3.204 \(\int \frac{1}{x \coth ^{\frac{5}{2}}(a+b \log (c x^n))} \, dx\)
Optimal. Leaf size=72 \[ -\frac{2}{3 b n \coth ^{\frac{3}{2}}\left (a+b \log \left (c x^n\right )\right )}+\frac{\tan ^{-1}\left (\sqrt{\coth \left (a+b \log \left (c x^n\right )\right )}\right )}{b n}+\frac{\tanh ^{-1}\left (\sqrt{\coth \left (a+b \log \left (c x^n\right )\right )}\right )}{b n} \]
[Out]
ArcTan[Sqrt[Coth[a + b*Log[c*x^n]]]]/(b*n) + ArcTanh[Sqrt[Coth[a + b*Log[c*x^n]]]]/(b*n) - 2/(3*b*n*Coth[a + b
*Log[c*x^n]]^(3/2))
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Rubi [A] time = 0.0510296, antiderivative size = 72, normalized size of antiderivative = 1.,
number of steps used = 7, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316, Rules used =
{3474, 3476, 329, 212, 206, 203} \[ -\frac{2}{3 b n \coth ^{\frac{3}{2}}\left (a+b \log \left (c x^n\right )\right )}+\frac{\tan ^{-1}\left (\sqrt{\coth \left (a+b \log \left (c x^n\right )\right )}\right )}{b n}+\frac{\tanh ^{-1}\left (\sqrt{\coth \left (a+b \log \left (c x^n\right )\right )}\right )}{b n} \]
Antiderivative was successfully verified.
[In]
Int[1/(x*Coth[a + b*Log[c*x^n]]^(5/2)),x]
[Out]
ArcTan[Sqrt[Coth[a + b*Log[c*x^n]]]]/(b*n) + ArcTanh[Sqrt[Coth[a + b*Log[c*x^n]]]]/(b*n) - 2/(3*b*n*Coth[a + b
*Log[c*x^n]]^(3/2))
Rule 3474
Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*Tan[c + d*x])^(n + 1)/(b*d*(n + 1)), x] - Dist[
1/b^2, Int[(b*Tan[c + d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[n, -1]
Rule 3476
Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Dist[b/d, Subst[Int[x^n/(b^2 + x^2), x], x, b*Tan[c + d
*x]], x] /; FreeQ[{b, c, d, n}, x] && !IntegerQ[n]
Rule 329
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + (b*x^(k*n))/c^n)^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]
Rule 212
Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-(a/b), 2]], s = Denominator[Rt[-(a/b), 2]
]}, Dist[r/(2*a), Int[1/(r - s*x^2), x], x] + Dist[r/(2*a), Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] &&
!GtQ[a/b, 0]
Rule 206
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])
Rule 203
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])
Rubi steps
\begin{align*} \int \frac{1}{x \coth ^{\frac{5}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{\coth ^{\frac{5}{2}}(a+b x)} \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=-\frac{2}{3 b n \coth ^{\frac{3}{2}}\left (a+b \log \left (c x^n\right )\right )}+\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt{\coth (a+b x)}} \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=-\frac{2}{3 b n \coth ^{\frac{3}{2}}\left (a+b \log \left (c x^n\right )\right )}-\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt{x} \left (-1+x^2\right )} \, dx,x,\coth \left (a+b \log \left (c x^n\right )\right )\right )}{b n}\\ &=-\frac{2}{3 b n \coth ^{\frac{3}{2}}\left (a+b \log \left (c x^n\right )\right )}-\frac{2 \operatorname{Subst}\left (\int \frac{1}{-1+x^4} \, dx,x,\sqrt{\coth \left (a+b \log \left (c x^n\right )\right )}\right )}{b n}\\ &=-\frac{2}{3 b n \coth ^{\frac{3}{2}}\left (a+b \log \left (c x^n\right )\right )}+\frac{\operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\sqrt{\coth \left (a+b \log \left (c x^n\right )\right )}\right )}{b n}+\frac{\operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\sqrt{\coth \left (a+b \log \left (c x^n\right )\right )}\right )}{b n}\\ &=\frac{\tan ^{-1}\left (\sqrt{\coth \left (a+b \log \left (c x^n\right )\right )}\right )}{b n}+\frac{\tanh ^{-1}\left (\sqrt{\coth \left (a+b \log \left (c x^n\right )\right )}\right )}{b n}-\frac{2}{3 b n \coth ^{\frac{3}{2}}\left (a+b \log \left (c x^n\right )\right )}\\ \end{align*}
Mathematica [C] time = 0.200434, size = 46, normalized size = 0.64 \[ -\frac{2 \, _2F_1\left (-\frac{3}{4},1;\frac{1}{4};\coth ^2\left (a+b \log \left (c x^n\right )\right )\right )}{3 b n \coth ^{\frac{3}{2}}\left (a+b \log \left (c x^n\right )\right )} \]
Antiderivative was successfully verified.
[In]
Integrate[1/(x*Coth[a + b*Log[c*x^n]]^(5/2)),x]
[Out]
(-2*Hypergeometric2F1[-3/4, 1, 1/4, Coth[a + b*Log[c*x^n]]^2])/(3*b*n*Coth[a + b*Log[c*x^n]]^(3/2))
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Maple [A] time = 0.013, size = 92, normalized size = 1.3 \begin{align*}{\frac{1}{2\,bn}\ln \left ( \sqrt{{\rm coth} \left (a+b\ln \left ( c{x}^{n} \right ) \right )}+1 \right ) }-{\frac{2}{3\,bn} \left ({\rm coth} \left (a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) ^{-{\frac{3}{2}}}}-{\frac{1}{2\,bn}\ln \left ( \sqrt{{\rm coth} \left (a+b\ln \left ( c{x}^{n} \right ) \right )}-1 \right ) }+{\frac{1}{bn}\arctan \left ( \sqrt{{\rm coth} \left (a+b\ln \left ( c{x}^{n} \right ) \right )} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/x/coth(a+b*ln(c*x^n))^(5/2),x)
[Out]
1/2/b/n*ln(coth(a+b*ln(c*x^n))^(1/2)+1)-2/3/b/n/coth(a+b*ln(c*x^n))^(3/2)-1/2/b/n*ln(coth(a+b*ln(c*x^n))^(1/2)
-1)+arctan(coth(a+b*ln(c*x^n))^(1/2))/b/n
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x \coth \left (b \log \left (c x^{n}\right ) + a\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x/coth(a+b*log(c*x^n))^(5/2),x, algorithm="maxima")
[Out]
integrate(1/(x*coth(b*log(c*x^n) + a)^(5/2)), x)
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Fricas [B] time = 2.88856, size = 3644, normalized size = 50.61 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x/coth(a+b*log(c*x^n))^(5/2),x, algorithm="fricas")
[Out]
-1/6*(4*cosh(b*n*log(x) + b*log(c) + a)^4 + 16*cosh(b*n*log(x) + b*log(c) + a)*sinh(b*n*log(x) + b*log(c) + a)
^3 + 4*sinh(b*n*log(x) + b*log(c) + a)^4 + 8*(3*cosh(b*n*log(x) + b*log(c) + a)^2 + 1)*sinh(b*n*log(x) + b*log
(c) + a)^2 + 6*(cosh(b*n*log(x) + b*log(c) + a)^4 + 4*cosh(b*n*log(x) + b*log(c) + a)*sinh(b*n*log(x) + b*log(
c) + a)^3 + sinh(b*n*log(x) + b*log(c) + a)^4 + 2*(3*cosh(b*n*log(x) + b*log(c) + a)^2 + 1)*sinh(b*n*log(x) +
b*log(c) + a)^2 + 2*cosh(b*n*log(x) + b*log(c) + a)^2 + 4*(cosh(b*n*log(x) + b*log(c) + a)^3 + cosh(b*n*log(x)
+ b*log(c) + a))*sinh(b*n*log(x) + b*log(c) + a) + 1)*arctan(-cosh(b*n*log(x) + b*log(c) + a)^2 - 2*cosh(b*n*
log(x) + b*log(c) + a)*sinh(b*n*log(x) + b*log(c) + a) - sinh(b*n*log(x) + b*log(c) + a)^2 + (cosh(b*n*log(x)
+ b*log(c) + a)^2 + 2*cosh(b*n*log(x) + b*log(c) + a)*sinh(b*n*log(x) + b*log(c) + a) + sinh(b*n*log(x) + b*lo
g(c) + a)^2 - 1)*sqrt(cosh(b*n*log(x) + b*log(c) + a)/sinh(b*n*log(x) + b*log(c) + a))) + 8*cosh(b*n*log(x) +
b*log(c) + a)^2 + 3*(cosh(b*n*log(x) + b*log(c) + a)^4 + 4*cosh(b*n*log(x) + b*log(c) + a)*sinh(b*n*log(x) + b
*log(c) + a)^3 + sinh(b*n*log(x) + b*log(c) + a)^4 + 2*(3*cosh(b*n*log(x) + b*log(c) + a)^2 + 1)*sinh(b*n*log(
x) + b*log(c) + a)^2 + 2*cosh(b*n*log(x) + b*log(c) + a)^2 + 4*(cosh(b*n*log(x) + b*log(c) + a)^3 + cosh(b*n*l
og(x) + b*log(c) + a))*sinh(b*n*log(x) + b*log(c) + a) + 1)*log(-cosh(b*n*log(x) + b*log(c) + a)^2 - 2*cosh(b*
n*log(x) + b*log(c) + a)*sinh(b*n*log(x) + b*log(c) + a) - sinh(b*n*log(x) + b*log(c) + a)^2 + (cosh(b*n*log(x
) + b*log(c) + a)^2 + 2*cosh(b*n*log(x) + b*log(c) + a)*sinh(b*n*log(x) + b*log(c) + a) + sinh(b*n*log(x) + b*
log(c) + a)^2 - 1)*sqrt(cosh(b*n*log(x) + b*log(c) + a)/sinh(b*n*log(x) + b*log(c) + a))) + 16*(cosh(b*n*log(x
) + b*log(c) + a)^3 + cosh(b*n*log(x) + b*log(c) + a))*sinh(b*n*log(x) + b*log(c) + a) + 4*(cosh(b*n*log(x) +
b*log(c) + a)^4 + 4*cosh(b*n*log(x) + b*log(c) + a)*sinh(b*n*log(x) + b*log(c) + a)^3 + sinh(b*n*log(x) + b*lo
g(c) + a)^4 + 2*(3*cosh(b*n*log(x) + b*log(c) + a)^2 - 1)*sinh(b*n*log(x) + b*log(c) + a)^2 - 2*cosh(b*n*log(x
) + b*log(c) + a)^2 + 4*(cosh(b*n*log(x) + b*log(c) + a)^3 - cosh(b*n*log(x) + b*log(c) + a))*sinh(b*n*log(x)
+ b*log(c) + a) + 1)*sqrt(cosh(b*n*log(x) + b*log(c) + a)/sinh(b*n*log(x) + b*log(c) + a)) + 4)/(b*n*cosh(b*n*
log(x) + b*log(c) + a)^4 + 4*b*n*cosh(b*n*log(x) + b*log(c) + a)*sinh(b*n*log(x) + b*log(c) + a)^3 + b*n*sinh(
b*n*log(x) + b*log(c) + a)^4 + 2*b*n*cosh(b*n*log(x) + b*log(c) + a)^2 + 2*(3*b*n*cosh(b*n*log(x) + b*log(c) +
a)^2 + b*n)*sinh(b*n*log(x) + b*log(c) + a)^2 + b*n + 4*(b*n*cosh(b*n*log(x) + b*log(c) + a)^3 + b*n*cosh(b*n
*log(x) + b*log(c) + a))*sinh(b*n*log(x) + b*log(c) + a))
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x/coth(a+b*ln(c*x**n))**(5/2),x)
[Out]
Timed out
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x \coth \left (b \log \left (c x^{n}\right ) + a\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x/coth(a+b*log(c*x^n))^(5/2),x, algorithm="giac")
[Out]
integrate(1/(x*coth(b*log(c*x^n) + a)^(5/2)), x)