Optimal. Leaf size=15 \[ -\frac {4 \cos ^3(a+b x)}{3 b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {4288, 2565, 30} \[ -\frac {4 \cos ^3(a+b x)}{3 b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 30
Rule 2565
Rule 4288
Rubi steps
\begin {align*} \int \csc (a+b x) \sin ^2(2 a+2 b x) \, dx &=4 \int \cos ^2(a+b x) \sin (a+b x) \, dx\\ &=-\frac {4 \operatorname {Subst}\left (\int x^2 \, dx,x,\cos (a+b x)\right )}{b}\\ &=-\frac {4 \cos ^3(a+b x)}{3 b}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 15, normalized size = 1.00 \[ -\frac {4 \cos ^3(a+b x)}{3 b} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.44, size = 13, normalized size = 0.87 \[ -\frac {4 \, \cos \left (b x + a\right )^{3}}{3 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.37, size = 52, normalized size = 3.47 \[ \frac {8 \, {\left (\frac {3 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{2}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{2}} + 1\right )}}{3 \, b {\left (\frac {\cos \left (b x + a\right ) - 1}{\cos \left (b x + a\right ) + 1} - 1\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.50, size = 14, normalized size = 0.93 \[ -\frac {4 \left (\cos ^{3}\left (b x +a \right )\right )}{3 b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.34, size = 23, normalized size = 1.53 \[ -\frac {\cos \left (3 \, b x + 3 \, a\right ) + 3 \, \cos \left (b x + a\right )}{3 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.03, size = 13, normalized size = 0.87 \[ -\frac {4\,{\cos \left (a+b\,x\right )}^3}{3\,b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________