Optimal. Leaf size=42 \[ \frac {\tan (a+b x)}{16 b}-\frac {\cot ^3(a+b x)}{48 b}-\frac {\cot (a+b x)}{8 b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.06, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {4287, 2620, 270} \[ \frac {\tan (a+b x)}{16 b}-\frac {\cot ^3(a+b x)}{48 b}-\frac {\cot (a+b x)}{8 b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 270
Rule 2620
Rule 4287
Rubi steps
\begin {align*} \int \cos ^2(a+b x) \csc ^4(2 a+2 b x) \, dx &=\frac {1}{16} \int \csc ^4(a+b x) \sec ^2(a+b x) \, dx\\ &=\frac {\operatorname {Subst}\left (\int \frac {\left (1+x^2\right )^2}{x^4} \, dx,x,\tan (a+b x)\right )}{16 b}\\ &=\frac {\operatorname {Subst}\left (\int \left (1+\frac {1}{x^4}+\frac {2}{x^2}\right ) \, dx,x,\tan (a+b x)\right )}{16 b}\\ &=-\frac {\cot (a+b x)}{8 b}-\frac {\cot ^3(a+b x)}{48 b}+\frac {\tan (a+b x)}{16 b}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.05, size = 48, normalized size = 1.14 \[ \frac {\tan (a+b x)}{16 b}-\frac {5 \cot (a+b x)}{48 b}-\frac {\cot (a+b x) \csc ^2(a+b x)}{48 b} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.46, size = 54, normalized size = 1.29 \[ -\frac {8 \, \cos \left (b x + a\right )^{4} - 12 \, \cos \left (b x + a\right )^{2} + 3}{48 \, {\left (b \cos \left (b x + a\right )^{3} - b \cos \left (b x + a\right )\right )} \sin \left (b x + a\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.50, size = 35, normalized size = 0.83 \[ -\frac {\frac {6 \, \tan \left (b x + a\right )^{2} + 1}{\tan \left (b x + a\right )^{3}} - 3 \, \tan \left (b x + a\right )}{48 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 1.62, size = 51, normalized size = 1.21 \[ \frac {-\frac {1}{3 \sin \left (b x +a \right )^{3} \cos \left (b x +a \right )}+\frac {4}{3 \sin \left (b x +a \right ) \cos \left (b x +a \right )}-\frac {8 \cot \left (b x +a \right )}{3}}{16 b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.34, size = 308, normalized size = 7.33 \[ \frac {{\left (2 \, \cos \left (2 \, b x + 2 \, a\right ) - 1\right )} \sin \left (8 \, b x + 8 \, a\right ) - 2 \, {\left (2 \, \cos \left (2 \, b x + 2 \, a\right ) - 1\right )} \sin \left (6 \, b x + 6 \, a\right ) - 2 \, \cos \left (8 \, b x + 8 \, a\right ) \sin \left (2 \, b x + 2 \, a\right ) + 4 \, \cos \left (6 \, b x + 6 \, a\right ) \sin \left (2 \, b x + 2 \, a\right )}{3 \, {\left (b \cos \left (8 \, b x + 8 \, a\right )^{2} + 4 \, b \cos \left (6 \, b x + 6 \, a\right )^{2} + 4 \, b \cos \left (2 \, b x + 2 \, a\right )^{2} + b \sin \left (8 \, b x + 8 \, a\right )^{2} + 4 \, b \sin \left (6 \, b x + 6 \, a\right )^{2} - 8 \, b \sin \left (6 \, b x + 6 \, a\right ) \sin \left (2 \, b x + 2 \, a\right ) + 4 \, b \sin \left (2 \, b x + 2 \, a\right )^{2} - 2 \, {\left (2 \, b \cos \left (6 \, b x + 6 \, a\right ) - 2 \, b \cos \left (2 \, b x + 2 \, a\right ) + b\right )} \cos \left (8 \, b x + 8 \, a\right ) - 4 \, {\left (2 \, b \cos \left (2 \, b x + 2 \, a\right ) - b\right )} \cos \left (6 \, b x + 6 \, a\right ) - 4 \, b \cos \left (2 \, b x + 2 \, a\right ) - 4 \, {\left (b \sin \left (6 \, b x + 6 \, a\right ) - b \sin \left (2 \, b x + 2 \, a\right )\right )} \sin \left (8 \, b x + 8 \, a\right ) + b\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.18, size = 37, normalized size = 0.88 \[ \frac {\mathrm {tan}\left (a+b\,x\right )}{16\,b}-\frac {\frac {{\mathrm {tan}\left (a+b\,x\right )}^2}{8}+\frac {1}{48}}{b\,{\mathrm {tan}\left (a+b\,x\right )}^3} \]
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]
________________________________________________________________________________________