3.77 \(\int \sin (x) \tan (5 x) \, dx\)

Optimal. Leaf size=112 \[ -\sin (x)-\frac{1}{20} \left (1-\sqrt{5}\right ) \log \left (-4 \sin (x)-\sqrt{5}+1\right )-\frac{1}{20} \left (1+\sqrt{5}\right ) \log \left (-4 \sin (x)+\sqrt{5}+1\right )+\frac{1}{20} \left (1-\sqrt{5}\right ) \log \left (4 \sin (x)-\sqrt{5}+1\right )+\frac{1}{20} \left (1+\sqrt{5}\right ) \log \left (4 \sin (x)+\sqrt{5}+1\right )+\frac{1}{5} \tanh ^{-1}(\sin (x)) \]

[Out]

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

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Rubi [A]  time = 0.169567, antiderivative size = 112, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 4, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.571, Rules used = {2075, 207, 632, 31} \[ -\sin (x)-\frac{1}{20} \left (1-\sqrt{5}\right ) \log \left (-4 \sin (x)-\sqrt{5}+1\right )-\frac{1}{20} \left (1+\sqrt{5}\right ) \log \left (-4 \sin (x)+\sqrt{5}+1\right )+\frac{1}{20} \left (1-\sqrt{5}\right ) \log \left (4 \sin (x)-\sqrt{5}+1\right )+\frac{1}{20} \left (1+\sqrt{5}\right ) \log \left (4 \sin (x)+\sqrt{5}+1\right )+\frac{1}{5} \tanh ^{-1}(\sin (x)) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2075

Int[(P_)^(p_)*(Qm_), x_Symbol] :> With[{PP = Factor[P]}, Int[ExpandIntegrand[PP^p*Qm, x], x] /; QuadraticProdu
ctQ[PP, x]] /; PolyQ[Qm, x] && PolyQ[P, x] && ILtQ[p, 0]

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 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 \sin (x) \tan (5 x) \, dx &=\operatorname{Subst}\left (\int \frac{x^2 \left (5-20 x^2+16 x^4\right )}{1-13 x^2+28 x^4-16 x^6} \, dx,x,\sin (x)\right )\\ &=\operatorname{Subst}\left (\int \left (-1-\frac{1}{5 \left (-1+x^2\right )}-\frac{2 (1+x)}{5 \left (-1-2 x+4 x^2\right )}+\frac{2 (-1+x)}{5 \left (-1+2 x+4 x^2\right )}\right ) \, dx,x,\sin (x)\right )\\ &=-\sin (x)-\frac{1}{5} \operatorname{Subst}\left (\int \frac{1}{-1+x^2} \, dx,x,\sin (x)\right )-\frac{2}{5} \operatorname{Subst}\left (\int \frac{1+x}{-1-2 x+4 x^2} \, dx,x,\sin (x)\right )+\frac{2}{5} \operatorname{Subst}\left (\int \frac{-1+x}{-1+2 x+4 x^2} \, dx,x,\sin (x)\right )\\ &=\frac{1}{5} \tanh ^{-1}(\sin (x))-\sin (x)+\frac{1}{5} \left (1-\sqrt{5}\right ) \operatorname{Subst}\left (\int \frac{1}{1-\sqrt{5}+4 x} \, dx,x,\sin (x)\right )-\frac{1}{5} \left (1-\sqrt{5}\right ) \operatorname{Subst}\left (\int \frac{1}{-1+\sqrt{5}+4 x} \, dx,x,\sin (x)\right )-\frac{1}{5} \left (1+\sqrt{5}\right ) \operatorname{Subst}\left (\int \frac{1}{-1-\sqrt{5}+4 x} \, dx,x,\sin (x)\right )+\frac{1}{5} \left (1+\sqrt{5}\right ) \operatorname{Subst}\left (\int \frac{1}{1+\sqrt{5}+4 x} \, dx,x,\sin (x)\right )\\ &=\frac{1}{5} \tanh ^{-1}(\sin (x))-\frac{1}{20} \left (1-\sqrt{5}\right ) \log \left (1-\sqrt{5}-4 \sin (x)\right )-\frac{1}{20} \left (1+\sqrt{5}\right ) \log \left (1+\sqrt{5}-4 \sin (x)\right )+\frac{1}{20} \left (1-\sqrt{5}\right ) \log \left (1-\sqrt{5}+4 \sin (x)\right )+\frac{1}{20} \left (1+\sqrt{5}\right ) \log \left (1+\sqrt{5}+4 \sin (x)\right )-\sin (x)\\ \end{align*}

Mathematica [A]  time = 0.150762, size = 100, normalized size = 0.89 \[ \frac{1}{20} \left (-20 \sin (x)+\left (\sqrt{5}-1\right ) \log \left (-4 \sin (x)-\sqrt{5}+1\right )-\left (1+\sqrt{5}\right ) \log \left (-4 \sin (x)+\sqrt{5}+1\right )-\left (\sqrt{5}-1\right ) \log \left (4 \sin (x)-\sqrt{5}+1\right )+\left (1+\sqrt{5}\right ) \log \left (4 \sin (x)+\sqrt{5}+1\right )+4 \tanh ^{-1}(\sin (x))\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.141, size = 84, normalized size = 0.8 \begin{align*} -{\frac{\ln \left ( 4\, \left ( \sin \left ( x \right ) \right ) ^{2}-2\,\sin \left ( x \right ) -1 \right ) }{20}}+{\frac{\sqrt{5}}{10}{\it Artanh} \left ({\frac{ \left ( 8\,\sin \left ( x \right ) -2 \right ) \sqrt{5}}{10}} \right ) }+{\frac{\ln \left ( 1+\sin \left ( x \right ) \right ) }{10}}-{\frac{\ln \left ( \sin \left ( x \right ) -1 \right ) }{10}}+{\frac{\ln \left ( 4\, \left ( \sin \left ( x \right ) \right ) ^{2}+2\,\sin \left ( x \right ) -1 \right ) }{20}}+{\frac{\sqrt{5}}{10}{\it Artanh} \left ({\frac{ \left ( 8\,\sin \left ( x \right ) +2 \right ) \sqrt{5}}{10}} \right ) }-\sin \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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(sin(x)*tan(5*x),x, algorithm="maxima")

[Out]

integrate(-1/5*((3*cos(7*x) - cos(5*x) - cos(3*x) + 3*cos(x))*cos(8*x) - 3*(cos(6*x) - cos(4*x) + cos(2*x) - 1
)*cos(7*x) + (cos(5*x) + cos(3*x) - 3*cos(x))*cos(6*x) - (cos(4*x) - cos(2*x) + 1)*cos(5*x) - (cos(3*x) - 3*co
s(x))*cos(4*x) + (cos(2*x) - 1)*cos(3*x) - 3*cos(2*x)*cos(x) + (3*sin(7*x) - sin(5*x) - sin(3*x) + 3*sin(x))*s
in(8*x) - 3*(sin(6*x) - sin(4*x) + sin(2*x))*sin(7*x) + (sin(5*x) + sin(3*x) - 3*sin(x))*sin(6*x) - (sin(4*x)
- sin(2*x))*sin(5*x) - (sin(3*x) - 3*sin(x))*sin(4*x) + sin(3*x)*sin(2*x) - 3*sin(2*x)*sin(x) + 3*cos(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)*cos(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) - sin(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) + 1/10*log(cos(x)^2 + sin(x)^2 + 2*sin(x) + 1) - 1/10*log(cos(x)^2 + sin(x)^2 - 2*sin(x) + 1) - sin
(x)

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Fricas [A]  time = 2.73607, size = 451, normalized size = 4.03 \begin{align*} \frac{1}{20} \, \sqrt{5} \log \left (\frac{8 \, \cos \left (x\right )^{2} - 4 \,{\left (\sqrt{5} - 1\right )} \sin \left (x\right ) + \sqrt{5} - 11}{4 \, \cos \left (x\right )^{2} + 2 \, \sin \left (x\right ) - 3}\right ) + \frac{1}{20} \, \sqrt{5} \log \left (-\frac{8 \, \cos \left (x\right )^{2} - 4 \,{\left (\sqrt{5} + 1\right )} \sin \left (x\right ) - \sqrt{5} - 11}{4 \, \cos \left (x\right )^{2} - 2 \, \sin \left (x\right ) - 3}\right ) - \frac{1}{20} \, \log \left (4 \, \cos \left (x\right )^{2} + 2 \, \sin \left (x\right ) - 3\right ) + \frac{1}{20} \, \log \left (4 \, \cos \left (x\right )^{2} - 2 \, \sin \left (x\right ) - 3\right ) + \frac{1}{10} \, \log \left (\sin \left (x\right ) + 1\right ) - \frac{1}{10} \, \log \left (-\sin \left (x\right ) + 1\right ) - \sin \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/20*sqrt(5)*log((8*cos(x)^2 - 4*(sqrt(5) - 1)*sin(x) + sqrt(5) - 11)/(4*cos(x)^2 + 2*sin(x) - 3)) + 1/20*sqrt
(5)*log(-(8*cos(x)^2 - 4*(sqrt(5) + 1)*sin(x) - sqrt(5) - 11)/(4*cos(x)^2 - 2*sin(x) - 3)) - 1/20*log(4*cos(x)
^2 + 2*sin(x) - 3) + 1/20*log(4*cos(x)^2 - 2*sin(x) - 3) + 1/10*log(sin(x) + 1) - 1/10*log(-sin(x) + 1) - sin(
x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sin \left (x\right ) \tan \left (5 \, x\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sin(x)*tan(5*x), x)