3.158 \(\int x^3 \tan (d (a+b \log (c x^n))) \, dx\)

Optimal. Leaf size=71 \[ \frac{1}{2} i x^4 \text{Hypergeometric2F1}\left (1,-\frac{2 i}{b d n},1-\frac{2 i}{b d n},-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )-\frac{i x^4}{4} \]

[Out]

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

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Rubi [F]  time = 0.0433235, 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 = {} \[ \int x^3 \tan \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx \]

Verification is Not applicable to the result.

[In]

Int[x^3*Tan[d*(a + b*Log[c*x^n])],x]

[Out]

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

Rubi steps

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

Mathematica [B]  time = 6.50766, size = 146, normalized size = 2.06 \[ \frac{x^4 \left (2 i e^{2 i d \left (a+b \log \left (c x^n\right )\right )} \text{Hypergeometric2F1}\left (1,1-\frac{2 i}{b d n},2-\frac{2 i}{b d n},-e^{2 i d \left (a+b \log \left (c x^n\right )\right )}\right )+(b d n-2 i) \text{Hypergeometric2F1}\left (1,-\frac{2 i}{b d n},1-\frac{2 i}{b d n},-e^{2 i d \left (a+b \log \left (c x^n\right )\right )}\right )\right )}{-8-4 i b d n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 1.566, size = 0, normalized size = 0. \begin{align*} \int{x}^{3}\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(x^3*tan(d*(a+b*ln(c*x^n))),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{3} \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(x^3*tan(d*(a+b*log(c*x^n))),x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (x^{3} \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(x^3*tan(d*(a+b*log(c*x^n))),x, algorithm="fricas")

[Out]

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

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Sympy [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(x**3*tan(d*(a+b*ln(c*x**n))),x)

[Out]

Timed out

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

[Out]

Timed out