3.94 \(\int \csc (6 x) \sin (x) \, dx\)

Optimal. Leaf size=36 \[ \frac {1}{6} \tanh ^{-1}(\sin (x))+\frac {1}{6} \tanh ^{-1}(2 \sin (x))-\frac {\tanh ^{-1}\left (\frac {2 \sin (x)}{\sqrt {3}}\right )}{2 \sqrt {3}} \]

[Out]

1/6*arctanh(sin(x))+1/6*arctanh(2*sin(x))-1/6*arctanh(2/3*sin(x)*3^(1/2))*3^(1/2)

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Rubi [A]  time = 0.05, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 3, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {12, 2057, 207} \[ \frac {1}{6} \tanh ^{-1}(\sin (x))+\frac {1}{6} \tanh ^{-1}(2 \sin (x))-\frac {\tanh ^{-1}\left (\frac {2 \sin (x)}{\sqrt {3}}\right )}{2 \sqrt {3}} \]

Antiderivative was successfully verified.

[In]

Int[Csc[6*x]*Sin[x],x]

[Out]

ArcTanh[Sin[x]]/6 + ArcTanh[2*Sin[x]]/6 - ArcTanh[(2*Sin[x])/Sqrt[3]]/(2*Sqrt[3])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 2057

Int[(P_)^(p_), x_Symbol] :> With[{u = Factor[P /. x -> Sqrt[x]]}, Int[ExpandIntegrand[(u /. x -> x^2)^p, x], x
] /;  !SumQ[NonfreeFactors[u, x]]] /; PolyQ[P, x^2] && ILtQ[p, 0]

Rubi steps

\begin {align*} \int \csc (6 x) \sin (x) \, dx &=\operatorname {Subst}\left (\int \frac {1}{2 \left (3-19 x^2+32 x^4-16 x^6\right )} \, dx,x,\sin (x)\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{3-19 x^2+32 x^4-16 x^6} \, dx,x,\sin (x)\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (-\frac {1}{3 \left (-1+x^2\right )}+\frac {2}{-3+4 x^2}-\frac {2}{3 \left (-1+4 x^2\right )}\right ) \, dx,x,\sin (x)\right )\\ &=-\left (\frac {1}{6} \operatorname {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sin (x)\right )\right )-\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{-1+4 x^2} \, dx,x,\sin (x)\right )+\operatorname {Subst}\left (\int \frac {1}{-3+4 x^2} \, dx,x,\sin (x)\right )\\ &=\frac {1}{6} \tanh ^{-1}(\sin (x))+\frac {1}{6} \tanh ^{-1}(2 \sin (x))-\frac {\tanh ^{-1}\left (\frac {2 \sin (x)}{\sqrt {3}}\right )}{2 \sqrt {3}}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 30, normalized size = 0.83 \[ \frac {1}{6} \left (\tanh ^{-1}(\sin (x))+\tanh ^{-1}(2 \sin (x))-\sqrt {3} \tanh ^{-1}\left (\frac {2 \sin (x)}{\sqrt {3}}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[6*x]*Sin[x],x]

[Out]

(ArcTanh[Sin[x]] + ArcTanh[2*Sin[x]] - Sqrt[3]*ArcTanh[(2*Sin[x])/Sqrt[3]])/6

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fricas [B]  time = 1.24, size = 68, normalized size = 1.89 \[ \frac {1}{12} \, \sqrt {3} \log \left (-\frac {4 \, \cos \relax (x)^{2} + 4 \, \sqrt {3} \sin \relax (x) - 7}{4 \, \cos \relax (x)^{2} - 1}\right ) + \frac {1}{12} \, \log \left (2 \, \sin \relax (x) + 1\right ) + \frac {1}{12} \, \log \left (\sin \relax (x) + 1\right ) - \frac {1}{12} \, \log \left (-\sin \relax (x) + 1\right ) - \frac {1}{12} \, \log \left (-2 \, \sin \relax (x) + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(6*x)*sin(x),x, algorithm="fricas")

[Out]

1/12*sqrt(3)*log(-(4*cos(x)^2 + 4*sqrt(3)*sin(x) - 7)/(4*cos(x)^2 - 1)) + 1/12*log(2*sin(x) + 1) + 1/12*log(si
n(x) + 1) - 1/12*log(-sin(x) + 1) - 1/12*log(-2*sin(x) + 1)

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giac [B]  time = 0.15, size = 68, normalized size = 1.89 \[ \frac {1}{12} \, \sqrt {3} \log \left (\frac {{\left | -4 \, \sqrt {3} + 8 \, \sin \relax (x) \right |}}{{\left | 4 \, \sqrt {3} + 8 \, \sin \relax (x) \right |}}\right ) + \frac {1}{12} \, \log \left (\sin \relax (x) + 1\right ) - \frac {1}{12} \, \log \left (-\sin \relax (x) + 1\right ) + \frac {1}{12} \, \log \left ({\left | 2 \, \sin \relax (x) + 1 \right |}\right ) - \frac {1}{12} \, \log \left ({\left | 2 \, \sin \relax (x) - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(6*x)*sin(x),x, algorithm="giac")

[Out]

1/12*sqrt(3)*log(abs(-4*sqrt(3) + 8*sin(x))/abs(4*sqrt(3) + 8*sin(x))) + 1/12*log(sin(x) + 1) - 1/12*log(-sin(
x) + 1) + 1/12*log(abs(2*sin(x) + 1)) - 1/12*log(abs(2*sin(x) - 1))

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maple [A]  time = 0.26, size = 47, normalized size = 1.31 \[ -\frac {\ln \left (-1+2 \sin \relax (x )\right )}{12}+\frac {\ln \left (1+2 \sin \relax (x )\right )}{12}-\frac {\arctanh \left (\frac {2 \sin \relax (x ) \sqrt {3}}{3}\right ) \sqrt {3}}{6}-\frac {\ln \left (\sin \relax (x )-1\right )}{12}+\frac {\ln \left (1+\sin \relax (x )\right )}{12} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(6*x)*sin(x),x)

[Out]

-1/12*ln(-1+2*sin(x))+1/12*ln(1+2*sin(x))-1/6*arctanh(2/3*sin(x)*3^(1/2))*3^(1/2)-1/12*ln(sin(x)-1)+1/12*ln(1+
sin(x))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {1}{24} \, \sqrt {3} \log \left (\frac {4}{3} \, \cos \relax (x)^{2} + \frac {4}{3} \, \sin \relax (x)^{2} + \frac {4}{3} \, \sqrt {3} \sin \relax (x) + \frac {4}{3} \, \cos \relax (x) + \frac {4}{3}\right ) - \frac {1}{24} \, \sqrt {3} \log \left (\frac {4}{3} \, \cos \relax (x)^{2} + \frac {4}{3} \, \sin \relax (x)^{2} + \frac {4}{3} \, \sqrt {3} \sin \relax (x) - \frac {4}{3} \, \cos \relax (x) + \frac {4}{3}\right ) + \frac {1}{24} \, \sqrt {3} \log \left (\frac {4}{3} \, \cos \relax (x)^{2} + \frac {4}{3} \, \sin \relax (x)^{2} - \frac {4}{3} \, \sqrt {3} \sin \relax (x) + \frac {4}{3} \, \cos \relax (x) + \frac {4}{3}\right ) + \frac {1}{24} \, \sqrt {3} \log \left (\frac {4}{3} \, \cos \relax (x)^{2} + \frac {4}{3} \, \sin \relax (x)^{2} - \frac {4}{3} \, \sqrt {3} \sin \relax (x) - \frac {4}{3} \, \cos \relax (x) + \frac {4}{3}\right ) + \int -\frac {{\left (\cos \left (3 \, x\right ) + \cos \relax (x)\right )} \cos \left (4 \, x\right ) - {\left (\cos \left (2 \, x\right ) - 1\right )} \cos \left (3 \, x\right ) - \cos \left (2 \, x\right ) \cos \relax (x) + {\left (\sin \left (3 \, x\right ) + \sin \relax (x)\right )} \sin \left (4 \, x\right ) - \sin \left (3 \, x\right ) \sin \left (2 \, x\right ) - \sin \left (2 \, x\right ) \sin \relax (x) + \cos \relax (x)}{6 \, {\left (2 \, {\left (\cos \left (2 \, x\right ) - 1\right )} \cos \left (4 \, x\right ) - \cos \left (4 \, x\right )^{2} - \cos \left (2 \, x\right )^{2} - \sin \left (4 \, x\right )^{2} + 2 \, \sin \left (4 \, x\right ) \sin \left (2 \, x\right ) - \sin \left (2 \, x\right )^{2} + 2 \, \cos \left (2 \, x\right ) - 1\right )}}\,{d x} + \frac {1}{12} \, \log \left (\cos \relax (x)^{2} + \sin \relax (x)^{2} + 2 \, \sin \relax (x) + 1\right ) - \frac {1}{12} \, \log \left (\cos \relax (x)^{2} + \sin \relax (x)^{2} - 2 \, \sin \relax (x) + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(6*x)*sin(x),x, algorithm="maxima")

[Out]

-1/24*sqrt(3)*log(4/3*cos(x)^2 + 4/3*sin(x)^2 + 4/3*sqrt(3)*sin(x) + 4/3*cos(x) + 4/3) - 1/24*sqrt(3)*log(4/3*
cos(x)^2 + 4/3*sin(x)^2 + 4/3*sqrt(3)*sin(x) - 4/3*cos(x) + 4/3) + 1/24*sqrt(3)*log(4/3*cos(x)^2 + 4/3*sin(x)^
2 - 4/3*sqrt(3)*sin(x) + 4/3*cos(x) + 4/3) + 1/24*sqrt(3)*log(4/3*cos(x)^2 + 4/3*sin(x)^2 - 4/3*sqrt(3)*sin(x)
- 4/3*cos(x) + 4/3) + integrate(-1/6*((cos(3*x) + cos(x))*cos(4*x) - (cos(2*x) - 1)*cos(3*x) - cos(2*x)*cos(x
) + (sin(3*x) + sin(x))*sin(4*x) - sin(3*x)*sin(2*x) - sin(2*x)*sin(x) + cos(x))/(2*(cos(2*x) - 1)*cos(4*x) -
cos(4*x)^2 - cos(2*x)^2 - sin(4*x)^2 + 2*sin(4*x)*sin(2*x) - sin(2*x)^2 + 2*cos(2*x) - 1), x) + 1/12*log(cos(x
)^2 + sin(x)^2 + 2*sin(x) + 1) - 1/12*log(cos(x)^2 + sin(x)^2 - 2*sin(x) + 1)

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mupad [B]  time = 2.46, size = 35, normalized size = 0.97 \[ \frac {\mathrm {atanh}\left (\frac {\sin \left (\frac {x}{2}\right )}{\cos \left (\frac {x}{2}\right )}\right )}{3}+\frac {\mathrm {atanh}\left (2\,\sin \relax (x)\right )}{6}-\frac {\sqrt {3}\,\mathrm {atanh}\left (\frac {2\,\sqrt {3}\,\sin \relax (x)}{3}\right )}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(x)/sin(6*x),x)

[Out]

atanh(sin(x/2)/cos(x/2))/3 + atanh(2*sin(x))/6 - (3^(1/2)*atanh((2*3^(1/2)*sin(x))/3))/6

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sin {\relax (x )} \csc {\left (6 x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(6*x)*sin(x),x)

[Out]

Integral(sin(x)*csc(6*x), x)

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