Optimal. Leaf size=196 \[ -\frac {8 (19 B-9 C) \tan (c+d x)}{15 a^3 d}+\frac {(13 B-6 C) \tanh ^{-1}(\sin (c+d x))}{2 a^3 d}+\frac {(13 B-6 C) \tan (c+d x) \sec (c+d x)}{2 a^3 d}-\frac {4 (19 B-9 C) \tan (c+d x) \sec (c+d x)}{15 d \left (a^3 \cos (c+d x)+a^3\right )}-\frac {(11 B-6 C) \tan (c+d x) \sec (c+d x)}{15 a d (a \cos (c+d x)+a)^2}-\frac {(B-C) \tan (c+d x) \sec (c+d x)}{5 d (a \cos (c+d x)+a)^3} \]
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Rubi [A] time = 0.57, antiderivative size = 196, normalized size of antiderivative = 1.00, number of steps used = 9, 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} \[ -\frac {8 (19 B-9 C) \tan (c+d x)}{15 a^3 d}+\frac {(13 B-6 C) \tanh ^{-1}(\sin (c+d x))}{2 a^3 d}+\frac {(13 B-6 C) \tan (c+d x) \sec (c+d x)}{2 a^3 d}-\frac {4 (19 B-9 C) \tan (c+d x) \sec (c+d x)}{15 d \left (a^3 \cos (c+d x)+a^3\right )}-\frac {(11 B-6 C) \tan (c+d x) \sec (c+d x)}{15 a d (a \cos (c+d x)+a)^2}-\frac {(B-C) \tan (c+d x) \sec (c+d x)}{5 d (a \cos (c+d x)+a)^3} \]
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))^3} \, dx &=\int \frac {(B+C \cos (c+d x)) \sec ^3(c+d x)}{(a+a \cos (c+d x))^3} \, dx\\ &=-\frac {(B-C) \sec (c+d x) \tan (c+d x)}{5 d (a+a \cos (c+d x))^3}+\frac {\int \frac {(a (7 B-2 C)-4 a (B-C) \cos (c+d x)) \sec ^3(c+d x)}{(a+a \cos (c+d x))^2} \, dx}{5 a^2}\\ &=-\frac {(B-C) \sec (c+d x) \tan (c+d x)}{5 d (a+a \cos (c+d x))^3}-\frac {(11 B-6 C) \sec (c+d x) \tan (c+d x)}{15 a d (a+a \cos (c+d x))^2}+\frac {\int \frac {\left (a^2 (43 B-18 C)-3 a^2 (11 B-6 C) \cos (c+d x)\right ) \sec ^3(c+d x)}{a+a \cos (c+d x)} \, dx}{15 a^4}\\ &=-\frac {(B-C) \sec (c+d x) \tan (c+d x)}{5 d (a+a \cos (c+d x))^3}-\frac {(11 B-6 C) \sec (c+d x) \tan (c+d x)}{15 a d (a+a \cos (c+d x))^2}-\frac {4 (19 B-9 C) \sec (c+d x) \tan (c+d x)}{15 d \left (a^3+a^3 \cos (c+d x)\right )}+\frac {\int \left (15 a^3 (13 B-6 C)-8 a^3 (19 B-9 C) \cos (c+d x)\right ) \sec ^3(c+d x) \, dx}{15 a^6}\\ &=-\frac {(B-C) \sec (c+d x) \tan (c+d x)}{5 d (a+a \cos (c+d x))^3}-\frac {(11 B-6 C) \sec (c+d x) \tan (c+d x)}{15 a d (a+a \cos (c+d x))^2}-\frac {4 (19 B-9 C) \sec (c+d x) \tan (c+d x)}{15 d \left (a^3+a^3 \cos (c+d x)\right )}-\frac {(8 (19 B-9 C)) \int \sec ^2(c+d x) \, dx}{15 a^3}+\frac {(13 B-6 C) \int \sec ^3(c+d x) \, dx}{a^3}\\ &=\frac {(13 B-6 C) \sec (c+d x) \tan (c+d x)}{2 a^3 d}-\frac {(B-C) \sec (c+d x) \tan (c+d x)}{5 d (a+a \cos (c+d x))^3}-\frac {(11 B-6 C) \sec (c+d x) \tan (c+d x)}{15 a d (a+a \cos (c+d x))^2}-\frac {4 (19 B-9 C) \sec (c+d x) \tan (c+d x)}{15 d \left (a^3+a^3 \cos (c+d x)\right )}+\frac {(13 B-6 C) \int \sec (c+d x) \, dx}{2 a^3}+\frac {(8 (19 B-9 C)) \operatorname {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{15 a^3 d}\\ &=\frac {(13 B-6 C) \tanh ^{-1}(\sin (c+d x))}{2 a^3 d}-\frac {8 (19 B-9 C) \tan (c+d x)}{15 a^3 d}+\frac {(13 B-6 C) \sec (c+d x) \tan (c+d x)}{2 a^3 d}-\frac {(B-C) \sec (c+d x) \tan (c+d x)}{5 d (a+a \cos (c+d x))^3}-\frac {(11 B-6 C) \sec (c+d x) \tan (c+d x)}{15 a d (a+a \cos (c+d x))^2}-\frac {4 (19 B-9 C) \sec (c+d x) \tan (c+d x)}{15 d \left (a^3+a^3 \cos (c+d x)\right )}\\ \end {align*}
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Mathematica [B] time = 4.77, size = 610, normalized size = 3.11 \[ -\frac {1920 (13 B-6 C) \cos ^6\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 (-4329 B \sin \left (c-\frac {d x}{2}\right )+1989 B \sin \left (c+\frac {d x}{2}\right )-3575 B \sin \left (2 c+\frac {d x}{2}\right )-475 B \sin \left (c+\frac {3 d x}{2}\right )+2005 B \sin \left (2 c+\frac {3 d x}{2}\right )-2275 B \sin \left (3 c+\frac {3 d x}{2}\right )+2673 B \sin \left (c+\frac {5 d x}{2}\right )+105 B \sin \left (2 c+\frac {5 d x}{2}\right )+1593 B \sin \left (3 c+\frac {5 d x}{2}\right )-975 B \sin \left (4 c+\frac {5 d x}{2}\right )+1325 B \sin \left (2 c+\frac {7 d x}{2}\right )+255 B \sin \left (3 c+\frac {7 d x}{2}\right )+875 B \sin \left (4 c+\frac {7 d x}{2}\right )-195 B \sin \left (5 c+\frac {7 d x}{2}\right )+304 B \sin \left (3 c+\frac {9 d x}{2}\right )+90 B \sin \left (4 c+\frac {9 d x}{2}\right )+214 B \sin \left (5 c+\frac {9 d x}{2}\right )+(870 C-1235 B) \sin \left (\frac {d x}{2}\right )+5 (761 B-366 C) \sin \left (\frac {3 d x}{2}\right )+2094 C \sin \left (c-\frac {d x}{2}\right )-1314 C \sin \left (c+\frac {d x}{2}\right )+1650 C \sin \left (2 c+\frac {d x}{2}\right )+450 C \sin \left (c+\frac {3 d x}{2}\right )-1230 C \sin \left (2 c+\frac {3 d x}{2}\right )+1050 C \sin \left (3 c+\frac {3 d x}{2}\right )-1278 C \sin \left (c+\frac {5 d x}{2}\right )+90 C \sin \left (2 c+\frac {5 d x}{2}\right )-918 C \sin \left (3 c+\frac {5 d x}{2}\right )+450 C \sin \left (4 c+\frac {5 d x}{2}\right )-630 C \sin \left (2 c+\frac {7 d x}{2}\right )-60 C \sin \left (3 c+\frac {7 d x}{2}\right )-480 C \sin \left (4 c+\frac {7 d x}{2}\right )+90 C \sin \left (5 c+\frac {7 d x}{2}\right )-144 C \sin \left (3 c+\frac {9 d x}{2}\right )-30 C \sin \left (4 c+\frac {9 d x}{2}\right )-114 C \sin \left (5 c+\frac {9 d x}{2}\right )\right )}{480 a^3 d (\cos (c+d x)+1)^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 295, normalized size = 1.51 \[ \frac {15 \, {\left ({\left (13 \, B - 6 \, C\right )} \cos \left (d x + c\right )^{5} + 3 \, {\left (13 \, B - 6 \, C\right )} \cos \left (d x + c\right )^{4} + 3 \, {\left (13 \, B - 6 \, C\right )} \cos \left (d x + c\right )^{3} + {\left (13 \, B - 6 \, C\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 15 \, {\left ({\left (13 \, B - 6 \, C\right )} \cos \left (d x + c\right )^{5} + 3 \, {\left (13 \, B - 6 \, C\right )} \cos \left (d x + c\right )^{4} + 3 \, {\left (13 \, B - 6 \, C\right )} \cos \left (d x + c\right )^{3} + {\left (13 \, B - 6 \, C\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (16 \, {\left (19 \, B - 9 \, C\right )} \cos \left (d x + c\right )^{4} + 3 \, {\left (239 \, B - 114 \, C\right )} \cos \left (d x + c\right )^{3} + {\left (479 \, B - 234 \, C\right )} \cos \left (d x + c\right )^{2} + 15 \, {\left (3 \, B - 2 \, C\right )} \cos \left (d x + c\right ) - 15 \, B\right )} \sin \left (d x + c\right )}{60 \, {\left (a^{3} d \cos \left (d x + c\right )^{5} + 3 \, a^{3} d \cos \left (d x + c\right )^{4} + 3 \, a^{3} d \cos \left (d x + c\right )^{3} + a^{3} d \cos \left (d x + c\right )^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.49, size = 233, normalized size = 1.19 \[ \frac {\frac {30 \, {\left (13 \, B - 6 \, C\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right )}{a^{3}} - \frac {30 \, {\left (13 \, B - 6 \, C\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right )}{a^{3}} + \frac {60 \, {\left (7 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 2 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 5 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 2 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{2} a^{3}} - \frac {3 \, B a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 3 \, C a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 40 \, B a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 30 \, C a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 465 \, B a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 255 \, C a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{15}}}{60 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.25, size = 334, normalized size = 1.70 \[ -\frac {B \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{20 d \,a^{3}}+\frac {C \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{20 d \,a^{3}}-\frac {2 B \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d \,a^{3}}+\frac {C \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d \,a^{3}}-\frac {31 B \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d \,a^{3}}+\frac {17 C \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d \,a^{3}}-\frac {13 B \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 d \,a^{3}}+\frac {3 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) C}{d \,a^{3}}+\frac {7 B}{2 d \,a^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {C}{d \,a^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {B}{2 d \,a^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {7 B}{2 d \,a^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {C}{d \,a^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {13 B \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 d \,a^{3}}-\frac {3 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) C}{d \,a^{3}}-\frac {B}{2 d \,a^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.36, size = 377, normalized size = 1.92 \[ -\frac {B {\left (\frac {60 \, {\left (\frac {5 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {7 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}\right )}}{a^{3} - \frac {2 \, a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {a^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}} + \frac {\frac {465 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {40 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {3 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}}{a^{3}} - \frac {390 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a^{3}} + \frac {390 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a^{3}}\right )} - 3 \, C {\left (\frac {40 \, \sin \left (d x + c\right )}{{\left (a^{3} - \frac {a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )} {\left (\cos \left (d x + c\right ) + 1\right )}} + \frac {\frac {85 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {10 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {\sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}}{a^{3}} - \frac {60 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a^{3}} + \frac {60 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a^{3}}\right )}}{60 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.09, size = 216, normalized size = 1.10 \[ \frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (7\,B-2\,C\right )-\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (5\,B-2\,C\right )}{d\,\left (a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-2\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+a^3\right )}-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {3\,\left (B-C\right )}{2\,a^3}+\frac {3\,\left (5\,B-3\,C\right )}{4\,a^3}+\frac {10\,B-2\,C}{4\,a^3}\right )}{d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {B-C}{4\,a^3}+\frac {5\,B-3\,C}{12\,a^3}\right )}{d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (B-C\right )}{20\,a^3\,d}+\frac {\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (13\,B-6\,C\right )}{a^3\,d} \]
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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