Optimal. Leaf size=93 \[ \frac{\sqrt{5} \sqrt{-\cos (c+d x)} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\sec (c+d x)-1} \sqrt{\sec (c+d x)+1} E\left (\sin ^{-1}\left (\frac{\sqrt{2-3 \cos (c+d x)}}{\sqrt{-\cos (c+d x)}}\right )|\frac{1}{5}\right )}{d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.206488, antiderivative size = 93, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.061, Rules used = {2995, 2994} \[ \frac{\sqrt{5} \sqrt{-\cos (c+d x)} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\sec (c+d x)-1} \sqrt{\sec (c+d x)+1} E\left (\sin ^{-1}\left (\frac{\sqrt{2-3 \cos (c+d x)}}{\sqrt{-\cos (c+d x)}}\right )|\frac{1}{5}\right )}{d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2995
Rule 2994
Rubi steps
\begin{align*} \int \frac{1+\cos (c+d x)}{\sqrt{2-3 \cos (c+d x)} \cos ^{\frac{3}{2}}(c+d x)} \, dx &=-\frac{\sqrt{-\cos (c+d x)} \int \frac{1+\cos (c+d x)}{\sqrt{2-3 \cos (c+d x)} (-\cos (c+d x))^{3/2}} \, dx}{\sqrt{\cos (c+d x)}}\\ &=\frac{\sqrt{5} \sqrt{-\cos (c+d x)} \sqrt{\cos (c+d x)} \csc (c+d x) E\left (\sin ^{-1}\left (\frac{\sqrt{2-3 \cos (c+d x)}}{\sqrt{-\cos (c+d x)}}\right )|\frac{1}{5}\right ) \sqrt{-1+\sec (c+d x)} \sqrt{1+\sec (c+d x)}}{d}\\ \end{align*}
Mathematica [F] time = 32.6107, size = 0, normalized size = 0. \[ \int \frac{1+\cos (c+d x)}{\sqrt{2-3 \cos (c+d x)} \cos ^{\frac{3}{2}}(c+d x)} \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.469, size = 611, normalized size = 6.6 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cos \left (d x + c\right ) + 1}{\sqrt{-3 \, \cos \left (d x + c\right ) + 2} \cos \left (d x + c\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{{\left (\cos \left (d x + c\right ) + 1\right )} \sqrt{-3 \, \cos \left (d x + c\right ) + 2} \sqrt{\cos \left (d x + c\right )}}{3 \, \cos \left (d x + c\right )^{3} - 2 \, \cos \left (d x + c\right )^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cos{\left (c + d x \right )} + 1}{\sqrt{2 - 3 \cos{\left (c + d x \right )}} \cos ^{\frac{3}{2}}{\left (c + d x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cos \left (d x + c\right ) + 1}{\sqrt{-3 \, \cos \left (d x + c\right ) + 2} \cos \left (d x + c\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]