3.16 \(\int (\frac{x^2}{\tan ^{\frac{3}{2}}(a+b x)}-\frac{4 x}{b \sqrt{\tan (a+b x)}}+x^2 \sqrt{\tan (a+b x)}) \, dx\)

Optimal. Leaf size=18 \[ -\frac{2 x^2}{b \sqrt{\tan (a+b x)}} \]

[Out]

(-2*x^2)/(b*Sqrt[Tan[a + b*x]])

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Rubi [A]  time = 0.122961, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 45, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.022, Rules used = {3721} \[ -\frac{2 x^2}{b \sqrt{\tan (a+b x)}} \]

Antiderivative was successfully verified.

[In]

Int[x^2/Tan[a + b*x]^(3/2) - (4*x)/(b*Sqrt[Tan[a + b*x]]) + x^2*Sqrt[Tan[a + b*x]],x]

[Out]

(-2*x^2)/(b*Sqrt[Tan[a + b*x]])

Rule 3721

Int[((c_.) + (d_.)*(x_))^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[((c + d*x)^m*(b*Tan[e +
f*x])^(n + 1))/(b*f*(n + 1)), x] + (-Dist[(d*m)/(b*f*(n + 1)), Int[(c + d*x)^(m - 1)*(b*Tan[e + f*x])^(n + 1)
, x], x] - Dist[1/b^2, Int[(c + d*x)^m*(b*Tan[e + f*x])^(n + 2), x], x]) /; FreeQ[{b, c, d, e, f}, x] && LtQ[n
, -1] && GtQ[m, 0]

Rubi steps

\begin{align*} \int \left (\frac{x^2}{\tan ^{\frac{3}{2}}(a+b x)}-\frac{4 x}{b \sqrt{\tan (a+b x)}}+x^2 \sqrt{\tan (a+b x)}\right ) \, dx &=-\frac{4 \int \frac{x}{\sqrt{\tan (a+b x)}} \, dx}{b}+\int \frac{x^2}{\tan ^{\frac{3}{2}}(a+b x)} \, dx+\int x^2 \sqrt{\tan (a+b x)} \, dx\\ &=-\frac{2 x^2}{b \sqrt{\tan (a+b x)}}\\ \end{align*}

Mathematica [A]  time = 0.885471, size = 18, normalized size = 1. \[ -\frac{2 x^2}{b \sqrt{\tan (a+b x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2/Tan[a + b*x]^(3/2) - (4*x)/(b*Sqrt[Tan[a + b*x]]) + x^2*Sqrt[Tan[a + b*x]],x]

[Out]

(-2*x^2)/(b*Sqrt[Tan[a + b*x]])

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Maple [F]  time = 0.218, size = 0, normalized size = 0. \begin{align*} \int -4\,{\frac{x}{b\sqrt{\tan \left ( bx+a \right ) }}}+{x}^{2}\sqrt{\tan \left ( bx+a \right ) }+{{x}^{2} \left ( \tan \left ( bx+a \right ) \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-4*x/b/tan(b*x+a)^(1/2)+x^2*tan(b*x+a)^(1/2)+x^2/tan(b*x+a)^(3/2),x)

[Out]

int(-4*x/b/tan(b*x+a)^(1/2)+x^2*tan(b*x+a)^(1/2)+x^2/tan(b*x+a)^(3/2),x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-4*x/b/tan(b*x+a)^(1/2)+x^2*tan(b*x+a)^(1/2)+x^2/tan(b*x+a)^(3/2),x, algorithm="maxima")

[Out]

integrate(x^2*sqrt(tan(b*x + a)) + x^2/tan(b*x + a)^(3/2) - 4*x/(b*sqrt(tan(b*x + a))), x)

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Fricas [A]  time = 1.61129, size = 42, normalized size = 2.33 \begin{align*} -\frac{2 \, x^{2}}{b \sqrt{\tan \left (b x + a\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-4*x/b/tan(b*x+a)^(1/2)+x^2*tan(b*x+a)^(1/2)+x^2/tan(b*x+a)^(3/2),x, algorithm="fricas")

[Out]

-2*x^2/(b*sqrt(tan(b*x + a)))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int - \frac{4 x}{\sqrt{\tan{\left (a + b x \right )}}}\, dx + \int \frac{b x^{2}}{\tan ^{\frac{3}{2}}{\left (a + b x \right )}}\, dx + \int b x^{2} \sqrt{\tan{\left (a + b x \right )}}\, dx}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-4*x/b/tan(b*x+a)**(1/2)+x**2*tan(b*x+a)**(1/2)+x**2/tan(b*x+a)**(3/2),x)

[Out]

(Integral(-4*x/sqrt(tan(a + b*x)), x) + Integral(b*x**2/tan(a + b*x)**(3/2), x) + Integral(b*x**2*sqrt(tan(a +
b*x)), x))/b

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-4*x/b/tan(b*x+a)^(1/2)+x^2*tan(b*x+a)^(1/2)+x^2/tan(b*x+a)^(3/2),x, algorithm="giac")

[Out]

Exception raised: RuntimeError