Optimal. Leaf size=98 \[ \frac {(6 A+5 C) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac {(6 A+5 C) \tan (c+d x) \sec ^3(c+d x)}{24 d}+\frac {(6 A+5 C) \tan (c+d x) \sec (c+d x)}{16 d}+\frac {C \tan (c+d x) \sec ^5(c+d x)}{6 d} \]
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Rubi [A] time = 0.06, antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {4046, 3768, 3770} \[ \frac {(6 A+5 C) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac {(6 A+5 C) \tan (c+d x) \sec ^3(c+d x)}{24 d}+\frac {(6 A+5 C) \tan (c+d x) \sec (c+d x)}{16 d}+\frac {C \tan (c+d x) \sec ^5(c+d x)}{6 d} \]
Antiderivative was successfully verified.
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Rule 3768
Rule 3770
Rule 4046
Rubi steps
\begin {align*} \int \sec ^5(c+d x) \left (A+C \sec ^2(c+d x)\right ) \, dx &=\frac {C \sec ^5(c+d x) \tan (c+d x)}{6 d}+\frac {1}{6} (6 A+5 C) \int \sec ^5(c+d x) \, dx\\ &=\frac {(6 A+5 C) \sec ^3(c+d x) \tan (c+d x)}{24 d}+\frac {C \sec ^5(c+d x) \tan (c+d x)}{6 d}+\frac {1}{8} (6 A+5 C) \int \sec ^3(c+d x) \, dx\\ &=\frac {(6 A+5 C) \sec (c+d x) \tan (c+d x)}{16 d}+\frac {(6 A+5 C) \sec ^3(c+d x) \tan (c+d x)}{24 d}+\frac {C \sec ^5(c+d x) \tan (c+d x)}{6 d}+\frac {1}{16} (6 A+5 C) \int \sec (c+d x) \, dx\\ &=\frac {(6 A+5 C) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac {(6 A+5 C) \sec (c+d x) \tan (c+d x)}{16 d}+\frac {(6 A+5 C) \sec ^3(c+d x) \tan (c+d x)}{24 d}+\frac {C \sec ^5(c+d x) \tan (c+d x)}{6 d}\\ \end {align*}
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Mathematica [A] time = 0.39, size = 75, normalized size = 0.77 \[ \frac {3 (6 A+5 C) \tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \sec (c+d x) \left (2 (6 A+5 C) \sec ^2(c+d x)+3 (6 A+5 C)+8 C \sec ^4(c+d x)\right )}{48 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.90, size = 114, normalized size = 1.16 \[ \frac {3 \, {\left (6 \, A + 5 \, C\right )} \cos \left (d x + c\right )^{6} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, {\left (6 \, A + 5 \, C\right )} \cos \left (d x + c\right )^{6} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (3 \, {\left (6 \, A + 5 \, C\right )} \cos \left (d x + c\right )^{4} + 2 \, {\left (6 \, A + 5 \, C\right )} \cos \left (d x + c\right )^{2} + 8 \, C\right )} \sin \left (d x + c\right )}{96 \, d \cos \left (d x + c\right )^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.78, size = 121, normalized size = 1.23 \[ \frac {3 \, {\left (6 \, A + 5 \, C\right )} \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right ) - 3 \, {\left (6 \, A + 5 \, C\right )} \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (18 \, A \sin \left (d x + c\right )^{5} + 15 \, C \sin \left (d x + c\right )^{5} - 48 \, A \sin \left (d x + c\right )^{3} - 40 \, C \sin \left (d x + c\right )^{3} + 30 \, A \sin \left (d x + c\right ) + 33 \, C \sin \left (d x + c\right )\right )}}{{\left (\sin \left (d x + c\right )^{2} - 1\right )}^{3}}}{96 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.44, size = 138, normalized size = 1.41 \[ \frac {A \left (\sec ^{3}\left (d x +c \right )\right ) \tan \left (d x +c \right )}{4 d}+\frac {3 A \sec \left (d x +c \right ) \tan \left (d x +c \right )}{8 d}+\frac {3 A \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8 d}+\frac {C \left (\sec ^{5}\left (d x +c \right )\right ) \tan \left (d x +c \right )}{6 d}+\frac {5 C \tan \left (d x +c \right ) \left (\sec ^{3}\left (d x +c \right )\right )}{24 d}+\frac {5 C \tan \left (d x +c \right ) \sec \left (d x +c \right )}{16 d}+\frac {5 C \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 126, normalized size = 1.29 \[ \frac {3 \, {\left (6 \, A + 5 \, C\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, {\left (6 \, A + 5 \, C\right )} \log \left (\sin \left (d x + c\right ) - 1\right ) - \frac {2 \, {\left (3 \, {\left (6 \, A + 5 \, C\right )} \sin \left (d x + c\right )^{5} - 8 \, {\left (6 \, A + 5 \, C\right )} \sin \left (d x + c\right )^{3} + 3 \, {\left (10 \, A + 11 \, C\right )} \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{6} - 3 \, \sin \left (d x + c\right )^{4} + 3 \, \sin \left (d x + c\right )^{2} - 1}}{96 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.48, size = 102, normalized size = 1.04 \[ \frac {\mathrm {atanh}\left (\sin \left (c+d\,x\right )\right )\,\left (\frac {3\,A}{8}+\frac {5\,C}{16}\right )}{d}-\frac {\left (\frac {3\,A}{8}+\frac {5\,C}{16}\right )\,{\sin \left (c+d\,x\right )}^5+\left (-A-\frac {5\,C}{6}\right )\,{\sin \left (c+d\,x\right )}^3+\left (\frac {5\,A}{8}+\frac {11\,C}{16}\right )\,\sin \left (c+d\,x\right )}{d\,\left ({\sin \left (c+d\,x\right )}^6-3\,{\sin \left (c+d\,x\right )}^4+3\,{\sin \left (c+d\,x\right )}^2-1\right )} \]
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 \[ \int \left (A + C \sec ^{2}{\left (c + d x \right )}\right ) \sec ^{5}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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