3.399 \(\int \frac {1}{(4+3 \tan (2 x))^{3/2}} \, dx\)
Optimal. Leaf size=87 \[ -\frac {9 \tan ^{-1}\left (\frac {1-3 \tan (2 x)}{\sqrt {2} \sqrt {3 \tan (2 x)+4}}\right )}{250 \sqrt {2}}-\frac {3}{25 \sqrt {3 \tan (2 x)+4}}+\frac {13 \tanh ^{-1}\left (\frac {\tan (2 x)+3}{\sqrt {2} \sqrt {3 \tan (2 x)+4}}\right )}{250 \sqrt {2}} \]
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Rubi [A] time = 0.11, antiderivative size = 87, normalized size of antiderivative = 1.00,
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 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])
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 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 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])
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*}
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Mathematica [C] time = 0.10, 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]
-1/50*((3 + 4*I)*Hypergeometric2F1[-1/2, 1, 1/2, (4/25 - (3*I)/25)*(4 + 3*Tan[2*x])] + (3 - 4*I)*Hypergeometri
c2F1[-1/2, 1, 1/2, (4/25 + (3*I)/25)*(4 + 3*Tan[2*x])])/Sqrt[4 + 3*Tan[2*x]]
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{(4+3 \tan (2 x))^{3/2}} \, dx \]
Verification is Not applicable to the result.
[In]
IntegrateAlgebraic[(4 + 3*Tan[2*x])^(-3/2),x]
[Out]
Could not integrate
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fricas [B] time = 0.92, size = 541, normalized size = 6.22 \[ -\frac {36 \, {\left (7 \, \sqrt {10} \sqrt {5} \cos \left (2 \, x\right )^{2} + 24 \, \sqrt {10} \sqrt {5} \cos \left (2 \, x\right ) \sin \left (2 \, x\right ) + 9 \, \sqrt {10} \sqrt {5}\right )} \arctan \left (\frac {1}{25} \, \sqrt {15} \sqrt {10} \sqrt {5} \sqrt {\frac {\sqrt {10} \sqrt {5} \sqrt {\frac {4 \, \cos \left (2 \, x\right ) + 3 \, \sin \left (2 \, x\right )}{\cos \left (2 \, x\right )}} \cos \left (2 \, x\right ) + 15 \, \cos \left (2 \, x\right ) + 5 \, \sin \left (2 \, x\right )}{\cos \left (2 \, x\right )}} - \frac {1}{5} \, \sqrt {10} \sqrt {5} \sqrt {\frac {4 \, \cos \left (2 \, x\right ) + 3 \, \sin \left (2 \, x\right )}{\cos \left (2 \, x\right )}} - 3\right ) + 36 \, {\left (7 \, \sqrt {10} \sqrt {5} \cos \left (2 \, x\right )^{2} + 24 \, \sqrt {10} \sqrt {5} \cos \left (2 \, x\right ) \sin \left (2 \, x\right ) + 9 \, \sqrt {10} \sqrt {5}\right )} \arctan \left (\frac {1}{25} \, \sqrt {15} \sqrt {10} \sqrt {5} \sqrt {-\frac {\sqrt {10} \sqrt {5} \sqrt {\frac {4 \, \cos \left (2 \, x\right ) + 3 \, \sin \left (2 \, x\right )}{\cos \left (2 \, x\right )}} \cos \left (2 \, x\right ) - 15 \, \cos \left (2 \, x\right ) - 5 \, \sin \left (2 \, x\right )}{\cos \left (2 \, x\right )}} - \frac {1}{5} \, \sqrt {10} \sqrt {5} \sqrt {\frac {4 \, \cos \left (2 \, x\right ) + 3 \, \sin \left (2 \, x\right )}{\cos \left (2 \, x\right )}} + 3\right ) - 13 \, {\left (7 \, \sqrt {10} \sqrt {5} \cos \left (2 \, x\right )^{2} + 24 \, \sqrt {10} \sqrt {5} \cos \left (2 \, x\right ) \sin \left (2 \, x\right ) + 9 \, \sqrt {10} \sqrt {5}\right )} \log \left (\frac {9375 \, {\left (\sqrt {10} \sqrt {5} \sqrt {\frac {4 \, \cos \left (2 \, x\right ) + 3 \, \sin \left (2 \, x\right )}{\cos \left (2 \, x\right )}} \cos \left (2 \, x\right ) + 15 \, \cos \left (2 \, x\right ) + 5 \, \sin \left (2 \, x\right )\right )}}{\cos \left (2 \, x\right )}\right ) + 13 \, {\left (7 \, \sqrt {10} \sqrt {5} \cos \left (2 \, x\right )^{2} + 24 \, \sqrt {10} \sqrt {5} \cos \left (2 \, x\right ) \sin \left (2 \, x\right ) + 9 \, \sqrt {10} \sqrt {5}\right )} \log \left (-\frac {9375 \, {\left (\sqrt {10} \sqrt {5} \sqrt {\frac {4 \, \cos \left (2 \, x\right ) + 3 \, \sin \left (2 \, x\right )}{\cos \left (2 \, x\right )}} \cos \left (2 \, x\right ) - 15 \, \cos \left (2 \, x\right ) - 5 \, \sin \left (2 \, x\right )\right )}}{\cos \left (2 \, x\right )}\right ) + 600 \, {\left (4 \, \cos \left (2 \, x\right )^{2} + 3 \, \cos \left (2 \, x\right ) \sin \left (2 \, x\right )\right )} \sqrt {\frac {4 \, \cos \left (2 \, x\right ) + 3 \, \sin \left (2 \, x\right )}{\cos \left (2 \, x\right )}}}{5000 \, {\left (7 \, \cos \left (2 \, x\right )^{2} + 24 \, \cos \left (2 \, x\right ) \sin \left (2 \, x\right ) + 9\right )}} \]
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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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (3 \, \tan \left (2 \, x\right ) + 4\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(4+3*tan(2*x))^(3/2),x, algorithm="giac")
[Out]
integrate((3*tan(2*x) + 4)^(-3/2), x)
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maple [A] time = 0.23, size = 130, normalized size = 1.49
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result |
size |
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| derivativedivides |
\(-\frac {13 \sqrt {2}\, \ln \left (9+3 \tan \left (2 x \right )-3 \sqrt {4+3 \tan \left (2 x \right )}\, \sqrt {2}\right )}{1000}+\frac {9 \sqrt {2}\, \arctan \left (\frac {\left (2 \sqrt {4+3 \tan \left (2 x \right )}-3 \sqrt {2}\right ) \sqrt {2}}{2}\right )}{500}+\frac {13 \sqrt {2}\, \ln \left (9+3 \tan \left (2 x \right )+3 \sqrt {4+3 \tan \left (2 x \right )}\, \sqrt {2}\right )}{1000}+\frac {9 \sqrt {2}\, \arctan \left (\frac {\left (2 \sqrt {4+3 \tan \left (2 x \right )}+3 \sqrt {2}\right ) \sqrt {2}}{2}\right )}{500}-\frac {3}{25 \sqrt {4+3 \tan \left (2 x \right )}}\) |
\(130\) |
| default |
\(-\frac {13 \sqrt {2}\, \ln \left (9+3 \tan \left (2 x \right )-3 \sqrt {4+3 \tan \left (2 x \right )}\, \sqrt {2}\right )}{1000}+\frac {9 \sqrt {2}\, \arctan \left (\frac {\left (2 \sqrt {4+3 \tan \left (2 x \right )}-3 \sqrt {2}\right ) \sqrt {2}}{2}\right )}{500}+\frac {13 \sqrt {2}\, \ln \left (9+3 \tan \left (2 x \right )+3 \sqrt {4+3 \tan \left (2 x \right )}\, \sqrt {2}\right )}{1000}+\frac {9 \sqrt {2}\, \arctan \left (\frac {\left (2 \sqrt {4+3 \tan \left (2 x \right )}+3 \sqrt {2}\right ) \sqrt {2}}{2}\right )}{500}-\frac {3}{25 \sqrt {4+3 \tan \left (2 x \right )}}\) |
\(130\) |
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Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(4+3*tan(2*x))^(3/2),x,method=_RETURNVERBOSE)
[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)*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))-3/25/(4+3*tan(2*x))^(1/2)
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maxima [B] time = 2.14, size = 3213, normalized size = 36.93 \[ \text {result too large to display} \]
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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mupad [B] time = 0.44, size = 63, normalized size = 0.72 \[ -\frac {3}{25\,\sqrt {3\,\mathrm {tan}\left (2\,x\right )+4}}+\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,\sqrt {3\,\mathrm {tan}\left (2\,x\right )+4}\,\left (\frac {1}{10}-\frac {3}{10}{}\mathrm {i}\right )\right )\,\left (\frac {9}{500}+\frac {13}{500}{}\mathrm {i}\right )+\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,\sqrt {3\,\mathrm {tan}\left (2\,x\right )+4}\,\left (\frac {1}{10}+\frac {3}{10}{}\mathrm {i}\right )\right )\,\left (\frac {9}{500}-\frac {13}{500}{}\mathrm {i}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(3*tan(2*x) + 4)^(3/2),x)
[Out]
2^(1/2)*atan(2^(1/2)*(3*tan(2*x) + 4)^(1/2)*(1/10 - 3i/10))*(9/500 + 13i/500) + 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))
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (3 \tan {\left (2 x \right )} + 4\right )^{\frac {3}{2}}}\, dx \]
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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