3.10 \(\int \frac {1}{-1+3 \sin ^2(2+3 x)} \, dx\)
Optimal. Leaf size=60 \[ \frac {\log \left (\cos (3 x+2)-\sqrt {2} \sin (3 x+2)\right )}{6 \sqrt {2}}-\frac {\log \left (\sqrt {2} \sin (3 x+2)+\cos (3 x+2)\right )}{6 \sqrt {2}} \]
[Out]
1/12*ln(cos(2+3*x)-sin(2+3*x)*2^(1/2))*2^(1/2)-1/12*ln(cos(2+3*x)+sin(2+3*x)*2^(1/2))*2^(1/2)
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Rubi [A] time = 0.02, antiderivative size = 60, normalized size of antiderivative = 1.00,
number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used =
{3181, 207} \[ \frac {\log \left (\cos (3 x+2)-\sqrt {2} \sin (3 x+2)\right )}{6 \sqrt {2}}-\frac {\log \left (\sqrt {2} \sin (3 x+2)+\cos (3 x+2)\right )}{6 \sqrt {2}} \]
Antiderivative was successfully verified.
[In]
Int[(-1 + 3*Sin[2 + 3*x]^2)^(-1),x]
[Out]
Log[Cos[2 + 3*x] - Sqrt[2]*Sin[2 + 3*x]]/(6*Sqrt[2]) - Log[Cos[2 + 3*x] + Sqrt[2]*Sin[2 + 3*x]]/(6*Sqrt[2])
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 3181
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(-1), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist
[ff/f, Subst[Int[1/(a + (a + b)*ff^2*x^2), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x]
Rubi steps
\begin {align*} \int \frac {1}{-1+3 \sin ^2(2+3 x)} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{-1+2 x^2} \, dx,x,\tan (2+3 x)\right )\\ &=\frac {\log \left (\cos (2+3 x)-\sqrt {2} \sin (2+3 x)\right )}{6 \sqrt {2}}-\frac {\log \left (\cos (2+3 x)+\sqrt {2} \sin (2+3 x)\right )}{6 \sqrt {2}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 22, normalized size = 0.37 \[ -\frac {\tanh ^{-1}\left (\sqrt {2} \tan (3 x+2)\right )}{3 \sqrt {2}} \]
Antiderivative was successfully verified.
[In]
Integrate[(-1 + 3*Sin[2 + 3*x]^2)^(-1),x]
[Out]
-1/3*ArcTanh[Sqrt[2]*Tan[2 + 3*x]]/Sqrt[2]
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fricas [A] time = 0.65, size = 86, normalized size = 1.43 \[ \frac {1}{24} \, \sqrt {2} \log \left (-\frac {7 \, \cos \left (3 \, x + 2\right )^{4} - 4 \, \cos \left (3 \, x + 2\right )^{2} - 4 \, {\left (\sqrt {2} \cos \left (3 \, x + 2\right )^{3} - 2 \, \sqrt {2} \cos \left (3 \, x + 2\right )\right )} \sin \left (3 \, x + 2\right ) - 4}{9 \, \cos \left (3 \, x + 2\right )^{4} - 12 \, \cos \left (3 \, x + 2\right )^{2} + 4}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(-1+3*sin(2+3*x)^2),x, algorithm="fricas")
[Out]
1/24*sqrt(2)*log(-(7*cos(3*x + 2)^4 - 4*cos(3*x + 2)^2 - 4*(sqrt(2)*cos(3*x + 2)^3 - 2*sqrt(2)*cos(3*x + 2))*s
in(3*x + 2) - 4)/(9*cos(3*x + 2)^4 - 12*cos(3*x + 2)^2 + 4))
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giac [A] time = 0.16, size = 39, normalized size = 0.65 \[ \frac {1}{12} \, \sqrt {2} \log \left (\frac {{\left | -2 \, \sqrt {2} + 4 \, \tan \left (3 \, x + 2\right ) \right |}}{{\left | 2 \, \sqrt {2} + 4 \, \tan \left (3 \, x + 2\right ) \right |}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(-1+3*sin(2+3*x)^2),x, algorithm="giac")
[Out]
1/12*sqrt(2)*log(abs(-2*sqrt(2) + 4*tan(3*x + 2))/abs(2*sqrt(2) + 4*tan(3*x + 2)))
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maple [A] time = 0.15, size = 17, normalized size = 0.28 \[ -\frac {\sqrt {2}\, \arctanh \left (\sqrt {2}\, \tan \left (2+3 x \right )\right )}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(-1+3*sin(2+3*x)^2),x)
[Out]
-1/6*2^(1/2)*arctanh(2^(1/2)*tan(2+3*x))
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maxima [A] time = 0.41, size = 34, normalized size = 0.57 \[ \frac {1}{12} \, \sqrt {2} \log \left (-\frac {\sqrt {2} - 2 \, \tan \left (3 \, x + 2\right )}{\sqrt {2} + 2 \, \tan \left (3 \, x + 2\right )}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(-1+3*sin(2+3*x)^2),x, algorithm="maxima")
[Out]
1/12*sqrt(2)*log(-(sqrt(2) - 2*tan(3*x + 2))/(sqrt(2) + 2*tan(3*x + 2)))
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mupad [B] time = 2.45, size = 16, normalized size = 0.27 \[ -\frac {\sqrt {2}\,\mathrm {atanh}\left (\sqrt {2}\,\mathrm {tan}\left (3\,x+2\right )\right )}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(3*sin(3*x + 2)^2 - 1),x)
[Out]
-(2^(1/2)*atanh(2^(1/2)*tan(3*x + 2)))/6
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sympy [B] time = 18.47, size = 1644, normalized size = 27.40 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(-1+3*sin(2+3*x)**2),x)
[Out]
-1387702511766624*sqrt(5 - 2*sqrt(6))*log(tan(3*x/2 + 1) - sqrt(5 - 2*sqrt(6)))/(-467972363532675 - 1910489173
96548*sqrt(6) + 13665597568857156*sqrt(6)*sqrt(5 - 2*sqrt(6))*sqrt(2*sqrt(6) + 5) + 33473741073918339*sqrt(5 -
2*sqrt(6))*sqrt(2*sqrt(6) + 5)) - 566527178101133*sqrt(6)*sqrt(5 - 2*sqrt(6))*log(tan(3*x/2 + 1) - sqrt(5 - 2
*sqrt(6)))/(-467972363532675 - 191048917396548*sqrt(6) + 13665597568857156*sqrt(6)*sqrt(5 - 2*sqrt(6))*sqrt(2*
sqrt(6) + 5) + 33473741073918339*sqrt(5 - 2*sqrt(6))*sqrt(2*sqrt(6) + 5)) + 1376499295618884*sqrt(2*sqrt(6) +
5)*log(tan(3*x/2 + 1) - sqrt(5 - 2*sqrt(6)))/(-467972363532675 - 191048917396548*sqrt(6) + 13665597568857156*s
qrt(6)*sqrt(5 - 2*sqrt(6))*sqrt(2*sqrt(6) + 5) + 33473741073918339*sqrt(5 - 2*sqrt(6))*sqrt(2*sqrt(6) + 5)) +
561953484261121*sqrt(6)*sqrt(2*sqrt(6) + 5)*log(tan(3*x/2 + 1) - sqrt(5 - 2*sqrt(6)))/(-467972363532675 - 1910
48917396548*sqrt(6) + 13665597568857156*sqrt(6)*sqrt(5 - 2*sqrt(6))*sqrt(2*sqrt(6) + 5) + 33473741073918339*sq
rt(5 - 2*sqrt(6))*sqrt(2*sqrt(6) + 5)) - 1247944371758796*sqrt(2*sqrt(6) + 5)*log(tan(3*x/2 + 1) + sqrt(5 - 2*
sqrt(6)))/(-467972363532675 - 191048917396548*sqrt(6) + 13665597568857156*sqrt(6)*sqrt(5 - 2*sqrt(6))*sqrt(2*s
qrt(6) + 5) + 33473741073918339*sqrt(5 - 2*sqrt(6))*sqrt(2*sqrt(6) + 5)) - 509471156364528*sqrt(6)*sqrt(2*sqrt
(6) + 5)*log(tan(3*x/2 + 1) + sqrt(5 - 2*sqrt(6)))/(-467972363532675 - 191048917396548*sqrt(6) + 1366559756885
7156*sqrt(6)*sqrt(5 - 2*sqrt(6))*sqrt(2*sqrt(6) + 5) + 33473741073918339*sqrt(5 - 2*sqrt(6))*sqrt(2*sqrt(6) +
5)) + 47005690897992*sqrt(6)*sqrt(5 - 2*sqrt(6))*log(tan(3*x/2 + 1) + sqrt(5 - 2*sqrt(6)))/(-467972363532675 -
191048917396548*sqrt(6) + 13665597568857156*sqrt(6)*sqrt(5 - 2*sqrt(6))*sqrt(2*sqrt(6) + 5) + 334737410739183
39*sqrt(5 - 2*sqrt(6))*sqrt(2*sqrt(6) + 5)) + 115139957707068*sqrt(5 - 2*sqrt(6))*log(tan(3*x/2 + 1) + sqrt(5
- 2*sqrt(6)))/(-467972363532675 - 191048917396548*sqrt(6) + 13665597568857156*sqrt(6)*sqrt(5 - 2*sqrt(6))*sqrt
(2*sqrt(6) + 5) + 33473741073918339*sqrt(5 - 2*sqrt(6))*sqrt(2*sqrt(6) + 5)) - 12353375735168316*sqrt(5 - 2*sq
rt(6))*log(tan(3*x/2 + 1) - sqrt(2*sqrt(6) + 5))/(-467972363532675 - 191048917396548*sqrt(6) + 136655975688571
56*sqrt(6)*sqrt(5 - 2*sqrt(6))*sqrt(2*sqrt(6) + 5) + 33473741073918339*sqrt(5 - 2*sqrt(6))*sqrt(2*sqrt(6) + 5)
) - 5043244525340232*sqrt(6)*sqrt(5 - 2*sqrt(6))*log(tan(3*x/2 + 1) - sqrt(2*sqrt(6) + 5))/(-467972363532675 -
191048917396548*sqrt(6) + 13665597568857156*sqrt(6)*sqrt(5 - 2*sqrt(6))*sqrt(2*sqrt(6) + 5) + 334737410739183
39*sqrt(5 - 2*sqrt(6))*sqrt(2*sqrt(6) + 5)) + 4748539075824*sqrt(6)*sqrt(2*sqrt(6) + 5)*log(tan(3*x/2 + 1) - s
qrt(2*sqrt(6) + 5))/(-467972363532675 - 191048917396548*sqrt(6) + 13665597568857156*sqrt(6)*sqrt(5 - 2*sqrt(6)
)*sqrt(2*sqrt(6) + 5) + 33473741073918339*sqrt(5 - 2*sqrt(6))*sqrt(2*sqrt(6) + 5)) + 11631497759436*sqrt(2*sqr
t(6) + 5)*log(tan(3*x/2 + 1) - sqrt(2*sqrt(6) + 5))/(-467972363532675 - 191048917396548*sqrt(6) + 136655975688
57156*sqrt(6)*sqrt(5 - 2*sqrt(6))*sqrt(2*sqrt(6) + 5) + 33473741073918339*sqrt(5 - 2*sqrt(6))*sqrt(2*sqrt(6) +
5)) - 140186421619524*sqrt(2*sqrt(6) + 5)*log(tan(3*x/2 + 1) + sqrt(2*sqrt(6) + 5))/(-467972363532675 - 19104
8917396548*sqrt(6) + 13665597568857156*sqrt(6)*sqrt(5 - 2*sqrt(6))*sqrt(2*sqrt(6) + 5) + 33473741073918339*sqr
t(5 - 2*sqrt(6))*sqrt(2*sqrt(6) + 5)) - 57230866972417*sqrt(6)*sqrt(2*sqrt(6) + 5)*log(tan(3*x/2 + 1) + sqrt(2
*sqrt(6) + 5))/(-467972363532675 - 191048917396548*sqrt(6) + 13665597568857156*sqrt(6)*sqrt(5 - 2*sqrt(6))*sqr
t(2*sqrt(6) + 5) + 33473741073918339*sqrt(5 - 2*sqrt(6))*sqrt(2*sqrt(6) + 5)) + 13625938289227872*sqrt(5 - 2*s
qrt(6))*log(tan(3*x/2 + 1) + sqrt(2*sqrt(6) + 5))/(-467972363532675 - 191048917396548*sqrt(6) + 13665597568857
156*sqrt(6)*sqrt(5 - 2*sqrt(6))*sqrt(2*sqrt(6) + 5) + 33473741073918339*sqrt(5 - 2*sqrt(6))*sqrt(2*sqrt(6) + 5
)) + 5562766012543373*sqrt(6)*sqrt(5 - 2*sqrt(6))*log(tan(3*x/2 + 1) + sqrt(2*sqrt(6) + 5))/(-467972363532675
- 191048917396548*sqrt(6) + 13665597568857156*sqrt(6)*sqrt(5 - 2*sqrt(6))*sqrt(2*sqrt(6) + 5) + 33473741073918
339*sqrt(5 - 2*sqrt(6))*sqrt(2*sqrt(6) + 5))
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