\(\int \frac {1-3 \tan (x)}{\sqrt {4+3 \tan (x)}} \, dx\) [376]

Optimal result

Integrand size = 17, antiderivative size = 27 \[ \int \frac {1-3 \tan (x)}{\sqrt {4+3 \tan (x)}} \, dx=\sqrt {2} \text {arctanh}\left (\frac {3+\tan (x)}{\sqrt {2} \sqrt {4+3 \tan (x)}}\right ) \]

[Out]

arctanh(1/2*(3+tan(x))*2^(1/2)/(4+3*tan(x))^(1/2))*2^(1/2)

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {3616, 213} \[ \int \frac {1-3 \tan (x)}{\sqrt {4+3 \tan (x)}} \, dx=\sqrt {2} \text {arctanh}\left (\frac {\tan (x)+3}{\sqrt {2} \sqrt {3 \tan (x)+4}}\right ) \]

[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 213

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

Rule 3616

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]

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.93 (sec) , antiderivative size = 61, normalized size of antiderivative = 2.26 \[ \int \frac {1-3 \tan (x)}{\sqrt {4+3 \tan (x)}} \, dx=\frac {(3-i) \text {arctanh}\left (\frac {\sqrt {4+3 \tan (x)}}{\sqrt {4-3 i}}\right )}{\sqrt {4-3 i}}+\frac {(3+i) \text {arctanh}\left (\frac {\sqrt {4+3 \tan (x)}}{\sqrt {4+3 i}}\right )}{\sqrt {4+3 i}} \]

[In]

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

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(51\) vs. \(2(22)=44\).

Time = 0.05 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.93

method	result	size
derivativedivides	\(-\frac {\sqrt {2}\, \ln \left (9+3 \tan \left (x \right )-3 \sqrt {4+3 \tan \left (x \right )}\, \sqrt {2}\right )}{2}+\frac {\sqrt {2}\, \ln \left (9+3 \tan \left (x \right )+3 \sqrt {4+3 \tan \left (x \right )}\, \sqrt {2}\right )}{2}\)	\(52\)
default	\(-\frac {\sqrt {2}\, \ln \left (9+3 \tan \left (x \right )-3 \sqrt {4+3 \tan \left (x \right )}\, \sqrt {2}\right )}{2}+\frac {\sqrt {2}\, \ln \left (9+3 \tan \left (x \right )+3 \sqrt {4+3 \tan \left (x \right )}\, \sqrt {2}\right )}{2}\)	\(52\)
parts	\(-\frac {\sqrt {2}\, \ln \left (9+3 \tan \left (x \right )-3 \sqrt {4+3 \tan \left (x \right )}\, \sqrt {2}\right )}{2}+\frac {\sqrt {2}\, \ln \left (9+3 \tan \left (x \right )+3 \sqrt {4+3 \tan \left (x \right )}\, \sqrt {2}\right )}{2}\)	\(52\)

[In]

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

[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
))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 47 vs. \(2 (22) = 44\).

Time = 0.25 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.74 \[ \int \frac {1-3 \tan (x)}{\sqrt {4+3 \tan (x)}} \, dx=\frac {1}{2} \, \sqrt {2} \log \left (\frac {\tan \left (x\right )^{2} + 2 \, {\left (\sqrt {2} \tan \left (x\right ) + 3 \, \sqrt {2}\right )} \sqrt {3 \, \tan \left (x\right ) + 4} + 12 \, \tan \left (x\right ) + 17}{\tan \left (x\right )^{2} + 1}\right ) \]

[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)
)

Sympy [F]

\[ \int \frac {1-3 \tan (x)}{\sqrt {4+3 \tan (x)}} \, dx=- \int \frac {3 \tan {\left (x \right )}}{\sqrt {3 \tan {\left (x \right )} + 4}}\, dx - \int \left (- \frac {1}{\sqrt {3 \tan {\left (x \right )} + 4}}\right )\, dx \]

[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)

Maxima [F]

\[ \int \frac {1-3 \tan (x)}{\sqrt {4+3 \tan (x)}} \, dx=\int { -\frac {3 \, \tan \left (x\right ) - 1}{\sqrt {3 \, \tan \left (x\right ) + 4}} \,d x } \]

[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)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 57 vs. \(2 (22) = 44\).

Time = 0.30 (sec) , antiderivative size = 57, normalized size of antiderivative = 2.11 \[ \int \frac {1-3 \tan (x)}{\sqrt {4+3 \tan (x)}} \, dx=\frac {1}{2} \, \sqrt {2} \log \left (\frac {3}{5} \cdot 25^{\frac {1}{4}} \sqrt {10} \sqrt {3 \, \tan \left (x\right ) + 4} + 3 \, \tan \left (x\right ) + 9\right ) - \frac {1}{2} \, \sqrt {2} \log \left (-\frac {3}{5} \cdot 25^{\frac {1}{4}} \sqrt {10} \sqrt {3 \, \tan \left (x\right ) + 4} + 3 \, \tan \left (x\right ) + 9\right ) \]

[In]

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

[Out]

1/2*sqrt(2)*log(3/5*25^(1/4)*sqrt(10)*sqrt(3*tan(x) + 4) + 3*tan(x) + 9) - 1/2*sqrt(2)*log(-3/5*25^(1/4)*sqrt(
10)*sqrt(3*tan(x) + 4) + 3*tan(x) + 9)

Mupad [B] (verification not implemented)

Time = 0.69 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.30 \[ \int \frac {1-3 \tan (x)}{\sqrt {4+3 \tan (x)}} \, dx=\sqrt {2}\,\left (\mathrm {atan}\left (\sqrt {6\,\mathrm {tan}\left (x\right )+8}\,\left (\frac {1}{10}-\frac {3}{10}{}\mathrm {i}\right )\right )-\mathrm {atan}\left (\sqrt {6\,\mathrm {tan}\left (x\right )+8}\,\left (\frac {1}{10}+\frac {3}{10}{}\mathrm {i}\right )\right )\right )\,1{}\mathrm {i} \]

[In]

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

[Out]

2^(1/2)*(atan((6*tan(x) + 8)^(1/2)*(1/10 - 3i/10)) - atan((6*tan(x) + 8)^(1/2)*(1/10 + 3i/10)))*1i