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

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 = 0.146582, 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, Rules used = {4342, 43} \[ \frac{2}{3} \sqrt{4-3 \tan (x)}+\frac{8}{3 \sqrt{4-3 \tan (x)}}+\frac{1}{3} \log (4-3 \tan (x)) \]

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

Rule 4342

Int[(u_)*(F_)[(c_.)*((a_.) + (b_.)*(x_))]^2, x_Symbol] :> With[{d = FreeFactors[Tan[c*(a + b*x)], x]}, Dist[d/
(b*c), Subst[Int[SubstFor[1, Tan[c*(a + b*x)]/d, u, x], x], x, Tan[c*(a + b*x)]/d], x] /; FunctionOfQ[Tan[c*(a
+ b*x)]/d, u, x, True]] /; FreeQ[{a, b, c}, x] && NonsumQ[u] && (EqQ[F, Sec] || EqQ[F, sec])

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{\sec ^2(x) \left (-\sqrt{4-3 \tan (x)}+3 \tan (x)\right )}{(4-3 \tan (x))^{3/2}} \, dx &=\operatorname{Subst}\left (\int \left (\frac{3 x}{(4-3 x)^{3/2}}+\frac{1}{-4+3 x}\right ) \, dx,x,\tan (x)\right )\\ &=\frac{1}{3} \log (4-3 \tan (x))+3 \operatorname{Subst}\left (\int \frac{x}{(4-3 x)^{3/2}} \, dx,x,\tan (x)\right )\\ &=\frac{1}{3} \log (4-3 \tan (x))+3 \operatorname{Subst}\left (\int \left (\frac{4}{3 (4-3 x)^{3/2}}-\frac{1}{3 \sqrt{4-3 x}}\right ) \, dx,x,\tan (x)\right )\\ &=\frac{1}{3} \log (4-3 \tan (x))+\frac{8}{3 \sqrt{4-3 \tan (x)}}+\frac{2}{3} \sqrt{4-3 \tan (x)}\\ \end{align*}

Mathematica [A]  time = 1.20141, 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 = 0.461, size = 219, normalized size = 5.5 \begin{align*}{\frac{ \left ( \cos \left ( x \right ) -1 \right ) ^{2} \left ( \cos \left ( x \right ) +1 \right ) ^{2}}{ \left ( 12\,\cos \left ( x \right ) -9\,\sin \left ( x \right ) \right ) \left ( \sin \left ( x \right ) \right ) ^{4}} \left ( 16\,\sqrt{{\frac{4\,\cos \left ( x \right ) -3\,\sin \left ( x \right ) }{\cos \left ( x \right ) }}}\cos \left ( x \right ) +4\,\cos \left ( x \right ) \ln \left ( -{\frac{\cos \left ( x \right ) -2\,\sin \left ( x \right ) -1}{\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 ) +\sin \left ( x \right ) -2}{\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 ) -6\,\sqrt{{\frac{4\,\cos \left ( x \right ) -3\,\sin \left ( x \right ) }{\cos \left ( x \right ) }}}\sin \left ( x \right ) -3\,\sin \left ( x \right ) \ln \left ( -{\frac{\cos \left ( x \right ) -2\,\sin \left ( x \right ) -1}{\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 ) +\sin \left ( x \right ) -2}{\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 ) \right ) } \end{align*}

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*(cos(x)+1)^2*(16*((4*cos(x)-3*sin(x))/cos(x))^(1/2)*cos(x)+4*cos(x)*ln(-(cos(x)-2*sin(x)-1)/s
in(x))-4*cos(x)*ln(-(-sin(x)-1+cos(x))/sin(x))+4*cos(x)*ln(-(2*cos(x)+sin(x)-2)/sin(x))-4*cos(x)*ln(-(sin(x)-1
+cos(x))/sin(x))-6*((4*cos(x)-3*sin(x))/cos(x))^(1/2)*sin(x)-3*sin(x)*ln(-(cos(x)-2*sin(x)-1)/sin(x))+3*sin(x)
*ln(-(-sin(x)-1+cos(x))/sin(x))-3*sin(x)*ln(-(2*cos(x)+sin(x)-2)/sin(x))+3*sin(x)*ln(-(sin(x)-1+cos(x))/sin(x)
))/(4*cos(x)-3*sin(x))/sin(x)^4

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Maxima [A]  time = 0.934669, size = 41, normalized size = 1.02 \begin{align*} \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 ) \end{align*}

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, 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 [B]  time = 2.07839, size = 259, normalized size = 6.48 \begin{align*} \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 )}} \end{align*}

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, 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. \begin{align*} \text{Timed out} \end{align*}

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 [A]  time = 1.20298, size = 42, normalized size = 1.05 \begin{align*} \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 ({\left | -3 \, \tan \left (x\right ) + 4 \right |}\right ) \end{align*}

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, algorithm="giac")

[Out]

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