3.376 \(\int \frac{1-3 \tan (x)}{\sqrt{4+3 \tan (x)}} \, dx\)

Optimal. Leaf size=27 \[ \sqrt{2} \tanh ^{-1}\left (\frac{\tan (x)+3}{\sqrt{2} \sqrt{3 \tan (x)+4}}\right ) \]

[Out]

Sqrt[2]*ArcTanh[(3 + Tan[x])/(Sqrt[2]*Sqrt[4 + 3*Tan[x]])]

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Rubi [A]  time = 0.0316424, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {3535, 207} \[ \sqrt{2} \tanh ^{-1}\left (\frac{\tan (x)+3}{\sqrt{2} \sqrt{3 \tan (x)+4}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[(1 - 3*Tan[x])/Sqrt[4 + 3*Tan[x]],x]

[Out]

Sqrt[2]*ArcTanh[(3 + Tan[x])/(Sqrt[2]*Sqrt[4 + 3*Tan[x]])]

Rule 3535

Int[((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])/Sqrt[(a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]], x_Symbol] :> Dist[(-2*
d^2)/f, Subst[Int[1/(2*b*c*d - 4*a*d^2 + x^2), x], x, (b*c - 2*a*d - b*d*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]
]], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && EqQ[2
*a*c*d - b*(c^2 - d^2), 0]

Rule 207

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTanh[(Rt[b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{1-3 \tan (x)}{\sqrt{4+3 \tan (x)}} \, dx &=-\left (18 \operatorname{Subst}\left (\int \frac{1}{-162+x^2} \, dx,x,\frac{27+9 \tan (x)}{\sqrt{4+3 \tan (x)}}\right )\right )\\ &=\sqrt{2} \tanh ^{-1}\left (\frac{3+\tan (x)}{\sqrt{2} \sqrt{4+3 \tan (x)}}\right )\\ \end{align*}

Mathematica [C]  time = 0.115579, size = 65, normalized size = 2.41 \[ \frac{1}{5} \left ((3+i) \sqrt{4-3 i} \tanh ^{-1}\left (\frac{\sqrt{3 \tan (x)+4}}{\sqrt{4-3 i}}\right )+(3-i) \sqrt{4+3 i} \tanh ^{-1}\left (\frac{\sqrt{3 \tan (x)+4}}{\sqrt{4+3 i}}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(1 - 3*Tan[x])/Sqrt[4 + 3*Tan[x]],x]

[Out]

((3 + I)*Sqrt[4 - 3*I]*ArcTanh[Sqrt[4 + 3*Tan[x]]/Sqrt[4 - 3*I]] + (3 - I)*Sqrt[4 + 3*I]*ArcTanh[Sqrt[4 + 3*Ta
n[x]]/Sqrt[4 + 3*I]])/5

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Maple [B]  time = 0.076, size = 52, normalized size = 1.9 \begin{align*}{\frac{\sqrt{2}}{2}\ln \left ( 9+3\,\tan \left ( x \right ) +3\,\sqrt{4+3\,\tan \left ( x \right ) }\sqrt{2} \right ) }-{\frac{\sqrt{2}}{2}\ln \left ( 9+3\,\tan \left ( x \right ) -3\,\sqrt{4+3\,\tan \left ( x \right ) }\sqrt{2} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*2^(1/2)*ln(9+3*tan(x)+3*(4+3*tan(x))^(1/2)*2^(1/2))-1/2*2^(1/2)*ln(9+3*tan(x)-3*(4+3*tan(x))^(1/2)*2^(1/2)
)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\int \frac{3 \, \tan \left (x\right ) - 1}{\sqrt{3 \, \tan \left (x\right ) + 4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-integrate((3*tan(x) - 1)/sqrt(3*tan(x) + 4), x)

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Fricas [B]  time = 1.01568, size = 153, normalized size = 5.67 \begin{align*} \frac{1}{2} \, \sqrt{2} \log \left (\frac{\tan \left (x\right )^{2} + 2 \,{\left (\sqrt{2} \tan \left (x\right ) + 3 \, \sqrt{2}\right )} \sqrt{3 \, \tan \left (x\right ) + 4} + 12 \, \tan \left (x\right ) + 17}{\tan \left (x\right )^{2} + 1}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*sqrt(2)*log((tan(x)^2 + 2*(sqrt(2)*tan(x) + 3*sqrt(2))*sqrt(3*tan(x) + 4) + 12*tan(x) + 17)/(tan(x)^2 + 1)
)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \int \frac{3 \tan{\left (x \right )}}{\sqrt{3 \tan{\left (x \right )} + 4}}\, dx - \int - \frac{1}{\sqrt{3 \tan{\left (x \right )} + 4}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-Integral(3*tan(x)/sqrt(3*tan(x) + 4), x) - Integral(-1/sqrt(3*tan(x) + 4), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{3 \, \tan \left (x\right ) - 1}{\sqrt{3 \, \tan \left (x\right ) + 4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(-(3*tan(x) - 1)/sqrt(3*tan(x) + 4), x)