}-2 c^{4} x^{4}+1\right )}{40 c^{6} \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4}-1}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6/csch(2*ln(c*x))^(3/2),x)

[Out]

1/40*2^(1/2)/c^6*x/(c^2*x^2/(c^4*x^4-1))^(1/2)*(c^8*x^8-2*c^4*x^4+1)

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maxima [A]  time = 0.47, size = 46, normalized size = 1.53 \[ \frac {{\left (\sqrt {2} c^{4} x^{4} - \sqrt {2}\right )} {\left (c^{2} x^{2} + 1\right )}^{\frac {3}{2}} {\left (c x + 1\right )}^{\frac {3}{2}} {\left (c x - 1\right )}^{\frac {3}{2}}}{40 \, c^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6/csch(2*log(c*x))^(3/2),x, algorithm="maxima")

[Out]

1/40*(sqrt(2)*c^4*x^4 - sqrt(2))*(c^2*x^2 + 1)^(3/2)*(c*x + 1)^(3/2)*(c*x - 1)^(3/2)/c^7

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mupad [B]  time = 1.56, size = 42, normalized size = 1.40 \[ \frac {{\left (c^4\,x^4-1\right )}^3\,\sqrt {\frac {2\,c^2\,x^2}{c^4\,x^4-1}}}{40\,c^8\,x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6/(1/sinh(2*log(c*x)))^(3/2),x)

[Out]

((c^4*x^4 - 1)^3*((2*c^2*x^2)/(c^4*x^4 - 1))^(1/2))/(40*c^8*x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{6}}{\operatorname {csch}^{\frac {3}{2}}{\left (2 \log {\left (c x \right )} \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**6/csch(2*ln(c*x))**(3/2),x)

[Out]

Integral(x**6/csch(2*log(c*x))**(3/2), x)

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