3.170 \(\int \frac {\tan ^2(d (a+b \log (c x^n)))}{x^2} \, dx\)

Optimal. Leaf size=157 \[ -\frac {2 i \, _2F_1\left (1,\frac {i}{2 b d n};1+\frac {i}{2 b d n};-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{b d n x}+\frac {i \left (1-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{b d n x \left (1+e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}+\frac {1+\frac {i}{b d n}}{x} \]

[Out]

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

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Rubi [F]  time = 0.06, 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 = {} \[ \int \frac {\tan ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 4.29, size = 184, normalized size = 1.17 \[ \frac {(2 b d n+i) \left (-i \, _2F_1\left (1,\frac {i}{2 b d n};1+\frac {i}{2 b d n};-e^{2 i d \left (a+b \log \left (c x^n\right )\right )}\right )+\tan \left (d \left (a+b \log \left (c x^n\right )\right )\right )+b d n\right )-e^{2 i d \left (a+b \log \left (c x^n\right )\right )} \, _2F_1\left (1,1+\frac {i}{2 b d n};2+\frac {i}{2 b d n};-e^{2 i d \left (a+b \log \left (c x^n\right )\right )}\right )}{b d n x (2 b d n+i)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [F]  time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\tan \left (b d \log \left (c x^{n}\right ) + a d\right )^{2}}{x^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(tan(b*d*log(c*x^n) + a*d)^2/x^2, 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(tan(d*(a+b*log(c*x^n)))^2/x^2,x, algorithm="giac")

[Out]

Timed out

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maple [F]  time = 0.23, size = 0, normalized size = 0.00 \[ \int \frac {\tan ^{2}\left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )}{x^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(tan(d*(a+b*ln(c*x^n)))^2/x^2,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(tan(d*(a+b*log(c*x^n)))^2/x^2,x, algorithm="maxima")

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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