\(\int \frac {\cos (x)}{\sin (x)+\sin ^{\sqrt {2}}(x)} \, dx\) [809]

Optimal result

Integrand size = 16, antiderivative size = 26 \[ \int \frac {\cos (x)}{\sin (x)+\sin ^{\sqrt {2}}(x)} \, dx=\log (\sin (x))-\left (1+\sqrt {2}\right ) \log \left (1+\sin ^{-1+\sqrt {2}}(x)\right ) \]

[Out]

ln(sin(x))-ln(1+sin(x)^(2^(1/2)-1))*(1+2^(1/2))

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {4419, 272, 36, 29, 31} \[ \int \frac {\cos (x)}{\sin (x)+\sin ^{\sqrt {2}}(x)} \, dx=\log (\sin (x))-\left (1+\sqrt {2}\right ) \log \left (\sin ^{\sqrt {2}-1}(x)+1\right ) \]

[In]

Int[Cos[x]/(Sin[x] + Sin[x]^Sqrt[2]),x]

[Out]

Log[Sin[x]] - (1 + Sqrt[2])*Log[1 + Sin[x]^(-1 + Sqrt[2])]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 4419

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

Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {1}{x \left (1+x^{-1+\sqrt {2}}\right )} \, dx,x,\sin (x)\right ) \\ & = \left (1+\sqrt {2}\right ) \text {Subst}\left (\int \frac {1}{x (1+x)} \, dx,x,\sin ^{-1+\sqrt {2}}(x)\right ) \\ & = \left (-1-\sqrt {2}\right ) \text {Subst}\left (\int \frac {1}{1+x} \, dx,x,\sin ^{-1+\sqrt {2}}(x)\right )+\left (1+\sqrt {2}\right ) \text {Subst}\left (\int \frac {1}{x} \, dx,x,\sin ^{-1+\sqrt {2}}(x)\right ) \\ & = \log (\sin (x))-\left (1+\sqrt {2}\right ) \log \left (1+\sin ^{-1+\sqrt {2}}(x)\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {\cos (x)}{\sin (x)+\sin ^{\sqrt {2}}(x)} \, dx=\log (\sin (x))-\left (1+\sqrt {2}\right ) \log \left (1+\sin ^{-1+\sqrt {2}}(x)\right ) \]

[In]

Integrate[Cos[x]/(Sin[x] + Sin[x]^Sqrt[2]),x]

[Out]

Log[Sin[x]] - (1 + Sqrt[2])*Log[1 + Sin[x]^(-1 + Sqrt[2])]

Maple [A] (verified)

Time = 1.43 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.19

method	result	size
derivativedivides	\(\left (2+\sqrt {2}\right ) \ln \left (\sin \left (x \right )\right )+\left (-1-\sqrt {2}\right ) \ln \left (\sin \left (x \right )+{\mathrm e}^{\sqrt {2}\, \ln \left (\sin \left (x \right )\right )}\right )\)	\(31\)
default	\(\left (2+\sqrt {2}\right ) \ln \left (\sin \left (x \right )\right )+\left (-1-\sqrt {2}\right ) \ln \left (\sin \left (x \right )+{\mathrm e}^{\sqrt {2}\, \ln \left (\sin \left (x \right )\right )}\right )\)	\(31\)
parallelrisch	\(\sqrt {2}\, \left (-\ln \left (\frac {\sin \left (x \right )+\sin \left (x \right )^{\sqrt {2}}}{\cos \left (x \right )+1}\right )+\ln \left (\frac {\csc \left (x \right )}{2}-\frac {\cot \left (x \right )}{2}\right )\right )-\ln \left (\frac {4 \sin \left (x \right )+4 \sin \left (x \right )^{\sqrt {2}}}{\cos \left (x \right )+1}\right )+2 \ln \left (\csc \left (x \right )-\cot \left (x \right )\right )-\ln \left (\frac {1}{\cos \left (x \right )+1}\right )\)	\(78\)
risch	\(\text {Expression too large to display}\)	\(674\)

[In]

int(cos(x)/(sin(x)+sin(x)^(2^(1/2))),x,method=_RETURNVERBOSE)

[Out]

(2+2^(1/2))*ln(sin(x))+(-1-2^(1/2))*ln(sin(x)+exp(2^(1/2)*ln(sin(x))))

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.04 \[ \int \frac {\cos (x)}{\sin (x)+\sin ^{\sqrt {2}}(x)} \, dx=-{\left (\sqrt {2} + 1\right )} \log \left (\sin \left (x\right )^{\left (\sqrt {2}\right )} + \sin \left (x\right )\right ) + {\left (\sqrt {2} + 2\right )} \log \left (\sin \left (x\right )\right ) \]

[In]

integrate(cos(x)/(sin(x)+sin(x)^(2^(1/2))),x, algorithm="fricas")

[Out]

-(sqrt(2) + 1)*log(sin(x)^sqrt(2) + sin(x)) + (sqrt(2) + 2)*log(sin(x))

Sympy [A] (verification not implemented)

Time = 0.44 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.38 \[ \int \frac {\cos (x)}{\sin (x)+\sin ^{\sqrt {2}}(x)} \, dx=- \frac {\log {\left (\sin {\left (x \right )} + \sin ^{\sqrt {2}}{\left (x \right )} \right )}}{-1 + \sqrt {2}} + \frac {\sqrt {2} \log {\left (\sin {\left (x \right )} \right )}}{-1 + \sqrt {2}} \]

[In]

integrate(cos(x)/(sin(x)+sin(x)**(2**(1/2))),x)

[Out]

-log(sin(x) + sin(x)**(sqrt(2)))/(-1 + sqrt(2)) + sqrt(2)*log(sin(x))/(-1 + sqrt(2))

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.31 \[ \int \frac {\cos (x)}{\sin (x)+\sin ^{\sqrt {2}}(x)} \, dx=\frac {\sqrt {2} \log \left (\sin \left (x\right )\right )}{\sqrt {2} - 1} - \frac {\log \left (\sin \left (x\right )^{\left (\sqrt {2}\right )} + \sin \left (x\right )\right )}{\sqrt {2} - 1} \]

[In]

integrate(cos(x)/(sin(x)+sin(x)^(2^(1/2))),x, algorithm="maxima")

[Out]

sqrt(2)*log(sin(x))/(sqrt(2) - 1) - log(sin(x)^sqrt(2) + sin(x))/(sqrt(2) - 1)

Giac [F]

\[ \int \frac {\cos (x)}{\sin (x)+\sin ^{\sqrt {2}}(x)} \, dx=\int { \frac {\cos \left (x\right )}{\sin \left (x\right )^{\left (\sqrt {2}\right )} + \sin \left (x\right )} \,d x } \]

[In]

integrate(cos(x)/(sin(x)+sin(x)^(2^(1/2))),x, algorithm="giac")

[Out]

integrate(cos(x)/(sin(x)^sqrt(2) + sin(x)), x)

Mupad [B] (verification not implemented)

Time = 27.36 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.12 \[ \int \frac {\cos (x)}{\sin (x)+\sin ^{\sqrt {2}}(x)} \, dx=\ln \left (\sin \left (x\right )\right )\,\left (\sqrt {2}+2\right )-\frac {\ln \left (\sin \left (x\right )+{\sin \left (x\right )}^{\sqrt {2}}\right )}{\sqrt {2}-1} \]

[In]

int(cos(x)/(sin(x) + sin(x)^(2^(1/2))),x)

[Out]

log(sin(x))*(2^(1/2) + 2) - log(sin(x) + sin(x)^(2^(1/2)))/(2^(1/2) - 1)