3.212 \(\int \cot (d (a+b \log (c x^n))) \, dx\)

Optimal. Leaf size=66 \[ i x-2 i x \, _2F_1\left (1,-\frac {i}{2 b d n};1-\frac {i}{2 b d n};e^{2 i a d} \left (c x^n\right )^{2 i b d}\right ) \]

[Out]

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

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Rubi [F]  time = 0.01, 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 = {} \[ \int \cot \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx \]

Verification is Not applicable to the result.

[In]

Int[Cot[d*(a + b*Log[c*x^n])],x]

[Out]

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

Rubi steps

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

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Mathematica [B]  time = 10.32, size = 141, normalized size = 2.14 \[ x \left (-\frac {e^{2 i d \left (a+b \log \left (c x^n\right )\right )} \, _2F_1\left (1,1-\frac {i}{2 b d n};2-\frac {i}{2 b d n};e^{2 i d \left (a+b \log \left (c x^n\right )\right )}\right )}{2 b d n-i}-i \, _2F_1\left (1,-\frac {i}{2 b d n};1-\frac {i}{2 b d n};e^{2 i d \left (a+b \log \left (c x^n\right )\right )}\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [F]  time = 1.18, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\cot \left (b d \log \left (c x^{n}\right ) + a d\right ), x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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maple [F]  time = 1.05, size = 0, normalized size = 0.00 \[ \int \cot \left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \mathrm {cot}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right ) \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(d*(a + b*log(c*x^n))),x)

[Out]

int(cot(d*(a + b*log(c*x^n))), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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