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

Optimal. Leaf size=156 \[ -\frac {2 i \, _2F_1\left (1,\frac {i}{b d n};1+\frac {i}{b d n};-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{b d n x^2}+\frac {i \left (1-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{b d n x^2 \left (1+e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}+\frac {1+\frac {2 i}{b d n}}{2 x^2} \]

[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

________________________________________________________________________________________

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

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 3.91, size = 179, normalized size = 1.15 \[ \frac {(b d n+i) \left (-2 i \, _2F_1\left (1,\frac {i}{b d n};1+\frac {i}{b d n};-e^{2 i d \left (a+b \log \left (c x^n\right )\right )}\right )+2 \tan \left (d \left (a+b \log \left (c x^n\right )\right )\right )+b d n\right )-2 e^{2 i d \left (a+b \log \left (c x^n\right )\right )} \, _2F_1\left (1,1+\frac {i}{b d n};2+\frac {i}{b d n};-e^{2 i d \left (a+b \log \left (c x^n\right )\right )}\right )}{2 b d n x^2 (b d n+i)} \]

Antiderivative was successfully verified.

[In]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [F]  time = 0.28, size = 0, normalized size = 0.00 \[ \int \frac {\tan ^{2}\left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )}{x^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(tan(d*(a+b*ln(c*x^n)))^2/x^3,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(tan(d*(a+b*log(c*x^n)))^2/x^3,x, algorithm="maxima")

[Out]

Timed out

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________