Optimal. Leaf size=76 \[ -\frac {(a+a \sec (e+f x)) \tan (e+f x)}{5 f (c-c \sec (e+f x))^3}-\frac {(a+a \sec (e+f x)) \tan (e+f x)}{15 c f (c-c \sec (e+f x))^2} \]
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Rubi [A]
time = 0.07, antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps
used = 2, 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 {\tan (e+f x) (a \sec (e+f x)+a)}{15 c f (c-c \sec (e+f x))^2}-\frac {\tan (e+f x) (a \sec (e+f x)+a)}{5 f (c-c \sec (e+f x))^3} \end {gather*}
Antiderivative was successfully verified.
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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))^3} \, dx &=-\frac {(a+a \sec (e+f x)) \tan (e+f x)}{5 f (c-c \sec (e+f x))^3}+\frac {\int \frac {\sec (e+f x) (a+a \sec (e+f x))}{(c-c \sec (e+f x))^2} \, dx}{5 c}\\ &=-\frac {(a+a \sec (e+f x)) \tan (e+f x)}{5 f (c-c \sec (e+f x))^3}-\frac {(a+a \sec (e+f x)) \tan (e+f x)}{15 c f (c-c \sec (e+f x))^2}\\ \end {align*}
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Mathematica [A]
time = 0.34, size = 87, normalized size = 1.14 \begin {gather*} -\frac {a \csc \left (\frac {e}{2}\right ) \csc ^5\left (\frac {1}{2} (e+f x)\right ) \left (25 \sin \left (\frac {f x}{2}\right )+15 \sin \left (e+\frac {f x}{2}\right )-5 \sin \left (e+\frac {3 f x}{2}\right )-15 \sin \left (2 e+\frac {3 f x}{2}\right )+4 \sin \left (2 e+\frac {5 f x}{2}\right )\right )}{240 c^3 f} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.15, size = 37, normalized size = 0.49
method | result | size |
derivativedivides | \(\frac {a \left (\frac {1}{5 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}-\frac {1}{3 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}\right )}{2 f \,c^{3}}\) | \(37\) |
default | \(\frac {a \left (\frac {1}{5 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}-\frac {1}{3 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}\right )}{2 f \,c^{3}}\) | \(37\) |
risch | \(\frac {2 i a \left (15 \,{\mathrm e}^{4 i \left (f x +e \right )}-15 \,{\mathrm e}^{3 i \left (f x +e \right )}+25 \,{\mathrm e}^{2 i \left (f x +e \right )}-5 \,{\mathrm e}^{i \left (f x +e \right )}+4\right )}{15 f \,c^{3} \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )^{5}}\) | \(70\) |
norman | \(\frac {-\frac {a}{10 c f}+\frac {4 a \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{15 c f}-\frac {a \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{6 c f}}{\left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )-1\right ) c^{2} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}\) | \(81\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 127, normalized size = 1.67 \begin {gather*} -\frac {\frac {a {\left (\frac {10 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {15 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} - 3\right )} {\left (\cos \left (f x + e\right ) + 1\right )}^{5}}{c^{3} \sin \left (f x + e\right )^{5}} + \frac {3 \, a {\left (\frac {5 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} - 1\right )} {\left (\cos \left (f x + e\right ) + 1\right )}^{5}}{c^{3} \sin \left (f x + e\right )^{5}}}{60 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.27, size = 84, normalized size = 1.11 \begin {gather*} \frac {4 \, a \cos \left (f x + e\right )^{3} + 7 \, a \cos \left (f x + e\right )^{2} + 2 \, a \cos \left (f x + e\right ) - a}{15 \, {\left (c^{3} f \cos \left (f x + e\right )^{2} - 2 \, c^{3} f \cos \left (f x + e\right ) + c^{3} f\right )} \sin \left (f x + e\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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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 ^{3}{\left (e + f x \right )} - 3 \sec ^{2}{\left (e + f x \right )} + 3 \sec {\left (e + f x \right )} - 1}\, dx + \int \frac {\sec ^{2}{\left (e + f x \right )}}{\sec ^{3}{\left (e + f x \right )} - 3 \sec ^{2}{\left (e + f x \right )} + 3 \sec {\left (e + f x \right )} - 1}\, dx\right )}{c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.49, size = 37, normalized size = 0.49 \begin {gather*} -\frac {5 \, a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 3 \, a}{30 \, c^{3} f \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.71, size = 35, normalized size = 0.46 \begin {gather*} \frac {a\,{\mathrm {cot}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left (3\,{\mathrm {cot}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2-5\right )}{30\,c^3\,f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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