Optimal. Leaf size=102 \[ \frac {(c-d) \tan (e+f x)}{5 f (a+a \sec (e+f x))^3}+\frac {(2 c+3 d) \tan (e+f x)}{15 a f (a+a \sec (e+f x))^2}+\frac {(2 c+3 d) \tan (e+f x)}{15 f \left (a^3+a^3 \sec (e+f x)\right )} \]
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Rubi [A]
time = 0.08, antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {4085, 3881,
3879} \begin {gather*} \frac {(2 c+3 d) \tan (e+f x)}{15 f \left (a^3 \sec (e+f x)+a^3\right )}+\frac {(2 c+3 d) \tan (e+f x)}{15 a f (a \sec (e+f x)+a)^2}+\frac {(c-d) \tan (e+f x)}{5 f (a \sec (e+f x)+a)^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 3879
Rule 3881
Rule 4085
Rubi steps
\begin {align*} \int \frac {\sec (e+f x) (c+d \sec (e+f x))}{(a+a \sec (e+f x))^3} \, dx &=\frac {(c-d) \tan (e+f x)}{5 f (a+a \sec (e+f x))^3}+\frac {(2 c+3 d) \int \frac {\sec (e+f x)}{(a+a \sec (e+f x))^2} \, dx}{5 a}\\ &=\frac {(c-d) \tan (e+f x)}{5 f (a+a \sec (e+f x))^3}+\frac {(2 c+3 d) \tan (e+f x)}{15 a f (a+a \sec (e+f x))^2}+\frac {(2 c+3 d) \int \frac {\sec (e+f x)}{a+a \sec (e+f x)} \, dx}{15 a^2}\\ &=\frac {(c-d) \tan (e+f x)}{5 f (a+a \sec (e+f x))^3}+\frac {(2 c+3 d) \tan (e+f x)}{15 a f (a+a \sec (e+f x))^2}+\frac {(2 c+3 d) \tan (e+f x)}{15 f \left (a^3+a^3 \sec (e+f x)\right )}\\ \end {align*}
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Mathematica [A]
time = 0.35, size = 135, normalized size = 1.32 \begin {gather*} \frac {\cos \left (\frac {1}{2} (e+f x)\right ) \sec \left (\frac {e}{2}\right ) \left (5 (8 c+3 d) \sin \left (\frac {f x}{2}\right )-15 (2 c+d) \sin \left (e+\frac {f x}{2}\right )+20 c \sin \left (e+\frac {3 f x}{2}\right )+15 d \sin \left (e+\frac {3 f x}{2}\right )-15 c \sin \left (2 e+\frac {3 f x}{2}\right )+7 c \sin \left (2 e+\frac {5 f x}{2}\right )+3 d \sin \left (2 e+\frac {5 f x}{2}\right )\right )}{30 a^3 f (1+\cos (e+f x))^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.22, size = 64, normalized size = 0.63
method | result | size |
derivativedivides | \(\frac {\frac {\left (c -d \right ) \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{5}-\frac {2 c \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3}+c \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+d \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{4 f \,a^{3}}\) | \(64\) |
default | \(\frac {\frac {\left (c -d \right ) \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{5}-\frac {2 c \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3}+c \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+d \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{4 f \,a^{3}}\) | \(64\) |
risch | \(\frac {2 i \left (15 c \,{\mathrm e}^{4 i \left (f x +e \right )}+30 c \,{\mathrm e}^{3 i \left (f x +e \right )}+15 d \,{\mathrm e}^{3 i \left (f x +e \right )}+40 \,{\mathrm e}^{2 i \left (f x +e \right )} c +15 d \,{\mathrm e}^{2 i \left (f x +e \right )}+20 \,{\mathrm e}^{i \left (f x +e \right )} c +15 d \,{\mathrm e}^{i \left (f x +e \right )}+7 c +3 d \right )}{15 f \,a^{3} \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )^{5}}\) | \(114\) |
norman | \(\frac {\frac {\left (c -d \right ) \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{20 a f}-\frac {\left (c +d \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{4 a f}+\frac {\left (5 c +3 d \right ) \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{12 f a}-\frac {\left (13 c -3 d \right ) \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{60 f a}}{\left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )-1\right ) a^{2}}\) | \(117\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 125, normalized size = 1.23 \begin {gather*} \frac {\frac {c {\left (\frac {15 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {10 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {3 \, \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}}\right )}}{a^{3}} + \frac {3 \, d {\left (\frac {5 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {\sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}}\right )}}{a^{3}}}{60 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.15, size = 99, normalized size = 0.97 \begin {gather*} \frac {{\left ({\left (7 \, c + 3 \, d\right )} \cos \left (f x + e\right )^{2} + 3 \, {\left (2 \, c + 3 \, d\right )} \cos \left (f x + e\right ) + 2 \, c + 3 \, d\right )} \sin \left (f x + e\right )}{15 \, {\left (a^{3} f \cos \left (f x + e\right )^{3} + 3 \, a^{3} f \cos \left (f x + e\right )^{2} + 3 \, a^{3} f \cos \left (f x + e\right ) + a^{3} f\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 {\int \frac {c \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 {d \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}{a^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.48, size = 75, normalized size = 0.74 \begin {gather*} \frac {3 \, c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 3 \, d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 10 \, c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 15 \, c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 15 \, d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )}{60 \, a^{3} f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.74, size = 66, normalized size = 0.65 \begin {gather*} \frac {\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (15\,c+15\,d-10\,c\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+3\,c\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4-3\,d\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\right )}{60\,a^3\,f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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