3.215 \(\int \frac {\cot (d (a+b \log (c x^n)))}{x^3} \, dx\)

Optimal. Leaf size=68 \[ \frac {i \, _2F_1\left (1,\frac {i}{b d n};1+\frac {i}{b d n};e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{x^2}-\frac {i}{2 x^2} \]

[Out]

-1/2*I/x^2+I*hypergeom([1, I/b/d/n],[1+I/b/d/n],exp(2*I*a*d)*(c*x^n)^(2*I*b*d))/x^2

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Rubi [F]  time = 0.03, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\cot \left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx \]

Verification is Not applicable to the result.

[In]

Int[Cot[d*(a + b*Log[c*x^n])]/x^3,x]

[Out]

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

Rubi steps

\begin {align*} \int \frac {\cot \left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx &=\int \frac {\cot \left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx\\ \end {align*}

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Mathematica [B]  time = 4.16, size = 211, normalized size = 3.10 \[ \frac {-\frac {e^{2 i d \left (a+b \log \left (c x^n\right )\right )} \, _2F_1\left (1,1+\frac {i}{b d n};2+\frac {i}{b d n};e^{2 i d \left (a+b \log \left (c x^n\right )\right )}\right )}{b d n+i}+i \, _2F_1\left (1,\frac {i}{b d n};1+\frac {i}{b d n};e^{2 i d \left (a+b \log \left (c x^n\right )\right )}\right )+\cot \left (d \left (a+b \log \left (c x^n\right )\right )\right )-\cot \left (d \left (a+b \log \left (c x^n\right )-b n \log (x)\right )\right )+\sin (b d n \log (x)) \csc \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \csc \left (d \left (a+b \log \left (c x^n\right )-b n \log (x)\right )\right )}{2 x^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Cot[d*(a + b*Log[c*x^n])] - Cot[d*(a - b*n*Log[x] + b*Log[c*x^n])] - (E^((2*I)*d*(a + b*Log[c*x^n]))*Hypergeo
metric2F1[1, 1 + I/(b*d*n), 2 + I/(b*d*n), E^((2*I)*d*(a + b*Log[c*x^n]))])/(I + b*d*n) + I*Hypergeometric2F1[
1, I/(b*d*n), 1 + I/(b*d*n), E^((2*I)*d*(a + b*Log[c*x^n]))] + Csc[d*(a + b*Log[c*x^n])]*Csc[d*(a - b*n*Log[x]
+ b*Log[c*x^n])]*Sin[b*d*n*Log[x]])/(2*x^2)

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fricas [F]  time = 1.51, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\cot \left (b d \log \left (c x^{n}\right ) + a d\right )}{x^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(cot(b*d*log(c*x^n) + a*d)/x^3, x)

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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maple [F]  time = 1.71, size = 0, normalized size = 0.00 \[ \int \frac {\cot \left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )}{x^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(d*(a+b*ln(c*x^n)))/x^3,x)

[Out]

int(cot(d*(a+b*ln(c*x^n)))/x^3,x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cot \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )}{x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\mathrm {cot}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right )}{x^3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(d*(a + b*log(c*x^n)))/x^3,x)

[Out]

int(cot(d*(a + b*log(c*x^n)))/x^3, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cot {\left (a d + b d \log {\left (c x^{n} \right )} \right )}}{x^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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