3.399 \(\int \frac{1}{(4+3 \tan (2 x))^{3/2}} \, dx\)

Optimal. Leaf size=87 \[ -\frac{9 \tan ^{-1}\left (\frac{1-3 \tan (2 x)}{\sqrt{2} \sqrt{3 \tan (2 x)+4}}\right )}{250 \sqrt{2}}-\frac{3}{25 \sqrt{3 \tan (2 x)+4}}+\frac{13 \tanh ^{-1}\left (\frac{\tan (2 x)+3}{\sqrt{2} \sqrt{3 \tan (2 x)+4}}\right )}{250 \sqrt{2}} \]

[Out]

(-9*ArcTan[(1 - 3*Tan[2*x])/(Sqrt[2]*Sqrt[4 + 3*Tan[2*x]])])/(250*Sqrt[2]) + (13*ArcTanh[(3 + Tan[2*x])/(Sqrt[
2]*Sqrt[4 + 3*Tan[2*x]])])/(250*Sqrt[2]) - 3/(25*Sqrt[4 + 3*Tan[2*x]])

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Rubi [A]  time = 0.113088, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417, Rules used = {3483, 3536, 3535, 203, 207} \[ -\frac{9 \tan ^{-1}\left (\frac{1-3 \tan (2 x)}{\sqrt{2} \sqrt{3 \tan (2 x)+4}}\right )}{250 \sqrt{2}}-\frac{3}{25 \sqrt{3 \tan (2 x)+4}}+\frac{13 \tanh ^{-1}\left (\frac{\tan (2 x)+3}{\sqrt{2} \sqrt{3 \tan (2 x)+4}}\right )}{250 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]

Int[(4 + 3*Tan[2*x])^(-3/2),x]

[Out]

(-9*ArcTan[(1 - 3*Tan[2*x])/(Sqrt[2]*Sqrt[4 + 3*Tan[2*x]])])/(250*Sqrt[2]) + (13*ArcTanh[(3 + Tan[2*x])/(Sqrt[
2]*Sqrt[4 + 3*Tan[2*x]])])/(250*Sqrt[2]) - 3/(25*Sqrt[4 + 3*Tan[2*x]])

Rule 3483

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] + Dist[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]

Rule 3536

Int[((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])/Sqrt[(a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]], x_Symbol] :> With[{q =
Rt[a^2 + b^2, 2]}, Dist[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] - Dist[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] && NeQ[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] && (PerfectSquareQ[a^2 + b^2] || RationalQ[a, b, c, d])

Rule 3535

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]

Rule 203

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

Rule 207

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

Rubi steps

\begin{align*} \int \frac{1}{(4+3 \tan (2 x))^{3/2}} \, dx &=-\frac{3}{25 \sqrt{4+3 \tan (2 x)}}+\frac{1}{25} \int \frac{4-3 \tan (2 x)}{\sqrt{4+3 \tan (2 x)}} \, dx\\ &=-\frac{3}{25 \sqrt{4+3 \tan (2 x)}}+\frac{1}{250} \int \frac{27+9 \tan (2 x)}{\sqrt{4+3 \tan (2 x)}} \, dx-\frac{1}{250} \int \frac{-13+39 \tan (2 x)}{\sqrt{4+3 \tan (2 x)}} \, dx\\ &=-\frac{3}{25 \sqrt{4+3 \tan (2 x)}}-\frac{81}{250} \operatorname{Subst}\left (\int \frac{1}{162+x^2} \, dx,x,\frac{9-27 \tan (2 x)}{\sqrt{4+3 \tan (2 x)}}\right )+\frac{1521}{250} \operatorname{Subst}\left (\int \frac{1}{-27378+x^2} \, dx,x,\frac{-351-117 \tan (2 x)}{\sqrt{4+3 \tan (2 x)}}\right )\\ &=-\frac{9 \tan ^{-1}\left (\frac{1-3 \tan (2 x)}{\sqrt{2} \sqrt{4+3 \tan (2 x)}}\right )}{250 \sqrt{2}}+\frac{13 \tanh ^{-1}\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)}}\\ \end{align*}

Mathematica [C]  time = 0.0925458, size = 73, normalized size = 0.84 \[ -\frac{(3+4 i) \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};\left (\frac{4}{25}-\frac{3 i}{25}\right ) (3 \tan (2 x)+4)\right )+(3-4 i) \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};\left (\frac{4}{25}+\frac{3 i}{25}\right ) (3 \tan (2 x)+4)\right )}{50 \sqrt{3 \tan (2 x)+4}} \]

Antiderivative was successfully verified.

[In]

Integrate[(4 + 3*Tan[2*x])^(-3/2),x]

[Out]

-((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])])/(50*Sqrt[4 + 3*Tan[2*x]])

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Maple [A]  time = 0.049, size = 130, normalized size = 1.5 \begin{align*} -{\frac{13\,\sqrt{2}}{1000}\ln \left ( 3\,\tan \left ( 2\,x \right ) +9-3\,\sqrt{4+3\,\tan \left ( 2\,x \right ) }\sqrt{2} \right ) }+{\frac{9\,\sqrt{2}}{500}\arctan \left ({\frac{\sqrt{2}}{2} \left ( 2\,\sqrt{4+3\,\tan \left ( 2\,x \right ) }-3\,\sqrt{2} \right ) } \right ) }+{\frac{13\,\sqrt{2}}{1000}\ln \left ( 3\,\tan \left ( 2\,x \right ) +9+3\,\sqrt{4+3\,\tan \left ( 2\,x \right ) }\sqrt{2} \right ) }+{\frac{9\,\sqrt{2}}{500}\arctan \left ({\frac{\sqrt{2}}{2} \left ( 2\,\sqrt{4+3\,\tan \left ( 2\,x \right ) }+3\,\sqrt{2} \right ) } \right ) }-{\frac{3}{25}{\frac{1}{\sqrt{4+3\,\tan \left ( 2\,x \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-13/1000*2^(1/2)*ln(3*tan(2*x)+9-3*(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(3*tan(2*x)+9+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))-3/25/(4+3*tan(2*x))^(1/2)

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Maxima [B]  time = 2.49861, size = 4338, normalized size = 49.86 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-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 + 2000*(3*cos(4*x) + sin(4*x))*cos(1/2*arctan2(-3*cos(8*x) + 4*sin(8*x) + 8*s
in(4*x) + 3, 4*cos(8*x) + 8*cos(4*x) + 3*sin(8*x) + 4))*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 - 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*(25*(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*s
in(4*x) + 3, 4*cos(8*x) + 8*cos(4*x) + 3*sin(8*x) + 4)) + 9*(18*(sqrt(2)*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 + sqrt(2)*sin(1/2*arctan2(-3*cos(8*x) + 4*s
in(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*arctan2(-8/3*cos(4*x)^2 + 2/9*
(7*cos(4*x) + 16)*sin(4*x) + 8/3*sin(4*x)^2 - 8/3*cos(4*x), 7/9*cos(4*x)^2 + 8/3*(2*cos(4*x) + 1)*sin(4*x) - 7
/9*sin(4*x)^2 + 32/9*cos(4*x) + 25/9)) + cos(4*x) - 4/3*sin(4*x), 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)*cos(1/2*arctan2(-8/3*cos(4*x)^2 + 2/9*(7*cos(4*x) +
16)*sin(4*x) + 8/3*sin(4*x)^2 - 8/3*cos(4*x), 7/9*cos(4*x)^2 + 8/3*(2*cos(4*x) + 1)*sin(4*x) - 7/9*sin(4*x)^2
+ 32/9*cos(4*x) + 25/9)) - 4/3*cos(4*x) - sin(4*x) - 4/3) + 18*(sqrt(2)*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 + sqrt(2)*sin(1/2*arctan2(-3*cos(8*x) + 4*si
n(8*x) + 8*sin(4*x) + 3, 4*cos(8*x) + 8*cos(4*x) + 3*sin(8*x) + 4))^2)*arctan2(2/3*4^(1/4)*(4*cos(4*x)^4 + 4*s
in(4*x)^4 + 16*cos(4*x)^3 + (8*cos(4*x)^2 + 16*cos(4*x) + 17)*sin(4*x)^2 + 12*sin(4*x)^3 + 33*cos(4*x)^2 + 12*
(cos(4*x)^2 + 2*cos(4*x) + 1)*sin(4*x) + 34*cos(4*x) + 13)^(1/4)*sin(1/2*arctan2(32/9*(cos(4*x) + 1)*sin(4*x)
+ 8/3*cos(4*x) + 8/3, 16/9*cos(4*x)^2 - 16/9*sin(4*x)^2 + 32/9*cos(4*x) - 8/3*sin(4*x) + 16/9)) + 4/3*sin(4*x)
+ 1, 2/3*4^(1/4)*(4*cos(4*x)^4 + 4*sin(4*x)^4 + 16*cos(4*x)^3 + (8*cos(4*x)^2 + 16*cos(4*x) + 17)*sin(4*x)^2
+ 12*sin(4*x)^3 + 33*cos(4*x)^2 + 12*(cos(4*x)^2 + 2*cos(4*x) + 1)*sin(4*x) + 34*cos(4*x) + 13)^(1/4)*cos(1/2*
arctan2(32/9*(cos(4*x) + 1)*sin(4*x) + 8/3*cos(4*x) + 8/3, 16/9*cos(4*x)^2 - 16/9*sin(4*x)^2 + 32/9*cos(4*x) -
8/3*sin(4*x) + 16/9)) + 4/3*cos(4*x) + 4/3) + 13*(sqrt(2)*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 + sqrt(2)*sin(1/2*arctan2(-3*cos(8*x) + 4*sin(8*x) + 8*si
n(4*x) + 3, 4*cos(8*x) + 8*cos(4*x) + 3*sin(8*x) + 4))^2)*log(-2/9*25^(1/4)*(25*cos(4*x)^4 + 25*sin(4*x)^4 + 6
4*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)*(4*cos(4*x) + 3*sin(4*x) + 4)*cos(1/2*arctan2(-8/3*cos(
4*x)^2 + 2/9*(7*cos(4*x) + 16)*sin(4*x) + 8/3*sin(4*x)^2 - 8/3*cos(4*x), 7/9*cos(4*x)^2 + 8/3*(2*cos(4*x) + 1)
*sin(4*x) - 7/9*sin(4*x)^2 + 32/9*cos(4*x) + 25/9)) + 5/9*sqrt(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)*cos(1/2*arctan2(-8/3*cos(4*x)^2 + 2/9*(7*cos(4*x) + 16)*sin(4*x) + 8/3*sin(
4*x)^2 - 8/3*cos(4*x), 7/9*cos(4*x)^2 + 8/3*(2*cos(4*x) + 1)*sin(4*x) - 7/9*sin(4*x)^2 + 32/9*cos(4*x) + 25/9)
)^2 + 2/9*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)*(3
*cos(4*x) - 4*sin(4*x))*sin(1/2*arctan2(-8/3*cos(4*x)^2 + 2/9*(7*cos(4*x) + 16)*sin(4*x) + 8/3*sin(4*x)^2 - 8/
3*cos(4*x), 7/9*cos(4*x)^2 + 8/3*(2*cos(4*x) + 1)*sin(4*x) - 7/9*sin(4*x)^2 + 32/9*cos(4*x) + 25/9)) + 5/9*sqr
t(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)*sin(1/2*arctan2(-8/3*cos(4
*x)^2 + 2/9*(7*cos(4*x) + 16)*sin(4*x) + 8/3*sin(4*x)^2 - 8/3*cos(4*x), 7/9*cos(4*x)^2 + 8/3*(2*cos(4*x) + 1)*
sin(4*x) - 7/9*sin(4*x)^2 + 32/9*cos(4*x) + 25/9))^2 + 25/9*cos(4*x)^2 + 25/9*sin(4*x)^2 + 32/9*cos(4*x) + 8/3
*sin(4*x) + 16/9) - 13*(sqrt(2)*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 + 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)*log(16/9*4^(1/4)*(4*cos(4*x)^4 + 4*sin(4*x)^4 + 16*cos(4*x)^3 + (8*cos(4*x)^2 +
16*cos(4*x) + 17)*sin(4*x)^2 + 12*sin(4*x)^3 + 33*cos(4*x)^2 + 12*(cos(4*x)^2 + 2*cos(4*x) + 1)*sin(4*x) + 34
*cos(4*x) + 13)^(1/4)*(cos(4*x) + 1)*cos(1/2*arctan2(32/9*(cos(4*x) + 1)*sin(4*x) + 8/3*cos(4*x) + 8/3, 16/9*c
os(4*x)^2 - 16/9*sin(4*x)^2 + 32/9*cos(4*x) - 8/3*sin(4*x) + 16/9)) + 8/9*sqrt(4*cos(4*x)^4 + 4*sin(4*x)^4 + 1
6*cos(4*x)^3 + (8*cos(4*x)^2 + 16*cos(4*x) + 17)*sin(4*x)^2 + 12*sin(4*x)^3 + 33*cos(4*x)^2 + 12*(cos(4*x)^2 +
2*cos(4*x) + 1)*sin(4*x) + 34*cos(4*x) + 13)*cos(1/2*arctan2(32/9*(cos(4*x) + 1)*sin(4*x) + 8/3*cos(4*x) + 8/
3, 16/9*cos(4*x)^2 - 16/9*sin(4*x)^2 + 32/9*cos(4*x) - 8/3*sin(4*x) + 16/9))^2 + 4/9*4^(1/4)*(4*cos(4*x)^4 + 4
*sin(4*x)^4 + 16*cos(4*x)^3 + (8*cos(4*x)^2 + 16*cos(4*x) + 17)*sin(4*x)^2 + 12*sin(4*x)^3 + 33*cos(4*x)^2 + 1
2*(cos(4*x)^2 + 2*cos(4*x) + 1)*sin(4*x) + 34*cos(4*x) + 13)^(1/4)*(4*sin(4*x) + 3)*sin(1/2*arctan2(32/9*(cos(
4*x) + 1)*sin(4*x) + 8/3*cos(4*x) + 8/3, 16/9*cos(4*x)^2 - 16/9*sin(4*x)^2 + 32/9*cos(4*x) - 8/3*sin(4*x) + 16
/9)) + 8/9*sqrt(4*cos(4*x)^4 + 4*sin(4*x)^4 + 16*cos(4*x)^3 + (8*cos(4*x)^2 + 16*cos(4*x) + 17)*sin(4*x)^2 + 1
2*sin(4*x)^3 + 33*cos(4*x)^2 + 12*(cos(4*x)^2 + 2*cos(4*x) + 1)*sin(4*x) + 34*cos(4*x) + 13)*sin(1/2*arctan2(3
2/9*(cos(4*x) + 1)*sin(4*x) + 8/3*cos(4*x) + 8/3, 16/9*cos(4*x)^2 - 16/9*sin(4*x)^2 + 32/9*cos(4*x) - 8/3*sin(
4*x) + 16/9))^2 + 16/9*cos(4*x)^2 + 16/9*sin(4*x)^2 + 32/9*cos(4*x) + 8/3*sin(4*x) + 25/9))*(2*(32*cos(4*x) -
24*sin(4*x) + 7)*cos(8*x) + 25*cos(8*x)^2 + 64*cos(4*x)^2 + 16*(3*cos(4*x) + 4*sin(4*x) + 3)*sin(8*x) + 25*sin
(8*x)^2 + 64*sin(4*x)^2 + 64*cos(4*x) + 48*sin(4*x) + 25)^(1/4))/((2*(32*cos(4*x) - 24*sin(4*x) + 7)*cos(8*x)
+ 25*cos(8*x)^2 + 64*cos(4*x)^2 + 16*(3*cos(4*x) + 4*sin(4*x) + 3)*sin(8*x) + 25*sin(8*x)^2 + 64*sin(4*x)^2 +
64*cos(4*x) + 48*sin(4*x) + 25)^(1/4)*(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 + sin(1/2*arctan2(-3*cos(8*x) + 4*sin(8*x) + 8*sin(4*x) + 3, 4*cos(8*x) + 8*c
os(4*x) + 3*sin(8*x) + 4))^2))

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Fricas [B]  time = 2.0897, size = 1613, normalized size = 18.54 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/5000*(36*(7*sqrt(10)*sqrt(5)*cos(2*x)^2 + 24*sqrt(10)*sqrt(5)*cos(2*x)*sin(2*x) + 9*sqrt(10)*sqrt(5))*arcta
n(1/25*sqrt(15)*sqrt(10)*sqrt(5)*sqrt((sqrt(10)*sqrt(5)*sqrt((4*cos(2*x) + 3*sin(2*x))/cos(2*x))*cos(2*x) + 15
*cos(2*x) + 5*sin(2*x))/cos(2*x)) - 1/5*sqrt(10)*sqrt(5)*sqrt((4*cos(2*x) + 3*sin(2*x))/cos(2*x)) - 3) + 36*(7
*sqrt(10)*sqrt(5)*cos(2*x)^2 + 24*sqrt(10)*sqrt(5)*cos(2*x)*sin(2*x) + 9*sqrt(10)*sqrt(5))*arctan(1/25*sqrt(15
)*sqrt(10)*sqrt(5)*sqrt(-(sqrt(10)*sqrt(5)*sqrt((4*cos(2*x) + 3*sin(2*x))/cos(2*x))*cos(2*x) - 15*cos(2*x) - 5
*sin(2*x))/cos(2*x)) - 1/5*sqrt(10)*sqrt(5)*sqrt((4*cos(2*x) + 3*sin(2*x))/cos(2*x)) + 3) - 13*(7*sqrt(10)*sqr
t(5)*cos(2*x)^2 + 24*sqrt(10)*sqrt(5)*cos(2*x)*sin(2*x) + 9*sqrt(10)*sqrt(5))*log(9375*(sqrt(10)*sqrt(5)*sqrt(
(4*cos(2*x) + 3*sin(2*x))/cos(2*x))*cos(2*x) + 15*cos(2*x) + 5*sin(2*x))/cos(2*x)) + 13*(7*sqrt(10)*sqrt(5)*co
s(2*x)^2 + 24*sqrt(10)*sqrt(5)*cos(2*x)*sin(2*x) + 9*sqrt(10)*sqrt(5))*log(-9375*(sqrt(10)*sqrt(5)*sqrt((4*cos
(2*x) + 3*sin(2*x))/cos(2*x))*cos(2*x) - 15*cos(2*x) - 5*sin(2*x))/cos(2*x)) + 600*(4*cos(2*x)^2 + 3*cos(2*x)*
sin(2*x))*sqrt((4*cos(2*x) + 3*sin(2*x))/cos(2*x)))/(7*cos(2*x)^2 + 24*cos(2*x)*sin(2*x) + 9)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (3 \tan{\left (2 x \right )} + 4\right )^{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(4+3*tan(2*x))**(3/2),x)

[Out]

Integral((3*tan(2*x) + 4)**(-3/2), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (3 \, \tan \left (2 \, x\right ) + 4\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((3*tan(2*x) + 4)^(-3/2), x)