### 3.10 $$\int \frac{\tan ^2(a+b x)}{x^2} \, dx$$

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

[Out]

Unintegrable[Tan[a + b*x]^2/x^2, x]

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Rubi [A]  time = 0.0304513, 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 = {} $\int \frac{\tan ^2(a+b x)}{x^2} \, dx$

Veriﬁcation is Not applicable to the result.

[In]

Int[Tan[a + b*x]^2/x^2,x]

[Out]

Defer[Int][Tan[a + b*x]^2/x^2, x]

Rubi steps

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

Mathematica [A]  time = 2.41713, size = 0, normalized size = 0. $\int \frac{\tan ^2(a+b x)}{x^2} \, dx$

Veriﬁcation is Not applicable to the result.

[In]

Integrate[Tan[a + b*x]^2/x^2,x]

[Out]

Integrate[Tan[a + b*x]^2/x^2, x]

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Maple [A]  time = 0.099, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( \tan \left ( bx+a \right ) \right ) ^{2}}{{x}^{2}}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b*x*cos(2*b*x + 2*a)^2 + b*x*sin(2*b*x + 2*a)^2 + 2*b*x*cos(2*b*x + 2*a) + b*x + 4*(b^2*x^2*cos(2*b*x + 2*a)^
2 + b^2*x^2*sin(2*b*x + 2*a)^2 + 2*b^2*x^2*cos(2*b*x + 2*a) + b^2*x^2)*integrate(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*sin(2*b*x + 2*a))
/(b*x^2*cos(2*b*x + 2*a)^2 + b*x^2*sin(2*b*x + 2*a)^2 + 2*b*x^2*cos(2*b*x + 2*a) + b*x^2)

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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 )^{2}}{x^{2}}, x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(tan(b*x + a)^2/x^2, x)

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\tan ^{2}{\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)**2/x**2,x)

[Out]

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(tan(b*x + a)^2/x^2, x)