Optimal. Leaf size=131 \[ \frac {(4 B-3 C) \tan ^3(c+d x)}{3 a d}+\frac {(4 B-3 C) \tan (c+d x)}{a d}-\frac {3 (B-C) \tanh ^{-1}(\sin (c+d x))}{2 a d}-\frac {3 (B-C) \tan (c+d x) \sec (c+d x)}{2 a d}-\frac {(B-C) \tan (c+d x) \sec ^2(c+d x)}{d (a \cos (c+d x)+a)} \]
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Rubi [A] time = 0.26, antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {3029, 2978, 2748, 3767, 3768, 3770} \[ \frac {(4 B-3 C) \tan ^3(c+d x)}{3 a d}+\frac {(4 B-3 C) \tan (c+d x)}{a d}-\frac {3 (B-C) \tanh ^{-1}(\sin (c+d x))}{2 a d}-\frac {3 (B-C) \tan (c+d x) \sec (c+d x)}{2 a d}-\frac {(B-C) \tan (c+d x) \sec ^2(c+d x)}{d (a \cos (c+d x)+a)} \]
Antiderivative was successfully verified.
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Rule 2748
Rule 2978
Rule 3029
Rule 3767
Rule 3768
Rule 3770
Rubi steps
\begin {align*} \int \frac {\left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^5(c+d x)}{a+a \cos (c+d x)} \, dx &=\int \frac {(B+C \cos (c+d x)) \sec ^4(c+d x)}{a+a \cos (c+d x)} \, dx\\ &=-\frac {(B-C) \sec ^2(c+d x) \tan (c+d x)}{d (a+a \cos (c+d x))}+\frac {\int (a (4 B-3 C)-3 a (B-C) \cos (c+d x)) \sec ^4(c+d x) \, dx}{a^2}\\ &=-\frac {(B-C) \sec ^2(c+d x) \tan (c+d x)}{d (a+a \cos (c+d x))}+\frac {(4 B-3 C) \int \sec ^4(c+d x) \, dx}{a}-\frac {(3 (B-C)) \int \sec ^3(c+d x) \, dx}{a}\\ &=-\frac {3 (B-C) \sec (c+d x) \tan (c+d x)}{2 a d}-\frac {(B-C) \sec ^2(c+d x) \tan (c+d x)}{d (a+a \cos (c+d x))}-\frac {(3 (B-C)) \int \sec (c+d x) \, dx}{2 a}-\frac {(4 B-3 C) \operatorname {Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (c+d x)\right )}{a d}\\ &=-\frac {3 (B-C) \tanh ^{-1}(\sin (c+d x))}{2 a d}+\frac {(4 B-3 C) \tan (c+d x)}{a d}-\frac {3 (B-C) \sec (c+d x) \tan (c+d x)}{2 a d}-\frac {(B-C) \sec ^2(c+d x) \tan (c+d x)}{d (a+a \cos (c+d x))}+\frac {(4 B-3 C) \tan ^3(c+d x)}{3 a d}\\ \end {align*}
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Mathematica [B] time = 4.48, size = 490, normalized size = 3.74 \[ \frac {\cos \left (\frac {1}{2} (c+d x)\right ) \left (\sec \left (\frac {c}{2}\right ) \sec (c) \sec ^3(c+d x) \left (-24 B \sin \left (c-\frac {d x}{2}\right )-6 B \sin \left (c+\frac {d x}{2}\right )-24 B \sin \left (2 c+\frac {d x}{2}\right )+21 B \sin \left (c+\frac {3 d x}{2}\right )+9 B \sin \left (2 c+\frac {3 d x}{2}\right )-9 B \sin \left (3 c+\frac {3 d x}{2}\right )+7 B \sin \left (c+\frac {5 d x}{2}\right )+B \sin \left (2 c+\frac {5 d x}{2}\right )-3 B \sin \left (3 c+\frac {5 d x}{2}\right )-9 B \sin \left (4 c+\frac {5 d x}{2}\right )+16 B \sin \left (2 c+\frac {7 d x}{2}\right )+10 B \sin \left (3 c+\frac {7 d x}{2}\right )+6 B \sin \left (4 c+\frac {7 d x}{2}\right )+6 (B+C) \sin \left (\frac {d x}{2}\right )+3 (13 B-9 C) \sin \left (\frac {3 d x}{2}\right )+12 C \sin \left (c-\frac {d x}{2}\right )+6 C \sin \left (c+\frac {d x}{2}\right )+24 C \sin \left (2 c+\frac {d x}{2}\right )-9 C \sin \left (c+\frac {3 d x}{2}\right )-9 C \sin \left (2 c+\frac {3 d x}{2}\right )+9 C \sin \left (3 c+\frac {3 d x}{2}\right )-3 C \sin \left (c+\frac {5 d x}{2}\right )+3 C \sin \left (2 c+\frac {5 d x}{2}\right )+3 C \sin \left (3 c+\frac {5 d x}{2}\right )+9 C \sin \left (4 c+\frac {5 d x}{2}\right )-12 C \sin \left (2 c+\frac {7 d x}{2}\right )-6 C \sin \left (3 c+\frac {7 d x}{2}\right )-6 C \sin \left (4 c+\frac {7 d x}{2}\right )\right )+144 (B-C) \cos \left (\frac {1}{2} (c+d x)\right ) \left (\log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )\right )\right )}{48 a d (\cos (c+d x)+1)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 168, normalized size = 1.28 \[ -\frac {9 \, {\left ({\left (B - C\right )} \cos \left (d x + c\right )^{4} + {\left (B - C\right )} \cos \left (d x + c\right )^{3}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 9 \, {\left ({\left (B - C\right )} \cos \left (d x + c\right )^{4} + {\left (B - C\right )} \cos \left (d x + c\right )^{3}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (4 \, {\left (4 \, B - 3 \, C\right )} \cos \left (d x + c\right )^{3} + {\left (7 \, B - 3 \, C\right )} \cos \left (d x + c\right )^{2} - {\left (B - 3 \, C\right )} \cos \left (d x + c\right ) + 2 \, B\right )} \sin \left (d x + c\right )}{12 \, {\left (a d \cos \left (d x + c\right )^{4} + a d \cos \left (d x + c\right )^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.39, size = 182, normalized size = 1.39 \[ -\frac {\frac {9 \, {\left (B - C\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right )}{a} - \frac {9 \, {\left (B - C\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right )}{a} - \frac {6 \, {\left (B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{a} + \frac {2 \, {\left (15 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 9 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 16 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 12 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 9 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 3 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{3} a}}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.24, size = 340, normalized size = 2.60 \[ \frac {B \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{a d}-\frac {C \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{a d}-\frac {B}{3 a d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}+\frac {C}{2 a d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {B}{a d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {3 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) B}{2 a d}-\frac {3 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) C}{2 a d}-\frac {5 B}{2 a d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {3 C}{2 a d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {B}{3 a d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {B}{a d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {C}{2 a d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {3 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) B}{2 a d}+\frac {3 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) C}{2 a d}-\frac {5 B}{2 a d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {3 C}{2 a d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.78, size = 368, normalized size = 2.81 \[ \frac {B {\left (\frac {2 \, {\left (\frac {9 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {16 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {15 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}\right )}}{a - \frac {3 \, a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {3 \, a \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {a \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}} - \frac {9 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a} + \frac {9 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a} + \frac {6 \, \sin \left (d x + c\right )}{a {\left (\cos \left (d x + c\right ) + 1\right )}}\right )} - 3 \, C {\left (\frac {2 \, {\left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {3 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}\right )}}{a - \frac {2 \, a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {a \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}} - \frac {3 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a} + \frac {3 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a} + \frac {2 \, \sin \left (d x + c\right )}{a {\left (\cos \left (d x + c\right ) + 1\right )}}\right )}}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.51, size = 152, normalized size = 1.16 \[ \frac {\left (5\,B-3\,C\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (4\,C-\frac {16\,B}{3}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (3\,B-C\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left (-a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+3\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-3\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+a\right )}-\frac {3\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (B-C\right )}{a\,d}+\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (B-C\right )}{a\,d} \]
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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