3.859 \(\int \frac{\csc (x) \sqrt{\cos (x)+\sin (x)}}{\cos ^{\frac{3}{2}}(x)} \, dx\)
Optimal. Leaf size=44 \[ -\log (\sin (x))+\frac{2 \sqrt{\sin (x)+\cos (x)}}{\sqrt{\cos (x)}}+2 \log \left (\sqrt{\sin (x)+\cos (x)}-\sqrt{\cos (x)}\right ) \]
[Out]
-Log[Sin[x]] + 2*Log[-Sqrt[Cos[x]] + Sqrt[Cos[x] + Sin[x]]] + (2*Sqrt[Cos[x] + Sin[x]])/Sqrt[Cos[x]]
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Rubi [F] time = 2.56892, antiderivative size = 0, normalized size of antiderivative = 0.,
number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {}
\[ \int \frac{\csc (x) \sqrt{\cos (x)+\sin (x)}}{\cos ^{\frac{3}{2}}(x)} \, dx \]
Verification is Not applicable to the result.
[In]
Int[(Csc[x]*Sqrt[Cos[x] + Sin[x]])/Cos[x]^(3/2),x]
[Out]
Defer[Int][(Csc[x]*Sqrt[Cos[x] + Sin[x]])/Cos[x]^(3/2), x]
Rubi steps
\begin{align*} \int \frac{\csc (x) \sqrt{\cos (x)+\sin (x)}}{\cos ^{\frac{3}{2}}(x)} \, dx &=\int \frac{\csc (x) \sqrt{\cos (x)+\sin (x)}}{\cos ^{\frac{3}{2}}(x)} \, dx\\ \end{align*}
Mathematica [A] time = 0.39238, size = 68, normalized size = 1.55 \[ \frac{2 \left (\sin (x)+\cos (x)-\sqrt{\cos (x)} \sqrt{\sqrt{\sin ^2(x)}+\cos (x)} \coth ^{-1}\left (\frac{\sqrt{\sqrt{\sin ^2(x)}+\cos (x)}}{\sqrt{\cos (x)}}\right )\right )}{\sqrt{\cos (x)} \sqrt{\sin (x)+\cos (x)}} \]
Warning: Unable to verify antiderivative.
[In]
Integrate[(Csc[x]*Sqrt[Cos[x] + Sin[x]])/Cos[x]^(3/2),x]
[Out]
(2*(Cos[x] + Sin[x] - ArcCoth[Sqrt[Cos[x] + Sqrt[Sin[x]^2]]/Sqrt[Cos[x]]]*Sqrt[Cos[x]]*Sqrt[Cos[x] + Sqrt[Sin[
x]^2]]))/(Sqrt[Cos[x]]*Sqrt[Cos[x] + Sin[x]])
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Maple [C] time = 0.435, size = 917, normalized size = 20.8 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(csc(x)*(cos(x)+sin(x))^(1/2)/cos(x)^(3/2),x)
[Out]
1/(2+2^(1/2))*(-1+cos(x))^2*(1+cos(x))^2*(EllipticPi(1/2*2^(1/2)*((2+2^(1/2))*2^(1/2)*(sin(x)-1)/cos(x))^(1/2)
,-2^(1/2)/(2+2^(1/2)),I/(2+2^(1/2))*((2-2^(1/2))*(2+2^(1/2)))^(1/2))*(2^(1/2)*(cos(x)*2^(1/2)-sin(x)*2^(1/2)+2
*sin(x)+2^(1/2)-2)/cos(x))^(1/2)*(2^(1/2)*(cos(x)*2^(1/2)-sin(x)*2^(1/2)-2*sin(x)+2^(1/2)+2)/cos(x))^(1/2)*((2
+2^(1/2))*2^(1/2)*(sin(x)-1)/cos(x))^(1/2)*sin(x)-EllipticF(1/2*2^(1/2)*((2+2^(1/2))*2^(1/2)*(sin(x)-1)/cos(x)
)^(1/2),I/(2+2^(1/2))*((2-2^(1/2))*(2+2^(1/2)))^(1/2))*(2^(1/2)*(cos(x)*2^(1/2)-sin(x)*2^(1/2)+2*sin(x)+2^(1/2
)-2)/cos(x))^(1/2)*(2^(1/2)*(cos(x)*2^(1/2)-sin(x)*2^(1/2)-2*sin(x)+2^(1/2)+2)/cos(x))^(1/2)*((2+2^(1/2))*2^(1
/2)*(sin(x)-1)/cos(x))^(1/2)*sin(x)+EllipticPi(1/2*2^(1/2)*((2+2^(1/2))*2^(1/2)*(sin(x)-1)/cos(x))^(1/2),2^(1/
2)/(2+2^(1/2)),I/(2+2^(1/2))*((2-2^(1/2))*(2+2^(1/2)))^(1/2))*(2^(1/2)*(cos(x)*2^(1/2)-sin(x)*2^(1/2)+2*sin(x)
+2^(1/2)-2)/cos(x))^(1/2)*(2^(1/2)*(cos(x)*2^(1/2)-sin(x)*2^(1/2)-2*sin(x)+2^(1/2)+2)/cos(x))^(1/2)*((2+2^(1/2
))*2^(1/2)*(sin(x)-1)/cos(x))^(1/2)*sin(x)+(2^(1/2)*(cos(x)*2^(1/2)-sin(x)*2^(1/2)+2*sin(x)+2^(1/2)-2)/cos(x))
^(1/2)*(2^(1/2)*(cos(x)*2^(1/2)-sin(x)*2^(1/2)-2*sin(x)+2^(1/2)+2)/cos(x))^(1/2)*EllipticPi(1/2*2^(1/2)*((2+2^
(1/2))*2^(1/2)*(sin(x)-1)/cos(x))^(1/2),-2^(1/2)/(2+2^(1/2)),I/(2+2^(1/2))*((2-2^(1/2))*(2+2^(1/2)))^(1/2))*((
2+2^(1/2))*2^(1/2)*(sin(x)-1)/cos(x))^(1/2)-(2^(1/2)*(cos(x)*2^(1/2)-sin(x)*2^(1/2)+2*sin(x)+2^(1/2)-2)/cos(x)
)^(1/2)*(2^(1/2)*(cos(x)*2^(1/2)-sin(x)*2^(1/2)-2*sin(x)+2^(1/2)+2)/cos(x))^(1/2)*EllipticF(1/2*2^(1/2)*((2+2^
(1/2))*2^(1/2)*(sin(x)-1)/cos(x))^(1/2),I/(2+2^(1/2))*((2-2^(1/2))*(2+2^(1/2)))^(1/2))*((2+2^(1/2))*2^(1/2)*(s
in(x)-1)/cos(x))^(1/2)+(2^(1/2)*(cos(x)*2^(1/2)-sin(x)*2^(1/2)+2*sin(x)+2^(1/2)-2)/cos(x))^(1/2)*(2^(1/2)*(cos
(x)*2^(1/2)-sin(x)*2^(1/2)-2*sin(x)+2^(1/2)+2)/cos(x))^(1/2)*EllipticPi(1/2*2^(1/2)*((2+2^(1/2))*2^(1/2)*(sin(
x)-1)/cos(x))^(1/2),2^(1/2)/(2+2^(1/2)),I/(2+2^(1/2))*((2-2^(1/2))*(2+2^(1/2)))^(1/2))*((2+2^(1/2))*2^(1/2)*(s
in(x)-1)/cos(x))^(1/2)+2*cos(x)*2^(1/2)+2*sin(x)*2^(1/2)+4*cos(x)+4*sin(x))/cos(x)^(1/2)/sin(x)^4/(cos(x)+sin(
x))^(1/2)
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Maxima [B] time = 2.44539, size = 699, normalized size = 15.89 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csc(x)*(cos(x)+sin(x))^(1/2)/cos(x)^(3/2),x, algorithm="maxima")
[Out]
4*((2*cos(2*x) + sin(2*x))*cos(1/2*arctan2(-cos(4*x) + sin(4*x) + 2*sin(2*x) + 1, cos(4*x) + 2*cos(2*x) + sin(
4*x) + 1))^3 + (2*cos(2*x) + sin(2*x))*cos(1/2*arctan2(-cos(4*x) + sin(4*x) + 2*sin(2*x) + 1, cos(4*x) + 2*cos
(2*x) + sin(4*x) + 1))*sin(1/2*arctan2(-cos(4*x) + sin(4*x) + 2*sin(2*x) + 1, cos(4*x) + 2*cos(2*x) + sin(4*x)
+ 1))^2 - (cos(2*x) - 2*sin(2*x) + 1)*sin(1/2*arctan2(-cos(4*x) + sin(4*x) + 2*sin(2*x) + 1, cos(4*x) + 2*cos
(2*x) + sin(4*x) + 1))^3 - (cos(2*x) - sin(2*x) - 1)*cos(1/2*arctan2(-cos(4*x) + sin(4*x) + 2*sin(2*x) + 1, co
s(4*x) + 2*cos(2*x) + sin(4*x) + 1)) - ((cos(2*x) - 2*sin(2*x) + 1)*cos(1/2*arctan2(-cos(4*x) + sin(4*x) + 2*s
in(2*x) + 1, cos(4*x) + 2*cos(2*x) + sin(4*x) + 1))^2 + cos(2*x) + sin(2*x) - 1)*sin(1/2*arctan2(-cos(4*x) + s
in(4*x) + 2*sin(2*x) + 1, cos(4*x) + 2*cos(2*x) + sin(4*x) + 1)))/((4*(cos(2*x) - sin(2*x))*cos(4*x) + 2*cos(4
*x)^2 + 4*cos(2*x)^2 + 4*(cos(2*x) + sin(2*x) + 1)*sin(4*x) + 2*sin(4*x)^2 + 4*sin(2*x)^2 + 4*cos(2*x) + 4*sin
(2*x) + 2)^(1/4)*(cos(1/2*arctan2(-cos(4*x) + sin(4*x) + 2*sin(2*x) + 1, cos(4*x) + 2*cos(2*x) + sin(4*x) + 1)
)^2 + sin(1/2*arctan2(-cos(4*x) + sin(4*x) + 2*sin(2*x) + 1, cos(4*x) + 2*cos(2*x) + sin(4*x) + 1))^2))
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Fricas [B] time = 2.15771, size = 363, normalized size = 8.25 \begin{align*} -\frac{\cos \left (x\right ) \log \left ({\left (2 \, \cos \left (x\right ) + \sin \left (x\right )\right )} \sqrt{\cos \left (x\right ) + \sin \left (x\right )} \sqrt{\cos \left (x\right )} + \frac{7}{4} \, \cos \left (x\right )^{2} + 2 \, \cos \left (x\right ) \sin \left (x\right ) + \frac{1}{4}\right ) - \cos \left (x\right ) \log \left (-{\left (2 \, \cos \left (x\right ) + \sin \left (x\right )\right )} \sqrt{\cos \left (x\right ) + \sin \left (x\right )} \sqrt{\cos \left (x\right )} + \frac{7}{4} \, \cos \left (x\right )^{2} + 2 \, \cos \left (x\right ) \sin \left (x\right ) + \frac{1}{4}\right ) - 8 \, \sqrt{\cos \left (x\right ) + \sin \left (x\right )} \sqrt{\cos \left (x\right )}}{4 \, \cos \left (x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csc(x)*(cos(x)+sin(x))^(1/2)/cos(x)^(3/2),x, algorithm="fricas")
[Out]
-1/4*(cos(x)*log((2*cos(x) + sin(x))*sqrt(cos(x) + sin(x))*sqrt(cos(x)) + 7/4*cos(x)^2 + 2*cos(x)*sin(x) + 1/4
) - cos(x)*log(-(2*cos(x) + sin(x))*sqrt(cos(x) + sin(x))*sqrt(cos(x)) + 7/4*cos(x)^2 + 2*cos(x)*sin(x) + 1/4)
- 8*sqrt(cos(x) + sin(x))*sqrt(cos(x)))/cos(x)
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{\sin{\left (x \right )} + \cos{\left (x \right )}} \csc{\left (x \right )}}{\cos ^{\frac{3}{2}}{\left (x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csc(x)*(cos(x)+sin(x))**(1/2)/cos(x)**(3/2),x)
[Out]
Integral(sqrt(sin(x) + cos(x))*csc(x)/cos(x)**(3/2), x)
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{\cos \left (x\right ) + \sin \left (x\right )} \csc \left (x\right )}{\cos \left (x\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csc(x)*(cos(x)+sin(x))^(1/2)/cos(x)^(3/2),x, algorithm="giac")
[Out]
integrate(sqrt(cos(x) + sin(x))*csc(x)/cos(x)^(3/2), x)