Integrand size = 32, antiderivative size = 40 \[ \int \frac {\sec ^2(x) \left (-\sqrt {4-3 \tan (x)}+3 \tan (x)\right )}{(4-3 \tan (x))^{3/2}} \, dx=\frac {1}{3} \log (4-3 \tan (x))+\frac {8}{3 \sqrt {4-3 \tan (x)}}+\frac {2}{3} \sqrt {4-3 \tan (x)} \] Output:
1/3*ln(4-3*tan(x))+8/3/(4-3*tan(x))^(1/2)+2/3*(4-3*tan(x))^(1/2)
Time = 5.12 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.75 \[ \int \frac {\sec ^2(x) \left (-\sqrt {4-3 \tan (x)}+3 \tan (x)\right )}{(4-3 \tan (x))^{3/2}} \, dx=\frac {1}{3} \left (\log (4-3 \tan (x))+\frac {2 (8-3 \tan (x))}{\sqrt {4-3 \tan (x)}}\right ) \] Input:
Integrate[(Sec[x]^2*(-Sqrt[4 - 3*Tan[x]] + 3*Tan[x]))/(4 - 3*Tan[x])^(3/2) ,x]
Output:
(Log[4 - 3*Tan[x]] + (2*(8 - 3*Tan[x]))/Sqrt[4 - 3*Tan[x]])/3
Time = 0.32 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.094, Rules used = {3042, 4842, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\left (3 \tan (x)-\sqrt {4-3 \tan (x)}\right ) \sec ^2(x)}{(4-3 \tan (x))^{3/2}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\left (3 \tan (x)-\sqrt {4-3 \tan (x)}\right ) \sec (x)^2}{(4-3 \tan (x))^{3/2}}dx\) |
\(\Big \downarrow \) 4842 |
\(\displaystyle \int \left (\frac {3 \tan (x)}{(4-3 \tan (x))^{3/2}}+\frac {1}{3 \tan (x)-4}\right )d\tan (x)\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {2}{3} \sqrt {4-3 \tan (x)}+\frac {8}{3 \sqrt {4-3 \tan (x)}}+\frac {1}{3} \log (4-3 \tan (x))\) |
Input:
Int[(Sec[x]^2*(-Sqrt[4 - 3*Tan[x]] + 3*Tan[x]))/(4 - 3*Tan[x])^(3/2),x]
Output:
Log[4 - 3*Tan[x]]/3 + 8/(3*Sqrt[4 - 3*Tan[x]]) + (2*Sqrt[4 - 3*Tan[x]])/3
Int[(u_)*(F_)[(c_.)*((a_.) + (b_.)*(x_))]^2, x_Symbol] :> With[{d = FreeFac tors[Tan[c*(a + b*x)], x]}, Simp[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])
Time = 0.57 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.78
method | result | size |
derivativedivides | \(\frac {\ln \left (4-3 \tan \left (x \right )\right )}{3}+\frac {8}{3 \sqrt {4-3 \tan \left (x \right )}}+\frac {2 \sqrt {4-3 \tan \left (x \right )}}{3}\) | \(31\) |
default | \(\frac {\ln \left (4-3 \tan \left (x \right )\right )}{3}+\frac {8}{3 \sqrt {4-3 \tan \left (x \right )}}+\frac {2 \sqrt {4-3 \tan \left (x \right )}}{3}\) | \(31\) |
Input:
int((-(4-3*tan(x))^(1/2)+3*tan(x))/cos(x)^2/(4-3*tan(x))^(3/2),x,method=_R ETURNVERBOSE)
Output:
1/3*ln(4-3*tan(x))+8/3/(4-3*tan(x))^(1/2)+2/3*(4-3*tan(x))^(1/2)
Leaf count of result is larger than twice the leaf count of optimal. 82 vs. \(2 (30) = 60\).
Time = 0.08 (sec) , antiderivative size = 82, normalized size of antiderivative = 2.05 \[ \int \frac {\sec ^2(x) \left (-\sqrt {4-3 \tan (x)}+3 \tan (x)\right )}{(4-3 \tan (x))^{3/2}} \, dx=\frac {{\left (4 \, \cos \left (x\right ) - 3 \, \sin \left (x\right )\right )} \log \left (\frac {7}{4} \, \cos \left (x\right )^{2} - 6 \, \cos \left (x\right ) \sin \left (x\right ) + \frac {9}{4}\right ) - {\left (4 \, \cos \left (x\right ) - 3 \, \sin \left (x\right )\right )} \log \left (\cos \left (x\right )^{2}\right ) + 4 \, \sqrt {\frac {4 \, \cos \left (x\right ) - 3 \, \sin \left (x\right )}{\cos \left (x\right )}} {\left (8 \, \cos \left (x\right ) - 3 \, \sin \left (x\right )\right )}}{6 \, {\left (4 \, \cos \left (x\right ) - 3 \, \sin \left (x\right )\right )}} \] Input:
integrate((-(4-3*tan(x))^(1/2)+3*tan(x))/cos(x)^2/(4-3*tan(x))^(3/2),x, al gorithm="fricas")
Output:
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))
\[ \int \frac {\sec ^2(x) \left (-\sqrt {4-3 \tan (x)}+3 \tan (x)\right )}{(4-3 \tan (x))^{3/2}} \, dx=- \int \frac {\sqrt {4 - 3 \tan {\left (x \right )}}}{- 3 \sqrt {4 - 3 \tan {\left (x \right )}} \cos ^{2}{\left (x \right )} \tan {\left (x \right )} + 4 \sqrt {4 - 3 \tan {\left (x \right )}} \cos ^{2}{\left (x \right )}}\, dx - \int \left (- \frac {3 \tan {\left (x \right )}}{- 3 \sqrt {4 - 3 \tan {\left (x \right )}} \cos ^{2}{\left (x \right )} \tan {\left (x \right )} + 4 \sqrt {4 - 3 \tan {\left (x \right )}} \cos ^{2}{\left (x \right )}}\right )\, dx \] Input:
integrate((-(4-3*tan(x))**(1/2)+3*tan(x))/cos(x)**2/(4-3*tan(x))**(3/2),x)
Output:
-Integral(sqrt(4 - 3*tan(x))/(-3*sqrt(4 - 3*tan(x))*cos(x)**2*tan(x) + 4*s qrt(4 - 3*tan(x))*cos(x)**2), x) - Integral(-3*tan(x)/(-3*sqrt(4 - 3*tan(x ))*cos(x)**2*tan(x) + 4*sqrt(4 - 3*tan(x))*cos(x)**2), x)
Time = 0.03 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.75 \[ \int \frac {\sec ^2(x) \left (-\sqrt {4-3 \tan (x)}+3 \tan (x)\right )}{(4-3 \tan (x))^{3/2}} \, dx=\frac {2}{3} \, \sqrt {-3 \, \tan \left (x\right ) + 4} + \frac {8}{3 \, \sqrt {-3 \, \tan \left (x\right ) + 4}} + \frac {1}{3} \, \log \left (-3 \, \tan \left (x\right ) + 4\right ) \] Input:
integrate((-(4-3*tan(x))^(1/2)+3*tan(x))/cos(x)^2/(4-3*tan(x))^(3/2),x, al gorithm="maxima")
Output:
2/3*sqrt(-3*tan(x) + 4) + 8/3/sqrt(-3*tan(x) + 4) + 1/3*log(-3*tan(x) + 4)
Time = 0.13 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.78 \[ \int \frac {\sec ^2(x) \left (-\sqrt {4-3 \tan (x)}+3 \tan (x)\right )}{(4-3 \tan (x))^{3/2}} \, dx=\frac {2}{3} \, \sqrt {-3 \, \tan \left (x\right ) + 4} + \frac {8}{3 \, \sqrt {-3 \, \tan \left (x\right ) + 4}} + \frac {1}{3} \, \log \left ({\left | -3 \, \tan \left (x\right ) + 4 \right |}\right ) \] Input:
integrate((-(4-3*tan(x))^(1/2)+3*tan(x))/cos(x)^2/(4-3*tan(x))^(3/2),x, al gorithm="giac")
Output:
2/3*sqrt(-3*tan(x) + 4) + 8/3/sqrt(-3*tan(x) + 4) + 1/3*log(abs(-3*tan(x) + 4))
Time = 0.79 (sec) , antiderivative size = 105, normalized size of antiderivative = 2.62 \[ \int \frac {\sec ^2(x) \left (-\sqrt {4-3 \tan (x)}+3 \tan (x)\right )}{(4-3 \tan (x))^{3/2}} \, dx=\frac {\ln \left ({\mathrm {e}}^{x\,2{}\mathrm {i}}\,\left (-\frac {16}{3}-4{}\mathrm {i}\right )-\frac {16}{3}+4{}\mathrm {i}\right )}{3}-\frac {\ln \left ({\mathrm {e}}^{x\,2{}\mathrm {i}}\,\left (\frac {16}{3}-4{}\mathrm {i}\right )+\frac {16}{3}-4{}\mathrm {i}\right )}{3}+\frac {2\,{\mathrm {e}}^{x\,1{}\mathrm {i}}\,\cos \left (x\right )\,\left (\frac {32\,{\mathrm {e}}^{x\,1{}\mathrm {i}}\,\cos \left (x\right )}{3}-4\,{\mathrm {e}}^{x\,1{}\mathrm {i}}\,\sin \left (x\right )\right )\,\sqrt {4-\frac {3\,\sin \left (x\right )}{\cos \left (x\right )}}}{8\,{\mathrm {e}}^{x\,2{}\mathrm {i}}+8\,\cos \left (2\,x\right )\,{\mathrm {e}}^{x\,2{}\mathrm {i}}-6\,\sin \left (2\,x\right )\,{\mathrm {e}}^{x\,2{}\mathrm {i}}} \] Input:
int((3*tan(x) - (4 - 3*tan(x))^(1/2))/(cos(x)^2*(4 - 3*tan(x))^(3/2)),x)
Output:
log(- exp(x*2i)*(16/3 + 4i) - (16/3 - 4i))/3 - log(exp(x*2i)*(16/3 - 4i) + (16/3 - 4i))/3 + (2*exp(x*1i)*cos(x)*((32*exp(x*1i)*cos(x))/3 - 4*exp(x*1 i)*sin(x))*(4 - (3*sin(x))/cos(x))^(1/2))/(8*exp(x*2i) + 8*cos(2*x)*exp(x* 2i) - 6*sin(2*x)*exp(x*2i))
\[ \int \frac {\sec ^2(x) \left (-\sqrt {4-3 \tan (x)}+3 \tan (x)\right )}{(4-3 \tan (x))^{3/2}} \, dx=3 \left (\int \frac {\sqrt {-3 \tan \left (x \right )+4}\, \tan \left (x \right )}{9 \cos \left (x \right )^{2} \tan \left (x \right )^{2}-24 \cos \left (x \right )^{2} \tan \left (x \right )+16 \cos \left (x \right )^{2}}d x \right )-\frac {\mathrm {log}\left (\tan \left (\frac {x}{2}\right )-1\right )}{3}+\frac {\mathrm {log}\left (\tan \left (\frac {x}{2}\right )+2\right )}{3}-\frac {\mathrm {log}\left (\tan \left (\frac {x}{2}\right )+1\right )}{3}+\frac {\mathrm {log}\left (2 \tan \left (\frac {x}{2}\right )-1\right )}{3} \] Input:
int((-(4-3*tan(x))^(1/2)+3*tan(x))/cos(x)^2/(4-3*tan(x))^(3/2),x)
Output:
(9*int((sqrt( - 3*tan(x) + 4)*tan(x))/(9*cos(x)**2*tan(x)**2 - 24*cos(x)** 2*tan(x) + 16*cos(x)**2),x) - log(tan(x/2) - 1) + log(tan(x/2) + 2) - log( tan(x/2) + 1) + log(2*tan(x/2) - 1))/3