Optimal. Leaf size=116 \[ -\frac {(a+a \sec (e+f x)) \tan (e+f x)}{7 f (c-c \sec (e+f x))^4}-\frac {2 (a+a \sec (e+f x)) \tan (e+f x)}{35 c f (c-c \sec (e+f x))^3}-\frac {2 (a+a \sec (e+f x)) \tan (e+f x)}{105 f \left (c^2-c^2 \sec (e+f x)\right )^2} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.11, antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {4036, 4035}
\begin {gather*} -\frac {2 \tan (e+f x) (a \sec (e+f x)+a)}{105 f \left (c^2-c^2 \sec (e+f x)\right )^2}-\frac {2 \tan (e+f x) (a \sec (e+f x)+a)}{35 c f (c-c \sec (e+f x))^3}-\frac {\tan (e+f x) (a \sec (e+f x)+a)}{7 f (c-c \sec (e+f x))^4} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 4035
Rule 4036
Rubi steps
\begin {align*} \int \frac {\sec (e+f x) (a+a \sec (e+f x))}{(c-c \sec (e+f x))^4} \, dx &=-\frac {(a+a \sec (e+f x)) \tan (e+f x)}{7 f (c-c \sec (e+f x))^4}+\frac {2 \int \frac {\sec (e+f x) (a+a \sec (e+f x))}{(c-c \sec (e+f x))^3} \, dx}{7 c}\\ &=-\frac {(a+a \sec (e+f x)) \tan (e+f x)}{7 f (c-c \sec (e+f x))^4}-\frac {2 (a+a \sec (e+f x)) \tan (e+f x)}{35 c f (c-c \sec (e+f x))^3}+\frac {2 \int \frac {\sec (e+f x) (a+a \sec (e+f x))}{(c-c \sec (e+f x))^2} \, dx}{35 c^2}\\ &=-\frac {(a+a \sec (e+f x)) \tan (e+f x)}{7 f (c-c \sec (e+f x))^4}-\frac {2 (a+a \sec (e+f x)) \tan (e+f x)}{35 c f (c-c \sec (e+f x))^3}-\frac {2 (a+a \sec (e+f x)) \tan (e+f x)}{105 f \left (c^2-c^2 \sec (e+f x)\right )^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.41, size = 113, normalized size = 0.97 \begin {gather*} -\frac {a \csc \left (\frac {e}{2}\right ) \csc ^7\left (\frac {1}{2} (e+f x)\right ) \left (350 \sin \left (\frac {f x}{2}\right )+455 \sin \left (e+\frac {f x}{2}\right )-273 \sin \left (e+\frac {3 f x}{2}\right )-210 \sin \left (2 e+\frac {3 f x}{2}\right )+56 \sin \left (2 e+\frac {5 f x}{2}\right )+105 \sin \left (3 e+\frac {5 f x}{2}\right )-23 \sin \left (3 e+\frac {7 f x}{2}\right )\right )}{6720 c^4 f} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.17, size = 50, normalized size = 0.43
method | result | size |
derivativedivides | \(\frac {a \left (-\frac {1}{7 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}}-\frac {1}{3 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}+\frac {2}{5 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}\right )}{4 f \,c^{4}}\) | \(50\) |
default | \(\frac {a \left (-\frac {1}{7 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}}-\frac {1}{3 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}+\frac {2}{5 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}\right )}{4 f \,c^{4}}\) | \(50\) |
risch | \(\frac {2 i a \left (105 \,{\mathrm e}^{6 i \left (f x +e \right )}-210 \,{\mathrm e}^{5 i \left (f x +e \right )}+455 \,{\mathrm e}^{4 i \left (f x +e \right )}-350 \,{\mathrm e}^{3 i \left (f x +e \right )}+273 \,{\mathrm e}^{2 i \left (f x +e \right )}-56 \,{\mathrm e}^{i \left (f x +e \right )}+23\right )}{105 f \,c^{4} \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )^{7}}\) | \(92\) |
norman | \(\frac {\frac {a}{28 c f}-\frac {19 a \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{140 c f}+\frac {11 a \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{60 c f}-\frac {a \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{12 c f}}{\left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )-1\right ) c^{3} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}}\) | \(101\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.28, size = 193, normalized size = 1.66 \begin {gather*} \frac {\frac {a {\left (\frac {21 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {35 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} - \frac {105 \, \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} - 15\right )} {\left (\cos \left (f x + e\right ) + 1\right )}^{7}}{c^{4} \sin \left (f x + e\right )^{7}} + \frac {3 \, a {\left (\frac {21 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {35 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {35 \, \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} - 5\right )} {\left (\cos \left (f x + e\right ) + 1\right )}^{7}}{c^{4} \sin \left (f x + e\right )^{7}}}{840 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 4.04, size = 112, normalized size = 0.97 \begin {gather*} \frac {23 \, a \cos \left (f x + e\right )^{4} + 36 \, a \cos \left (f x + e\right )^{3} + 5 \, a \cos \left (f x + e\right )^{2} - 6 \, a \cos \left (f x + e\right ) + 2 \, a}{105 \, {\left (c^{4} f \cos \left (f x + e\right )^{3} - 3 \, c^{4} f \cos \left (f x + e\right )^{2} + 3 \, c^{4} f \cos \left (f x + e\right ) - c^{4} f\right )} \sin \left (f x + e\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {a \left (\int \frac {\sec {\left (e + f x \right )}}{\sec ^{4}{\left (e + f x \right )} - 4 \sec ^{3}{\left (e + f x \right )} + 6 \sec ^{2}{\left (e + f x \right )} - 4 \sec {\left (e + f x \right )} + 1}\, dx + \int \frac {\sec ^{2}{\left (e + f x \right )}}{\sec ^{4}{\left (e + f x \right )} - 4 \sec ^{3}{\left (e + f x \right )} + 6 \sec ^{2}{\left (e + f x \right )} - 4 \sec {\left (e + f x \right )} + 1}\, dx\right )}{c^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.71, size = 51, normalized size = 0.44 \begin {gather*} -\frac {35 \, a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 42 \, a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 15 \, a}{420 \, c^{4} f \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 1.75, size = 61, normalized size = 0.53 \begin {gather*} \frac {a\,{\mathrm {cot}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5}{10\,c^4\,f}-\frac {a\,{\mathrm {cot}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3}{12\,c^4\,f}-\frac {a\,{\mathrm {cot}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7}{28\,c^4\,f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________