Optimal. Leaf size=19 \[ -\frac {\tan (e+f x) \sec ^5(e+f x)}{f} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 19, 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 = {4043} \[ -\frac {\tan (e+f x) \sec ^5(e+f x)}{f} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4043
Rubi steps
\begin {align*} \int \sec ^5(e+f x) \left (5-6 \sec ^2(e+f x)\right ) \, dx &=-\frac {\sec ^5(e+f x) \tan (e+f x)}{f}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.07, size = 19, normalized size = 1.00 \[ -\frac {\tan (e+f x) \sec ^5(e+f x)}{f} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.41, size = 19, normalized size = 1.00 \[ -\frac {\sin \left (f x + e\right )}{f \cos \left (f x + e\right )^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.28, size = 24, normalized size = 1.26 \[ \frac {\sin \left (f x + e\right )}{{\left (\sin \left (f x + e\right )^{2} - 1\right )}^{3} f} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 1.56, size = 70, normalized size = 3.68 \[ \frac {-5 \left (-\frac {\left (\sec ^{3}\left (f x +e \right )\right )}{4}-\frac {3 \sec \left (f x +e \right )}{8}\right ) \tan \left (f x +e \right )+6 \left (-\frac {\left (\sec ^{5}\left (f x +e \right )\right )}{6}-\frac {5 \left (\sec ^{3}\left (f x +e \right )\right )}{24}-\frac {5 \sec \left (f x +e \right )}{16}\right ) \tan \left (f x +e \right )}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.63, size = 42, normalized size = 2.21 \[ \frac {\sin \left (f x + e\right )}{{\left (\sin \left (f x + e\right )^{6} - 3 \, \sin \left (f x + e\right )^{4} + 3 \, \sin \left (f x + e\right )^{2} - 1\right )} f} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 2.46, size = 42, normalized size = 2.21 \[ \frac {\sin \left (e+f\,x\right )}{f\,\left ({\sin \left (e+f\,x\right )}^6-3\,{\sin \left (e+f\,x\right )}^4+3\,{\sin \left (e+f\,x\right )}^2-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ - \int \left (- 5 \sec ^{5}{\left (e + f x \right )}\right )\, dx - \int 6 \sec ^{7}{\left (e + f x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________