### 3.14 $$\int \frac{\tan ^3(a+b x)}{x} \, dx$$

Optimal. Leaf size=14 $\text{Unintegrable}\left (\frac{\tan ^3(a+b x)}{x},x\right )$

[Out]

Unintegrable[Tan[a + b*x]^3/x, x]

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Rubi [A]  time = 0.0267057, 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 = {} $\int \frac{\tan ^3(a+b x)}{x} \, dx$

Veriﬁcation is Not applicable to the result.

[In]

Int[Tan[a + b*x]^3/x,x]

[Out]

Defer[Int][Tan[a + b*x]^3/x, x]

Rubi steps

\begin{align*} \int \frac{\tan ^3(a+b x)}{x} \, dx &=\int \frac{\tan ^3(a+b x)}{x} \, dx\\ \end{align*}

Mathematica [A]  time = 4.57348, size = 0, normalized size = 0. $\int \frac{\tan ^3(a+b x)}{x} \, dx$

Veriﬁcation is Not applicable to the result.

[In]

Integrate[Tan[a + b*x]^3/x,x]

[Out]

Integrate[Tan[a + b*x]^3/x, x]

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Maple [A]  time = 0.552, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( \tan \left ( bx+a \right ) \right ) ^{3}}{x}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(tan(b*x+a)^3/x,x)

[Out]

int(tan(b*x+a)^3/x,x)

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{4 \, b x \cos \left (2 \, b x + 2 \, a\right )^{2} + 4 \, b x \sin \left (2 \, b x + 2 \, a\right )^{2} + 2 \, b x \cos \left (2 \, b x + 2 \, a\right ) +{\left (2 \, b x \cos \left (2 \, b x + 2 \, a\right ) - \sin \left (2 \, b x + 2 \, a\right )\right )} \cos \left (4 \, b x + 4 \, a\right ) - 2 \,{\left (b^{2} x^{2} \cos \left (4 \, b x + 4 \, a\right )^{2} + 4 \, b^{2} x^{2} \cos \left (2 \, b x + 2 \, a\right )^{2} + b^{2} x^{2} \sin \left (4 \, b x + 4 \, a\right )^{2} + 4 \, b^{2} x^{2} \sin \left (4 \, b x + 4 \, a\right ) \sin \left (2 \, b x + 2 \, a\right ) + 4 \, b^{2} x^{2} \sin \left (2 \, b x + 2 \, a\right )^{2} + 4 \, b^{2} x^{2} \cos \left (2 \, b x + 2 \, a\right ) + b^{2} x^{2} + 2 \,{\left (2 \, b^{2} x^{2} \cos \left (2 \, b x + 2 \, a\right ) + b^{2} x^{2}\right )} \cos \left (4 \, b x + 4 \, a\right )\right )} \int \frac{{\left (b^{2} x^{2} - 1\right )} \sin \left (2 \, b x + 2 \, a\right )}{b^{2} x^{3} \cos \left (2 \, b x + 2 \, a\right )^{2} + b^{2} x^{3} \sin \left (2 \, b x + 2 \, a\right )^{2} + 2 \, b^{2} x^{3} \cos \left (2 \, b x + 2 \, a\right ) + b^{2} x^{3}}\,{d x} +{\left (2 \, b x \sin \left (2 \, b x + 2 \, a\right ) + \cos \left (2 \, b x + 2 \, a\right ) + 1\right )} \sin \left (4 \, b x + 4 \, a\right ) + \sin \left (2 \, b x + 2 \, a\right )}{b^{2} x^{2} \cos \left (4 \, b x + 4 \, a\right )^{2} + 4 \, b^{2} x^{2} \cos \left (2 \, b x + 2 \, a\right )^{2} + b^{2} x^{2} \sin \left (4 \, b x + 4 \, a\right )^{2} + 4 \, b^{2} x^{2} \sin \left (4 \, b x + 4 \, a\right ) \sin \left (2 \, b x + 2 \, a\right ) + 4 \, b^{2} x^{2} \sin \left (2 \, b x + 2 \, a\right )^{2} + 4 \, b^{2} x^{2} \cos \left (2 \, b x + 2 \, a\right ) + b^{2} x^{2} + 2 \,{\left (2 \, b^{2} x^{2} \cos \left (2 \, b x + 2 \, a\right ) + b^{2} x^{2}\right )} \cos \left (4 \, b x + 4 \, a\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(b*x+a)^3/x,x, algorithm="maxima")

[Out]

(4*b*x*cos(2*b*x + 2*a)^2 + 4*b*x*sin(2*b*x + 2*a)^2 + 2*b*x*cos(2*b*x + 2*a) + (2*b*x*cos(2*b*x + 2*a) - sin(
2*b*x + 2*a))*cos(4*b*x + 4*a) - (b^2*x^2*cos(4*b*x + 4*a)^2 + 4*b^2*x^2*cos(2*b*x + 2*a)^2 + b^2*x^2*sin(4*b*
x + 4*a)^2 + 4*b^2*x^2*sin(4*b*x + 4*a)*sin(2*b*x + 2*a) + 4*b^2*x^2*sin(2*b*x + 2*a)^2 + 4*b^2*x^2*cos(2*b*x
+ 2*a) + b^2*x^2 + 2*(2*b^2*x^2*cos(2*b*x + 2*a) + b^2*x^2)*cos(4*b*x + 4*a))*integrate(2*(b^2*x^2 - 1)*sin(2*
b*x + 2*a)/(b^2*x^3*cos(2*b*x + 2*a)^2 + b^2*x^3*sin(2*b*x + 2*a)^2 + 2*b^2*x^3*cos(2*b*x + 2*a) + b^2*x^3), x
) + (2*b*x*sin(2*b*x + 2*a) + cos(2*b*x + 2*a) + 1)*sin(4*b*x + 4*a) + sin(2*b*x + 2*a))/(b^2*x^2*cos(4*b*x +
4*a)^2 + 4*b^2*x^2*cos(2*b*x + 2*a)^2 + b^2*x^2*sin(4*b*x + 4*a)^2 + 4*b^2*x^2*sin(4*b*x + 4*a)*sin(2*b*x + 2*
a) + 4*b^2*x^2*sin(2*b*x + 2*a)^2 + 4*b^2*x^2*cos(2*b*x + 2*a) + b^2*x^2 + 2*(2*b^2*x^2*cos(2*b*x + 2*a) + b^2
*x^2)*cos(4*b*x + 4*a))

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\tan \left (b x + a\right )^{3}}{x}, x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(b*x+a)^3/x,x, algorithm="fricas")

[Out]

integral(tan(b*x + a)^3/x, x)

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\tan ^{3}{\left (a + b x \right )}}{x}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(b*x+a)**3/x,x)

[Out]

Integral(tan(a + b*x)**3/x, x)

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\tan \left (b x + a\right )^{3}}{x}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(b*x+a)^3/x,x, algorithm="giac")

[Out]

integrate(tan(b*x + a)^3/x, x)