Optimal. Leaf size=120 \[ \frac {\tan (c+d x)}{5 d \left (a^4 \sec (c+d x)+a^4\right )}-\frac {8 \tan (c+d x)}{35 d \left (a^2 \sec (c+d x)+a^2\right )^2}+\frac {\tan (c+d x) \sec ^3(c+d x)}{7 d (a \sec (c+d x)+a)^4}+\frac {3 \tan (c+d x)}{35 a d (a \sec (c+d x)+a)^3} \]
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Rubi [A] time = 0.17, antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3810, 3799, 4000, 3794} \[ \frac {\tan (c+d x)}{5 d \left (a^4 \sec (c+d x)+a^4\right )}-\frac {8 \tan (c+d x)}{35 d \left (a^2 \sec (c+d x)+a^2\right )^2}+\frac {\tan (c+d x) \sec ^3(c+d x)}{7 d (a \sec (c+d x)+a)^4}+\frac {3 \tan (c+d x)}{35 a d (a \sec (c+d x)+a)^3} \]
Antiderivative was successfully verified.
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Rule 3794
Rule 3799
Rule 3810
Rule 4000
Rubi steps
\begin {align*} \int \frac {\sec ^4(c+d x)}{(a+a \sec (c+d x))^4} \, dx &=\frac {\sec ^3(c+d x) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}+\frac {3 \int \frac {\sec ^3(c+d x)}{(a+a \sec (c+d x))^3} \, dx}{7 a}\\ &=\frac {\sec ^3(c+d x) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}+\frac {3 \tan (c+d x)}{35 a d (a+a \sec (c+d x))^3}+\frac {3 \int \frac {\sec (c+d x) (-3 a+5 a \sec (c+d x))}{(a+a \sec (c+d x))^2} \, dx}{35 a^3}\\ &=\frac {\sec ^3(c+d x) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}+\frac {3 \tan (c+d x)}{35 a d (a+a \sec (c+d x))^3}-\frac {8 \tan (c+d x)}{35 d \left (a^2+a^2 \sec (c+d x)\right )^2}+\frac {\int \frac {\sec (c+d x)}{a+a \sec (c+d x)} \, dx}{5 a^3}\\ &=\frac {\sec ^3(c+d x) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}+\frac {3 \tan (c+d x)}{35 a d (a+a \sec (c+d x))^3}-\frac {8 \tan (c+d x)}{35 d \left (a^2+a^2 \sec (c+d x)\right )^2}+\frac {\tan (c+d x)}{5 d \left (a^4+a^4 \sec (c+d x)\right )}\\ \end {align*}
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Mathematica [A] time = 0.20, size = 69, normalized size = 0.58 \[ \frac {\left (35 \sin \left (\frac {1}{2} (c+d x)\right )+21 \sin \left (\frac {3}{2} (c+d x)\right )+7 \sin \left (\frac {5}{2} (c+d x)\right )+\sin \left (\frac {7}{2} (c+d x)\right )\right ) \sec ^7\left (\frac {1}{2} (c+d x)\right )}{1120 a^4 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.43, size = 99, normalized size = 0.82 \[ \frac {{\left (2 \, \cos \left (d x + c\right )^{3} + 8 \, \cos \left (d x + c\right )^{2} + 13 \, \cos \left (d x + c\right ) + 12\right )} \sin \left (d x + c\right )}{35 \, {\left (a^{4} d \cos \left (d x + c\right )^{4} + 4 \, a^{4} d \cos \left (d x + c\right )^{3} + 6 \, a^{4} d \cos \left (d x + c\right )^{2} + 4 \, a^{4} d \cos \left (d x + c\right ) + a^{4} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 2.87, size = 59, normalized size = 0.49 \[ \frac {5 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 21 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 35 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 35 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{280 \, a^{4} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.41, size = 56, normalized size = 0.47 \[ \frac {\frac {\left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}+\frac {3 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}+\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d \,a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.02, size = 87, normalized size = 0.72 \[ \frac {\frac {35 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {35 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {21 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {5 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{280 \, a^{4} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.67, size = 58, normalized size = 0.48 \[ \frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+21\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+35\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+35\right )}{280\,a^4\,d} \]
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 \[ \frac {\int \frac {\sec ^{4}{\left (c + d x \right )}}{\sec ^{4}{\left (c + d x \right )} + 4 \sec ^{3}{\left (c + d x \right )} + 6 \sec ^{2}{\left (c + d x \right )} + 4 \sec {\left (c + d x \right )} + 1}\, dx}{a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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