\(\int \cos ^{\cos (x)}(x) (1+\log (\cos (x))) \sin (x) \, dx\) [175]

Optimal result

Integrand size = 13, antiderivative size = 7 \[ \int \cos ^{\cos (x)}(x) (1+\log (\cos (x))) \sin (x) \, dx=-\cos ^{\cos (x)}(x) \]

[Out]

-cos(x)^cos(x)

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 7, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {4420, 6874, 2633} \[ \int \cos ^{\cos (x)}(x) (1+\log (\cos (x))) \sin (x) \, dx=-\cos ^{\cos (x)}(x) \]

[In]

Int[Cos[x]^Cos[x]*(1 + Log[Cos[x]])*Sin[x],x]

[Out]

-Cos[x]^Cos[x]

Rule 2633

Int[Log[u_]*(u_)^((a_.)*(x_)), x_Symbol] :> Simp[u^(a*x)/a, x] - Int[SimplifyIntegrand[x*u^(a*x - 1)*D[u, x],
x], x] /; FreeQ[a, x] && InverseFunctionFreeQ[u, x]

Rule 4420

Int[(u_)*(F_)[(c_.)*((a_.) + (b_.)*(x_))], x_Symbol] :> With[{d = FreeFactors[Cos[c*(a + b*x)], x]}, Dist[-d/(
b*c), Subst[Int[SubstFor[1, Cos[c*(a + b*x)]/d, u, x], x], x, Cos[c*(a + b*x)]/d], x] /; FunctionOfQ[Cos[c*(a
+ b*x)]/d, u, x, True]] /; FreeQ[{a, b, c}, x] && (EqQ[F, Sin] || EqQ[F, sin])

Rule 6874

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int x^x (1+\log (x)) \, dx,x,\cos (x)\right ) \\ & = -\text {Subst}\left (\int \left (x^x+x^x \log (x)\right ) \, dx,x,\cos (x)\right ) \\ & = -\text {Subst}\left (\int x^x \, dx,x,\cos (x)\right )-\text {Subst}\left (\int x^x \log (x) \, dx,x,\cos (x)\right ) \\ & = -\cos ^{\cos (x)}(x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 7, normalized size of antiderivative = 1.00 \[ \int \cos ^{\cos (x)}(x) (1+\log (\cos (x))) \sin (x) \, dx=-\cos ^{\cos (x)}(x) \]

[In]

Integrate[Cos[x]^Cos[x]*(1 + Log[Cos[x]])*Sin[x],x]

[Out]

-Cos[x]^Cos[x]

Maple [F]

\[\int \cos \left (x \right )^{1+\cos \left (x \right )} \tan \left (x \right ) \left (1+\ln \left (\cos \left (x \right )\right )\right )d x\]

[In]

int(cos(x)^(1+cos(x))*tan(x)*(1+ln(cos(x))),x)

[Out]

int(cos(x)^(1+cos(x))*tan(x)*(1+ln(cos(x))),x)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.86 \[ \int \cos ^{\cos (x)}(x) (1+\log (\cos (x))) \sin (x) \, dx=-\frac {\cos \left (x\right )^{\cos \left (x\right ) + 1}}{\cos \left (x\right )} \]

[In]

integrate(cos(x)^(1+cos(x))*tan(x)*(1+log(cos(x))),x, algorithm="fricas")

[Out]

-cos(x)^(cos(x) + 1)/cos(x)

Sympy [F]

\[ \int \cos ^{\cos (x)}(x) (1+\log (\cos (x))) \sin (x) \, dx=\int \left (\log {\left (\cos {\left (x \right )} \right )} + 1\right ) \cos ^{\cos {\left (x \right )} + 1}{\left (x \right )} \tan {\left (x \right )}\, dx \]

[In]

integrate(cos(x)**(1+cos(x))*tan(x)*(1+ln(cos(x))),x)

[Out]

Integral((log(cos(x)) + 1)*cos(x)**(cos(x) + 1)*tan(x), x)

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 7, normalized size of antiderivative = 1.00 \[ \int \cos ^{\cos (x)}(x) (1+\log (\cos (x))) \sin (x) \, dx=-\cos \left (x\right )^{\cos \left (x\right )} \]

[In]

integrate(cos(x)^(1+cos(x))*tan(x)*(1+log(cos(x))),x, algorithm="maxima")

[Out]

-cos(x)^cos(x)

Giac [F]

\[ \int \cos ^{\cos (x)}(x) (1+\log (\cos (x))) \sin (x) \, dx=\int { \cos \left (x\right )^{\cos \left (x\right ) + 1} {\left (\log \left (\cos \left (x\right )\right ) + 1\right )} \tan \left (x\right ) \,d x } \]

[In]

integrate(cos(x)^(1+cos(x))*tan(x)*(1+log(cos(x))),x, algorithm="giac")

[Out]

integrate(cos(x)^(cos(x) + 1)*(log(cos(x)) + 1)*tan(x), x)

Mupad [B] (verification not implemented)

Time = 17.65 (sec) , antiderivative size = 7, normalized size of antiderivative = 1.00 \[ \int \cos ^{\cos (x)}(x) (1+\log (\cos (x))) \sin (x) \, dx=-{\cos \left (x\right )}^{\cos \left (x\right )} \]

[In]

int(cos(x)^(cos(x) + 1)*tan(x)*(log(cos(x)) + 1),x)

[Out]

-cos(x)^cos(x)