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*S
qrt[4 + 3*Tan[2*x]])
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Rubi [A] time = 0.181672, 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
\[ -\frac{9 \tan ^{-1}\left (\frac{1-3 \tan (2 x)}{\sqrt{2} \sqrt{3 \tan (2 x)+4}}\right )}{250 \sqrt{2}}-\frac{3}{25 \sqrt{3 \tan (2 x)+4}}+\frac{13 \tanh ^{-1}\left (\frac{\tan (2 x)+3}{\sqrt{2} \sqrt{3 \tan (2 x)+4}}\right )}{250 \sqrt{2}} \]
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*S
qrt[4 + 3*Tan[2*x]])
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Rubi in Sympy [A] time = 6.77787, size = 92, normalized size = 1.06 \[ - \frac{9 \sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} \left (- 27 \tan{\left (2 x \right )} + 9\right )}{18 \sqrt{3 \tan{\left (2 x \right )} + 4}} \right )}}{500} - \frac{13 \sqrt{2} \operatorname{atanh}{\left (\frac{\sqrt{2} \left (- 117 \tan{\left (2 x \right )} - 351\right )}{234 \sqrt{3 \tan{\left (2 x \right )} + 4}} \right )}}{500} - \frac{3}{25 \sqrt{3 \tan{\left (2 x \right )} + 4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(4+3*tan(2*x))**(3/2),x)
[Out]
-9*sqrt(2)*atan(sqrt(2)*(-27*tan(2*x) + 9)/(18*sqrt(3*tan(2*x) + 4)))/500 - 13*s
qrt(2)*atanh(sqrt(2)*(-117*tan(2*x) - 351)/(234*sqrt(3*tan(2*x) + 4)))/500 - 3/(
25*sqrt(3*tan(2*x) + 4))
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Mathematica [C] time = 0.342002, size = 83, normalized size = 0.95 \[ \frac{-\frac{150}{\sqrt{3 \tan (2 x)+4}}+(24-7 i) \sqrt{4-3 i} \tanh ^{-1}\left (\frac{\sqrt{3 \tan (2 x)+4}}{\sqrt{4-3 i}}\right )+(24+7 i) \sqrt{4+3 i} \tanh ^{-1}\left (\frac{\sqrt{3 \tan (2 x)+4}}{\sqrt{4+3 i}}\right )}{1250} \]
Antiderivative was successfully verified.
[In] Integrate[(4 + 3*Tan[2*x])^(-3/2),x]
[Out]
((24 - 7*I)*Sqrt[4 - 3*I]*ArcTanh[Sqrt[4 + 3*Tan[2*x]]/Sqrt[4 - 3*I]] + (24 + 7*
I)*Sqrt[4 + 3*I]*ArcTanh[Sqrt[4 + 3*Tan[2*x]]/Sqrt[4 + 3*I]] - 150/Sqrt[4 + 3*Ta
n[2*x]])/1250
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Maple [A] time = 0.072, size = 130, normalized size = 1.5 \[ -{\frac{13\,\sqrt{2}}{1000}\ln \left ( 9+3\,\tan \left ( 2\,x \right ) -3\,\sqrt{4+3\,\tan \left ( 2\,x \right ) }\sqrt{2} \right ) }+{\frac{9\,\sqrt{2}}{500}\arctan \left ({\frac{\sqrt{2}}{2} \left ( 2\,\sqrt{4+3\,\tan \left ( 2\,x \right ) }-3\,\sqrt{2} \right ) } \right ) }+{\frac{13\,\sqrt{2}}{1000}\ln \left ( 9+3\,\tan \left ( 2\,x \right ) +3\,\sqrt{4+3\,\tan \left ( 2\,x \right ) }\sqrt{2} \right ) }+{\frac{9\,\sqrt{2}}{500}\arctan \left ({\frac{\sqrt{2}}{2} \left ( 2\,\sqrt{4+3\,\tan \left ( 2\,x \right ) }+3\,\sqrt{2} \right ) } \right ) }-{\frac{3}{25}{\frac{1}{\sqrt{4+3\,\tan \left ( 2\,x \right ) }}}} \]
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(9+3*tan(2*x)-3*(4+3*tan(2*x))^(1/2)*2^(1/2))+9/500*2^(1/2)*a
rctan(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))-3/25/(4+3*tan(2*x))^(1/2)
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Maxima [A] time = 2.51699, size = 4338, normalized size = 49.86 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((3*tan(2*x) + 4)^(-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*sin(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*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)*s
in(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*c
os(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 + sq
rt(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(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)*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*arc
tan2(-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(-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)*(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*c
os(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*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)*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*arcta
n2(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 + 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*co
s(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 + 12*(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*c
os(4*x)^3 + (8*cos(4*x)^2 + 16*cos(4*x) + 17)*sin(4*x)^2 + 12*sin(4*x)^3 + 33*co
s(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(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 + 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*cos(4*x) + 3*sin(8*x)
+ 4))^2))
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Fricas [A] time = 0.236857, size = 618, normalized size = 7.1 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((3*tan(2*x) + 4)^(-3/2),x, algorithm="fricas")
[Out]
-1/5000*sqrt(10)*(36*sqrt(5)*sqrt((4*cos(2*x) + 3*sin(2*x))/cos(2*x))*arctan(5*s
qrt(5)/(sqrt(15)*sqrt(10)*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)) + 5*sqrt(10)*sqrt((4*co
s(2*x) + 3*sin(2*x))/cos(2*x)) + 15*sqrt(5))) + 36*sqrt(5)*sqrt((4*cos(2*x) + 3*
sin(2*x))/cos(2*x))*arctan(5*sqrt(5)/(sqrt(15)*sqrt(10)*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))/co
s(2*x)) + 5*sqrt(10)*sqrt((4*cos(2*x) + 3*sin(2*x))/cos(2*x)) - 15*sqrt(5))) - 1
3*sqrt(5)*sqrt((4*cos(2*x) + 3*sin(2*x))/cos(2*x))*log(9375*(sqrt(10)*sqrt(5)*sq
rt((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*sqrt(5)*sqrt((4*cos(2*x) + 3*sin(2*x))/cos(2*x))*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)) + 60*sqrt(10))/sqrt((4*cos(2*x) + 3*sin(2*x))/cos(2*x))
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (3 \tan{\left (2 x \right )} + 4\right )^{\frac{3}{2}}}\, dx \]
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/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (3 \, \tan \left (2 \, x\right ) + 4\right )}^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((3*tan(2*x) + 4)^(-3/2),x, algorithm="giac")
[Out]
integrate((3*tan(2*x) + 4)^(-3/2), x)