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

Optimal. Leaf size=159 \[ -\frac{2 i x^4 \text{Hypergeometric2F1}\left (1,-\frac{2 i}{b d n},1-\frac{2 i}{b d n},-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{b d n}+\frac{i x^4 \left (1-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{b d n \left (1+e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}+\frac{x^4 (-b d n+4 i)}{4 b d n} \]

[Out]

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

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Rubi [F]  time = 0.0900549, 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 = {} \[ \int x^3 \tan ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 6.58613, size = 179, normalized size = 1.13 \[ -\frac{x^4 \left ((b d n-2 i) \left (4 i \text{Hypergeometric2F1}\left (1,-\frac{2 i}{b d n},1-\frac{2 i}{b d n},-e^{2 i d \left (a+b \log \left (c x^n\right )\right )}\right )-4 \tan \left (d \left (a+b \log \left (c x^n\right )\right )\right )+b d n\right )-8 e^{2 i d \left (a+b \log \left (c x^n\right )\right )} \text{Hypergeometric2F1}\left (1,1-\frac{2 i}{b d n},2-\frac{2 i}{b d n},-e^{2 i d \left (a+b \log \left (c x^n\right )\right )}\right )\right )}{4 b d n (b d n-2 i)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 1.464, size = 0, normalized size = 0. \begin{align*} \int{x}^{3} \left ( \tan \left ( d \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) \right ) ^{2}\, 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)))^2,x)

[Out]

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

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Maxima [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)))^2,x, algorithm="maxima")

[Out]

Timed out

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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 )^{2}, 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)))^2,x, algorithm="fricas")

[Out]

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

[Out]

Timed out