Optimal. Leaf size=46 \[ -\frac{E\left (\left .a+b x-\frac{\pi }{4}\right |2\right )}{2 b}-\frac{\cos ^2(a+b x)}{b \sqrt{\sin (2 a+2 b x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.035921, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {4295, 2639} \[ -\frac{E\left (\left .a+b x-\frac{\pi }{4}\right |2\right )}{2 b}-\frac{\cos ^2(a+b x)}{b \sqrt{\sin (2 a+2 b x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4295
Rule 2639
Rubi steps
\begin{align*} \int \frac{\cos ^2(a+b x)}{\sin ^{\frac{3}{2}}(2 a+2 b x)} \, dx &=-\frac{\cos ^2(a+b x)}{b \sqrt{\sin (2 a+2 b x)}}-\frac{1}{2} \int \sqrt{\sin (2 a+2 b x)} \, dx\\ &=-\frac{E\left (\left .a-\frac{\pi }{4}+b x\right |2\right )}{2 b}-\frac{\cos ^2(a+b x)}{b \sqrt{\sin (2 a+2 b x)}}\\ \end{align*}
Mathematica [A] time = 0.113925, size = 39, normalized size = 0.85 \[ -\frac{E\left (\left .a+b x-\frac{\pi }{4}\right |2\right )+\sqrt{\sin (2 (a+b x))} \cot (a+b x)}{2 b} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 9.786, size = 94273592, normalized size = 2049425.9 \begin{align*} \text{output 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 (b x + a\right )^{2}}{\sin \left (2 \, b x + 2 \, a\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{\cos \left (b x + a\right )^{2} \sqrt{\sin \left (2 \, b x + 2 \, a\right )}}{\cos \left (2 \, b x + 2 \, a\right )^{2} - 1}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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 (b x + a\right )^{2}}{\sin \left (2 \, b x + 2 \, a\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]