Optimal. Leaf size=97 \[ \frac {(3 A+4 C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac {(3 A+4 C) \tan (c+d x) \sec (c+d x)}{8 d}+\frac {A \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac {B \tan ^3(c+d x)}{3 d}+\frac {B \tan (c+d x)}{d} \]
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Rubi [A] time = 0.10, antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {3021, 2748, 3767, 3768, 3770} \[ \frac {(3 A+4 C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac {(3 A+4 C) \tan (c+d x) \sec (c+d x)}{8 d}+\frac {A \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac {B \tan ^3(c+d x)}{3 d}+\frac {B \tan (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 2748
Rule 3021
Rule 3767
Rule 3768
Rule 3770
Rubi steps
\begin {align*} \int \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx &=\frac {A \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac {1}{4} \int (4 B+(3 A+4 C) \cos (c+d x)) \sec ^4(c+d x) \, dx\\ &=\frac {A \sec ^3(c+d x) \tan (c+d x)}{4 d}+B \int \sec ^4(c+d x) \, dx+\frac {1}{4} (3 A+4 C) \int \sec ^3(c+d x) \, dx\\ &=\frac {(3 A+4 C) \sec (c+d x) \tan (c+d x)}{8 d}+\frac {A \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac {1}{8} (3 A+4 C) \int \sec (c+d x) \, dx-\frac {B \operatorname {Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (c+d x)\right )}{d}\\ &=\frac {(3 A+4 C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac {B \tan (c+d x)}{d}+\frac {(3 A+4 C) \sec (c+d x) \tan (c+d x)}{8 d}+\frac {A \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac {B \tan ^3(c+d x)}{3 d}\\ \end {align*}
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Mathematica [A] time = 0.27, size = 71, normalized size = 0.73 \[ \frac {\tan (c+d x) \left (3 (3 A+4 C) \sec (c+d x)+6 A \sec ^3(c+d x)+8 B \left (\tan ^2(c+d x)+3\right )\right )+3 (3 A+4 C) \tanh ^{-1}(\sin (c+d x))}{24 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 117, normalized size = 1.21 \[ \frac {3 \, {\left (3 \, A + 4 \, C\right )} \cos \left (d x + c\right )^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, {\left (3 \, A + 4 \, C\right )} \cos \left (d x + c\right )^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (16 \, B \cos \left (d x + c\right )^{3} + 3 \, {\left (3 \, A + 4 \, C\right )} \cos \left (d x + c\right )^{2} + 8 \, B \cos \left (d x + c\right ) + 6 \, A\right )} \sin \left (d x + c\right )}{48 \, d \cos \left (d x + c\right )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.46, size = 230, normalized size = 2.37 \[ \frac {3 \, {\left (3 \, A + 4 \, C\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 3 \, {\left (3 \, A + 4 \, C\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + \frac {2 \, {\left (15 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 24 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 12 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 9 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 40 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 12 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 9 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 40 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 12 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 15 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 24 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 12 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{4}}}{24 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.30, size = 130, normalized size = 1.34 \[ \frac {A \left (\sec ^{3}\left (d x +c \right )\right ) \tan \left (d x +c \right )}{4 d}+\frac {3 A \sec \left (d x +c \right ) \tan \left (d x +c \right )}{8 d}+\frac {3 A \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8 d}+\frac {2 B \tan \left (d x +c \right )}{3 d}+\frac {B \tan \left (d x +c \right ) \left (\sec ^{2}\left (d x +c \right )\right )}{3 d}+\frac {C \tan \left (d x +c \right ) \sec \left (d x +c \right )}{2 d}+\frac {C \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 139, normalized size = 1.43 \[ \frac {16 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} B - 3 \, A {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 12 \, C {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )}}{48 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.56, size = 160, normalized size = 1.65 \[ \frac {\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (\frac {3\,A}{4}+C\right )}{d}+\frac {\left (\frac {5\,A}{4}-2\,B+C\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {3\,A}{4}+\frac {10\,B}{3}-C\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (\frac {3\,A}{4}-\frac {10\,B}{3}-C\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (\frac {5\,A}{4}+2\,B+C\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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