\(\int \frac {1}{\sec (x)+\sin (x)} \, dx\) [20]

Optimal result

Integrand size = 7, antiderivative size = 28 \[ \int \frac {1}{\sec (x)+\sin (x)} \, dx=\arctan (\cos (x)+\sin (x))-\frac {\text {arctanh}\left (\frac {\cos (x)-\sin (x)}{\sqrt {3}}\right )}{\sqrt {3}} \]

[Out]

arctan(cos(x)+sin(x))-1/3*3^(1/2)*arctanh(1/3*(cos(x)-sin(x))*3^(1/2))

Rubi [F]

\[ \int \frac {1}{\sec (x)+\sin (x)} \, dx=\int \frac {1}{\sec (x)+\sin (x)} \, dx \]

[In]

Int[(Sec[x] + Sin[x])^(-1),x]

[Out]

Defer[Int][(Sec[x] + Sin[x])^(-1), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{\sec (x)+\sin (x)} \, dx \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(93\) vs. \(2(28)=56\).

Time = 1.25 (sec) , antiderivative size = 93, normalized size of antiderivative = 3.32 \[ \int \frac {1}{\sec (x)+\sin (x)} \, dx=\arctan \left (1-\left (-1+\sqrt {3}\right ) \tan \left (\frac {x}{2}\right )\right )+\arctan \left (1+\left (1+\sqrt {3}\right ) \tan \left (\frac {x}{2}\right )\right )+\frac {-\log \left (\sec ^2\left (\frac {x}{2}\right ) \left (\sqrt {3}+\cos (x)-\sin (x)\right )\right )+\log \left (-\sec ^2\left (\frac {x}{2}\right ) \left (\sqrt {3}-\cos (x)+\sin (x)\right )\right )}{2 \sqrt {3}} \]

[In]

Integrate[(Sec[x] + Sin[x])^(-1),x]

[Out]

ArcTan[1 - (-1 + Sqrt[3])*Tan[x/2]] + ArcTan[1 + (1 + Sqrt[3])*Tan[x/2]] + (-Log[Sec[x/2]^2*(Sqrt[3] + Cos[x]
- Sin[x])] + Log[-(Sec[x/2]^2*(Sqrt[3] - Cos[x] + Sin[x]))])/(2*Sqrt[3])

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.21 (sec) , antiderivative size = 57, normalized size of antiderivative = 2.04

method	result	size
default	\(\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}-2 \textit {\_Z}^{3}+2 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )}{\sum }\frac {\left (-\textit {\_R}^{2}+1\right ) \ln \left (\tan \left (\frac {x}{2}\right )-\textit {\_R} \right )}{2 \textit {\_R}^{3}-3 \textit {\_R}^{2}+2 \textit {\_R} +1}\)	\(57\)
risch	\(-\frac {i \ln \left ({\mathrm e}^{i x}+\frac {1}{2}-\frac {i}{2}+\frac {i \sqrt {3}}{2}-\frac {\sqrt {3}}{2}\right )}{2}+\frac {\ln \left ({\mathrm e}^{i x}+\frac {1}{2}-\frac {i}{2}+\frac {i \sqrt {3}}{2}-\frac {\sqrt {3}}{2}\right ) \sqrt {3}}{6}-\frac {i \ln \left ({\mathrm e}^{i x}+\frac {1}{2}-\frac {i}{2}-\frac {i \sqrt {3}}{2}+\frac {\sqrt {3}}{2}\right )}{2}-\frac {\ln \left ({\mathrm e}^{i x}+\frac {1}{2}-\frac {i}{2}-\frac {i \sqrt {3}}{2}+\frac {\sqrt {3}}{2}\right ) \sqrt {3}}{6}+\frac {i \ln \left ({\mathrm e}^{i x}-\frac {1}{2}+\frac {i}{2}+\frac {i \sqrt {3}}{2}-\frac {\sqrt {3}}{2}\right )}{2}+\frac {\ln \left ({\mathrm e}^{i x}-\frac {1}{2}+\frac {i}{2}+\frac {i \sqrt {3}}{2}-\frac {\sqrt {3}}{2}\right ) \sqrt {3}}{6}+\frac {i \ln \left ({\mathrm e}^{i x}-\frac {1}{2}+\frac {i}{2}-\frac {i \sqrt {3}}{2}+\frac {\sqrt {3}}{2}\right )}{2}-\frac {\ln \left ({\mathrm e}^{i x}-\frac {1}{2}+\frac {i}{2}-\frac {i \sqrt {3}}{2}+\frac {\sqrt {3}}{2}\right ) \sqrt {3}}{6}\)	\(202\)

[In]

int(1/(sin(x)+sec(x)),x,method=_RETURNVERBOSE)

[Out]

sum((-_R^2+1)/(2*_R^3-3*_R^2+2*_R+1)*ln(tan(1/2*x)-_R),_R=RootOf(_Z^4-2*_Z^3+2*_Z^2+2*_Z+1))

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.29 (sec) , antiderivative size = 165, normalized size of antiderivative = 5.89 \[ \int \frac {1}{\sec (x)+\sin (x)} \, dx=-\frac {1}{12} \, \sqrt {3} \sqrt {2 i \, \sqrt {3} - 2} \log \left (-\frac {1}{2} \, {\left ({\left (\sqrt {3} - i\right )} \sin \left (x\right ) + 2 i \, \cos \left (x\right )\right )} \sqrt {2 i \, \sqrt {3} - 2} + 2\right ) + \frac {1}{12} \, \sqrt {3} \sqrt {2 i \, \sqrt {3} - 2} \log \left (-\frac {1}{2} \, {\left ({\left (\sqrt {3} - i\right )} \sin \left (x\right ) + 2 i \, \cos \left (x\right )\right )} \sqrt {2 i \, \sqrt {3} - 2} - 2\right ) + \frac {1}{12} \, \sqrt {3} \sqrt {-2 i \, \sqrt {3} - 2} \log \left (\frac {1}{2} \, {\left ({\left (\sqrt {3} + i\right )} \sin \left (x\right ) - 2 i \, \cos \left (x\right )\right )} \sqrt {-2 i \, \sqrt {3} - 2} + 2\right ) - \frac {1}{12} \, \sqrt {3} \sqrt {-2 i \, \sqrt {3} - 2} \log \left (\frac {1}{2} \, {\left ({\left (\sqrt {3} + i\right )} \sin \left (x\right ) - 2 i \, \cos \left (x\right )\right )} \sqrt {-2 i \, \sqrt {3} - 2} - 2\right ) \]

[In]

integrate(1/(sin(x)+sec(x)),x, algorithm="fricas")

[Out]

-1/12*sqrt(3)*sqrt(2*I*sqrt(3) - 2)*log(-1/2*((sqrt(3) - I)*sin(x) + 2*I*cos(x))*sqrt(2*I*sqrt(3) - 2) + 2) +
1/12*sqrt(3)*sqrt(2*I*sqrt(3) - 2)*log(-1/2*((sqrt(3) - I)*sin(x) + 2*I*cos(x))*sqrt(2*I*sqrt(3) - 2) - 2) + 1
/12*sqrt(3)*sqrt(-2*I*sqrt(3) - 2)*log(1/2*((sqrt(3) + I)*sin(x) - 2*I*cos(x))*sqrt(-2*I*sqrt(3) - 2) + 2) - 1
/12*sqrt(3)*sqrt(-2*I*sqrt(3) - 2)*log(1/2*((sqrt(3) + I)*sin(x) - 2*I*cos(x))*sqrt(-2*I*sqrt(3) - 2) - 2)

Sympy [F]

\[ \int \frac {1}{\sec (x)+\sin (x)} \, dx=\int \frac {1}{\sin {\left (x \right )} + \sec {\left (x \right )}}\, dx \]

[In]

integrate(1/(sin(x)+sec(x)),x)

[Out]

Integral(1/(sin(x) + sec(x)), x)

Maxima [F]

\[ \int \frac {1}{\sec (x)+\sin (x)} \, dx=\int { \frac {1}{\sec \left (x\right ) + \sin \left (x\right )} \,d x } \]

[In]

integrate(1/(sin(x)+sec(x)),x, algorithm="maxima")

[Out]

integrate(1/(sec(x) + sin(x)), x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 81 vs. \(2 (25) = 50\).

Time = 0.29 (sec) , antiderivative size = 81, normalized size of antiderivative = 2.89 \[ \int \frac {1}{\sec (x)+\sin (x)} \, dx=\frac {1}{2} \, \pi + \frac {1}{6} \, \sqrt {3} \log \left ({\left (\sqrt {3} + \tan \left (\frac {1}{2} \, x\right ) - 1\right )}^{2} + \tan \left (\frac {1}{2} \, x\right )^{2}\right ) - \frac {1}{6} \, \sqrt {3} \log \left ({\left (\sqrt {3} - \tan \left (\frac {1}{2} \, x\right ) + 1\right )}^{2} + \tan \left (\frac {1}{2} \, x\right )^{2}\right ) + \arctan \left ({\left (\sqrt {3} + 1\right )} \tan \left (\frac {1}{2} \, x\right ) + 1\right ) + \arctan \left (-{\left (\sqrt {3} - 1\right )} \tan \left (\frac {1}{2} \, x\right ) + 1\right ) \]

[In]

integrate(1/(sin(x)+sec(x)),x, algorithm="giac")

[Out]

1/2*pi + 1/6*sqrt(3)*log((sqrt(3) + tan(1/2*x) - 1)^2 + tan(1/2*x)^2) - 1/6*sqrt(3)*log((sqrt(3) - tan(1/2*x)
+ 1)^2 + tan(1/2*x)^2) + arctan((sqrt(3) + 1)*tan(1/2*x) + 1) + arctan(-(sqrt(3) - 1)*tan(1/2*x) + 1)

Mupad [B] (verification not implemented)

Time = 16.06 (sec) , antiderivative size = 233, normalized size of antiderivative = 8.32 \[ \int \frac {1}{\sec (x)+\sin (x)} \, dx=-\mathrm {atan}\left (\frac {96\,\mathrm {tan}\left (\frac {x}{2}\right )}{64\,\mathrm {tan}\left (\frac {x}{2}\right )+64-\sqrt {3}\,\mathrm {tan}\left (\frac {x}{2}\right )\,64{}\mathrm {i}}+\frac {\sqrt {3}\,32{}\mathrm {i}}{64\,\mathrm {tan}\left (\frac {x}{2}\right )+64-\sqrt {3}\,\mathrm {tan}\left (\frac {x}{2}\right )\,64{}\mathrm {i}}+\frac {32}{64\,\mathrm {tan}\left (\frac {x}{2}\right )+64-\sqrt {3}\,\mathrm {tan}\left (\frac {x}{2}\right )\,64{}\mathrm {i}}+\frac {\sqrt {3}\,\mathrm {tan}\left (\frac {x}{2}\right )\,32{}\mathrm {i}}{64\,\mathrm {tan}\left (\frac {x}{2}\right )+64-\sqrt {3}\,\mathrm {tan}\left (\frac {x}{2}\right )\,64{}\mathrm {i}}\right )\,\left (-1+\frac {\sqrt {3}\,1{}\mathrm {i}}{3}\right )-\mathrm {atan}\left (-\frac {96\,\mathrm {tan}\left (\frac {x}{2}\right )}{64\,\mathrm {tan}\left (\frac {x}{2}\right )+64+\sqrt {3}\,\mathrm {tan}\left (\frac {x}{2}\right )\,64{}\mathrm {i}}+\frac {\sqrt {3}\,32{}\mathrm {i}}{64\,\mathrm {tan}\left (\frac {x}{2}\right )+64+\sqrt {3}\,\mathrm {tan}\left (\frac {x}{2}\right )\,64{}\mathrm {i}}-\frac {32}{64\,\mathrm {tan}\left (\frac {x}{2}\right )+64+\sqrt {3}\,\mathrm {tan}\left (\frac {x}{2}\right )\,64{}\mathrm {i}}+\frac {\sqrt {3}\,\mathrm {tan}\left (\frac {x}{2}\right )\,32{}\mathrm {i}}{64\,\mathrm {tan}\left (\frac {x}{2}\right )+64+\sqrt {3}\,\mathrm {tan}\left (\frac {x}{2}\right )\,64{}\mathrm {i}}\right )\,\left (1+\frac {\sqrt {3}\,1{}\mathrm {i}}{3}\right ) \]

[In]

int(1/(sin(x) + 1/cos(x)),x)

[Out]

- atan((96*tan(x/2))/(64*tan(x/2) - 3^(1/2)*tan(x/2)*64i + 64) + (3^(1/2)*32i)/(64*tan(x/2) - 3^(1/2)*tan(x/2)
*64i + 64) + 32/(64*tan(x/2) - 3^(1/2)*tan(x/2)*64i + 64) + (3^(1/2)*tan(x/2)*32i)/(64*tan(x/2) - 3^(1/2)*tan(
x/2)*64i + 64))*((3^(1/2)*1i)/3 - 1) - atan((3^(1/2)*32i)/(64*tan(x/2) + 3^(1/2)*tan(x/2)*64i + 64) - (96*tan(
x/2))/(64*tan(x/2) + 3^(1/2)*tan(x/2)*64i + 64) - 32/(64*tan(x/2) + 3^(1/2)*tan(x/2)*64i + 64) + (3^(1/2)*tan(
x/2)*32i)/(64*tan(x/2) + 3^(1/2)*tan(x/2)*64i + 64))*((3^(1/2)*1i)/3 + 1)