3.216 \(\int x^3 \cot ^2(d (a+b \log (c x^n))) \, dx\)

Optimal. Leaf size=158 \[ -\frac {2 i x^4 \, _2F_1\left (1,-\frac {2 i}{b d n};1-\frac {2 i}{b d n};e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{b d n}+\frac {i x^4 \left (1+e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{b d n \left (1-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}+\frac {x^4 (-b d n+4 i)}{4 b d n} \]

[Out]

1/4*(4*I-b*d*n)*x^4/b/d/n+I*x^4*(1+exp(2*I*a*d)*(c*x^n)^(2*I*b*d))/b/d/n/(1-exp(2*I*a*d)*(c*x^n)^(2*I*b*d))-2*
I*x^4*hypergeom([1, -2*I/b/d/n],[1-2*I/b/d/n],exp(2*I*a*d)*(c*x^n)^(2*I*b*d))/b/d/n

________________________________________________________________________________________

Rubi [F]  time = 0.09, 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 = {} \[ \int x^3 \cot ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 4.64, size = 175, normalized size = 1.11 \[ -\frac {x^4 \left (8 e^{2 i d \left (a+b \log \left (c x^n\right )\right )} \, _2F_1\left (1,1-\frac {2 i}{b d n};2-\frac {2 i}{b d n};e^{2 i d \left (a+b \log \left (c x^n\right )\right )}\right )+(b d n-2 i) \left (4 i \, _2F_1\left (1,-\frac {2 i}{b d n};1-\frac {2 i}{b d n};e^{2 i d \left (a+b \log \left (c x^n\right )\right )}\right )+4 \cot \left (d \left (a+b \log \left (c x^n\right )\right )\right )+b d n\right )\right )}{4 b d n (b d n-2 i)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/4*(x^4*(8*E^((2*I)*d*(a + b*Log[c*x^n]))*Hypergeometric2F1[1, 1 - (2*I)/(b*d*n), 2 - (2*I)/(b*d*n), E^((2*I
)*d*(a + b*Log[c*x^n]))] + (-2*I + b*d*n)*(b*d*n + 4*Cot[d*(a + b*Log[c*x^n])] + (4*I)*Hypergeometric2F1[1, (-
2*I)/(b*d*n), 1 - (2*I)/(b*d*n), E^((2*I)*d*(a + b*Log[c*x^n]))])))/(b*d*n*(-2*I + b*d*n))

________________________________________________________________________________________

fricas [F]  time = 0.54, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x^{3} \cot \left (b d \log \left (c x^{n}\right ) + a d\right )^{2}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

maple [F]  time = 0.42, size = 0, normalized size = 0.00 \[ \int x^{3} \left (\cot ^{2}\left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________