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

Optimal. Leaf size=19 \[ -\frac{\log \left (\cos \left (a+b x+c x^2\right )\right )}{2 c} \]

[Out]

-Log[Cos[a + b*x + c*x^2]]/(2*c)

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Rubi [A]  time = 0.0176489, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.03, Rules used = {3763} \[ -\frac{\log \left (\cos \left (a+b x+c x^2\right )\right )}{2 c} \]

Antiderivative was successfully verified.

[In]

Int[(b*Tan[a + b*x + c*x^2])/(2*c) + x*Tan[a + b*x + c*x^2],x]

[Out]

-Log[Cos[a + b*x + c*x^2]]/(2*c)

Rule 3763

Int[((d_.) + (e_.)*(x_))*Tan[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> -Simp[(e*Log[Cos[a + b*x + c*x^2
]])/(2*c), x] + Dist[(2*c*d - b*e)/(2*c), Int[Tan[a + b*x + c*x^2], x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ
[2*c*d - b*e, 0]

Rubi steps

\begin{align*} \int \left (\frac{b \tan \left (a+b x+c x^2\right )}{2 c}+x \tan \left (a+b x+c x^2\right )\right ) \, dx &=\frac{b \int \tan \left (a+b x+c x^2\right ) \, dx}{2 c}+\int x \tan \left (a+b x+c x^2\right ) \, dx\\ &=-\frac{\log \left (\cos \left (a+b x+c x^2\right )\right )}{2 c}\\ \end{align*}

Mathematica [A]  time = 0.676132, size = 18, normalized size = 0.95 \[ -\frac{\log (\cos (a+x (b+c x)))}{2 c} \]

Antiderivative was successfully verified.

[In]

Integrate[(b*Tan[a + b*x + c*x^2])/(2*c) + x*Tan[a + b*x + c*x^2],x]

[Out]

-Log[Cos[a + x*(b + c*x)]]/(2*c)

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Maple [A]  time = 0.03, size = 18, normalized size = 1. \begin{align*} -{\frac{\ln \left ( \cos \left ( c{x}^{2}+bx+a \right ) \right ) }{2\,c}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/2*b*tan(c*x^2+b*x+a)/c+x*tan(c*x^2+b*x+a),x)

[Out]

-1/2*ln(cos(c*x^2+b*x+a))/c

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Maxima [B]  time = 1.20718, size = 112, normalized size = 5.89 \begin{align*} -\frac{\log \left (\cos \left (2 \, c x^{2}\right )^{2} + 2 \, \cos \left (2 \, c x^{2}\right ) \cos \left (2 \, b x + 2 \, a\right ) + \cos \left (2 \, b x + 2 \, a\right )^{2} + \sin \left (2 \, c x^{2}\right )^{2} - 2 \, \sin \left (2 \, c x^{2}\right ) \sin \left (2 \, b x + 2 \, a\right ) + \sin \left (2 \, b x + 2 \, a\right )^{2}\right )}{4 \, c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*b*tan(c*x^2+b*x+a)/c+x*tan(c*x^2+b*x+a),x, algorithm="maxima")

[Out]

-1/4*log(cos(2*c*x^2)^2 + 2*cos(2*c*x^2)*cos(2*b*x + 2*a) + cos(2*b*x + 2*a)^2 + sin(2*c*x^2)^2 - 2*sin(2*c*x^
2)*sin(2*b*x + 2*a) + sin(2*b*x + 2*a)^2)/c

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Fricas [A]  time = 2.37484, size = 59, normalized size = 3.11 \begin{align*} -\frac{\log \left (\frac{1}{\tan \left (c x^{2} + b x + a\right )^{2} + 1}\right )}{4 \, c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*b*tan(c*x^2+b*x+a)/c+x*tan(c*x^2+b*x+a),x, algorithm="fricas")

[Out]

-1/4*log(1/(tan(c*x^2 + b*x + a)^2 + 1))/c

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int b \tan{\left (a + b x + c x^{2} \right )}\, dx + \int 2 c x \tan{\left (a + b x + c x^{2} \right )}\, dx}{2 c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*b*tan(c*x**2+b*x+a)/c+x*tan(c*x**2+b*x+a),x)

[Out]

(Integral(b*tan(a + b*x + c*x**2), x) + Integral(2*c*x*tan(a + b*x + c*x**2), x))/(2*c)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x \tan \left (c x^{2} + b x + a\right ) + \frac{b \tan \left (c x^{2} + b x + a\right )}{2 \, c}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*b*tan(c*x^2+b*x+a)/c+x*tan(c*x^2+b*x+a),x, algorithm="giac")

[Out]

integrate(x*tan(c*x^2 + b*x + a) + 1/2*b*tan(c*x^2 + b*x + a)/c, x)