Optimal. Leaf size=13 \[ -\frac{\cot (a+b x)}{4 b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0329684, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {4287, 3767, 8} \[ -\frac{\cot (a+b x)}{4 b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4287
Rule 3767
Rule 8
Rubi steps
\begin{align*} \int \cos ^2(a+b x) \csc ^2(2 a+2 b x) \, dx &=\frac{1}{4} \int \csc ^2(a+b x) \, dx\\ &=-\frac{\operatorname{Subst}(\int 1 \, dx,x,\cot (a+b x))}{4 b}\\ &=-\frac{\cot (a+b x)}{4 b}\\ \end{align*}
Mathematica [A] time = 0.012905, size = 13, normalized size = 1. \[ -\frac{\cot (a+b x)}{4 b} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.02, size = 12, normalized size = 0.9 \begin{align*} -{\frac{\cot \left ( bx+a \right ) }{4\,b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.1505, size = 72, normalized size = 5.54 \begin{align*} -\frac{\sin \left (2 \, b x + 2 \, a\right )}{2 \,{\left (b \cos \left (2 \, b x + 2 \, a\right )^{2} + b \sin \left (2 \, b x + 2 \, a\right )^{2} - 2 \, b \cos \left (2 \, b x + 2 \, a\right ) + b\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 0.458278, size = 49, normalized size = 3.77 \begin{align*} -\frac{\cos \left (b x + a\right )}{4 \, b \sin \left (b x + a\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 [A] time = 1.19405, size = 18, normalized size = 1.38 \begin{align*} -\frac{1}{4 \, b \tan \left (b x + a\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]