3.119 \(\int \cos (x) \sec (5 x) \, dx\)

Optimal. Leaf size=163 \[ \frac {1}{10} \sqrt {\frac {1}{2} \left (5-\sqrt {5}\right )} \log \left (\cos (x)-\sqrt {5-2 \sqrt {5}} \sin (x)\right )-\frac {1}{10} \sqrt {\frac {1}{2} \left (5-\sqrt {5}\right )} \log \left (\sqrt {5-2 \sqrt {5}} \sin (x)+\cos (x)\right )-\frac {1}{10} \sqrt {\frac {1}{2} \left (5+\sqrt {5}\right )} \log \left (\cos (x)-\sqrt {5+2 \sqrt {5}} \sin (x)\right )+\frac {1}{10} \sqrt {\frac {1}{2} \left (5+\sqrt {5}\right )} \log \left (\sqrt {5+2 \sqrt {5}} \sin (x)+\cos (x)\right ) \]

[Out]

1/20*ln(cos(x)-sin(x)*(5-2*5^(1/2))^(1/2))*(10-2*5^(1/2))^(1/2)-1/20*ln(cos(x)+sin(x)*(5-2*5^(1/2))^(1/2))*(10
-2*5^(1/2))^(1/2)-1/20*ln(cos(x)-sin(x)*(5+2*5^(1/2))^(1/2))*(10+2*5^(1/2))^(1/2)+1/20*ln(cos(x)+sin(x)*(5+2*5
^(1/2))^(1/2))*(10+2*5^(1/2))^(1/2)

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Rubi [A]  time = 0.13, antiderivative size = 163, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {1166, 207} \[ \frac {1}{10} \sqrt {\frac {1}{2} \left (5-\sqrt {5}\right )} \log \left (\cos (x)-\sqrt {5-2 \sqrt {5}} \sin (x)\right )-\frac {1}{10} \sqrt {\frac {1}{2} \left (5-\sqrt {5}\right )} \log \left (\sqrt {5-2 \sqrt {5}} \sin (x)+\cos (x)\right )-\frac {1}{10} \sqrt {\frac {1}{2} \left (5+\sqrt {5}\right )} \log \left (\cos (x)-\sqrt {5+2 \sqrt {5}} \sin (x)\right )+\frac {1}{10} \sqrt {\frac {1}{2} \left (5+\sqrt {5}\right )} \log \left (\sqrt {5+2 \sqrt {5}} \sin (x)+\cos (x)\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[(5 - Sqrt[5])/2]*Log[Cos[x] - Sqrt[5 - 2*Sqrt[5]]*Sin[x]])/10 - (Sqrt[(5 - Sqrt[5])/2]*Log[Cos[x] + Sqrt
[5 - 2*Sqrt[5]]*Sin[x]])/10 - (Sqrt[(5 + Sqrt[5])/2]*Log[Cos[x] - Sqrt[5 + 2*Sqrt[5]]*Sin[x]])/10 + (Sqrt[(5 +
Sqrt[5])/2]*Log[Cos[x] + Sqrt[5 + 2*Sqrt[5]]*Sin[x]])/10

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 1166

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

Rubi steps

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

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Mathematica [A]  time = 0.10, size = 84, normalized size = 0.52 \[ \frac {\sqrt {5+\sqrt {5}} \tanh ^{-1}\left (\frac {\left (5+\sqrt {5}\right ) \tan (x)}{\sqrt {10-2 \sqrt {5}}}\right )+\sqrt {5-\sqrt {5}} \tanh ^{-1}\left (\frac {\left (\sqrt {5}-5\right ) \tan (x)}{\sqrt {2 \left (5+\sqrt {5}\right )}}\right )}{5 \sqrt {2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[5 + Sqrt[5]]*ArcTanh[((5 + Sqrt[5])*Tan[x])/Sqrt[10 - 2*Sqrt[5]]] + Sqrt[5 - Sqrt[5]]*ArcTanh[((-5 + Sqr
t[5])*Tan[x])/Sqrt[2*(5 + Sqrt[5])]])/(5*Sqrt[2])

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fricas [B]  time = 2.07, size = 231, normalized size = 1.42 \[ -\frac {1}{40} \, \sqrt {2} \sqrt {\sqrt {5} + 5} \log \left ({\left (\sqrt {5} \sqrt {2} - \sqrt {2}\right )} \sqrt {\sqrt {5} + 5} \cos \relax (x) \sin \relax (x) + 2 \, {\left (\sqrt {5} + 1\right )} \cos \relax (x)^{2} - \sqrt {5} - 5\right ) + \frac {1}{40} \, \sqrt {2} \sqrt {\sqrt {5} + 5} \log \left (-{\left (\sqrt {5} \sqrt {2} - \sqrt {2}\right )} \sqrt {\sqrt {5} + 5} \cos \relax (x) \sin \relax (x) + 2 \, {\left (\sqrt {5} + 1\right )} \cos \relax (x)^{2} - \sqrt {5} - 5\right ) - \frac {1}{40} \, \sqrt {2} \sqrt {-\sqrt {5} + 5} \log \left ({\left (\sqrt {5} \sqrt {2} + \sqrt {2}\right )} \sqrt {-\sqrt {5} + 5} \cos \relax (x) \sin \relax (x) + 2 \, {\left (\sqrt {5} - 1\right )} \cos \relax (x)^{2} - \sqrt {5} + 5\right ) + \frac {1}{40} \, \sqrt {2} \sqrt {-\sqrt {5} + 5} \log \left (-{\left (\sqrt {5} \sqrt {2} + \sqrt {2}\right )} \sqrt {-\sqrt {5} + 5} \cos \relax (x) \sin \relax (x) + 2 \, {\left (\sqrt {5} - 1\right )} \cos \relax (x)^{2} - \sqrt {5} + 5\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/40*sqrt(2)*sqrt(sqrt(5) + 5)*log((sqrt(5)*sqrt(2) - sqrt(2))*sqrt(sqrt(5) + 5)*cos(x)*sin(x) + 2*(sqrt(5) +
1)*cos(x)^2 - sqrt(5) - 5) + 1/40*sqrt(2)*sqrt(sqrt(5) + 5)*log(-(sqrt(5)*sqrt(2) - sqrt(2))*sqrt(sqrt(5) + 5
)*cos(x)*sin(x) + 2*(sqrt(5) + 1)*cos(x)^2 - sqrt(5) - 5) - 1/40*sqrt(2)*sqrt(-sqrt(5) + 5)*log((sqrt(5)*sqrt(
2) + sqrt(2))*sqrt(-sqrt(5) + 5)*cos(x)*sin(x) + 2*(sqrt(5) - 1)*cos(x)^2 - sqrt(5) + 5) + 1/40*sqrt(2)*sqrt(-
sqrt(5) + 5)*log(-(sqrt(5)*sqrt(2) + sqrt(2))*sqrt(-sqrt(5) + 5)*cos(x)*sin(x) + 2*(sqrt(5) - 1)*cos(x)^2 - sq
rt(5) + 5)

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giac [A]  time = 0.28, size = 105, normalized size = 0.64 \[ -\frac {1}{20} \, \sqrt {-2 \, \sqrt {5} + 10} \log \left ({\left | \sqrt {\frac {2}{5} \, \sqrt {5} + 1} + \tan \relax (x) \right |}\right ) + \frac {1}{20} \, \sqrt {-2 \, \sqrt {5} + 10} \log \left ({\left | -\sqrt {\frac {2}{5} \, \sqrt {5} + 1} + \tan \relax (x) \right |}\right ) + \frac {1}{20} \, \sqrt {2 \, \sqrt {5} + 10} \log \left ({\left | \sqrt {-\frac {2}{5} \, \sqrt {5} + 1} + \tan \relax (x) \right |}\right ) - \frac {1}{20} \, \sqrt {2 \, \sqrt {5} + 10} \log \left ({\left | -\sqrt {-\frac {2}{5} \, \sqrt {5} + 1} + \tan \relax (x) \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/20*sqrt(-2*sqrt(5) + 10)*log(abs(sqrt(2/5*sqrt(5) + 1) + tan(x))) + 1/20*sqrt(-2*sqrt(5) + 10)*log(abs(-sqr
t(2/5*sqrt(5) + 1) + tan(x))) + 1/20*sqrt(2*sqrt(5) + 10)*log(abs(sqrt(-2/5*sqrt(5) + 1) + tan(x))) - 1/20*sqr
t(2*sqrt(5) + 10)*log(abs(-sqrt(-2/5*sqrt(5) + 1) + tan(x)))

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maple [A]  time = 0.26, size = 68, normalized size = 0.42 \[ -\frac {\left (5+\sqrt {5}\right ) \sqrt {5}\, \arctanh \left (\frac {5 \tan \relax (x )}{\sqrt {25+10 \sqrt {5}}}\right )}{10 \sqrt {25+10 \sqrt {5}}}-\frac {\sqrt {5}\, \left (\sqrt {5}-5\right ) \arctanh \left (\frac {5 \tan \relax (x )}{\sqrt {25-10 \sqrt {5}}}\right )}{10 \sqrt {25-10 \sqrt {5}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/10*(5+5^(1/2))*5^(1/2)/(25+10*5^(1/2))^(1/2)*arctanh(5*tan(x)/(25+10*5^(1/2))^(1/2))-1/10*5^(1/2)*(5^(1/2)-
5)/(25-10*5^(1/2))^(1/2)*arctanh(5*tan(x)/(25-10*5^(1/2))^(1/2))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \cos \relax (x) \sec \left (5 \, x\right )\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(cos(x)*sec(5*x), x)

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mupad [B]  time = 2.67, size = 217, normalized size = 1.33 \[ \frac {\sqrt {2}\,\mathrm {atanh}\left (-\frac {34359738368\,\sqrt {2}\,\mathrm {tan}\left (\frac {x}{2}\right )\,\sqrt {5-\sqrt {5}}}{5\,\left (\frac {124554051584\,\sqrt {5}}{25}-\frac {124554051584\,\sqrt {5}\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{25}-\frac {55834574848\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{5}+\frac {55834574848}{5}\right )}-\frac {77309411328\,\sqrt {2}\,\sqrt {5}\,\mathrm {tan}\left (\frac {x}{2}\right )\,\sqrt {5-\sqrt {5}}}{25\,\left (\frac {124554051584\,\sqrt {5}}{25}-\frac {124554051584\,\sqrt {5}\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{25}-\frac {55834574848\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{5}+\frac {55834574848}{5}\right )}\right )\,\sqrt {5-\sqrt {5}}}{10}-\frac {\sqrt {2}\,\mathrm {atanh}\left (\frac {77309411328\,\sqrt {2}\,\sqrt {5}\,\mathrm {tan}\left (\frac {x}{2}\right )\,\sqrt {\sqrt {5}+5}}{25\,\left (\frac {124554051584\,\sqrt {5}}{25}-\frac {124554051584\,\sqrt {5}\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{25}+\frac {55834574848\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{5}-\frac {55834574848}{5}\right )}-\frac {34359738368\,\sqrt {2}\,\mathrm {tan}\left (\frac {x}{2}\right )\,\sqrt {\sqrt {5}+5}}{5\,\left (\frac {124554051584\,\sqrt {5}}{25}-\frac {124554051584\,\sqrt {5}\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{25}+\frac {55834574848\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{5}-\frac {55834574848}{5}\right )}\right )\,\sqrt {\sqrt {5}+5}}{10} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2^(1/2)*atanh(- (34359738368*2^(1/2)*tan(x/2)*(5 - 5^(1/2))^(1/2))/(5*((124554051584*5^(1/2))/25 - (124554051
584*5^(1/2)*tan(x/2)^2)/25 - (55834574848*tan(x/2)^2)/5 + 55834574848/5)) - (77309411328*2^(1/2)*5^(1/2)*tan(x
/2)*(5 - 5^(1/2))^(1/2))/(25*((124554051584*5^(1/2))/25 - (124554051584*5^(1/2)*tan(x/2)^2)/25 - (55834574848*
tan(x/2)^2)/5 + 55834574848/5)))*(5 - 5^(1/2))^(1/2))/10 - (2^(1/2)*atanh((77309411328*2^(1/2)*5^(1/2)*tan(x/2
)*(5^(1/2) + 5)^(1/2))/(25*((124554051584*5^(1/2))/25 - (124554051584*5^(1/2)*tan(x/2)^2)/25 + (55834574848*ta
n(x/2)^2)/5 - 55834574848/5)) - (34359738368*2^(1/2)*tan(x/2)*(5^(1/2) + 5)^(1/2))/(5*((124554051584*5^(1/2))/
25 - (124554051584*5^(1/2)*tan(x/2)^2)/25 + (55834574848*tan(x/2)^2)/5 - 55834574848/5)))*(5^(1/2) + 5)^(1/2))
/10

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \cos {\relax (x )} \sec {\left (5 x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(cos(x)*sec(5*x), x)

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