3.126 \(\int \cos (x) \csc (5 x) \, dx\)

Optimal. Leaf size=62 \[ -\frac{1}{20} \left (1+\sqrt{5}\right ) \log \left (-8 \sin ^2(x)-\sqrt{5}+5\right )-\frac{1}{20} \left (1-\sqrt{5}\right ) \log \left (-8 \sin ^2(x)+\sqrt{5}+5\right )+\frac{1}{5} \log (\sin (x)) \]

[Out]

Log[Sin[x]]/5 - ((1 + Sqrt[5])*Log[5 - Sqrt[5] - 8*Sin[x]^2])/20 - ((1 - Sqrt[5])*Log[5 + Sqrt[5] - 8*Sin[x]^2
])/20

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Rubi [A]  time = 0.0699113, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.857, Rules used = {4356, 1114, 705, 29, 632, 31} \[ -\frac{1}{20} \left (1+\sqrt{5}\right ) \log \left (-8 \sin ^2(x)-\sqrt{5}+5\right )-\frac{1}{20} \left (1-\sqrt{5}\right ) \log \left (-8 \sin ^2(x)+\sqrt{5}+5\right )+\frac{1}{5} \log (\sin (x)) \]

Antiderivative was successfully verified.

[In]

Int[Cos[x]*Csc[5*x],x]

[Out]

Log[Sin[x]]/5 - ((1 + Sqrt[5])*Log[5 - Sqrt[5] - 8*Sin[x]^2])/20 - ((1 - Sqrt[5])*Log[5 + Sqrt[5] - 8*Sin[x]^2
])/20

Rule 4356

Int[(u_)*(F_)[(c_.)*((a_.) + (b_.)*(x_))], x_Symbol] :> With[{d = FreeFactors[Sin[c*(a + b*x)], x]}, Dist[d/(b
*c), Subst[Int[SubstFor[1, Sin[c*(a + b*x)]/d, u, x], x], x, Sin[c*(a + b*x)]/d], x] /; FunctionOfQ[Sin[c*(a +
b*x)]/d, u, x]] /; FreeQ[{a, b, c}, x] && (EqQ[F, Cos] || EqQ[F, cos])

Rule 1114

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[x^((m - 1)/2)*(a +
b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, p}, x] && IntegerQ[(m - 1)/2]

Rule 705

Int[1/(((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[e^2/(c*d^2 - b*d*e + a*e^2
), Int[1/(d + e*x), x], x] + Dist[1/(c*d^2 - b*d*e + a*e^2), Int[(c*d - b*e - c*e*x)/(a + b*x + c*x^2), x], x]
/; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 632

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[
(c*d - e*(b/2 - q/2))/q, Int[1/(b/2 - q/2 + c*x), x], x] - Dist[(c*d - e*(b/2 + q/2))/q, Int[1/(b/2 + q/2 + c*
x), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] && NiceSqrtQ[b^2 - 4*a*
c]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

\begin{align*} \int \cos (x) \csc (5 x) \, dx &=\operatorname{Subst}\left (\int \frac{1}{x \left (5-20 x^2+16 x^4\right )} \, dx,x,\sin (x)\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{x \left (5-20 x+16 x^2\right )} \, dx,x,\sin ^2(x)\right )\\ &=\frac{1}{10} \operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,\sin ^2(x)\right )+\frac{1}{10} \operatorname{Subst}\left (\int \frac{20-16 x}{5-20 x+16 x^2} \, dx,x,\sin ^2(x)\right )\\ &=\frac{1}{5} \log (\sin (x))-\frac{1}{5} \left (4 \left (1-\sqrt{5}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-10-2 \sqrt{5}+16 x} \, dx,x,\sin ^2(x)\right )-\frac{1}{5} \left (4 \left (1+\sqrt{5}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-10+2 \sqrt{5}+16 x} \, dx,x,\sin ^2(x)\right )\\ &=\frac{1}{5} \log (\sin (x))-\frac{1}{20} \left (1+\sqrt{5}\right ) \log \left (5-\sqrt{5}-8 \sin ^2(x)\right )-\frac{1}{20} \left (1-\sqrt{5}\right ) \log \left (5+\sqrt{5}-8 \sin ^2(x)\right )\\ \end{align*}

Mathematica [A]  time = 0.0623529, size = 57, normalized size = 0.92 \[ \frac{1}{20} \left (4 \log (\sin (x))-\left (1+\sqrt{5}\right ) \log \left (4 \cos (2 x)-\sqrt{5}+1\right )+\left (\sqrt{5}-1\right ) \log \left (4 \cos (2 x)+\sqrt{5}+1\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[x]*Csc[5*x],x]

[Out]

(-((1 + Sqrt[5])*Log[1 - Sqrt[5] + 4*Cos[2*x]]) + (-1 + Sqrt[5])*Log[1 + Sqrt[5] + 4*Cos[2*x]] + 4*Log[Sin[x]]
)/20

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Maple [A]  time = 0.097, size = 80, normalized size = 1.3 \begin{align*} -{\frac{\ln \left ( 4\, \left ( \cos \left ( x \right ) \right ) ^{2}-2\,\cos \left ( x \right ) -1 \right ) }{20}}+{\frac{\sqrt{5}}{10}{\it Artanh} \left ({\frac{ \left ( 8\,\cos \left ( x \right ) -2 \right ) \sqrt{5}}{10}} \right ) }+{\frac{\ln \left ( 1+\cos \left ( x \right ) \right ) }{10}}+{\frac{\ln \left ( -1+\cos \left ( x \right ) \right ) }{10}}-{\frac{\ln \left ( 4\, \left ( \cos \left ( x \right ) \right ) ^{2}+2\,\cos \left ( x \right ) -1 \right ) }{20}}-{\frac{\sqrt{5}}{10}{\it Artanh} \left ({\frac{ \left ( 8\,\cos \left ( x \right ) +2 \right ) \sqrt{5}}{10}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(x)*csc(5*x),x)

[Out]

-1/20*ln(4*cos(x)^2-2*cos(x)-1)+1/10*5^(1/2)*arctanh(1/10*(8*cos(x)-2)*5^(1/2))+1/10*ln(1+cos(x))+1/10*ln(-1+c
os(x))-1/20*ln(4*cos(x)^2+2*cos(x)-1)-1/10*5^(1/2)*arctanh(1/10*(8*cos(x)+2)*5^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)*csc(5*x),x, algorithm="maxima")

[Out]

-1/10*integrate(-(cos(2*x)*sin(4*x) - cos(4*x)*sin(2*x) + cos(3/2*arctan2(sin(2*x), cos(2*x)))*sin(2*x) + cos(
1/2*arctan2(sin(2*x), cos(2*x)))*sin(2*x) - cos(2*x)*sin(3/2*arctan2(sin(2*x), cos(2*x))) - cos(2*x)*sin(1/2*a
rctan2(sin(2*x), cos(2*x))) - sin(2*x))/(2*(cos(2*x) + 1)*cos(4*x) + cos(4*x)^2 + cos(2*x)^2 - 2*(cos(4*x) + c
os(2*x) - cos(1/2*arctan2(sin(2*x), cos(2*x))) + 1)*cos(3/2*arctan2(sin(2*x), cos(2*x))) + cos(3/2*arctan2(sin
(2*x), cos(2*x)))^2 - 2*(cos(4*x) + cos(2*x) + 1)*cos(1/2*arctan2(sin(2*x), cos(2*x))) + cos(1/2*arctan2(sin(2
*x), cos(2*x)))^2 + sin(4*x)^2 + 2*sin(4*x)*sin(2*x) + sin(2*x)^2 - 2*(sin(4*x) + sin(2*x) - sin(1/2*arctan2(s
in(2*x), cos(2*x))))*sin(3/2*arctan2(sin(2*x), cos(2*x))) + sin(3/2*arctan2(sin(2*x), cos(2*x)))^2 - 2*(sin(4*
x) + sin(2*x))*sin(1/2*arctan2(sin(2*x), cos(2*x))) + sin(1/2*arctan2(sin(2*x), cos(2*x)))^2 + 2*cos(2*x) + 1)
, x) + 1/10*integrate((cos(2*x)*sin(4*x) - cos(4*x)*sin(2*x) - cos(3/2*arctan2(sin(2*x), cos(2*x)))*sin(2*x) -
cos(1/2*arctan2(sin(2*x), cos(2*x)))*sin(2*x) + cos(2*x)*sin(3/2*arctan2(sin(2*x), cos(2*x))) + cos(2*x)*sin(
1/2*arctan2(sin(2*x), cos(2*x))) - sin(2*x))/(2*(cos(2*x) + 1)*cos(4*x) + cos(4*x)^2 + cos(2*x)^2 + 2*(cos(4*x
) + cos(2*x) + cos(1/2*arctan2(sin(2*x), cos(2*x))) + 1)*cos(3/2*arctan2(sin(2*x), cos(2*x))) + cos(3/2*arctan
2(sin(2*x), cos(2*x)))^2 + 2*(cos(4*x) + cos(2*x) + 1)*cos(1/2*arctan2(sin(2*x), cos(2*x))) + cos(1/2*arctan2(
sin(2*x), cos(2*x)))^2 + sin(4*x)^2 + 2*sin(4*x)*sin(2*x) + sin(2*x)^2 + 2*(sin(4*x) + sin(2*x) + sin(1/2*arct
an2(sin(2*x), cos(2*x))))*sin(3/2*arctan2(sin(2*x), cos(2*x))) + sin(3/2*arctan2(sin(2*x), cos(2*x)))^2 + 2*(s
in(4*x) + sin(2*x))*sin(1/2*arctan2(sin(2*x), cos(2*x))) + sin(1/2*arctan2(sin(2*x), cos(2*x)))^2 + 2*cos(2*x)
+ 1), x) - 1/10*integrate((cos(x)*sin(4*x) + cos(x)*sin(3*x) + cos(x)*sin(2*x) - cos(4*x)*sin(x) - cos(3*x)*s
in(x) - cos(2*x)*sin(x) - sin(x))/(2*(cos(3*x) + cos(2*x) + cos(x) + 1)*cos(4*x) + cos(4*x)^2 + 2*(cos(2*x) +
cos(x) + 1)*cos(3*x) + cos(3*x)^2 + 2*(cos(x) + 1)*cos(2*x) + cos(2*x)^2 + cos(x)^2 + 2*(sin(3*x) + sin(2*x) +
sin(x))*sin(4*x) + sin(4*x)^2 + 2*(sin(2*x) + sin(x))*sin(3*x) + sin(3*x)^2 + sin(2*x)^2 + 2*sin(2*x)*sin(x)
+ sin(x)^2 + 2*cos(x) + 1), x) + 1/10*integrate(-(cos(x)*sin(4*x) - cos(x)*sin(3*x) + cos(x)*sin(2*x) - cos(4*
x)*sin(x) + cos(3*x)*sin(x) - cos(2*x)*sin(x) - sin(x))/(2*(cos(3*x) - cos(2*x) + cos(x) - 1)*cos(4*x) - cos(4
*x)^2 + 2*(cos(2*x) - cos(x) + 1)*cos(3*x) - cos(3*x)^2 + 2*(cos(x) - 1)*cos(2*x) - cos(2*x)^2 - cos(x)^2 + 2*
(sin(3*x) - sin(2*x) + sin(x))*sin(4*x) - sin(4*x)^2 + 2*(sin(2*x) - sin(x))*sin(3*x) - sin(3*x)^2 - sin(2*x)^
2 + 2*sin(2*x)*sin(x) - sin(x)^2 + 2*cos(x) - 1), x) + 3/10*integrate(-(cos(4/3*arctan2(sin(3*x), cos(3*x)))*s
in(3*x) + cos(2/3*arctan2(sin(3*x), cos(3*x)))*sin(3*x) + cos(1/3*arctan2(sin(3*x), cos(3*x)))*sin(3*x) - cos(
3*x)*sin(4/3*arctan2(sin(3*x), cos(3*x))) - cos(3*x)*sin(2/3*arctan2(sin(3*x), cos(3*x))) - cos(3*x)*sin(1/3*a
rctan2(sin(3*x), cos(3*x))) + sin(3*x))/(cos(3*x)^2 + 2*(cos(3*x) + cos(2/3*arctan2(sin(3*x), cos(3*x))) + cos
(1/3*arctan2(sin(3*x), cos(3*x))) + 1)*cos(4/3*arctan2(sin(3*x), cos(3*x))) + cos(4/3*arctan2(sin(3*x), cos(3*
x)))^2 + 2*(cos(3*x) + cos(1/3*arctan2(sin(3*x), cos(3*x))) + 1)*cos(2/3*arctan2(sin(3*x), cos(3*x))) + cos(2/
3*arctan2(sin(3*x), cos(3*x)))^2 + 2*(cos(3*x) + 1)*cos(1/3*arctan2(sin(3*x), cos(3*x))) + cos(1/3*arctan2(sin
(3*x), cos(3*x)))^2 + sin(3*x)^2 + 2*(sin(3*x) + sin(2/3*arctan2(sin(3*x), cos(3*x))) + sin(1/3*arctan2(sin(3*
x), cos(3*x))))*sin(4/3*arctan2(sin(3*x), cos(3*x))) + sin(4/3*arctan2(sin(3*x), cos(3*x)))^2 + 2*(sin(3*x) +
sin(1/3*arctan2(sin(3*x), cos(3*x))))*sin(2/3*arctan2(sin(3*x), cos(3*x))) + sin(2/3*arctan2(sin(3*x), cos(3*x
)))^2 + 2*sin(3*x)*sin(1/3*arctan2(sin(3*x), cos(3*x))) + sin(1/3*arctan2(sin(3*x), cos(3*x)))^2 + 2*cos(3*x)
+ 1), x) - 3/10*integrate(-(cos(4/3*arctan2(sin(3*x), cos(3*x)))*sin(3*x) + cos(2/3*arctan2(sin(3*x), cos(3*x)
))*sin(3*x) - cos(1/3*arctan2(sin(3*x), cos(3*x)))*sin(3*x) - cos(3*x)*sin(4/3*arctan2(sin(3*x), cos(3*x))) -
cos(3*x)*sin(2/3*arctan2(sin(3*x), cos(3*x))) + cos(3*x)*sin(1/3*arctan2(sin(3*x), cos(3*x))) + sin(3*x))/(cos
(3*x)^2 - 2*(cos(3*x) - cos(2/3*arctan2(sin(3*x), cos(3*x))) + cos(1/3*arctan2(sin(3*x), cos(3*x))) - 1)*cos(4
/3*arctan2(sin(3*x), cos(3*x))) + cos(4/3*arctan2(sin(3*x), cos(3*x)))^2 - 2*(cos(3*x) + cos(1/3*arctan2(sin(3
*x), cos(3*x))) - 1)*cos(2/3*arctan2(sin(3*x), cos(3*x))) + cos(2/3*arctan2(sin(3*x), cos(3*x)))^2 + 2*(cos(3*
x) - 1)*cos(1/3*arctan2(sin(3*x), cos(3*x))) + cos(1/3*arctan2(sin(3*x), cos(3*x)))^2 + sin(3*x)^2 - 2*(sin(3*
x) - sin(2/3*arctan2(sin(3*x), cos(3*x))) + sin(1/3*arctan2(sin(3*x), cos(3*x))))*sin(4/3*arctan2(sin(3*x), co
s(3*x))) + sin(4/3*arctan2(sin(3*x), cos(3*x)))^2 - 2*(sin(3*x) + sin(1/3*arctan2(sin(3*x), cos(3*x))))*sin(2/
3*arctan2(sin(3*x), cos(3*x))) + sin(2/3*arctan2(sin(3*x), cos(3*x)))^2 + 2*sin(3*x)*sin(1/3*arctan2(sin(3*x),
cos(3*x))) + sin(1/3*arctan2(sin(3*x), cos(3*x)))^2 - 2*cos(3*x) + 1), x) + 1/5*integrate((sin(4*x) + sin(3*x
) + sin(2*x) + sin(x))/(2*(cos(3*x) + cos(2*x) + cos(x) + 1)*cos(4*x) + cos(4*x)^2 + 2*(cos(2*x) + cos(x) + 1)
*cos(3*x) + cos(3*x)^2 + 2*(cos(x) + 1)*cos(2*x) + cos(2*x)^2 + cos(x)^2 + 2*(sin(3*x) + sin(2*x) + sin(x))*si
n(4*x) + sin(4*x)^2 + 2*(sin(2*x) + sin(x))*sin(3*x) + sin(3*x)^2 + sin(2*x)^2 + 2*sin(2*x)*sin(x) + sin(x)^2
+ 2*cos(x) + 1), x) + 1/5*integrate(-(sin(4*x) - sin(3*x) + sin(2*x) - sin(x))/(2*(cos(3*x) - cos(2*x) + cos(x
) - 1)*cos(4*x) - cos(4*x)^2 + 2*(cos(2*x) - cos(x) + 1)*cos(3*x) - cos(3*x)^2 + 2*(cos(x) - 1)*cos(2*x) - cos
(2*x)^2 - cos(x)^2 + 2*(sin(3*x) - sin(2*x) + sin(x))*sin(4*x) - sin(4*x)^2 + 2*(sin(2*x) - sin(x))*sin(3*x) -
sin(3*x)^2 - sin(2*x)^2 + 2*sin(2*x)*sin(x) - sin(x)^2 + 2*cos(x) - 1), x) + 1/10*log(cos(x)^2 + sin(x)^2 + 2
*cos(x) + 1) + 1/10*log(cos(x)^2 + sin(x)^2 - 2*cos(x) + 1)

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Fricas [A]  time = 2.56656, size = 232, normalized size = 3.74 \begin{align*} \frac{1}{20} \, \sqrt{5} \log \left (\frac{32 \, \cos \left (x\right )^{4} + 8 \,{\left (\sqrt{5} - 3\right )} \cos \left (x\right )^{2} - 3 \, \sqrt{5} + 7}{16 \, \cos \left (x\right )^{4} - 12 \, \cos \left (x\right )^{2} + 1}\right ) - \frac{1}{20} \, \log \left (16 \, \cos \left (x\right )^{4} - 12 \, \cos \left (x\right )^{2} + 1\right ) + \frac{1}{5} \, \log \left (\frac{1}{2} \, \sin \left (x\right )\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)*csc(5*x),x, algorithm="fricas")

[Out]

1/20*sqrt(5)*log((32*cos(x)^4 + 8*(sqrt(5) - 3)*cos(x)^2 - 3*sqrt(5) + 7)/(16*cos(x)^4 - 12*cos(x)^2 + 1)) - 1
/20*log(16*cos(x)^4 - 12*cos(x)^2 + 1) + 1/5*log(1/2*sin(x))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \cos{\left (x \right )} \csc{\left (5 x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)*csc(5*x),x)

[Out]

Integral(cos(x)*csc(5*x), x)

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Giac [B]  time = 1.20177, size = 177, normalized size = 2.85 \begin{align*} -\frac{1}{20} \, \sqrt{5} \log \left (\frac{{\left | -16 \, \sqrt{5} - \frac{10 \,{\left (\cos \left (x\right ) + 1\right )}}{\cos \left (x\right ) - 1} - \frac{10 \,{\left (\cos \left (x\right ) - 1\right )}}{\cos \left (x\right ) + 1} - 60 \right |}}{{\left | 16 \, \sqrt{5} - \frac{10 \,{\left (\cos \left (x\right ) + 1\right )}}{\cos \left (x\right ) - 1} - \frac{10 \,{\left (\cos \left (x\right ) - 1\right )}}{\cos \left (x\right ) + 1} - 60 \right |}}\right ) - \frac{1}{20} \, \log \left ({\left | 5 \,{\left (\frac{\cos \left (x\right ) + 1}{\cos \left (x\right ) - 1} + \frac{\cos \left (x\right ) - 1}{\cos \left (x\right ) + 1}\right )}^{2} + \frac{60 \,{\left (\cos \left (x\right ) + 1\right )}}{\cos \left (x\right ) - 1} + \frac{60 \,{\left (\cos \left (x\right ) - 1\right )}}{\cos \left (x\right ) + 1} + 116 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)*csc(5*x),x, algorithm="giac")

[Out]

-1/20*sqrt(5)*log(abs(-16*sqrt(5) - 10*(cos(x) + 1)/(cos(x) - 1) - 10*(cos(x) - 1)/(cos(x) + 1) - 60)/abs(16*s
qrt(5) - 10*(cos(x) + 1)/(cos(x) - 1) - 10*(cos(x) - 1)/(cos(x) + 1) - 60)) - 1/20*log(abs(5*((cos(x) + 1)/(co
s(x) - 1) + (cos(x) - 1)/(cos(x) + 1))^2 + 60*(cos(x) + 1)/(cos(x) - 1) + 60*(cos(x) - 1)/(cos(x) + 1) + 116))