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

Optimal. Leaf size=351 \[ \frac{i (e x)^{m+1} \left (-2 b^2 d^2 n^2+m^2+2 m+1\right ) \text{Hypergeometric2F1}\left (1,-\frac{i (m+1)}{2 b d n},1-\frac{i (m+1)}{2 b d n},-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{b^2 d^2 e (m+1) n^2}-\frac{i e^{-2 i a d} (e x)^{m+1} \left (\frac{e^{2 i a d} (-2 i b d n+m+1)}{n}-\frac{e^{4 i a d} (2 i b d n+m+1) \left (c x^n\right )^{2 i b d}}{n}\right )}{2 b^2 d^2 e n \left (1+e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}-\frac{(e x)^{m+1} \left (1-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )^2}{2 b d e n \left (1+e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )^2}-\frac{(e x)^{m+1} (-b d n+i (m+1)) (2 i b d n+m+1)}{2 b^2 d^2 e (m+1) n^2} \]

[Out]

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

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Rubi [F]  time = 0.0759238, 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 = {} \[ \int (e x)^m \tan ^3\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 17.7815, size = 642, normalized size = 1.83 \[ -\frac{x^{-m} (e x)^m \left (2 b^2 d^2 n^2-m^2-2 m-1\right ) \sec \left (d \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )\right ) \left (\frac{x^{m+1} \sin (b d n \log (x)) \sec \left (d \left (a+b \log \left (c x^n\right )\right )\right )}{m+1}-\frac{i \cos \left (d \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )\right ) \exp \left (-\frac{(2 m+1) \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )}{b n}\right ) \left ((2 i b d n+m+1) \left (-\exp \left (\frac{2 a m+a+b (2 m+1) \left (\log \left (c x^n\right )-n \log (x)\right )+b (m+1) n \log (x)}{b n}\right )\right ) \text{Hypergeometric2F1}\left (1,-\frac{i (m+1)}{2 b d n},1-\frac{i (m+1)}{2 b d n},-e^{2 i d \left (a+b \log \left (c x^n\right )\right )}\right )+(m+1) \exp \left (\frac{a (2 i b d n+2 m+1)}{b n}+\frac{(2 i b d n+2 m+1) \left (\log \left (c x^n\right )-n \log (x)\right )}{n}+\log (x) (2 i b d n+m+1)\right ) \text{Hypergeometric2F1}\left (1,-\frac{i (2 i b d n+m+1)}{2 b d n},-\frac{i (4 i b d n+m+1)}{2 b d n},-e^{2 i d \left (a+b \log \left (c x^n\right )\right )}\right )-i (2 i b d n+m+1) \tan \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \exp \left (\frac{2 a m+a+b (2 m+1) \left (\log \left (c x^n\right )-n \log (x)\right )+b (m+1) n \log (x)}{b n}\right )\right )}{(m+1) (2 i b d n+m+1)}\right )}{2 b^2 d^2 n^2}-\frac{(m+1) x (e x)^m \sin (b d n \log (x)) \sec \left (d \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )\right ) \sec \left (d \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )+b d n \log (x)\right )}{2 b^2 d^2 n^2}-\frac{x (e x)^m \tan \left (d \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )\right )}{m+1}+\frac{x (e x)^m \sec ^2\left (d \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )+b d n \log (x)\right )}{2 b d n} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(x*(e*x)^m*Sec[b*d*n*Log[x] + d*(a + b*(-(n*Log[x]) + Log[c*x^n]))]^2)/(2*b*d*n) - ((1 + m)*x*(e*x)^m*Sec[d*(a
+ b*(-(n*Log[x]) + Log[c*x^n]))]*Sec[b*d*n*Log[x] + d*(a + b*(-(n*Log[x]) + Log[c*x^n]))]*Sin[b*d*n*Log[x]])/
(2*b^2*d^2*n^2) - ((-1 - 2*m - m^2 + 2*b^2*d^2*n^2)*(e*x)^m*Sec[d*(a + b*(-(n*Log[x]) + Log[c*x^n]))]*((x^(1 +
m)*Sec[d*(a + b*Log[c*x^n])]*Sin[b*d*n*Log[x]])/(1 + m) - (I*Cos[d*(a + b*(-(n*Log[x]) + Log[c*x^n]))]*(-(E^(
(a + 2*a*m + b*(1 + m)*n*Log[x] + b*(1 + 2*m)*(-(n*Log[x]) + Log[c*x^n]))/(b*n))*(1 + m + (2*I)*b*d*n)*Hyperge
ometric2F1[1, ((-I/2)*(1 + m))/(b*d*n), 1 - ((I/2)*(1 + m))/(b*d*n), -E^((2*I)*d*(a + b*Log[c*x^n]))]) + E^((a
*(1 + 2*m + (2*I)*b*d*n))/(b*n) + (1 + m + (2*I)*b*d*n)*Log[x] + ((1 + 2*m + (2*I)*b*d*n)*(-(n*Log[x]) + Log[c
*x^n]))/n)*(1 + m)*Hypergeometric2F1[1, ((-I/2)*(1 + m + (2*I)*b*d*n))/(b*d*n), ((-I/2)*(1 + m + (4*I)*b*d*n))
/(b*d*n), -E^((2*I)*d*(a + b*Log[c*x^n]))] - I*E^((a + 2*a*m + b*(1 + m)*n*Log[x] + b*(1 + 2*m)*(-(n*Log[x]) +
Log[c*x^n]))/(b*n))*(1 + m + (2*I)*b*d*n)*Tan[d*(a + b*Log[c*x^n])]))/(E^(((1 + 2*m)*(a + b*(-(n*Log[x]) + Lo
g[c*x^n])))/(b*n))*(1 + m)*(1 + m + (2*I)*b*d*n))))/(2*b^2*d^2*n^2*x^m) - (x*(e*x)^m*Tan[d*(a + b*(-(n*Log[x])
+ Log[c*x^n]))])/(1 + m)

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Maple [F]  time = 1.796, size = 0, normalized size = 0. \begin{align*} \int \left ( ex \right ) ^{m} \left ( \tan \left ( d \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) \right ) ^{3}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out