Optimal. Leaf size=83 \[ \frac {7 \tan (c+d x)}{15 d \left (a^3 \sec (c+d x)+a^3\right )}-\frac {8 \tan (c+d x)}{15 a d (a \sec (c+d x)+a)^2}+\frac {\tan (c+d x)}{5 d (a \sec (c+d x)+a)^3} \]
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Rubi [A] time = 0.12, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3799, 4000, 3794} \[ \frac {7 \tan (c+d x)}{15 d \left (a^3 \sec (c+d x)+a^3\right )}-\frac {8 \tan (c+d x)}{15 a d (a \sec (c+d x)+a)^2}+\frac {\tan (c+d x)}{5 d (a \sec (c+d x)+a)^3} \]
Antiderivative was successfully verified.
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Rule 3794
Rule 3799
Rule 4000
Rubi steps
\begin {align*} \int \frac {\sec ^3(c+d x)}{(a+a \sec (c+d x))^3} \, dx &=\frac {\tan (c+d x)}{5 d (a+a \sec (c+d x))^3}+\frac {\int \frac {\sec (c+d x) (-3 a+5 a \sec (c+d x))}{(a+a \sec (c+d x))^2} \, dx}{5 a^2}\\ &=\frac {\tan (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac {8 \tan (c+d x)}{15 a d (a+a \sec (c+d x))^2}+\frac {7 \int \frac {\sec (c+d x)}{a+a \sec (c+d x)} \, dx}{15 a^2}\\ &=\frac {\tan (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac {8 \tan (c+d x)}{15 a d (a+a \sec (c+d x))^2}+\frac {7 \tan (c+d x)}{15 d \left (a^3+a^3 \sec (c+d x)\right )}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 57, normalized size = 0.69 \[ \frac {\left (10 \sin \left (\frac {1}{2} (c+d x)\right )+5 \sin \left (\frac {3}{2} (c+d x)\right )+\sin \left (\frac {5}{2} (c+d x)\right )\right ) \sec ^5\left (\frac {1}{2} (c+d x)\right )}{120 a^3 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.58, size = 75, normalized size = 0.90 \[ \frac {{\left (2 \, \cos \left (d x + c\right )^{2} + 6 \, \cos \left (d x + c\right ) + 7\right )} \sin \left (d x + c\right )}{15 \, {\left (a^{3} d \cos \left (d x + c\right )^{3} + 3 \, a^{3} d \cos \left (d x + c\right )^{2} + 3 \, a^{3} d \cos \left (d x + c\right ) + a^{3} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.57, size = 46, normalized size = 0.55 \[ \frac {3 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 10 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 15 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{60 \, a^{3} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.41, size = 45, normalized size = 0.54 \[ \frac {\frac {\left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}+\frac {2 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d \,a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.89, size = 67, normalized size = 0.81 \[ \frac {\frac {15 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {10 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {3 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}}{60 \, a^{3} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.62, size = 45, normalized size = 0.54 \[ \frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+15\right )}{60\,a^3\,d} \]
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 \[ \frac {\int \frac {\sec ^{3}{\left (c + d x \right )}}{\sec ^{3}{\left (c + d x \right )} + 3 \sec ^{2}{\left (c + d x \right )} + 3 \sec {\left (c + d x \right )} + 1}\, dx}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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