Optimal. Leaf size=18 \[ \frac {\cot (e+f x) \csc ^3(e+f x)}{f} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {3011} \[ \frac {\cot (e+f x) \csc ^3(e+f x)}{f} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3011
Rubi steps
\begin {align*} \int \csc ^5(e+f x) \left (-4+3 \sin ^2(e+f x)\right ) \, dx &=\frac {\cot (e+f x) \csc ^3(e+f x)}{f}\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] time = 0.03, size = 39, normalized size = 2.17 \[ \frac {\csc ^4\left (\frac {1}{2} (e+f x)\right )}{16 f}-\frac {\sec ^4\left (\frac {1}{2} (e+f x)\right )}{16 f} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.40, size = 32, normalized size = 1.78 \[ \frac {\cos \left (f x + e\right )}{f \cos \left (f x + e\right )^{4} - 2 \, f \cos \left (f x + e\right )^{2} + f} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.18, size = 98, normalized size = 5.44 \[ -\frac {\frac {{\left (\frac {2 \, {\left (\cos \left (f x + e\right ) - 1\right )}}{\cos \left (f x + e\right ) + 1} - 1\right )} {\left (\cos \left (f x + e\right ) + 1\right )}^{2}}{{\left (\cos \left (f x + e\right ) - 1\right )}^{2}} - \frac {2 \, {\left (\cos \left (f x + e\right ) - 1\right )}}{\cos \left (f x + e\right ) + 1} + \frac {{\left (\cos \left (f x + e\right ) - 1\right )}^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}}}{16 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.58, size = 47, normalized size = 2.61 \[ \frac {-4 \left (-\frac {\left (\csc ^{3}\left (f x +e \right )\right )}{4}-\frac {3 \csc \left (f x +e \right )}{8}\right ) \cot \left (f x +e \right )-\frac {3 \csc \left (f x +e \right ) \cot \left (f x +e \right )}{2}}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.51, size = 32, normalized size = 1.78 \[ \frac {\cos \left (f x + e\right )}{{\left (\cos \left (f x + e\right )^{4} - 2 \, \cos \left (f x + e\right )^{2} + 1\right )} f} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 13.23, size = 22, normalized size = 1.22 \[ \frac {\cos \left (e+f\,x\right )}{f\,{\left ({\cos \left (e+f\,x\right )}^2-1\right )}^2} \]
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]
________________________________________________________________________________________