Optimal. Leaf size=87 \[ \frac {(7 A+6 C) \tan (c+d x)}{7 d}+\frac {C \sec ^6(c+d x) \tan (c+d x)}{7 d}+\frac {2 (7 A+6 C) \tan ^3(c+d x)}{21 d}+\frac {(7 A+6 C) \tan ^5(c+d x)}{35 d} \]
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Rubi [A]
time = 0.04, antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {4131, 3852}
\begin {gather*} \frac {(7 A+6 C) \tan ^5(c+d x)}{35 d}+\frac {2 (7 A+6 C) \tan ^3(c+d x)}{21 d}+\frac {(7 A+6 C) \tan (c+d x)}{7 d}+\frac {C \tan (c+d x) \sec ^6(c+d x)}{7 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 3852
Rule 4131
Rubi steps
\begin {align*} \int \sec ^6(c+d x) \left (A+C \sec ^2(c+d x)\right ) \, dx &=\frac {C \sec ^6(c+d x) \tan (c+d x)}{7 d}+\frac {1}{7} (7 A+6 C) \int \sec ^6(c+d x) \, dx\\ &=\frac {C \sec ^6(c+d x) \tan (c+d x)}{7 d}-\frac {(7 A+6 C) \text {Subst}\left (\int \left (1+2 x^2+x^4\right ) \, dx,x,-\tan (c+d x)\right )}{7 d}\\ &=\frac {(7 A+6 C) \tan (c+d x)}{7 d}+\frac {C \sec ^6(c+d x) \tan (c+d x)}{7 d}+\frac {2 (7 A+6 C) \tan ^3(c+d x)}{21 d}+\frac {(7 A+6 C) \tan ^5(c+d x)}{35 d}\\ \end {align*}
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Mathematica [A]
time = 0.21, size = 81, normalized size = 0.93 \begin {gather*} \frac {A \left (\tan (c+d x)+\frac {2}{3} \tan ^3(c+d x)+\frac {1}{5} \tan ^5(c+d x)\right )}{d}+\frac {C \left (\tan (c+d x)+\tan ^3(c+d x)+\frac {3}{5} \tan ^5(c+d x)+\frac {1}{7} \tan ^7(c+d x)\right )}{d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.39, size = 78, normalized size = 0.90
method | result | size |
derivativedivides | \(\frac {-A \left (-\frac {8}{15}-\frac {\left (\sec ^{4}\left (d x +c \right )\right )}{5}-\frac {4 \left (\sec ^{2}\left (d x +c \right )\right )}{15}\right ) \tan \left (d x +c \right )-C \left (-\frac {16}{35}-\frac {\left (\sec ^{6}\left (d x +c \right )\right )}{7}-\frac {6 \left (\sec ^{4}\left (d x +c \right )\right )}{35}-\frac {8 \left (\sec ^{2}\left (d x +c \right )\right )}{35}\right ) \tan \left (d x +c \right )}{d}\) | \(78\) |
default | \(\frac {-A \left (-\frac {8}{15}-\frac {\left (\sec ^{4}\left (d x +c \right )\right )}{5}-\frac {4 \left (\sec ^{2}\left (d x +c \right )\right )}{15}\right ) \tan \left (d x +c \right )-C \left (-\frac {16}{35}-\frac {\left (\sec ^{6}\left (d x +c \right )\right )}{7}-\frac {6 \left (\sec ^{4}\left (d x +c \right )\right )}{35}-\frac {8 \left (\sec ^{2}\left (d x +c \right )\right )}{35}\right ) \tan \left (d x +c \right )}{d}\) | \(78\) |
risch | \(\frac {16 i \left (70 A \,{\mathrm e}^{8 i \left (d x +c \right )}+175 A \,{\mathrm e}^{6 i \left (d x +c \right )}+210 C \,{\mathrm e}^{6 i \left (d x +c \right )}+147 A \,{\mathrm e}^{4 i \left (d x +c \right )}+126 C \,{\mathrm e}^{4 i \left (d x +c \right )}+49 A \,{\mathrm e}^{2 i \left (d x +c \right )}+42 C \,{\mathrm e}^{2 i \left (d x +c \right )}+7 A +6 C \right )}{105 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{7}}\) | \(111\) |
norman | \(\frac {-\frac {2 \left (A +C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}-\frac {2 \left (A +C \right ) \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {4 \left (5 A +3 C \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}+\frac {4 \left (5 A +3 C \right ) \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}+\frac {8 \left (91 A +53 C \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{35 d}-\frac {2 \left (113 A +129 C \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15 d}-\frac {2 \left (113 A +129 C \right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15 d}}{\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{7}}\) | \(169\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 60, normalized size = 0.69 \begin {gather*} \frac {15 \, C \tan \left (d x + c\right )^{7} + 21 \, {\left (A + 3 \, C\right )} \tan \left (d x + c\right )^{5} + 35 \, {\left (2 \, A + 3 \, C\right )} \tan \left (d x + c\right )^{3} + 105 \, {\left (A + C\right )} \tan \left (d x + c\right )}{105 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.79, size = 74, normalized size = 0.85 \begin {gather*} \frac {{\left (8 \, {\left (7 \, A + 6 \, C\right )} \cos \left (d x + c\right )^{6} + 4 \, {\left (7 \, A + 6 \, C\right )} \cos \left (d x + c\right )^{4} + 3 \, {\left (7 \, A + 6 \, C\right )} \cos \left (d x + c\right )^{2} + 15 \, C\right )} \sin \left (d x + c\right )}{105 \, d \cos \left (d x + c\right )^{7}} \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*} \int \left (A + C \sec ^{2}{\left (c + d x \right )}\right ) \sec ^{6}{\left (c + d x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.47, size = 79, normalized size = 0.91 \begin {gather*} \frac {15 \, C \tan \left (d x + c\right )^{7} + 21 \, A \tan \left (d x + c\right )^{5} + 63 \, C \tan \left (d x + c\right )^{5} + 70 \, A \tan \left (d x + c\right )^{3} + 105 \, C \tan \left (d x + c\right )^{3} + 105 \, A \tan \left (d x + c\right ) + 105 \, C \tan \left (d x + c\right )}{105 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.44, size = 56, normalized size = 0.64 \begin {gather*} \frac {\frac {C\,{\mathrm {tan}\left (c+d\,x\right )}^7}{7}+\left (\frac {A}{5}+\frac {3\,C}{5}\right )\,{\mathrm {tan}\left (c+d\,x\right )}^5+\left (\frac {2\,A}{3}+C\right )\,{\mathrm {tan}\left (c+d\,x\right )}^3+\left (A+C\right )\,\mathrm {tan}\left (c+d\,x\right )}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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