Optimal. Leaf size=152 \[ -\frac {2 (8 B-5 C) \tan (c+d x)}{3 a^2 d}+\frac {(7 B-4 C) \tanh ^{-1}(\sin (c+d x))}{2 a^2 d}+\frac {(7 B-4 C) \tan (c+d x) \sec (c+d x)}{2 a^2 d}-\frac {(8 B-5 C) \tan (c+d x) \sec (c+d x)}{3 a^2 d (\cos (c+d x)+1)}-\frac {(B-C) \tan (c+d x) \sec (c+d x)}{3 d (a \cos (c+d x)+a)^2} \]
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Rubi [A] time = 0.40, antiderivative size = 152, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.175, Rules used = {3029, 2978, 2748, 3768, 3770, 3767, 8} \[ -\frac {2 (8 B-5 C) \tan (c+d x)}{3 a^2 d}+\frac {(7 B-4 C) \tanh ^{-1}(\sin (c+d x))}{2 a^2 d}+\frac {(7 B-4 C) \tan (c+d x) \sec (c+d x)}{2 a^2 d}-\frac {(8 B-5 C) \tan (c+d x) \sec (c+d x)}{3 a^2 d (\cos (c+d x)+1)}-\frac {(B-C) \tan (c+d x) \sec (c+d x)}{3 d (a \cos (c+d x)+a)^2} \]
Antiderivative was successfully verified.
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Rule 8
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 ^4(c+d x)}{(a+a \cos (c+d x))^2} \, dx &=\int \frac {(B+C \cos (c+d x)) \sec ^3(c+d x)}{(a+a \cos (c+d x))^2} \, dx\\ &=-\frac {(B-C) \sec (c+d x) \tan (c+d x)}{3 d (a+a \cos (c+d x))^2}+\frac {\int \frac {(a (5 B-2 C)-3 a (B-C) \cos (c+d x)) \sec ^3(c+d x)}{a+a \cos (c+d x)} \, dx}{3 a^2}\\ &=-\frac {(8 B-5 C) \sec (c+d x) \tan (c+d x)}{3 a^2 d (1+\cos (c+d x))}-\frac {(B-C) \sec (c+d x) \tan (c+d x)}{3 d (a+a \cos (c+d x))^2}+\frac {\int \left (3 a^2 (7 B-4 C)-2 a^2 (8 B-5 C) \cos (c+d x)\right ) \sec ^3(c+d x) \, dx}{3 a^4}\\ &=-\frac {(8 B-5 C) \sec (c+d x) \tan (c+d x)}{3 a^2 d (1+\cos (c+d x))}-\frac {(B-C) \sec (c+d x) \tan (c+d x)}{3 d (a+a \cos (c+d x))^2}-\frac {(2 (8 B-5 C)) \int \sec ^2(c+d x) \, dx}{3 a^2}+\frac {(7 B-4 C) \int \sec ^3(c+d x) \, dx}{a^2}\\ &=\frac {(7 B-4 C) \sec (c+d x) \tan (c+d x)}{2 a^2 d}-\frac {(8 B-5 C) \sec (c+d x) \tan (c+d x)}{3 a^2 d (1+\cos (c+d x))}-\frac {(B-C) \sec (c+d x) \tan (c+d x)}{3 d (a+a \cos (c+d x))^2}+\frac {(7 B-4 C) \int \sec (c+d x) \, dx}{2 a^2}+\frac {(2 (8 B-5 C)) \operatorname {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{3 a^2 d}\\ &=\frac {(7 B-4 C) \tanh ^{-1}(\sin (c+d x))}{2 a^2 d}-\frac {2 (8 B-5 C) \tan (c+d x)}{3 a^2 d}+\frac {(7 B-4 C) \sec (c+d x) \tan (c+d x)}{2 a^2 d}-\frac {(8 B-5 C) \sec (c+d x) \tan (c+d x)}{3 a^2 d (1+\cos (c+d x))}-\frac {(B-C) \sec (c+d x) \tan (c+d x)}{3 d (a+a \cos (c+d x))^2}\\ \end {align*}
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Mathematica [B] time = 3.19, size = 496, normalized size = 3.26 \[ -\frac {96 (7 B-4 C) \cos ^4\left (\frac {1}{2} (c+d x)\right ) \left (\log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )\right )+\sec \left (\frac {c}{2}\right ) \sec (c) \cos \left (\frac {1}{2} (c+d x)\right ) \sec ^2(c+d x) \left (-126 B \sin \left (c-\frac {d x}{2}\right )+42 B \sin \left (c+\frac {d x}{2}\right )-98 B \sin \left (2 c+\frac {d x}{2}\right )-3 B \sin \left (c+\frac {3 d x}{2}\right )+37 B \sin \left (2 c+\frac {3 d x}{2}\right )-63 B \sin \left (3 c+\frac {3 d x}{2}\right )+75 B \sin \left (c+\frac {5 d x}{2}\right )+15 B \sin \left (2 c+\frac {5 d x}{2}\right )+39 B \sin \left (3 c+\frac {5 d x}{2}\right )-21 B \sin \left (4 c+\frac {5 d x}{2}\right )+32 B \sin \left (2 c+\frac {7 d x}{2}\right )+12 B \sin \left (3 c+\frac {7 d x}{2}\right )+20 B \sin \left (4 c+\frac {7 d x}{2}\right )-14 (B-C) \sin \left (\frac {d x}{2}\right )+(97 B-64 C) \sin \left (\frac {3 d x}{2}\right )+84 C \sin \left (c-\frac {d x}{2}\right )-42 C \sin \left (c+\frac {d x}{2}\right )+56 C \sin \left (2 c+\frac {d x}{2}\right )+6 C \sin \left (c+\frac {3 d x}{2}\right )-34 C \sin \left (2 c+\frac {3 d x}{2}\right )+36 C \sin \left (3 c+\frac {3 d x}{2}\right )-48 C \sin \left (c+\frac {5 d x}{2}\right )-6 C \sin \left (2 c+\frac {5 d x}{2}\right )-30 C \sin \left (3 c+\frac {5 d x}{2}\right )+12 C \sin \left (4 c+\frac {5 d x}{2}\right )-20 C \sin \left (2 c+\frac {7 d x}{2}\right )-6 C \sin \left (3 c+\frac {7 d x}{2}\right )-14 C \sin \left (4 c+\frac {7 d x}{2}\right )\right )}{48 a^2 d (\cos (c+d x)+1)^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 228, normalized size = 1.50 \[ \frac {3 \, {\left ({\left (7 \, B - 4 \, C\right )} \cos \left (d x + c\right )^{4} + 2 \, {\left (7 \, B - 4 \, C\right )} \cos \left (d x + c\right )^{3} + {\left (7 \, B - 4 \, C\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, {\left ({\left (7 \, B - 4 \, C\right )} \cos \left (d x + c\right )^{4} + 2 \, {\left (7 \, B - 4 \, C\right )} \cos \left (d x + c\right )^{3} + {\left (7 \, B - 4 \, C\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (4 \, {\left (8 \, B - 5 \, C\right )} \cos \left (d x + c\right )^{3} + {\left (43 \, B - 28 \, C\right )} \cos \left (d x + c\right )^{2} + 6 \, {\left (B - C\right )} \cos \left (d x + c\right ) - 3 \, B\right )} \sin \left (d x + c\right )}{12 \, {\left (a^{2} d \cos \left (d x + c\right )^{4} + 2 \, a^{2} d \cos \left (d x + c\right )^{3} + a^{2} d \cos \left (d x + c\right )^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.39, size = 198, normalized size = 1.30 \[ \frac {\frac {3 \, {\left (7 \, B - 4 \, C\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right )}{a^{2}} - \frac {3 \, {\left (7 \, B - 4 \, C\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right )}{a^{2}} + \frac {6 \, {\left (5 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 2 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 3 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 2 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{2} a^{2}} - \frac {B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 21 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 15 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{6}}}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.25, size = 294, normalized size = 1.93 \[ -\frac {B \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 d \,a^{2}}+\frac {C \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 d \,a^{2}}-\frac {7 B \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 d \,a^{2}}+\frac {5 C \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 d \,a^{2}}-\frac {7 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) B}{2 d \,a^{2}}+\frac {2 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) C}{d \,a^{2}}+\frac {5 B}{2 d \,a^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {C}{d \,a^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {B}{2 d \,a^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {5 B}{2 d \,a^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {C}{d \,a^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {7 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) B}{2 d \,a^{2}}-\frac {2 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) C}{d \,a^{2}}-\frac {B}{2 d \,a^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.42, size = 336, normalized size = 2.21 \[ -\frac {B {\left (\frac {6 \, {\left (\frac {3 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {5 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}\right )}}{a^{2} - \frac {2 \, a^{2} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {a^{2} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}} + \frac {\frac {21 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {\sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}}{a^{2}} - \frac {21 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a^{2}} + \frac {21 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a^{2}}\right )} - C {\left (\frac {\frac {15 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {\sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}}{a^{2}} - \frac {12 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a^{2}} + \frac {12 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a^{2}} + \frac {12 \, \sin \left (d x + c\right )}{{\left (a^{2} - \frac {a^{2} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )} {\left (\cos \left (d x + c\right ) + 1\right )}}\right )}}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.17, size = 165, normalized size = 1.09 \[ \frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (5\,B-2\,C\right )-\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (3\,B-2\,C\right )}{d\,\left (a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-2\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+a^2\right )}-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {3\,\left (B-C\right )}{2\,a^2}+\frac {4\,B-2\,C}{2\,a^2}\right )}{d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (B-C\right )}{6\,a^2\,d}+\frac {\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (7\,B-4\,C\right )}{a^2\,d} \]
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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