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

Optimal. Leaf size=15 \[ \text {Int}\left (\frac {\tan ^2(a+b x)}{x},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.03, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\tan ^2(a+b x)}{x} \, dx \]

Verification is Not applicable to the result.

[In]

Int[Tan[a + b*x]^2/x,x]

[Out]

Defer[Int][Tan[a + b*x]^2/x, x]

Rubi steps

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

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Mathematica [A]  time = 2.76, size = 0, normalized size = 0.00 \[ \int \frac {\tan ^2(a+b x)}{x} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[Tan[a + b*x]^2/x,x]

[Out]

Integrate[Tan[a + b*x]^2/x, x]

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fricas [A]  time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\tan \left (b x + a\right )^{2}}{x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(tan(b*x + a)^2/x, x)

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\tan \left (b x + a\right )^{2}}{x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(tan(b*x + a)^2/x, x)

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maple [A]  time = 0.36, size = 0, normalized size = 0.00 \[ \int \frac {\tan ^{2}\left (b x +a \right )}{x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {b x \cos \left (2 \, b x + 2 \, a\right )^{2} \log \relax (x) + b x \log \relax (x) \sin \left (2 \, b x + 2 \, a\right )^{2} + 2 \, b x \cos \left (2 \, b x + 2 \, a\right ) \log \relax (x) + b x \log \relax (x) - \frac {2 \, {\left (b^{2} x \cos \left (2 \, b x + 2 \, a\right )^{2} + b^{2} x \sin \left (2 \, b x + 2 \, a\right )^{2} + 2 \, b^{2} x \cos \left (2 \, b x + 2 \, a\right ) + b^{2} x\right )} \int \frac {\sin \left (2 \, b x + 2 \, a\right )}{{\left (\cos \left (2 \, b x + 2 \, a\right )^{2} + \sin \left (2 \, b x + 2 \, a\right )^{2} + 2 \, \cos \left (2 \, b x + 2 \, a\right ) + 1\right )} x^{2}}\,{d x}}{b^{2}} - 2 \, \sin \left (2 \, b x + 2 \, a\right )}{b x \cos \left (2 \, b x + 2 \, a\right )^{2} + b x \sin \left (2 \, b x + 2 \, a\right )^{2} + 2 \, b x \cos \left (2 \, b x + 2 \, a\right ) + b x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(b*x*cos(2*b*x + 2*a)^2*log(x) + b*x*log(x)*sin(2*b*x + 2*a)^2 + 2*b*x*cos(2*b*x + 2*a)*log(x) + b*x*log(x) -
2*(b^2*x*cos(2*b*x + 2*a)^2 + b^2*x*sin(2*b*x + 2*a)^2 + 2*b^2*x*cos(2*b*x + 2*a) + b^2*x)*integrate(sin(2*b*
x + 2*a)/(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), x)
- 2*sin(2*b*x + 2*a))/(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)

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mupad [A]  time = 0.00, size = -1, normalized size = -0.07 \[ \int \frac {{\mathrm {tan}\left (a+b\,x\right )}^2}{x} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(a + b*x)^2/x,x)

[Out]

int(tan(a + b*x)^2/x, x)

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\tan ^{2}{\left (a + b x \right )}}{x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(tan(a + b*x)**2/x, x)

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