Integrand size = 12, antiderivative size = 87 \[ \int \frac {1}{(4+3 \tan (2 x))^{3/2}} \, dx=-\frac {9 \arctan \left (\frac {1-3 \tan (2 x)}{\sqrt {2} \sqrt {4+3 \tan (2 x)}}\right )}{250 \sqrt {2}}+\frac {13 \text {arctanh}\left (\frac {3+\tan (2 x)}{\sqrt {2} \sqrt {4+3 \tan (2 x)}}\right )}{250 \sqrt {2}}-\frac {3}{25 \sqrt {4+3 \tan (2 x)}} \]
-9/500*arctan(1/2*(1-3*tan(2*x))*2^(1/2)/(4+3*tan(2*x))^(1/2))*2^(1/2)+13/ 500*arctanh(1/2*(3+tan(2*x))*2^(1/2)/(4+3*tan(2*x))^(1/2))*2^(1/2)-3/25/(4 +3*tan(2*x))^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.11 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.84 \[ \int \frac {1}{(4+3 \tan (2 x))^{3/2}} \, dx=-\frac {(3+4 i) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},\left (\frac {4}{25}-\frac {3 i}{25}\right ) (4+3 \tan (2 x))\right )+(3-4 i) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},\left (\frac {4}{25}+\frac {3 i}{25}\right ) (4+3 \tan (2 x))\right )}{50 \sqrt {4+3 \tan (2 x)}} \]
-1/50*((3 + 4*I)*Hypergeometric2F1[-1/2, 1, 1/2, (4/25 - (3*I)/25)*(4 + 3* Tan[2*x])] + (3 - 4*I)*Hypergeometric2F1[-1/2, 1, 1/2, (4/25 + (3*I)/25)*( 4 + 3*Tan[2*x])])/Sqrt[4 + 3*Tan[2*x]]
Time = 0.42 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.06, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {3042, 3964, 3042, 4019, 27, 3042, 4018, 216, 220}
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.
| \(\displaystyle \int \frac {1}{(3 \tan (2 x)+4)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{(3 \tan (2 x)+4)^{3/2}}dx\) |
| \(\Big \downarrow \) 3964 |
| \(\displaystyle \frac {1}{25} \int \frac {4-3 \tan (2 x)}{\sqrt {3 \tan (2 x)+4}}dx-\frac {3}{25 \sqrt {3 \tan (2 x)+4}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{25} \int \frac {4-3 \tan (2 x)}{\sqrt {3 \tan (2 x)+4}}dx-\frac {3}{25 \sqrt {3 \tan (2 x)+4}}\) |
| \(\Big \downarrow \) 4019 |
| \(\displaystyle \frac {1}{25} \left (\frac {1}{10} \int \frac {9 (\tan (2 x)+3)}{\sqrt {3 \tan (2 x)+4}}dx-\frac {1}{10} \int -\frac {13 (1-3 \tan (2 x))}{\sqrt {3 \tan (2 x)+4}}dx\right )-\frac {3}{25 \sqrt {3 \tan (2 x)+4}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{25} \left (\frac {13}{10} \int \frac {1-3 \tan (2 x)}{\sqrt {3 \tan (2 x)+4}}dx+\frac {9}{10} \int \frac {\tan (2 x)+3}{\sqrt {3 \tan (2 x)+4}}dx\right )-\frac {3}{25 \sqrt {3 \tan (2 x)+4}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{25} \left (\frac {13}{10} \int \frac {1-3 \tan (2 x)}{\sqrt {3 \tan (2 x)+4}}dx+\frac {9}{10} \int \frac {\tan (2 x)+3}{\sqrt {3 \tan (2 x)+4}}dx\right )-\frac {3}{25 \sqrt {3 \tan (2 x)+4}}\) |
| \(\Big \downarrow \) 4018 |
| \(\displaystyle \frac {1}{25} \left (-\frac {9}{10} \int \frac {1}{\frac {(1-3 \tan (2 x))^2}{3 \tan (2 x)+4}+2}d\frac {1-3 \tan (2 x)}{\sqrt {3 \tan (2 x)+4}}-\frac {117}{10} \int \frac {1}{\frac {81 (\tan (2 x)+3)^2}{3 \tan (2 x)+4}-162}d\frac {9 (\tan (2 x)+3)}{\sqrt {3 \tan (2 x)+4}}\right )-\frac {3}{25 \sqrt {3 \tan (2 x)+4}}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {1}{25} \left (-\frac {117}{10} \int \frac {1}{\frac {81 (\tan (2 x)+3)^2}{3 \tan (2 x)+4}-162}d\frac {9 (\tan (2 x)+3)}{\sqrt {3 \tan (2 x)+4}}-\frac {9 \arctan \left (\frac {1-3 \tan (2 x)}{\sqrt {2} \sqrt {3 \tan (2 x)+4}}\right )}{10 \sqrt {2}}\right )-\frac {3}{25 \sqrt {3 \tan (2 x)+4}}\) |
| \(\Big \downarrow \) 220 |
| \(\displaystyle \frac {1}{25} \left (\frac {13 \text {arctanh}\left (\frac {\tan (2 x)+3}{\sqrt {2} \sqrt {3 \tan (2 x)+4}}\right )}{10 \sqrt {2}}-\frac {9 \arctan \left (\frac {1-3 \tan (2 x)}{\sqrt {2} \sqrt {3 \tan (2 x)+4}}\right )}{10 \sqrt {2}}\right )-\frac {3}{25 \sqrt {3 \tan (2 x)+4}}\) |
((-9*ArcTan[(1 - 3*Tan[2*x])/(Sqrt[2]*Sqrt[4 + 3*Tan[2*x]])])/(10*Sqrt[2]) + (13*ArcTanh[(3 + Tan[2*x])/(Sqrt[2]*Sqrt[4 + 3*Tan[2*x]])])/(10*Sqrt[2] ))/25 - 3/(25*Sqrt[4 + 3*Tan[2*x]])
3.4.99.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
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])
Int[((a_) + (b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b*((a + b*Tan[c + d*x])^(n + 1)/(d*(n + 1)*(a^2 + b^2))), x] + Simp[1/(a^2 + b^2) Int[(a - b*Tan[c + d*x])*(a + b*Tan[c + d*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0] && LtQ[n, -1]
Int[((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])/Sqrt[(a_) + (b_.)*tan[(e_.) + ( f_.)*(x_)]], x_Symbol] :> Simp[-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]
Int[((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])/Sqrt[(a_) + (b_.)*tan[(e_.) + ( f_.)*(x_)]], x_Symbol] :> With[{q = Rt[a^2 + b^2, 2]}, Simp[1/(2*q) Int[( a*c + b*d + c*q + (b*c - a*d + d*q)*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]], x], x] - Simp[1/(2*q) Int[(a*c + b*d - c*q + (b*c - a*d - d*q)*Tan[e + f *x])/Sqrt[a + b*Tan[e + f*x]], x], x]] /; FreeQ[{a, b, c, d, e, f}, x] && N eQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && NeQ[2*a*c*d - b*(c^2 - d^2), 0] && NiceSqrtQ[a^2 + b^2]
Time = 0.29 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.49
| method | result | size |
| derivativedivides | \(-\frac {3}{25 \sqrt {4+3 \tan \left (2 x \right )}}+\frac {13 \sqrt {2}\, \ln \left (9+3 \tan \left (2 x \right )+3 \sqrt {4+3 \tan \left (2 x \right )}\, \sqrt {2}\right )}{1000}+\frac {9 \sqrt {2}\, \arctan \left (\frac {\left (2 \sqrt {4+3 \tan \left (2 x \right )}+3 \sqrt {2}\right ) \sqrt {2}}{2}\right )}{500}-\frac {13 \sqrt {2}\, \ln \left (9+3 \tan \left (2 x \right )-3 \sqrt {4+3 \tan \left (2 x \right )}\, \sqrt {2}\right )}{1000}+\frac {9 \sqrt {2}\, \arctan \left (\frac {\left (2 \sqrt {4+3 \tan \left (2 x \right )}-3 \sqrt {2}\right ) \sqrt {2}}{2}\right )}{500}\) | \(130\) |
| default | \(-\frac {3}{25 \sqrt {4+3 \tan \left (2 x \right )}}+\frac {13 \sqrt {2}\, \ln \left (9+3 \tan \left (2 x \right )+3 \sqrt {4+3 \tan \left (2 x \right )}\, \sqrt {2}\right )}{1000}+\frac {9 \sqrt {2}\, \arctan \left (\frac {\left (2 \sqrt {4+3 \tan \left (2 x \right )}+3 \sqrt {2}\right ) \sqrt {2}}{2}\right )}{500}-\frac {13 \sqrt {2}\, \ln \left (9+3 \tan \left (2 x \right )-3 \sqrt {4+3 \tan \left (2 x \right )}\, \sqrt {2}\right )}{1000}+\frac {9 \sqrt {2}\, \arctan \left (\frac {\left (2 \sqrt {4+3 \tan \left (2 x \right )}-3 \sqrt {2}\right ) \sqrt {2}}{2}\right )}{500}\) | \(130\) |
-3/25/(4+3*tan(2*x))^(1/2)+13/1000*2^(1/2)*ln(9+3*tan(2*x)+3*(4+3*tan(2*x) )^(1/2)*2^(1/2))+9/500*2^(1/2)*arctan(1/2*(2*(4+3*tan(2*x))^(1/2)+3*2^(1/2 ))*2^(1/2))-13/1000*2^(1/2)*ln(9+3*tan(2*x)-3*(4+3*tan(2*x))^(1/2)*2^(1/2) )+9/500*2^(1/2)*arctan(1/2*(2*(4+3*tan(2*x))^(1/2)-3*2^(1/2))*2^(1/2))
Result contains complex when optimal does not.
Time = 0.26 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.74 \[ \int \frac {1}{(4+3 \tan (2 x))^{3/2}} \, dx=\frac {\sqrt {117 i + 44} {\left (3 \, \tan \left (2 \, x\right ) + 4\right )} \log \left (-\left (7 i - 24\right ) \, \sqrt {117 i + 44} + 125 \, \sqrt {3 \, \tan \left (2 \, x\right ) + 4}\right ) - \sqrt {117 i + 44} {\left (3 \, \tan \left (2 \, x\right ) + 4\right )} \log \left (\left (7 i - 24\right ) \, \sqrt {117 i + 44} + 125 \, \sqrt {3 \, \tan \left (2 \, x\right ) + 4}\right ) + \sqrt {-117 i + 44} {\left (3 \, \tan \left (2 \, x\right ) + 4\right )} \log \left (\left (7 i + 24\right ) \, \sqrt {-117 i + 44} + 125 \, \sqrt {3 \, \tan \left (2 \, x\right ) + 4}\right ) - \sqrt {-117 i + 44} {\left (3 \, \tan \left (2 \, x\right ) + 4\right )} \log \left (-\left (7 i + 24\right ) \, \sqrt {-117 i + 44} + 125 \, \sqrt {3 \, \tan \left (2 \, x\right ) + 4}\right ) - 60 \, \sqrt {3 \, \tan \left (2 \, x\right ) + 4}}{500 \, {\left (3 \, \tan \left (2 \, x\right ) + 4\right )}} \]
1/500*(sqrt(117*I + 44)*(3*tan(2*x) + 4)*log(-(7*I - 24)*sqrt(117*I + 44) + 125*sqrt(3*tan(2*x) + 4)) - sqrt(117*I + 44)*(3*tan(2*x) + 4)*log((7*I - 24)*sqrt(117*I + 44) + 125*sqrt(3*tan(2*x) + 4)) + sqrt(-117*I + 44)*(3*t an(2*x) + 4)*log((7*I + 24)*sqrt(-117*I + 44) + 125*sqrt(3*tan(2*x) + 4)) - sqrt(-117*I + 44)*(3*tan(2*x) + 4)*log(-(7*I + 24)*sqrt(-117*I + 44) + 1 25*sqrt(3*tan(2*x) + 4)) - 60*sqrt(3*tan(2*x) + 4))/(3*tan(2*x) + 4)
\[ \int \frac {1}{(4+3 \tan (2 x))^{3/2}} \, dx=\int \frac {1}{\left (3 \tan {\left (2 x \right )} + 4\right )^{\frac {3}{2}}}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 3213 vs. \(2 (69) = 138\).
Time = 0.52 (sec) , antiderivative size = 3213, normalized size of antiderivative = 36.93 \[ \int \frac {1}{(4+3 \tan (2 x))^{3/2}} \, dx=\text {Too large to display} \]
-1/18000*(2000*(3*cos(4*x) + sin(4*x))*cos(1/2*arctan2(-3*cos(8*x) + 4*sin (8*x) + 8*sin(4*x) + 3, 4*cos(8*x) + 8*cos(4*x) + 3*sin(8*x) + 4))^3 + 200 0*(3*cos(4*x) + sin(4*x))*cos(1/2*arctan2(-3*cos(8*x) + 4*sin(8*x) + 8*sin (4*x) + 3, 4*cos(8*x) + 8*cos(4*x) + 3*sin(8*x) + 4))*sin(1/2*arctan2(-3*c os(8*x) + 4*sin(8*x) + 8*sin(4*x) + 3, 4*cos(8*x) + 8*cos(4*x) + 3*sin(8*x ) + 4))^2 - 2000*(cos(4*x) - 3*sin(4*x) - 3)*sin(1/2*arctan2(-3*cos(8*x) + 4*sin(8*x) + 8*sin(4*x) + 3, 4*cos(8*x) + 8*cos(4*x) + 3*sin(8*x) + 4))^3 - 80*(48*cos(4*x) + 25*sin(4*x) - 27)*cos(1/2*arctan2(-3*cos(8*x) + 4*sin (8*x) + 8*sin(4*x) + 3, 4*cos(8*x) + 8*cos(4*x) + 3*sin(8*x) + 4)) - 80*(2 5*(cos(4*x) - 3*sin(4*x) - 3)*cos(1/2*arctan2(-3*cos(8*x) + 4*sin(8*x) + 8 *sin(4*x) + 3, 4*cos(8*x) + 8*cos(4*x) + 3*sin(8*x) + 4))^2 - 25*cos(4*x) + 48*sin(4*x) + 75)*sin(1/2*arctan2(-3*cos(8*x) + 4*sin(8*x) + 8*sin(4*x) + 3, 4*cos(8*x) + 8*cos(4*x) + 3*sin(8*x) + 4)) + 9*(18*(sqrt(2)*cos(1/2*a rctan2(-3*cos(8*x) + 4*sin(8*x) + 8*sin(4*x) + 3, 4*cos(8*x) + 8*cos(4*x) + 3*sin(8*x) + 4))^2 + sqrt(2)*sin(1/2*arctan2(-3*cos(8*x) + 4*sin(8*x) + 8*sin(4*x) + 3, 4*cos(8*x) + 8*cos(4*x) + 3*sin(8*x) + 4))^2)*arctan2(1/3* 25^(1/4)*(25*cos(4*x)^4 + 25*sin(4*x)^4 + 64*cos(4*x)^3 + 2*(25*cos(4*x)^2 + 32*cos(4*x) + 25)*sin(4*x)^2 + 48*sin(4*x)^3 + 78*cos(4*x)^2 + 48*(cos( 4*x)^2 + 2*cos(4*x) + 1)*sin(4*x) + 64*cos(4*x) + 25)^(1/4)*sin(1/2*arctan 2(-8/3*cos(4*x)^2 + 2/9*(7*cos(4*x) + 16)*sin(4*x) + 8/3*sin(4*x)^2 - 8...
\[ \int \frac {1}{(4+3 \tan (2 x))^{3/2}} \, dx=\int { \frac {1}{{\left (3 \, \tan \left (2 \, x\right ) + 4\right )}^{\frac {3}{2}}} \,d x } \]
Time = 0.47 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.72 \[ \int \frac {1}{(4+3 \tan (2 x))^{3/2}} \, dx=-\frac {3}{25\,\sqrt {3\,\mathrm {tan}\left (2\,x\right )+4}}+\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,\sqrt {3\,\mathrm {tan}\left (2\,x\right )+4}\,\left (\frac {1}{10}-\frac {3}{10}{}\mathrm {i}\right )\right )\,\left (\frac {9}{500}+\frac {13}{500}{}\mathrm {i}\right )+\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,\sqrt {3\,\mathrm {tan}\left (2\,x\right )+4}\,\left (\frac {1}{10}+\frac {3}{10}{}\mathrm {i}\right )\right )\,\left (\frac {9}{500}-\frac {13}{500}{}\mathrm {i}\right ) \]
2^(1/2)*atan(2^(1/2)*(3*tan(2*x) + 4)^(1/2)*(1/10 - 3i/10))*(9/500 + 13i/5 00) + 2^(1/2)*atan(2^(1/2)*(3*tan(2*x) + 4)^(1/2)*(1/10 + 3i/10))*(9/500 - 13i/500) - 3/(25*(3*tan(2*x) + 4)^(1/2))
\[ \int \frac {1}{(4+3 \tan (2 x))^{3/2}} \, dx=\frac {-\sqrt {3 \tan \left (2 x \right )+4}-9 \left (\int \frac {\sqrt {3 \tan \left (2 x \right )+4}\, \tan \left (2 x \right )^{2}}{9 \tan \left (2 x \right )^{2}+24 \tan \left (2 x \right )+16}d x \right ) \tan \left (2 x \right )-12 \left (\int \frac {\sqrt {3 \tan \left (2 x \right )+4}\, \tan \left (2 x \right )^{2}}{9 \tan \left (2 x \right )^{2}+24 \tan \left (2 x \right )+16}d x \right )}{9 \tan \left (2 x \right )+12} \]