3.222 \(\int \frac {\cot ^2(d (a+b \log (c x^n)))}{x^3} \, dx\)

Optimal. Leaf size=155 \[ -\frac {2 i \, _2F_1\left (1,\frac {i}{b d n};1+\frac {i}{b d n};e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{b d n x^2}+\frac {i \left (1+e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{b d n x^2 \left (1-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}+\frac {1+\frac {2 i}{b d n}}{2 x^2} \]

[Out]

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

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Rubi [F]  time = 0.05, 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 = {} \[ \int \frac {\cot ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

\begin {align*} \int \frac {\cot ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx &=\int \frac {\cot ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx\\ \end {align*}

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Mathematica [A]  time = 3.91, size = 175, normalized size = 1.13 \[ \frac {2 e^{2 i d \left (a+b \log \left (c x^n\right )\right )} \, _2F_1\left (1,1+\frac {i}{b d n};2+\frac {i}{b d n};e^{2 i d \left (a+b \log \left (c x^n\right )\right )}\right )+(b d n+i) \left (-2 i \, _2F_1\left (1,\frac {i}{b d n};1+\frac {i}{b d n};e^{2 i d \left (a+b \log \left (c x^n\right )\right )}\right )-2 \cot \left (d \left (a+b \log \left (c x^n\right )\right )\right )+b d n\right )}{2 b d n x^2 (b d n+i)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [F]  time = 1.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\cot \left (b d \log \left (c x^{n}\right ) + a d\right )^{2}}{x^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cot \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )^{2}}{x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.40, size = 0, normalized size = 0.00 \[ \int \frac {\cot ^{2}\left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )}{x^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [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)))^2/x^3,x, algorithm="maxima")

[Out]

Timed out

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\mathrm {cot}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right )}^2}{x^3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cot ^{2}{\left (a d + b d \log {\left (c x^{n} \right )} \right )}}{x^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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