3.108 \(\int \cos (x) \tan (5 x) \, dx\)

Optimal. Leaf size=84 \[ -\cos (x)+\frac {1}{5} \sqrt {\frac {1}{2} \left (5+\sqrt {5}\right )} \tanh ^{-1}\left (2 \sqrt {\frac {2}{5+\sqrt {5}}} \cos (x)\right )+\frac {1}{5} \sqrt {\frac {1}{2} \left (5-\sqrt {5}\right )} \tanh ^{-1}\left (\sqrt {\frac {2}{5} \left (5+\sqrt {5}\right )} \cos (x)\right ) \]

[Out]

-cos(x)+1/10*arctanh(1/5*cos(x)*(50+10*5^(1/2))^(1/2))*(10-2*5^(1/2))^(1/2)+1/10*arctanh(2*cos(x)*2^(1/2)/(5+5
^(1/2))^(1/2))*(10+2*5^(1/2))^(1/2)

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Rubi [A]  time = 0.10, antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {1676, 1166, 207} \[ -\cos (x)+\frac {1}{5} \sqrt {\frac {1}{2} \left (5+\sqrt {5}\right )} \tanh ^{-1}\left (2 \sqrt {\frac {2}{5+\sqrt {5}}} \cos (x)\right )+\frac {1}{5} \sqrt {\frac {1}{2} \left (5-\sqrt {5}\right )} \tanh ^{-1}\left (\sqrt {\frac {2}{5} \left (5+\sqrt {5}\right )} \cos (x)\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[(5 + Sqrt[5])/2]*ArcTanh[2*Sqrt[2/(5 + Sqrt[5])]*Cos[x]])/5 + (Sqrt[(5 - Sqrt[5])/2]*ArcTanh[Sqrt[(2*(5
+ Sqrt[5]))/5]*Cos[x]])/5 - Cos[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 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]

Rule 1676

Int[(Pq_)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> Int[ExpandIntegrand[Pq/(a + b*x^2 + c*x^4), x], x
] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x^2] && Expon[Pq, x^2] > 1

Rubi steps

\begin {align*} \int \cos (x) \tan (5 x) \, dx &=-\operatorname {Subst}\left (\int \frac {1-12 x^2+16 x^4}{5-20 x^2+16 x^4} \, dx,x,\cos (x)\right )\\ &=-\operatorname {Subst}\left (\int \left (1-\frac {4 \left (1-2 x^2\right )}{5-20 x^2+16 x^4}\right ) \, dx,x,\cos (x)\right )\\ &=-\cos (x)+4 \operatorname {Subst}\left (\int \frac {1-2 x^2}{5-20 x^2+16 x^4} \, dx,x,\cos (x)\right )\\ &=-\cos (x)-\frac {1}{5} \left (4 \left (5-\sqrt {5}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-10+2 \sqrt {5}+16 x^2} \, dx,x,\cos (x)\right )-\frac {1}{5} \left (4 \left (5+\sqrt {5}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-10-2 \sqrt {5}+16 x^2} \, dx,x,\cos (x)\right )\\ &=\frac {1}{5} \sqrt {\frac {1}{2} \left (5+\sqrt {5}\right )} \tanh ^{-1}\left (2 \sqrt {\frac {2}{5+\sqrt {5}}} \cos (x)\right )+\frac {1}{5} \sqrt {\frac {1}{2} \left (5-\sqrt {5}\right )} \tanh ^{-1}\left (\sqrt {\frac {2}{5} \left (5+\sqrt {5}\right )} \cos (x)\right )-\cos (x)\\ \end {align*}

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Mathematica [B]  time = 0.59, size = 215, normalized size = 2.56 \[ -\cos (x)+\frac {\left (1+\sqrt {5}\right ) \tanh ^{-1}\left (\frac {4-\left (\sqrt {5}-1\right ) \tan \left (\frac {x}{2}\right )}{\sqrt {2 \left (5+\sqrt {5}\right )}}\right )}{\sqrt {10 \left (5+\sqrt {5}\right )}}+\frac {\left (1+\sqrt {5}\right ) \tanh ^{-1}\left (\frac {\left (\sqrt {5}-1\right ) \tan \left (\frac {x}{2}\right )+4}{\sqrt {2 \left (5+\sqrt {5}\right )}}\right )}{\sqrt {10 \left (5+\sqrt {5}\right )}}+\frac {\left (\sqrt {5}-1\right ) \tanh ^{-1}\left (\frac {4-\left (1+\sqrt {5}\right ) \tan \left (\frac {x}{2}\right )}{\sqrt {10-2 \sqrt {5}}}\right )}{\sqrt {50-10 \sqrt {5}}}+\frac {\left (\sqrt {5}-1\right ) \tanh ^{-1}\left (\frac {\left (1+\sqrt {5}\right ) \tan \left (\frac {x}{2}\right )+4}{\sqrt {10-2 \sqrt {5}}}\right )}{\sqrt {50-10 \sqrt {5}}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((1 + Sqrt[5])*ArcTanh[(4 - (-1 + Sqrt[5])*Tan[x/2])/Sqrt[2*(5 + Sqrt[5])]])/Sqrt[10*(5 + Sqrt[5])] + ((1 + Sq
rt[5])*ArcTanh[(4 + (-1 + Sqrt[5])*Tan[x/2])/Sqrt[2*(5 + Sqrt[5])]])/Sqrt[10*(5 + Sqrt[5])] + ((-1 + Sqrt[5])*
ArcTanh[(4 - (1 + Sqrt[5])*Tan[x/2])/Sqrt[10 - 2*Sqrt[5]]])/Sqrt[50 - 10*Sqrt[5]] + ((-1 + Sqrt[5])*ArcTanh[(4
+ (1 + Sqrt[5])*Tan[x/2])/Sqrt[10 - 2*Sqrt[5]]])/Sqrt[50 - 10*Sqrt[5]] - Cos[x]

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fricas [B]  time = 1.20, size = 129, normalized size = 1.54 \[ \frac {1}{20} \, \sqrt {2} \sqrt {\sqrt {5} + 5} \log \left (\sqrt {2} \sqrt {\sqrt {5} + 5} + 4 \, \cos \relax (x)\right ) - \frac {1}{20} \, \sqrt {2} \sqrt {\sqrt {5} + 5} \log \left (\sqrt {2} \sqrt {\sqrt {5} + 5} - 4 \, \cos \relax (x)\right ) + \frac {1}{20} \, \sqrt {2} \sqrt {-\sqrt {5} + 5} \log \left (\sqrt {2} \sqrt {-\sqrt {5} + 5} + 4 \, \cos \relax (x)\right ) - \frac {1}{20} \, \sqrt {2} \sqrt {-\sqrt {5} + 5} \log \left (\sqrt {2} \sqrt {-\sqrt {5} + 5} - 4 \, \cos \relax (x)\right ) - \cos \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/20*sqrt(2)*sqrt(sqrt(5) + 5)*log(sqrt(2)*sqrt(sqrt(5) + 5) + 4*cos(x)) - 1/20*sqrt(2)*sqrt(sqrt(5) + 5)*log(
sqrt(2)*sqrt(sqrt(5) + 5) - 4*cos(x)) + 1/20*sqrt(2)*sqrt(-sqrt(5) + 5)*log(sqrt(2)*sqrt(-sqrt(5) + 5) + 4*cos
(x)) - 1/20*sqrt(2)*sqrt(-sqrt(5) + 5)*log(sqrt(2)*sqrt(-sqrt(5) + 5) - 4*cos(x)) - cos(x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(cos(x)*tan(5*x), x)

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maple [A]  time = 0.07, size = 72, normalized size = 0.86 \[ -\cos \relax (x )+\frac {\left (\sqrt {5}-1\right ) \sqrt {5}\, \arctanh \left (\frac {4 \cos \relax (x )}{\sqrt {10-2 \sqrt {5}}}\right )}{5 \sqrt {10-2 \sqrt {5}}}+\frac {\left (\sqrt {5}+1\right ) \sqrt {5}\, \arctanh \left (\frac {4 \cos \relax (x )}{\sqrt {10+2 \sqrt {5}}}\right )}{5 \sqrt {10+2 \sqrt {5}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-cos(x)+1/5*(5^(1/2)-1)*5^(1/2)/(10-2*5^(1/2))^(1/2)*arctanh(4*cos(x)/(10-2*5^(1/2))^(1/2))+1/5*(5^(1/2)+1)*5^
(1/2)/(10+2*5^(1/2))^(1/2)*arctanh(4*cos(x)/(10+2*5^(1/2))^(1/2))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\cos \relax (x) - \int \frac {{\left (\sin \left (7 \, x\right ) - \sin \left (5 \, x\right ) + \sin \left (3 \, x\right ) - \sin \relax (x)\right )} \cos \left (8 \, x\right ) + {\left (\sin \left (6 \, x\right ) - \sin \left (4 \, x\right ) + \sin \left (2 \, x\right )\right )} \cos \left (7 \, x\right ) + {\left (\sin \left (5 \, x\right ) - \sin \left (3 \, x\right ) + \sin \relax (x)\right )} \cos \left (6 \, x\right ) + {\left (\sin \left (4 \, x\right ) - \sin \left (2 \, x\right )\right )} \cos \left (5 \, x\right ) + {\left (\sin \left (3 \, x\right ) - \sin \relax (x)\right )} \cos \left (4 \, x\right ) - {\left (\cos \left (7 \, x\right ) - \cos \left (5 \, x\right ) + \cos \left (3 \, x\right ) - \cos \relax (x)\right )} \sin \left (8 \, x\right ) - {\left (\cos \left (6 \, x\right ) - \cos \left (4 \, x\right ) + \cos \left (2 \, x\right ) - 1\right )} \sin \left (7 \, x\right ) - {\left (\cos \left (5 \, x\right ) - \cos \left (3 \, x\right ) + \cos \relax (x)\right )} \sin \left (6 \, x\right ) - {\left (\cos \left (4 \, x\right ) - \cos \left (2 \, x\right ) + 1\right )} \sin \left (5 \, x\right ) - {\left (\cos \left (3 \, x\right ) - \cos \relax (x)\right )} \sin \left (4 \, x\right ) - {\left (\cos \left (2 \, x\right ) - 1\right )} \sin \left (3 \, x\right ) + \cos \left (3 \, x\right ) \sin \left (2 \, x\right ) - \cos \relax (x) \sin \left (2 \, x\right ) + \cos \left (2 \, x\right ) \sin \relax (x) - \sin \relax (x)}{2 \, {\left (\cos \left (6 \, x\right ) - \cos \left (4 \, x\right ) + \cos \left (2 \, x\right ) - 1\right )} \cos \left (8 \, x\right ) - \cos \left (8 \, x\right )^{2} + 2 \, {\left (\cos \left (4 \, x\right ) - \cos \left (2 \, x\right ) + 1\right )} \cos \left (6 \, x\right ) - \cos \left (6 \, x\right )^{2} + 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} + 2 \, {\left (\sin \left (6 \, x\right ) - \sin \left (4 \, x\right ) + \sin \left (2 \, x\right )\right )} \sin \left (8 \, x\right ) - \sin \left (8 \, x\right )^{2} + 2 \, {\left (\sin \left (4 \, x\right ) - \sin \left (2 \, x\right )\right )} \sin \left (6 \, x\right ) - \sin \left (6 \, 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}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-cos(x) - integrate(((sin(7*x) - sin(5*x) + sin(3*x) - sin(x))*cos(8*x) + (sin(6*x) - sin(4*x) + sin(2*x))*cos
(7*x) + (sin(5*x) - sin(3*x) + sin(x))*cos(6*x) + (sin(4*x) - sin(2*x))*cos(5*x) + (sin(3*x) - sin(x))*cos(4*x
) - (cos(7*x) - cos(5*x) + cos(3*x) - cos(x))*sin(8*x) - (cos(6*x) - cos(4*x) + cos(2*x) - 1)*sin(7*x) - (cos(
5*x) - cos(3*x) + cos(x))*sin(6*x) - (cos(4*x) - cos(2*x) + 1)*sin(5*x) - (cos(3*x) - cos(x))*sin(4*x) - (cos(
2*x) - 1)*sin(3*x) + cos(3*x)*sin(2*x) - cos(x)*sin(2*x) + cos(2*x)*sin(x) - sin(x))/(2*(cos(6*x) - cos(4*x) +
cos(2*x) - 1)*cos(8*x) - cos(8*x)^2 + 2*(cos(4*x) - cos(2*x) + 1)*cos(6*x) - cos(6*x)^2 + 2*(cos(2*x) - 1)*co
s(4*x) - cos(4*x)^2 - cos(2*x)^2 + 2*(sin(6*x) - sin(4*x) + sin(2*x))*sin(8*x) - sin(8*x)^2 + 2*(sin(4*x) - si
n(2*x))*sin(6*x) - sin(6*x)^2 - sin(4*x)^2 + 2*sin(4*x)*sin(2*x) - sin(2*x)^2 + 2*cos(2*x) - 1), x)

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mupad [B]  time = 2.50, size = 407, normalized size = 4.85 \[ \frac {\sqrt {2}\,\mathrm {atanh}\left (\frac {18032420192256\,\sqrt {2}\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2\,\sqrt {\sqrt {5}+5}}{\frac {8851927597056\,\sqrt {5}}{25}-\frac {676375744741376\,\sqrt {5}\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{25}-\frac {333433343574016\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{5}[phone]}-\frac {867583393792\,\sqrt {2}\,\sqrt {5}\,\sqrt {\sqrt {5}+5}}{25\,\left (\frac {8851927597056\,\sqrt {5}}{25}-\frac {676375744741376\,\sqrt {5}\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{25}-\frac {333433343574016\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{5}[phone]\right )}-\frac {3805341024256\,\sqrt {2}\,\sqrt {\sqrt {5}+5}}{5\,\left (\frac {8851927597056\,\sqrt {5}}{25}-\frac {676375744741376\,\sqrt {5}\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{25}-\frac {333433343574016\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{5}[phone]\right )}+\frac {6886980059136\,\sqrt {2}\,\sqrt {5}\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2\,\sqrt {\sqrt {5}+5}}{\frac {8851927597056\,\sqrt {5}}{25}-\frac {676375744741376\,\sqrt {5}\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{25}-\frac {333433343574016\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{5}[phone]}\right )\,\sqrt {\sqrt {5}+5}}{10}-\frac {\sqrt {2}\,\mathrm {atanh}\left (\frac {867583393792\,\sqrt {2}\,\sqrt {5}\,\sqrt {5-\sqrt {5}}}{25\,\left (\frac {8851927597056\,\sqrt {5}}{25}-\frac {676375744741376\,\sqrt {5}\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{25}+\frac {333433343574016\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{5}-2398739234816\right )}-\frac {3805341024256\,\sqrt {2}\,\sqrt {5-\sqrt {5}}}{5\,\left (\frac {8851927597056\,\sqrt {5}}{25}-\frac {676375744741376\,\sqrt {5}\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{25}+\frac {333433343574016\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{5}-2398739234816\right )}+\frac {18032420192256\,\sqrt {2}\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2\,\sqrt {5-\sqrt {5}}}{\frac {8851927597056\,\sqrt {5}}{25}-\frac {676375744741376\,\sqrt {5}\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{25}+\frac {333433343574016\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{5}-2398739234816}-\frac {6886980059136\,\sqrt {2}\,\sqrt {5}\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2\,\sqrt {5-\sqrt {5}}}{\frac {8851927597056\,\sqrt {5}}{25}-\frac {676375744741376\,\sqrt {5}\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{25}+\frac {333433343574016\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{5}-2398739234816}\right )\,\sqrt {5-\sqrt {5}}}{10}-\frac {2}{{\mathrm {tan}\left (\frac {x}{2}\right )}^2+1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2^(1/2)*atanh((18032420192256*2^(1/2)*tan(x/2)^2*(5^(1/2) + 5)^(1/2))/((8851927597056*5^(1/2))/25 - (67637574
4741376*5^(1/2)*tan(x/2)^2)/25 - (333433343574016*tan(x/2)^2)/5 + 2398739234816) - (867583393792*2^(1/2)*5^(1/
2)*(5^(1/2) + 5)^(1/2))/(25*((8851927597056*5^(1/2))/25 - (676375744741376*5^(1/2)*tan(x/2)^2)/25 - (333433343
574016*tan(x/2)^2)/5 + 2398739234816)) - (3805341024256*2^(1/2)*(5^(1/2) + 5)^(1/2))/(5*((8851927597056*5^(1/2
))/25 - (676375744741376*5^(1/2)*tan(x/2)^2)/25 - (333433343574016*tan(x/2)^2)/5 + 2398739234816)) + (68869800
59136*2^(1/2)*5^(1/2)*tan(x/2)^2*(5^(1/2) + 5)^(1/2))/((8851927597056*5^(1/2))/25 - (676375744741376*5^(1/2)*t
an(x/2)^2)/25 - (333433343574016*tan(x/2)^2)/5 + 2398739234816))*(5^(1/2) + 5)^(1/2))/10 - (2^(1/2)*atanh((867
583393792*2^(1/2)*5^(1/2)*(5 - 5^(1/2))^(1/2))/(25*((8851927597056*5^(1/2))/25 - (676375744741376*5^(1/2)*tan(
x/2)^2)/25 + (333433343574016*tan(x/2)^2)/5 - 2398739234816)) - (3805341024256*2^(1/2)*(5 - 5^(1/2))^(1/2))/(5
*((8851927597056*5^(1/2))/25 - (676375744741376*5^(1/2)*tan(x/2)^2)/25 + (333433343574016*tan(x/2)^2)/5 - 2398
739234816)) + (18032420192256*2^(1/2)*tan(x/2)^2*(5 - 5^(1/2))^(1/2))/((8851927597056*5^(1/2))/25 - (676375744
741376*5^(1/2)*tan(x/2)^2)/25 + (333433343574016*tan(x/2)^2)/5 - 2398739234816) - (6886980059136*2^(1/2)*5^(1/
2)*tan(x/2)^2*(5 - 5^(1/2))^(1/2))/((8851927597056*5^(1/2))/25 - (676375744741376*5^(1/2)*tan(x/2)^2)/25 + (33
3433343574016*tan(x/2)^2)/5 - 2398739234816))*(5 - 5^(1/2))^(1/2))/10 - 2/(tan(x/2)^2 + 1)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(cos(x)*tan(5*x), x)

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