Optimal. Leaf size=136 \[ \frac {\tanh ^{-1}(\sin (c+d x))}{a^4 d}-\frac {43 \tan (c+d x)}{21 a^4 d (\sec (c+d x)+1)}+\frac {11 \tan (c+d x)}{21 a^4 d (\sec (c+d x)+1)^2}-\frac {\tan (c+d x) \sec ^3(c+d x)}{7 d (a \sec (c+d x)+a)^4}-\frac {2 \tan (c+d x) \sec ^2(c+d x)}{7 a d (a \sec (c+d x)+a)^3} \]
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Rubi [A] time = 0.32, antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3816, 4019, 4008, 3998, 3770, 3794} \[ \frac {\tanh ^{-1}(\sin (c+d x))}{a^4 d}-\frac {43 \tan (c+d x)}{21 a^4 d (\sec (c+d x)+1)}+\frac {11 \tan (c+d x)}{21 a^4 d (\sec (c+d x)+1)^2}-\frac {\tan (c+d x) \sec ^3(c+d x)}{7 d (a \sec (c+d x)+a)^4}-\frac {2 \tan (c+d x) \sec ^2(c+d x)}{7 a d (a \sec (c+d x)+a)^3} \]
Antiderivative was successfully verified.
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Rule 3770
Rule 3794
Rule 3816
Rule 3998
Rule 4008
Rule 4019
Rubi steps
\begin {align*} \int \frac {\sec ^5(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 {\int \frac {\sec ^3(c+d x) (3 a-7 a \sec (c+d x))}{(a+a \sec (c+d x))^3} \, dx}{7 a^2}\\ &=-\frac {\sec ^3(c+d x) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}-\frac {2 \sec ^2(c+d x) \tan (c+d x)}{7 a d (a+a \sec (c+d x))^3}-\frac {\int \frac {\sec ^2(c+d x) \left (20 a^2-35 a^2 \sec (c+d x)\right )}{(a+a \sec (c+d x))^2} \, dx}{35 a^4}\\ &=\frac {11 \tan (c+d x)}{21 a^4 d (1+\sec (c+d x))^2}-\frac {\sec ^3(c+d x) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}-\frac {2 \sec ^2(c+d x) \tan (c+d x)}{7 a d (a+a \sec (c+d x))^3}+\frac {\int \frac {\sec (c+d x) \left (-110 a^3+105 a^3 \sec (c+d x)\right )}{a+a \sec (c+d x)} \, dx}{105 a^6}\\ &=\frac {11 \tan (c+d x)}{21 a^4 d (1+\sec (c+d x))^2}-\frac {\sec ^3(c+d x) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}-\frac {2 \sec ^2(c+d x) \tan (c+d x)}{7 a d (a+a \sec (c+d x))^3}+\frac {\int \sec (c+d x) \, dx}{a^4}-\frac {43 \int \frac {\sec (c+d x)}{a+a \sec (c+d x)} \, dx}{21 a^3}\\ &=\frac {\tanh ^{-1}(\sin (c+d x))}{a^4 d}+\frac {11 \tan (c+d x)}{21 a^4 d (1+\sec (c+d x))^2}-\frac {\sec ^3(c+d x) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}-\frac {2 \sec ^2(c+d x) \tan (c+d x)}{7 a d (a+a \sec (c+d x))^3}-\frac {43 \tan (c+d x)}{21 d \left (a^4+a^4 \sec (c+d x)\right )}\\ \end {align*}
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Mathematica [A] time = 0.94, size = 193, normalized size = 1.42 \[ -\frac {\cos \left (\frac {1}{2} (c+d x)\right ) \sec ^4(c+d x) \left (\sec \left (\frac {c}{2}\right ) \left (-434 \sin \left (c+\frac {d x}{2}\right )+525 \sin \left (c+\frac {3 d x}{2}\right )-147 \sin \left (2 c+\frac {3 d x}{2}\right )+203 \sin \left (2 c+\frac {5 d x}{2}\right )-21 \sin \left (3 c+\frac {5 d x}{2}\right )+32 \sin \left (3 c+\frac {7 d x}{2}\right )+686 \sin \left (\frac {d x}{2}\right )\right )+1344 \cos ^7\left (\frac {1}{2} (c+d x)\right ) \left (\log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )\right )\right )}{84 a^4 d (\sec (c+d x)+1)^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.59, size = 202, normalized size = 1.49 \[ \frac {21 \, {\left (\cos \left (d x + c\right )^{4} + 4 \, \cos \left (d x + c\right )^{3} + 6 \, \cos \left (d x + c\right )^{2} + 4 \, \cos \left (d x + c\right ) + 1\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 21 \, {\left (\cos \left (d x + c\right )^{4} + 4 \, \cos \left (d x + c\right )^{3} + 6 \, \cos \left (d x + c\right )^{2} + 4 \, \cos \left (d x + c\right ) + 1\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (32 \, \cos \left (d x + c\right )^{3} + 107 \, \cos \left (d x + c\right )^{2} + 124 \, \cos \left (d x + c\right ) + 52\right )} \sin \left (d x + c\right )}{42 \, {\left (a^{4} d \cos \left (d x + c\right )^{4} + 4 \, a^{4} d \cos \left (d x + c\right )^{3} + 6 \, a^{4} d \cos \left (d x + c\right )^{2} + 4 \, a^{4} d \cos \left (d x + c\right ) + a^{4} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 2.87, size = 110, normalized size = 0.81 \[ \frac {\frac {168 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right )}{a^{4}} - \frac {168 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right )}{a^{4}} - \frac {3 \, a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 21 \, a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 77 \, a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 315 \, a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{28}}}{168 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.40, size = 115, normalized size = 0.85 \[ -\frac {\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )}{56 d \,a^{4}}-\frac {\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d \,a^{4}}-\frac {11 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d \,a^{4}}-\frac {15 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d \,a^{4}}-\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d \,a^{4}}+\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d \,a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.78, size = 139, normalized size = 1.02 \[ -\frac {\frac {\frac {315 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {77 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {21 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {3 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a^{4}} - \frac {168 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a^{4}} + \frac {168 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a^{4}}}{168 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.67, size = 83, normalized size = 0.61 \[ -\frac {\frac {11\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24\,a^4}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{8\,a^4}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{56\,a^4}-\frac {2\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{a^4}+\frac {15\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8\,a^4}}{d} \]
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 \[ \frac {\int \frac {\sec ^{5}{\left (c + d x \right )}}{\sec ^{4}{\left (c + d x \right )} + 4 \sec ^{3}{\left (c + d x \right )} + 6 \sec ^{2}{\left (c + d x \right )} + 4 \sec {\left (c + d x \right )} + 1}\, dx}{a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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