3.15 \(\int \frac{\tan ^3(a+b x)}{x^2} \, dx\)

Optimal. Leaf size=14 \[ \text{Unintegrable}\left (\frac{\tan ^3(a+b x)}{x^2},x\right ) \]

[Out]

Unintegrable[Tan[a + b*x]^3/x^2, x]

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Rubi [A]  time = 0.0277722, 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 = {} \[ \int \frac{\tan ^3(a+b x)}{x^2} \, dx \]

Verification is Not applicable to the result.

[In]

Int[Tan[a + b*x]^3/x^2,x]

[Out]

Defer[Int][Tan[a + b*x]^3/x^2, x]

Rubi steps

\begin{align*} \int \frac{\tan ^3(a+b x)}{x^2} \, dx &=\int \frac{\tan ^3(a+b x)}{x^2} \, dx\\ \end{align*}

Mathematica [A]  time = 3.16353, size = 0, normalized size = 0. \[ \int \frac{\tan ^3(a+b x)}{x^2} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[Tan[a + b*x]^3/x^2,x]

[Out]

Integrate[Tan[a + b*x]^3/x^2, x]

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Maple [A]  time = 0.722, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( \tan \left ( bx+a \right ) \right ) ^{3}}{{x}^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(b*x+a)^3/x^2,x)

[Out]

int(tan(b*x+a)^3/x^2,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 ) + 2 \,{\left (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^{3} \cos \left (4 \, b x + 4 \, a\right )^{2} + 4 \, b^{2} x^{3} \cos \left (2 \, b x + 2 \, a\right )^{2} + b^{2} x^{3} \sin \left (4 \, b x + 4 \, a\right )^{2} + 4 \, b^{2} x^{3} \sin \left (4 \, b x + 4 \, a\right ) \sin \left (2 \, b x + 2 \, a\right ) + 4 \, b^{2} x^{3} \sin \left (2 \, b x + 2 \, a\right )^{2} + 4 \, b^{2} x^{3} \cos \left (2 \, b x + 2 \, a\right ) + b^{2} x^{3} + 2 \,{\left (2 \, b^{2} x^{3} \cos \left (2 \, b x + 2 \, a\right ) + b^{2} x^{3}\right )} \cos \left (4 \, b x + 4 \, a\right )\right )} \int \frac{{\left (b^{2} x^{2} - 3\right )} \sin \left (2 \, b x + 2 \, a\right )}{b^{2} x^{4} \cos \left (2 \, b x + 2 \, a\right )^{2} + b^{2} x^{4} \sin \left (2 \, b x + 2 \, a\right )^{2} + 2 \, b^{2} x^{4} \cos \left (2 \, b x + 2 \, a\right ) + b^{2} x^{4}}\,{d x} + 2 \,{\left (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 ) + 2 \, \sin \left (2 \, b x + 2 \, a\right )}{b^{2} x^{3} \cos \left (4 \, b x + 4 \, a\right )^{2} + 4 \, b^{2} x^{3} \cos \left (2 \, b x + 2 \, a\right )^{2} + b^{2} x^{3} \sin \left (4 \, b x + 4 \, a\right )^{2} + 4 \, b^{2} x^{3} \sin \left (4 \, b x + 4 \, a\right ) \sin \left (2 \, b x + 2 \, a\right ) + 4 \, b^{2} x^{3} \sin \left (2 \, b x + 2 \, a\right )^{2} + 4 \, b^{2} x^{3} \cos \left (2 \, b x + 2 \, a\right ) + b^{2} x^{3} + 2 \,{\left (2 \, b^{2} x^{3} \cos \left (2 \, b x + 2 \, a\right ) + b^{2} x^{3}\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^2,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^3*cos(4*b*x + 4*a)^2 + 4*b^2*x^3*cos(2*b*x + 2*a)^2 + b^2*x^3*sin(4*b*
x + 4*a)^2 + 4*b^2*x^3*sin(4*b*x + 4*a)*sin(2*b*x + 2*a) + 4*b^2*x^3*sin(2*b*x + 2*a)^2 + 4*b^2*x^3*cos(2*b*x
+ 2*a) + b^2*x^3 + 2*(2*b^2*x^3*cos(2*b*x + 2*a) + b^2*x^3)*cos(4*b*x + 4*a))*integrate(2*(b^2*x^2 - 3)*sin(2*
b*x + 2*a)/(b^2*x^4*cos(2*b*x + 2*a)^2 + b^2*x^4*sin(2*b*x + 2*a)^2 + 2*b^2*x^4*cos(2*b*x + 2*a) + b^2*x^4), x
) + 2*(b*x*sin(2*b*x + 2*a) + cos(2*b*x + 2*a) + 1)*sin(4*b*x + 4*a) + 2*sin(2*b*x + 2*a))/(b^2*x^3*cos(4*b*x
+ 4*a)^2 + 4*b^2*x^3*cos(2*b*x + 2*a)^2 + b^2*x^3*sin(4*b*x + 4*a)^2 + 4*b^2*x^3*sin(4*b*x + 4*a)*sin(2*b*x +
2*a) + 4*b^2*x^3*sin(2*b*x + 2*a)^2 + 4*b^2*x^3*cos(2*b*x + 2*a) + b^2*x^3 + 2*(2*b^2*x^3*cos(2*b*x + 2*a) + b
^2*x^3)*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^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(b*x+a)^3/x^2,x, algorithm="fricas")

[Out]

integral(tan(b*x + a)^3/x^2, 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^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(b*x+a)**3/x**2,x)

[Out]

Integral(tan(a + b*x)**3/x**2, 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^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(b*x+a)^3/x^2,x, algorithm="giac")

[Out]

integrate(tan(b*x + a)^3/x^2, x)