\(\int \frac {1}{x \cot ^{\frac {3}{2}}(a+b \log (c x^n))} \, dx\) [235]

Optimal result

Integrand size = 19, antiderivative size = 199 \[ \int \frac {1}{x \cot ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}\right )}{\sqrt {2} b n}+\frac {\arctan \left (1+\sqrt {2} \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}\right )}{\sqrt {2} b n}+\frac {2}{b n \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}}+\frac {\log \left (1-\sqrt {2} \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}+\cot \left (a+b \log \left (c x^n\right )\right )\right )}{2 \sqrt {2} b n}-\frac {\log \left (1+\sqrt {2} \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}+\cot \left (a+b \log \left (c x^n\right )\right )\right )}{2 \sqrt {2} b n} \]

[Out]

1/2*arctan(-1+2^(1/2)*cot(a+b*ln(c*x^n))^(1/2))/b/n*2^(1/2)+1/2*arctan(1+2^(1/2)*cot(a+b*ln(c*x^n))^(1/2))/b/n
*2^(1/2)+1/4*ln(1+cot(a+b*ln(c*x^n))-2^(1/2)*cot(a+b*ln(c*x^n))^(1/2))/b/n*2^(1/2)-1/4*ln(1+cot(a+b*ln(c*x^n))
+2^(1/2)*cot(a+b*ln(c*x^n))^(1/2))/b/n*2^(1/2)+2/b/n/cot(a+b*ln(c*x^n))^(1/2)

Rubi [A] (verified)

Time = 0.16 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.474, Rules used = {3555, 3557, 335, 303, 1176, 631, 210, 1179, 642} \[ \int \frac {1}{x \cot ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}\right )}{\sqrt {2} b n}+\frac {\arctan \left (\sqrt {2} \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}+1\right )}{\sqrt {2} b n}+\frac {\log \left (\cot \left (a+b \log \left (c x^n\right )\right )-\sqrt {2} \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}+1\right )}{2 \sqrt {2} b n}-\frac {\log \left (\cot \left (a+b \log \left (c x^n\right )\right )+\sqrt {2} \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}+1\right )}{2 \sqrt {2} b n}+\frac {2}{b n \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}} \]

[In]

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

[Out]

-(ArcTan[1 - Sqrt[2]*Sqrt[Cot[a + b*Log[c*x^n]]]]/(Sqrt[2]*b*n)) + ArcTan[1 + Sqrt[2]*Sqrt[Cot[a + b*Log[c*x^n
]]]]/(Sqrt[2]*b*n) + 2/(b*n*Sqrt[Cot[a + b*Log[c*x^n]]]) + Log[1 - Sqrt[2]*Sqrt[Cot[a + b*Log[c*x^n]]] + Cot[a
+ b*Log[c*x^n]]]/(2*Sqrt[2]*b*n) - Log[1 + Sqrt[2]*Sqrt[Cot[a + b*Log[c*x^n]]] + Cot[a + b*Log[c*x^n]]]/(2*Sq
rt[2]*b*n)

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 303

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]},
Dist[1/(2*s), Int[(r + s*x^2)/(a + b*x^4), x], x] - Dist[1/(2*s), Int[(r - s*x^2)/(a + b*x^4), x], x]] /; Free
Q[{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ,
b]]))

Rule 335

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 631

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[a*(c/b^2)]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 1176

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[2*(d/e), 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 1179

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[-2*(d/e), 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rule 3555

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 3557

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]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1}{\cot ^{\frac {3}{2}}(a+b x)} \, dx,x,\log \left (c x^n\right )\right )}{n} \\ & = \frac {2}{b n \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}}-\frac {\text {Subst}\left (\int \sqrt {\cot (a+b x)} \, dx,x,\log \left (c x^n\right )\right )}{n} \\ & = \frac {2}{b n \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}}+\frac {\text {Subst}\left (\int \frac {\sqrt {x}}{1+x^2} \, dx,x,\cot \left (a+b \log \left (c x^n\right )\right )\right )}{b n} \\ & = \frac {2}{b n \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}}+\frac {2 \text {Subst}\left (\int \frac {x^2}{1+x^4} \, dx,x,\sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}\right )}{b n} \\ & = \frac {2}{b n \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}}-\frac {\text {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}\right )}{b n}+\frac {\text {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}\right )}{b n} \\ & = \frac {2}{b n \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}}+\frac {\text {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}\right )}{2 b n}+\frac {\text {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}\right )}{2 b n}+\frac {\text {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}\right )}{2 \sqrt {2} b n}+\frac {\text {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}\right )}{2 \sqrt {2} b n} \\ & = \frac {2}{b n \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}}+\frac {\log \left (1-\sqrt {2} \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}+\cot \left (a+b \log \left (c x^n\right )\right )\right )}{2 \sqrt {2} b n}-\frac {\log \left (1+\sqrt {2} \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}+\cot \left (a+b \log \left (c x^n\right )\right )\right )}{2 \sqrt {2} b n}+\frac {\text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}\right )}{\sqrt {2} b n}-\frac {\text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}\right )}{\sqrt {2} b n} \\ & = -\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}\right )}{\sqrt {2} b n}+\frac {\arctan \left (1+\sqrt {2} \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}\right )}{\sqrt {2} b n}+\frac {2}{b n \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}}+\frac {\log \left (1-\sqrt {2} \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}+\cot \left (a+b \log \left (c x^n\right )\right )\right )}{2 \sqrt {2} b n}-\frac {\log \left (1+\sqrt {2} \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}+\cot \left (a+b \log \left (c x^n\right )\right )\right )}{2 \sqrt {2} b n} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.53 \[ \int \frac {1}{x \cot ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\frac {2+\arctan \left (\sqrt [4]{-\cot ^2\left (a+b \log \left (c x^n\right )\right )}\right ) \sqrt [4]{-\cot ^2\left (a+b \log \left (c x^n\right )\right )}-\text {arctanh}\left (\sqrt [4]{-\cot ^2\left (a+b \log \left (c x^n\right )\right )}\right ) \sqrt [4]{-\cot ^2\left (a+b \log \left (c x^n\right )\right )}}{b n \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}} \]

[In]

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

[Out]

(2 + ArcTan[(-Cot[a + b*Log[c*x^n]]^2)^(1/4)]*(-Cot[a + b*Log[c*x^n]]^2)^(1/4) - ArcTanh[(-Cot[a + b*Log[c*x^n
]]^2)^(1/4)]*(-Cot[a + b*Log[c*x^n]]^2)^(1/4))/(b*n*Sqrt[Cot[a + b*Log[c*x^n]]])

Maple [A] (verified)

Time = 0.98 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.70

method	result	size
derivativedivides	\(\frac {\frac {2}{\sqrt {\cot \left (a +b \ln \left (c \,x^{n}\right )\right )}}+\frac {\sqrt {2}\, \left (\ln \left (\frac {1+\cot \left (a +b \ln \left (c \,x^{n}\right )\right )-\sqrt {2}\, \sqrt {\cot \left (a +b \ln \left (c \,x^{n}\right )\right )}}{1+\cot \left (a +b \ln \left (c \,x^{n}\right )\right )+\sqrt {2}\, \sqrt {\cot \left (a +b \ln \left (c \,x^{n}\right )\right )}}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\cot \left (a +b \ln \left (c \,x^{n}\right )\right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\cot \left (a +b \ln \left (c \,x^{n}\right )\right )}\right )\right )}{4}}{n b}\)	\(139\)
default	\(\frac {\frac {2}{\sqrt {\cot \left (a +b \ln \left (c \,x^{n}\right )\right )}}+\frac {\sqrt {2}\, \left (\ln \left (\frac {1+\cot \left (a +b \ln \left (c \,x^{n}\right )\right )-\sqrt {2}\, \sqrt {\cot \left (a +b \ln \left (c \,x^{n}\right )\right )}}{1+\cot \left (a +b \ln \left (c \,x^{n}\right )\right )+\sqrt {2}\, \sqrt {\cot \left (a +b \ln \left (c \,x^{n}\right )\right )}}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\cot \left (a +b \ln \left (c \,x^{n}\right )\right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\cot \left (a +b \ln \left (c \,x^{n}\right )\right )}\right )\right )}{4}}{n b}\)	\(139\)

[In]

int(1/x/cot(a+b*ln(c*x^n))^(3/2),x,method=_RETURNVERBOSE)

[Out]

1/n/b*(2/cot(a+b*ln(c*x^n))^(1/2)+1/4*2^(1/2)*(ln((1+cot(a+b*ln(c*x^n))-2^(1/2)*cot(a+b*ln(c*x^n))^(1/2))/(1+c
ot(a+b*ln(c*x^n))+2^(1/2)*cot(a+b*ln(c*x^n))^(1/2)))+2*arctan(1+2^(1/2)*cot(a+b*ln(c*x^n))^(1/2))+2*arctan(-1+
2^(1/2)*cot(a+b*ln(c*x^n))^(1/2))))

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.26 (sec) , antiderivative size = 756, normalized size of antiderivative = 3.80 \[ \int \frac {1}{x \cot ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-1/2*((b*n*(-1/(b^4*n^4))^(1/4)*cos(2*b*n*log(x) + 2*b*log(c) + 2*a) + b*n*(-1/(b^4*n^4))^(1/4))*log((b*n*(-1/
(b^4*n^4))^(1/4)*cos(2*b*n*log(x) + 2*b*log(c) + 2*a) + b*n*(-1/(b^4*n^4))^(1/4) + sqrt((cos(2*b*n*log(x) + 2*
b*log(c) + 2*a) + 1)/sin(2*b*n*log(x) + 2*b*log(c) + 2*a))*sin(2*b*n*log(x) + 2*b*log(c) + 2*a))/(cos(2*b*n*lo
g(x) + 2*b*log(c) + 2*a) + 1)) - (b*n*(-1/(b^4*n^4))^(1/4)*cos(2*b*n*log(x) + 2*b*log(c) + 2*a) + b*n*(-1/(b^4
*n^4))^(1/4))*log(-(b*n*(-1/(b^4*n^4))^(1/4)*cos(2*b*n*log(x) + 2*b*log(c) + 2*a) + b*n*(-1/(b^4*n^4))^(1/4) -
sqrt((cos(2*b*n*log(x) + 2*b*log(c) + 2*a) + 1)/sin(2*b*n*log(x) + 2*b*log(c) + 2*a))*sin(2*b*n*log(x) + 2*b*
log(c) + 2*a))/(cos(2*b*n*log(x) + 2*b*log(c) + 2*a) + 1)) + (I*b*n*(-1/(b^4*n^4))^(1/4)*cos(2*b*n*log(x) + 2*
b*log(c) + 2*a) + I*b*n*(-1/(b^4*n^4))^(1/4))*log((I*b*n*(-1/(b^4*n^4))^(1/4)*cos(2*b*n*log(x) + 2*b*log(c) +
2*a) + I*b*n*(-1/(b^4*n^4))^(1/4) + sqrt((cos(2*b*n*log(x) + 2*b*log(c) + 2*a) + 1)/sin(2*b*n*log(x) + 2*b*log
(c) + 2*a))*sin(2*b*n*log(x) + 2*b*log(c) + 2*a))/(cos(2*b*n*log(x) + 2*b*log(c) + 2*a) + 1)) + (-I*b*n*(-1/(b
^4*n^4))^(1/4)*cos(2*b*n*log(x) + 2*b*log(c) + 2*a) - I*b*n*(-1/(b^4*n^4))^(1/4))*log((-I*b*n*(-1/(b^4*n^4))^(
1/4)*cos(2*b*n*log(x) + 2*b*log(c) + 2*a) - I*b*n*(-1/(b^4*n^4))^(1/4) + sqrt((cos(2*b*n*log(x) + 2*b*log(c) +
2*a) + 1)/sin(2*b*n*log(x) + 2*b*log(c) + 2*a))*sin(2*b*n*log(x) + 2*b*log(c) + 2*a))/(cos(2*b*n*log(x) + 2*b
*log(c) + 2*a) + 1)) - 4*sqrt((cos(2*b*n*log(x) + 2*b*log(c) + 2*a) + 1)/sin(2*b*n*log(x) + 2*b*log(c) + 2*a))
*sin(2*b*n*log(x) + 2*b*log(c) + 2*a))/(b*n*cos(2*b*n*log(x) + 2*b*log(c) + 2*a) + b*n)

Sympy [F]

\[ \int \frac {1}{x \cot ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\int \frac {1}{x \cot ^{\frac {3}{2}}{\left (a + b \log {\left (c x^{n} \right )} \right )}}\, dx \]

[In]

integrate(1/x/cot(a+b*ln(c*x**n))**(3/2),x)

[Out]

Integral(1/(x*cot(a + b*log(c*x**n))**(3/2)), x)

Maxima [F]

\[ \int \frac {1}{x \cot ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\int { \frac {1}{x \cot \left (b \log \left (c x^{n}\right ) + a\right )^{\frac {3}{2}}} \,d x } \]

[In]

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

[Out]

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

Giac [F(-1)]

Timed out. \[ \int \frac {1}{x \cot ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Mupad [B] (verification not implemented)

Time = 28.25 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.40 \[ \int \frac {1}{x \cot ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\frac {2}{b\,n\,\sqrt {\mathrm {cot}\left (a+b\,\ln \left (c\,x^n\right )\right )}}+\frac {{\left (-1\right )}^{1/4}\,\mathrm {atan}\left ({\left (-1\right )}^{1/4}\,\sqrt {\mathrm {cot}\left (a+b\,\ln \left (c\,x^n\right )\right )}\right )}{b\,n}-\frac {{\left (-1\right )}^{1/4}\,\mathrm {atanh}\left ({\left (-1\right )}^{1/4}\,\sqrt {\mathrm {cot}\left (a+b\,\ln \left (c\,x^n\right )\right )}\right )}{b\,n} \]

[In]

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

[Out]

2/(b*n*cot(a + b*log(c*x^n))^(1/2)) + ((-1)^(1/4)*atan((-1)^(1/4)*cot(a + b*log(c*x^n))^(1/2)))/(b*n) - ((-1)^
(1/4)*atanh((-1)^(1/4)*cot(a + b*log(c*x^n))^(1/2)))/(b*n)