\(\int \csc (6 x) \sin (x) \, dx\) [94]

Optimal result

Integrand size = 7, antiderivative size = 36 \[ \int \csc (6 x) \sin (x) \, dx=\frac {1}{6} \text {arctanh}(\sin (x))+\frac {1}{6} \text {arctanh}(2 \sin (x))-\frac {\text {arctanh}\left (\frac {2 \sin (x)}{\sqrt {3}}\right )}{2 \sqrt {3}} \]

[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)

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {12, 2082, 213} \[ \int \csc (6 x) \sin (x) \, dx=\frac {1}{6} \text {arctanh}(\sin (x))+\frac {1}{6} \text {arctanh}(2 \sin (x))-\frac {\text {arctanh}\left (\frac {2 \sin (x)}{\sqrt {3}}\right )}{2 \sqrt {3}} \]

[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 213

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

Rule 2082

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

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.83 \[ \int \csc (6 x) \sin (x) \, dx=\frac {1}{6} \left (\text {arctanh}(\sin (x))+\text {arctanh}(2 \sin (x))-\sqrt {3} \text {arctanh}\left (\frac {2 \sin (x)}{\sqrt {3}}\right )\right ) \]

[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

Maple [A] (verified)

Time = 0.72 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.31

method	result	size
default	\(\frac {\ln \left (1+\sin \left (x \right )\right )}{12}-\frac {\ln \left (\sin \left (x \right )-1\right )}{12}-\frac {\operatorname {arctanh}\left (\frac {2 \sin \left (x \right ) \sqrt {3}}{3}\right ) \sqrt {3}}{6}-\frac {\ln \left (2 \sin \left (x \right )-1\right )}{12}+\frac {\ln \left (1+2 \sin \left (x \right )\right )}{12}\)	\(47\)
risch	\(\frac {\ln \left (i+{\mathrm e}^{i x}\right )}{6}-\frac {\ln \left ({\mathrm e}^{i x}-i\right )}{6}+\frac {\sqrt {3}\, \ln \left ({\mathrm e}^{2 i x}-i \sqrt {3}\, {\mathrm e}^{i x}-1\right )}{12}-\frac {\sqrt {3}\, \ln \left ({\mathrm e}^{2 i x}+i \sqrt {3}\, {\mathrm e}^{i x}-1\right )}{12}-\frac {\ln \left (-i {\mathrm e}^{i x}+{\mathrm e}^{2 i x}-1\right )}{12}+\frac {\ln \left (i {\mathrm e}^{i x}+{\mathrm e}^{2 i x}-1\right )}{12}\)	\(108\)

[In]

int(csc(6*x)*sin(x),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 68 vs. \(2 (26) = 52\).

Time = 0.27 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.89 \[ \int \csc (6 x) \sin (x) \, dx=\frac {1}{12} \, \sqrt {3} \log \left (-\frac {4 \, \cos \left (x\right )^{2} + 4 \, \sqrt {3} \sin \left (x\right ) - 7}{4 \, \cos \left (x\right )^{2} - 1}\right ) + \frac {1}{12} \, \log \left (2 \, \sin \left (x\right ) + 1\right ) + \frac {1}{12} \, \log \left (\sin \left (x\right ) + 1\right ) - \frac {1}{12} \, \log \left (-\sin \left (x\right ) + 1\right ) - \frac {1}{12} \, \log \left (-2 \, \sin \left (x\right ) + 1\right ) \]

[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)

Sympy [F]

\[ \int \csc (6 x) \sin (x) \, dx=\int \sin {\left (x \right )} \csc {\left (6 x \right )}\, dx \]

[In]

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

[Out]

Integral(sin(x)*csc(6*x), x)

Maxima [F]

\[ \int \csc (6 x) \sin (x) \, dx=\int { \csc \left (6 \, x\right ) \sin \left (x\right ) \,d x } \]

[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)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 68 vs. \(2 (26) = 52\).

Time = 0.28 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.89 \[ \int \csc (6 x) \sin (x) \, dx=\frac {1}{12} \, \sqrt {3} \log \left (\frac {{\left | -4 \, \sqrt {3} + 8 \, \sin \left (x\right ) \right |}}{{\left | 4 \, \sqrt {3} + 8 \, \sin \left (x\right ) \right |}}\right ) + \frac {1}{12} \, \log \left (\sin \left (x\right ) + 1\right ) - \frac {1}{12} \, \log \left (-\sin \left (x\right ) + 1\right ) + \frac {1}{12} \, \log \left ({\left | 2 \, \sin \left (x\right ) + 1 \right |}\right ) - \frac {1}{12} \, \log \left ({\left | 2 \, \sin \left (x\right ) - 1 \right |}\right ) \]

[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))

Mupad [B] (verification not implemented)

Time = 28.12 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.97 \[ \int \csc (6 x) \sin (x) \, dx=\frac {\mathrm {atanh}\left (\frac {\sin \left (\frac {x}{2}\right )}{\cos \left (\frac {x}{2}\right )}\right )}{3}+\frac {\mathrm {atanh}\left (2\,\sin \left (x\right )\right )}{6}-\frac {\sqrt {3}\,\mathrm {atanh}\left (\frac {2\,\sqrt {3}\,\sin \left (x\right )}{3}\right )}{6} \]

[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