Optimal. Leaf size=120 \[ \frac {\sec ^3(c+d x) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}+\frac {3 \tan (c+d x)}{35 a d (a+a \sec (c+d x))^3}-\frac {8 \tan (c+d x)}{35 d \left (a^2+a^2 \sec (c+d x)\right )^2}+\frac {\tan (c+d x)}{5 d \left (a^4+a^4 \sec (c+d x)\right )} \]
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Rubi [A]
time = 0.11, 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 = {3895, 3884,
4085, 3879} \begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 3879
Rule 3884
Rule 3895
Rule 4085
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 = 0.21, size = 69, normalized size = 0.58 \begin {gather*} \frac {\sec ^7\left (\frac {1}{2} (c+d x)\right ) \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 )}{1120 a^4 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 56, normalized size = 0.47
method | result | size |
derivativedivides | \(\frac {\frac {\left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}+\frac {3 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}+\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d \,a^{4}}\) | \(56\) |
default | \(\frac {\frac {\left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}+\frac {3 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}+\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d \,a^{4}}\) | \(56\) |
risch | \(\frac {4 i \left (35 \,{\mathrm e}^{3 i \left (d x +c \right )}+21 \,{\mathrm e}^{2 i \left (d x +c \right )}+7 \,{\mathrm e}^{i \left (d x +c \right )}+1\right )}{35 d \,a^{4} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{7}}\) | \(58\) |
norman | \(\frac {-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a d}+\frac {\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )}{4 a d}-\frac {3 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{40 a d}-\frac {3 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{70 a d}-\frac {13 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{280 a d}+\frac {3 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{140 a d}+\frac {\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )}{56 a d}}{\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3} a^{3}}\) | \(152\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 87, normalized size = 0.72 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 9.54, size = 99, normalized size = 0.82 \begin {gather*} \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 )}} \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 {\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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.49, size = 59, normalized size = 0.49 \begin {gather*} \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} \end {gather*}
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 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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