3.400 \(\int \frac{\sec ^2(x) \left (-\sqrt{4-3 \tan (x)}+3 \tan (x)\right )}{(4-3 \tan (x))^{3/2}} \, dx\)

Optimal. Leaf size=40 \[ \frac{2}{3} \sqrt{4-3 \tan (x)}+\frac{8}{3 \sqrt{4-3 \tan (x)}}+\frac{1}{3} \log (4-3 \tan (x)) \]

[Out]

Log[4 - 3*Tan[x]]/3 + 8/(3*Sqrt[4 - 3*Tan[x]]) + (2*Sqrt[4 - 3*Tan[x]])/3

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Rubi [A]  time = 0.223963, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062 \[ \frac{2}{3} \sqrt{4-3 \tan (x)}+\frac{8}{3 \sqrt{4-3 \tan (x)}}+\frac{1}{3} \log (4-3 \tan (x)) \]

Antiderivative was successfully verified.

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

[Out]

Log[4 - 3*Tan[x]]/3 + 8/(3*Sqrt[4 - 3*Tan[x]]) + (2*Sqrt[4 - 3*Tan[x]])/3

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Rubi in Sympy [A]  time = 7.678, size = 36, normalized size = 0.9 \[ \frac{2 \sqrt{- 3 \tan{\left (x \right )} + 4}}{3} + \frac{\log{\left (- 3 \tan{\left (x \right )} + 4 \right )}}{3} + \frac{8}{3 \sqrt{- 3 \tan{\left (x \right )} + 4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-(4-3*tan(x))**(1/2)+3*tan(x))/cos(x)**2/(4-3*tan(x))**(3/2),x)

[Out]

2*sqrt(-3*tan(x) + 4)/3 + log(-3*tan(x) + 4)/3 + 8/(3*sqrt(-3*tan(x) + 4))

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Mathematica [A]  time = 1.43235, size = 38, normalized size = 0.95 \[ \frac{-6 \tan (x)+\sqrt{4-3 \tan (x)} \log (4-3 \tan (x))+16}{3 \sqrt{4-3 \tan (x)}} \]

Antiderivative was successfully verified.

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

[Out]

(16 + Log[4 - 3*Tan[x]]*Sqrt[4 - 3*Tan[x]] - 6*Tan[x])/(3*Sqrt[4 - 3*Tan[x]])

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Maple [B]  time = 0.564, size = 227, normalized size = 5.7 \[{\frac{ \left ( \cos \left ( x \right ) -1 \right ) ^{2} \left ( 1+\cos \left ( x \right ) \right ) ^{2}}{ \left ( -12\,\cos \left ( x \right ) +9\,\sin \left ( x \right ) \right ) \left ( \sin \left ( x \right ) \right ) ^{4}} \left ( 3\,\sin \left ( x \right ) \sqrt{-2\,{\frac{-4\,\cos \left ( x \right ) +3\,\sin \left ( x \right ) }{\cos \left ( x \right ) }}}\sqrt{2}-8\,\cos \left ( x \right ) \sqrt{-2\,{\frac{-4\,\cos \left ( x \right ) +3\,\sin \left ( x \right ) }{\cos \left ( x \right ) }}}\sqrt{2}+3\,\sin \left ( x \right ) \ln \left ({\frac{-\cos \left ( x \right ) +1+2\,\sin \left ( x \right ) }{\sin \left ( x \right ) }} \right ) -3\,\sin \left ( x \right ) \ln \left ({\frac{1-\cos \left ( x \right ) +\sin \left ( x \right ) }{\sin \left ( x \right ) }} \right ) -3\,\sin \left ( x \right ) \ln \left ( -{\frac{\sin \left ( x \right ) -1+\cos \left ( x \right ) }{\sin \left ( x \right ) }} \right ) +3\,\sin \left ( x \right ) \ln \left ( -{\frac{2\,\cos \left ( x \right ) -2+\sin \left ( x \right ) }{\sin \left ( x \right ) }} \right ) -4\,\cos \left ( x \right ) \ln \left ({\frac{-\cos \left ( x \right ) +1+2\,\sin \left ( x \right ) }{\sin \left ( x \right ) }} \right ) +4\,\cos \left ( x \right ) \ln \left ({\frac{1-\cos \left ( x \right ) +\sin \left ( x \right ) }{\sin \left ( x \right ) }} \right ) +4\,\cos \left ( x \right ) \ln \left ( -{\frac{\sin \left ( x \right ) -1+\cos \left ( x \right ) }{\sin \left ( x \right ) }} \right ) -4\,\cos \left ( x \right ) \ln \left ( -{\frac{2\,\cos \left ( x \right ) -2+\sin \left ( x \right ) }{\sin \left ( x \right ) }} \right ) \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-(4-3*tan(x))^(1/2)+3*tan(x))/cos(x)^2/(4-3*tan(x))^(3/2),x)

[Out]

1/3*(cos(x)-1)^2*(1+cos(x))^2*(3*sin(x)*(-2*(-4*cos(x)+3*sin(x))/cos(x))^(1/2)*2
^(1/2)-8*cos(x)*(-2*(-4*cos(x)+3*sin(x))/cos(x))^(1/2)*2^(1/2)+3*sin(x)*ln((-cos
(x)+1+2*sin(x))/sin(x))-3*sin(x)*ln((1-cos(x)+sin(x))/sin(x))-3*sin(x)*ln(-(sin(
x)-1+cos(x))/sin(x))+3*sin(x)*ln(-(2*cos(x)-2+sin(x))/sin(x))-4*cos(x)*ln((-cos(
x)+1+2*sin(x))/sin(x))+4*cos(x)*ln((1-cos(x)+sin(x))/sin(x))+4*cos(x)*ln(-(sin(x
)-1+cos(x))/sin(x))-4*cos(x)*ln(-(2*cos(x)-2+sin(x))/sin(x)))/(-4*cos(x)+3*sin(x
))/sin(x)^4

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Maxima [A]  time = 1.35364, size = 41, normalized size = 1.02 \[ \frac{2}{3} \, \sqrt{-3 \, \tan \left (x\right ) + 4} + \frac{8}{3 \, \sqrt{-3 \, \tan \left (x\right ) + 4}} + \frac{1}{3} \, \log \left (-3 \, \tan \left (x\right ) + 4\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(sqrt(-3*tan(x) + 4) - 3*tan(x))/((-3*tan(x) + 4)^(3/2)*cos(x)^2),x, algorithm="maxima")

[Out]

2/3*sqrt(-3*tan(x) + 4) + 8/3/sqrt(-3*tan(x) + 4) + 1/3*log(-3*tan(x) + 4)

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Fricas [A]  time = 0.241023, size = 111, normalized size = 2.78 \[ \frac{{\left (4 \, \cos \left (x\right ) - 3 \, \sin \left (x\right )\right )} \log \left (\frac{7}{4} \, \cos \left (x\right )^{2} - 6 \, \cos \left (x\right ) \sin \left (x\right ) + \frac{9}{4}\right ) -{\left (4 \, \cos \left (x\right ) - 3 \, \sin \left (x\right )\right )} \log \left (\cos \left (x\right )^{2}\right ) + 4 \, \sqrt{\frac{4 \, \cos \left (x\right ) - 3 \, \sin \left (x\right )}{\cos \left (x\right )}}{\left (8 \, \cos \left (x\right ) - 3 \, \sin \left (x\right )\right )}}{6 \,{\left (4 \, \cos \left (x\right ) - 3 \, \sin \left (x\right )\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(sqrt(-3*tan(x) + 4) - 3*tan(x))/((-3*tan(x) + 4)^(3/2)*cos(x)^2),x, algorithm="fricas")

[Out]

1/6*((4*cos(x) - 3*sin(x))*log(7/4*cos(x)^2 - 6*cos(x)*sin(x) + 9/4) - (4*cos(x)
- 3*sin(x))*log(cos(x)^2) + 4*sqrt((4*cos(x) - 3*sin(x))/cos(x))*(8*cos(x) - 3*
sin(x)))/(4*cos(x) - 3*sin(x))

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-(4-3*tan(x))**(1/2)+3*tan(x))/cos(x)**2/(4-3*tan(x))**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.21784, size = 42, normalized size = 1.05 \[ \frac{2}{3} \, \sqrt{-3 \, \tan \left (x\right ) + 4} + \frac{8}{3 \, \sqrt{-3 \, \tan \left (x\right ) + 4}} + \frac{1}{3} \,{\rm ln}\left ({\left | -3 \, \tan \left (x\right ) + 4 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(sqrt(-3*tan(x) + 4) - 3*tan(x))/((-3*tan(x) + 4)^(3/2)*cos(x)^2),x, algorithm="giac")

[Out]

2/3*sqrt(-3*tan(x) + 4) + 8/3/sqrt(-3*tan(x) + 4) + 1/3*ln(abs(-3*tan(x) + 4))