3.93 \(\int \csc (5 x) \sin (x) \, dx\)

Optimal. Leaf size=165 \[ -\frac {1}{10} \sqrt {\frac {1}{2} \left (5-\sqrt {5}\right )} \log \left (\sqrt {5-2 \sqrt {5}} \cos (x)-\sin (x)\right )+\frac {1}{10} \sqrt {\frac {1}{2} \left (5+\sqrt {5}\right )} \log \left (\sqrt {5+2 \sqrt {5}} \cos (x)-\sin (x)\right )+\frac {1}{10} \sqrt {\frac {1}{2} \left (5-\sqrt {5}\right )} \log \left (\sin (x)+\sqrt {5-2 \sqrt {5}} \cos (x)\right )-\frac {1}{10} \sqrt {\frac {1}{2} \left (5+\sqrt {5}\right )} \log \left (\sin (x)+\sqrt {5+2 \sqrt {5}} \cos (x)\right ) \]

[Out]

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

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Rubi [A]  time = 0.14, antiderivative size = 165, 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} \[ -\frac {1}{10} \sqrt {\frac {1}{2} \left (5-\sqrt {5}\right )} \log \left (\sqrt {5-2 \sqrt {5}} \cos (x)-\sin (x)\right )+\frac {1}{10} \sqrt {\frac {1}{2} \left (5+\sqrt {5}\right )} \log \left (\sqrt {5+2 \sqrt {5}} \cos (x)-\sin (x)\right )+\frac {1}{10} \sqrt {\frac {1}{2} \left (5-\sqrt {5}\right )} \log \left (\sin (x)+\sqrt {5-2 \sqrt {5}} \cos (x)\right )-\frac {1}{10} \sqrt {\frac {1}{2} \left (5+\sqrt {5}\right )} \log \left (\sin (x)+\sqrt {5+2 \sqrt {5}} \cos (x)\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[(5 - Sqrt[5])/2]*Log[Sqrt[5 - 2*Sqrt[5]]*Cos[x] - Sin[x]])/10 + (Sqrt[(5 + Sqrt[5])/2]*Log[Sqrt[5 + 2*S
qrt[5]]*Cos[x] - Sin[x]])/10 + (Sqrt[(5 - Sqrt[5])/2]*Log[Sqrt[5 - 2*Sqrt[5]]*Cos[x] + Sin[x]])/10 - (Sqrt[(5
+ Sqrt[5])/2]*Log[Sqrt[5 + 2*Sqrt[5]]*Cos[x] + 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 \csc (5 x) \sin (x) \, dx &=\operatorname {Subst}\left (\int \frac {1+x^2}{5-10 x^2+x^4} \, dx,x,\tan (x)\right )\\ &=\frac {1}{10} \left (5-3 \sqrt {5}\right ) \operatorname {Subst}\left (\int \frac {1}{-5+2 \sqrt {5}+x^2} \, dx,x,\tan (x)\right )+\frac {1}{10} \left (5+3 \sqrt {5}\right ) \operatorname {Subst}\left (\int \frac {1}{-5-2 \sqrt {5}+x^2} \, dx,x,\tan (x)\right )\\ &=-\frac {1}{10} \sqrt {\frac {1}{2} \left (5-\sqrt {5}\right )} \log \left (\sqrt {5-2 \sqrt {5}} \cos (x)-\sin (x)\right )+\frac {1}{10} \sqrt {\frac {1}{2} \left (5+\sqrt {5}\right )} \log \left (\sqrt {5+2 \sqrt {5}} \cos (x)-\sin (x)\right )+\frac {1}{10} \sqrt {\frac {1}{2} \left (5-\sqrt {5}\right )} \log \left (\sqrt {5-2 \sqrt {5}} \cos (x)+\sin (x)\right )-\frac {1}{10} \sqrt {\frac {1}{2} \left (5+\sqrt {5}\right )} \log \left (\sqrt {5+2 \sqrt {5}} \cos (x)+\sin (x)\right )\\ \end {align*}

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Mathematica [A]  time = 0.11, size = 84, normalized size = 0.51 \[ \frac {\sqrt {5+\sqrt {5}} \tanh ^{-1}\left (\frac {\left (\sqrt {5}-3\right ) \tan (x)}{\sqrt {10-2 \sqrt {5}}}\right )+\sqrt {5-\sqrt {5}} \tanh ^{-1}\left (\frac {\left (3+\sqrt {5}\right ) \tan (x)}{\sqrt {2 \left (5+\sqrt {5}\right )}}\right )}{5 \sqrt {2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(Sqrt[5 + Sqrt[5]]*ArcTanh[((-3 + Sqrt[5])*Tan[x])/Sqrt[10 - 2*Sqrt[5]]] + Sqrt[5 - Sqrt[5]]*ArcTanh[((3 + Sqr
t[5])*Tan[x])/Sqrt[2*(5 + Sqrt[5])]])/(5*Sqrt[2])

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fricas [B]  time = 0.93, size = 231, normalized size = 1.40 \[ -\frac {1}{40} \, \sqrt {2} \sqrt {\sqrt {5} + 5} \log \left ({\left (\sqrt {5} \sqrt {2} - \sqrt {2}\right )} \sqrt {\sqrt {5} + 5} \cos \relax (x) \sin \relax (x) + 2 \, {\left (\sqrt {5} + 1\right )} \cos \relax (x)^{2} - \sqrt {5} + 3\right ) + \frac {1}{40} \, \sqrt {2} \sqrt {\sqrt {5} + 5} \log \left (-{\left (\sqrt {5} \sqrt {2} - \sqrt {2}\right )} \sqrt {\sqrt {5} + 5} \cos \relax (x) \sin \relax (x) + 2 \, {\left (\sqrt {5} + 1\right )} \cos \relax (x)^{2} - \sqrt {5} + 3\right ) - \frac {1}{40} \, \sqrt {2} \sqrt {-\sqrt {5} + 5} \log \left ({\left (\sqrt {5} \sqrt {2} + \sqrt {2}\right )} \sqrt {-\sqrt {5} + 5} \cos \relax (x) \sin \relax (x) + 2 \, {\left (\sqrt {5} - 1\right )} \cos \relax (x)^{2} - \sqrt {5} - 3\right ) + \frac {1}{40} \, \sqrt {2} \sqrt {-\sqrt {5} + 5} \log \left (-{\left (\sqrt {5} \sqrt {2} + \sqrt {2}\right )} \sqrt {-\sqrt {5} + 5} \cos \relax (x) \sin \relax (x) + 2 \, {\left (\sqrt {5} - 1\right )} \cos \relax (x)^{2} - \sqrt {5} - 3\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(5*x)*sin(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) + 3) + 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) + 3) - 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) - 3) + 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) - 3)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.29, size = 66, normalized size = 0.40 \[ -\frac {\left (3+\sqrt {5}\right ) \sqrt {5}\, \arctanh \left (\frac {\tan \relax (x )}{\sqrt {5+2 \sqrt {5}}}\right )}{10 \sqrt {5+2 \sqrt {5}}}-\frac {\left (\sqrt {5}-3\right ) \sqrt {5}\, \arctanh \left (\frac {\tan \relax (x )}{\sqrt {5-2 \sqrt {5}}}\right )}{10 \sqrt {5-2 \sqrt {5}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/10*(3+5^(1/2))*5^(1/2)/(5+2*5^(1/2))^(1/2)*arctanh(tan(x)/(5+2*5^(1/2))^(1/2))-1/10*(5^(1/2)-3)*5^(1/2)/(5-
2*5^(1/2))^(1/2)*arctanh(tan(x)/(5-2*5^(1/2))^(1/2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(csc(5*x)*sin(x), x)

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mupad [B]  time = 2.59, size = 217, normalized size = 1.32 \[ \frac {\sqrt {2}\,\mathrm {atanh}\left (-\frac {34359738368\,\sqrt {2}\,\mathrm {tan}\left (\frac {x}{2}\right )\,\sqrt {\sqrt {5}+5}}{1953125\,\left (\frac {90194313216\,\sqrt {5}}{1953125}-\frac {90194313216\,\sqrt {5}\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{1953125}-\frac {201863462912\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{1953125}+\frac {201863462912}{1953125}\right )}-\frac {77309411328\,\sqrt {2}\,\sqrt {5}\,\mathrm {tan}\left (\frac {x}{2}\right )\,\sqrt {\sqrt {5}+5}}{9765625\,\left (\frac {90194313216\,\sqrt {5}}{1953125}-\frac {90194313216\,\sqrt {5}\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{1953125}-\frac {201863462912\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{1953125}+\frac {201863462912}{1953125}\right )}\right )\,\sqrt {\sqrt {5}+5}}{10}-\frac {\sqrt {2}\,\mathrm {atanh}\left (\frac {77309411328\,\sqrt {2}\,\sqrt {5}\,\mathrm {tan}\left (\frac {x}{2}\right )\,\sqrt {5-\sqrt {5}}}{9765625\,\left (\frac {90194313216\,\sqrt {5}}{1953125}-\frac {90194313216\,\sqrt {5}\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{1953125}+\frac {201863462912\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{1953125}-\frac {201863462912}{1953125}\right )}-\frac {34359738368\,\sqrt {2}\,\mathrm {tan}\left (\frac {x}{2}\right )\,\sqrt {5-\sqrt {5}}}{1953125\,\left (\frac {90194313216\,\sqrt {5}}{1953125}-\frac {90194313216\,\sqrt {5}\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{1953125}+\frac {201863462912\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{1953125}-\frac {201863462912}{1953125}\right )}\right )\,\sqrt {5-\sqrt {5}}}{10} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2^(1/2)*atanh(- (34359738368*2^(1/2)*tan(x/2)*(5^(1/2) + 5)^(1/2))/(1953125*((90194313216*5^(1/2))/1953125 -
(90194313216*5^(1/2)*tan(x/2)^2)/1953125 - (201863462912*tan(x/2)^2)/1953125 + 201863462912/1953125)) - (77309
411328*2^(1/2)*5^(1/2)*tan(x/2)*(5^(1/2) + 5)^(1/2))/(9765625*((90194313216*5^(1/2))/1953125 - (90194313216*5^
(1/2)*tan(x/2)^2)/1953125 - (201863462912*tan(x/2)^2)/1953125 + 201863462912/1953125)))*(5^(1/2) + 5)^(1/2))/1
0 - (2^(1/2)*atanh((77309411328*2^(1/2)*5^(1/2)*tan(x/2)*(5 - 5^(1/2))^(1/2))/(9765625*((90194313216*5^(1/2))/
1953125 - (90194313216*5^(1/2)*tan(x/2)^2)/1953125 + (201863462912*tan(x/2)^2)/1953125 - 201863462912/1953125)
) - (34359738368*2^(1/2)*tan(x/2)*(5 - 5^(1/2))^(1/2))/(1953125*((90194313216*5^(1/2))/1953125 - (90194313216*
5^(1/2)*tan(x/2)^2)/1953125 + (201863462912*tan(x/2)^2)/1953125 - 201863462912/1953125)))*(5 - 5^(1/2))^(1/2))
/10

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sin(x)*csc(5*x), x)

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