\(\int \cos (x) \csc (5 x) \, dx\) [126]
Optimal result
Integrand size = 7, antiderivative size = 62 \[
\int \cos (x) \csc (5 x) \, dx=\frac {1}{5} \log (\sin (x))-\frac {1}{20} \left (1+\sqrt {5}\right ) \log \left (5-\sqrt {5}-8 \sin ^2(x)\right )-\frac {1}{20} \left (1-\sqrt {5}\right ) \log \left (5+\sqrt {5}-8 \sin ^2(x)\right )
\]
[Out]
1/5*ln(sin(x))-1/20*ln(5-8*sin(x)^2+5^(1/2))*(-5^(1/2)+1)-1/20*ln(5-8*sin(x)^2-5^(1/2))*(5^(1/2)+1)
Rubi [A] (verified)
Time = 0.09 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.00, number of
steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.857, Rules used = {4441, 1128, 719, 29, 646, 31}
\[
\int \cos (x) \csc (5 x) \, dx=-\frac {1}{20} \left (1+\sqrt {5}\right ) \log \left (-8 \sin ^2(x)-\sqrt {5}+5\right )-\frac {1}{20} \left (1-\sqrt {5}\right ) \log \left (-8 \sin ^2(x)+\sqrt {5}+5\right )+\frac {1}{5} \log (\sin (x))
\]
[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 29
Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]
Rule 31
Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]
Rule 646
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 719
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 1128
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 4441
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])
Rubi steps \begin{align*}
\text {integral}& = \text {Subst}\left (\int \frac {1}{x \left (5-20 x^2+16 x^4\right )} \, dx,x,\sin (x)\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \frac {1}{x \left (5-20 x+16 x^2\right )} \, dx,x,\sin ^2(x)\right ) \\ & = \frac {1}{10} \text {Subst}\left (\int \frac {1}{x} \, dx,x,\sin ^2(x)\right )+\frac {1}{10} \text {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 ) \text {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 ) \text {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] (verified)
Time = 0.08 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.92
\[
\int \cos (x) \csc (5 x) \, dx=\frac {1}{20} \left (-\left (\left (1+\sqrt {5}\right ) \log \left (1-\sqrt {5}+4 \cos (2 x)\right )\right )+\left (-1+\sqrt {5}\right ) \log \left (1+\sqrt {5}+4 \cos (2 x)\right )+4 \log (\sin (x))\right )
\]
[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
Maple [A] (verified)
Time = 1.81 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.29
| | |
| method | result | size |
| | |
| default |
\(-\frac {\ln \left (4 \cos \left (x \right )^{2}+2 \cos \left (x \right )-1\right )}{20}-\frac {\sqrt {5}\, \operatorname {arctanh}\left (\frac {\left (8 \cos \left (x \right )+2\right ) \sqrt {5}}{10}\right )}{10}+\frac {\ln \left (\cos \left (x \right )-1\right )}{10}-\frac {\ln \left (4 \cos \left (x \right )^{2}-2 \cos \left (x \right )-1\right )}{20}+\frac {\sqrt {5}\, \operatorname {arctanh}\left (\frac {\left (8 \cos \left (x \right )-2\right ) \sqrt {5}}{10}\right )}{10}+\frac {\ln \left (\cos \left (x \right )+1\right )}{10}\) |
\(80\) |
| risch |
\(\frac {\ln \left ({\mathrm e}^{2 i x}-1\right )}{5}-\frac {\ln \left ({\mathrm e}^{4 i x}+\left (\frac {1}{2}-\frac {\sqrt {5}}{2}\right ) {\mathrm e}^{2 i x}+1\right )}{20}-\frac {\ln \left ({\mathrm e}^{4 i x}+\left (\frac {1}{2}-\frac {\sqrt {5}}{2}\right ) {\mathrm e}^{2 i x}+1\right ) \sqrt {5}}{20}-\frac {\ln \left ({\mathrm e}^{4 i x}+\left (\frac {1}{2}+\frac {\sqrt {5}}{2}\right ) {\mathrm e}^{2 i x}+1\right )}{20}+\frac {\ln \left ({\mathrm e}^{4 i x}+\left (\frac {1}{2}+\frac {\sqrt {5}}{2}\right ) {\mathrm e}^{2 i x}+1\right ) \sqrt {5}}{20}\) |
\(110\) |
| | |
|
|
|
|
[In]
int(cos(x)*csc(5*x),x,method=_RETURNVERBOSE)
[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(cos(x)-1)-1/20*ln(4*co
s(x)^2-2*cos(x)-1)+1/10*5^(1/2)*arctanh(1/10*(8*cos(x)-2)*5^(1/2))+1/10*ln(cos(x)+1)
Fricas [A] (verification not implemented)
none
Time = 0.25 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.16
\[
\int \cos (x) \csc (5 x) \, dx=\frac {1}{20} \, \sqrt {5} \log \left (\frac {32 \, \cos \left (x\right )^{4} + 8 \, {\left (\sqrt {5} - 3\right )} \cos \left (x\right )^{2} - 3 \, \sqrt {5} + 7}{16 \, \cos \left (x\right )^{4} - 12 \, \cos \left (x\right )^{2} + 1}\right ) - \frac {1}{20} \, \log \left (16 \, \cos \left (x\right )^{4} - 12 \, \cos \left (x\right )^{2} + 1\right ) + \frac {1}{5} \, \log \left (\frac {1}{2} \, \sin \left (x\right )\right )
\]
[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))
Sympy [F]
\[
\int \cos (x) \csc (5 x) \, dx=\int \cos {\left (x \right )} \csc {\left (5 x \right )}\, dx
\]
[In]
integrate(cos(x)*csc(5*x),x)
[Out]
Integral(cos(x)*csc(5*x), x)
Maxima [F]
\[
\int \cos (x) \csc (5 x) \, dx=\int { \cos \left (x\right ) \csc \left (5 \, x\right ) \,d x }
\]
[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)
Giac [A] (verification not implemented)
none
Time = 0.26 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.08
\[
\int \cos (x) \csc (5 x) \, dx=-\frac {1}{20} \, \sqrt {5} \log \left (\frac {{\left | 32 \, \cos \left (x\right )^{2} - 4 \, \sqrt {5} - 12 \right |}}{{\left | 32 \, \cos \left (x\right )^{2} + 4 \, \sqrt {5} - 12 \right |}}\right ) + \frac {1}{10} \, \log \left (-\cos \left (x\right )^{2} + 1\right ) - \frac {1}{20} \, \log \left ({\left | 16 \, \cos \left (x\right )^{4} - 12 \, \cos \left (x\right )^{2} + 1 \right |}\right )
\]
[In]
integrate(cos(x)*csc(5*x),x, algorithm="giac")
[Out]
-1/20*sqrt(5)*log(abs(32*cos(x)^2 - 4*sqrt(5) - 12)/abs(32*cos(x)^2 + 4*sqrt(5) - 12)) + 1/10*log(-cos(x)^2 +
1) - 1/20*log(abs(16*cos(x)^4 - 12*cos(x)^2 + 1))
Mupad [B] (verification not implemented)
Time = 27.45 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.82
\[
\int \cos (x) \csc (5 x) \, dx=\frac {\ln \left (\sin \left (x\right )\right )}{5}+\ln \left (-{\cos \left (x\right )}^2-\frac {\sqrt {5}}{8}+\frac {3}{8}\right )\,\left (\frac {\sqrt {5}}{20}-\frac {1}{20}\right )-\ln \left (-{\cos \left (x\right )}^2+\frac {\sqrt {5}}{8}+\frac {3}{8}\right )\,\left (\frac {\sqrt {5}}{20}+\frac {1}{20}\right )
\]
[In]
int(cos(x)/sin(5*x),x)
[Out]
log(sin(x))/5 + log(3/8 - 5^(1/2)/8 - cos(x)^2)*(5^(1/2)/20 - 1/20) - log(5^(1/2)/8 - cos(x)^2 + 3/8)*(5^(1/2)
/20 + 1/20)