3.161 \(\int \tan (d (a+b \log (c x^n))) \, dx\)

Optimal. Leaf size=67 \[ 2 i x \text{Hypergeometric2F1}\left (1,-\frac{i}{2 b d n},1-\frac{i}{2 b d n},-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )-i x \]

[Out]

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

________________________________________________________________________________________

Rubi [F]  time = 0.0117656, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \tan \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx \]

Verification is Not applicable to the result.

[In]

Int[Tan[d*(a + b*Log[c*x^n])],x]

[Out]

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

Rubi steps

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

Mathematica [B]  time = 10.9698, size = 151, normalized size = 2.25 \[ \frac{x \left ((1+2 i b d n) \text{Hypergeometric2F1}\left (1,-\frac{i}{2 b d n},1-\frac{i}{2 b d n},-e^{2 i d \left (a+b \log \left (c x^n\right )\right )}\right )-e^{2 i d \left (a+b \log \left (c x^n\right )\right )} \text{Hypergeometric2F1}\left (1,1-\frac{i}{2 b d n},2-\frac{i}{2 b d n},-e^{2 i d \left (a+b \log \left (c x^n\right )\right )}\right )\right )}{2 b d n-i} \]

Antiderivative was successfully verified.

[In]

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

________________________________________________________________________________________

Maple [F]  time = 1.083, size = 0, normalized size = 0. \begin{align*} \int \tan \left ( d \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \tan \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\tan \left (b d \log \left (c x^{n}\right ) + a d\right ), x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \tan{\left (d \left (a + b \log{\left (c x^{n} \right )}\right ) \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out