Optimal. Leaf size=157 \[ -\frac {2 (80 A-3 B-4 C) \sin (c+d x)}{105 a^4 d (\cos (c+d x)+1)}-\frac {(55 A-6 B-8 C) \sin (c+d x)}{105 a^4 d (\cos (c+d x)+1)^2}+\frac {A \tanh ^{-1}(\sin (c+d x))}{a^4 d}-\frac {(10 A-3 B-4 C) \sin (c+d x)}{35 a d (a \cos (c+d x)+a)^3}-\frac {(A-B+C) \sin (c+d x)}{7 d (a \cos (c+d x)+a)^4} \]
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Rubi [A] time = 0.49, antiderivative size = 157, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {3041, 2978, 12, 3770} \[ -\frac {2 (80 A-3 B-4 C) \sin (c+d x)}{105 a^4 d (\cos (c+d x)+1)}-\frac {(55 A-6 B-8 C) \sin (c+d x)}{105 a^4 d (\cos (c+d x)+1)^2}+\frac {A \tanh ^{-1}(\sin (c+d x))}{a^4 d}-\frac {(10 A-3 B-4 C) \sin (c+d x)}{35 a d (a \cos (c+d x)+a)^3}-\frac {(A-B+C) \sin (c+d x)}{7 d (a \cos (c+d x)+a)^4} \]
Antiderivative was successfully verified.
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Rule 12
Rule 2978
Rule 3041
Rule 3770
Rubi steps
\begin {align*} \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec (c+d x)}{(a+a \cos (c+d x))^4} \, dx &=-\frac {(A-B+C) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}+\frac {\int \frac {(7 a A-a (3 A-3 B-4 C) \cos (c+d x)) \sec (c+d x)}{(a+a \cos (c+d x))^3} \, dx}{7 a^2}\\ &=-\frac {(A-B+C) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac {(10 A-3 B-4 C) \sin (c+d x)}{35 a d (a+a \cos (c+d x))^3}+\frac {\int \frac {\left (35 a^2 A-2 a^2 (10 A-3 B-4 C) \cos (c+d x)\right ) \sec (c+d x)}{(a+a \cos (c+d x))^2} \, dx}{35 a^4}\\ &=-\frac {(55 A-6 B-8 C) \sin (c+d x)}{105 a^4 d (1+\cos (c+d x))^2}-\frac {(A-B+C) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac {(10 A-3 B-4 C) \sin (c+d x)}{35 a d (a+a \cos (c+d x))^3}+\frac {\int \frac {\left (105 a^3 A-a^3 (55 A-6 B-8 C) \cos (c+d x)\right ) \sec (c+d x)}{a+a \cos (c+d x)} \, dx}{105 a^6}\\ &=-\frac {(55 A-6 B-8 C) \sin (c+d x)}{105 a^4 d (1+\cos (c+d x))^2}-\frac {(A-B+C) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac {(10 A-3 B-4 C) \sin (c+d x)}{35 a d (a+a \cos (c+d x))^3}-\frac {2 (80 A-3 B-4 C) \sin (c+d x)}{105 d \left (a^4+a^4 \cos (c+d x)\right )}+\frac {\int 105 a^4 A \sec (c+d x) \, dx}{105 a^8}\\ &=-\frac {(55 A-6 B-8 C) \sin (c+d x)}{105 a^4 d (1+\cos (c+d x))^2}-\frac {(A-B+C) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac {(10 A-3 B-4 C) \sin (c+d x)}{35 a d (a+a \cos (c+d x))^3}-\frac {2 (80 A-3 B-4 C) \sin (c+d x)}{105 d \left (a^4+a^4 \cos (c+d x)\right )}+\frac {A \int \sec (c+d x) \, dx}{a^4}\\ &=\frac {A \tanh ^{-1}(\sin (c+d x))}{a^4 d}-\frac {(55 A-6 B-8 C) \sin (c+d x)}{105 a^4 d (1+\cos (c+d x))^2}-\frac {(A-B+C) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac {(10 A-3 B-4 C) \sin (c+d x)}{35 a d (a+a \cos (c+d x))^3}-\frac {2 (80 A-3 B-4 C) \sin (c+d x)}{105 d \left (a^4+a^4 \cos (c+d x)\right )}\\ \end {align*}
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Mathematica [B] time = 2.62, size = 334, normalized size = 2.13 \[ -\frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \left (\sec \left (\frac {c}{2}\right ) \cos \left (\frac {1}{2} (c+d x)\right ) \left (70 (49 A-3 B-2 C) \sin \left (\frac {d x}{2}\right )-70 (31 A-2 C) \sin \left (c+\frac {d x}{2}\right )+2625 A \sin \left (c+\frac {3 d x}{2}\right )-735 A \sin \left (2 c+\frac {3 d x}{2}\right )+1015 A \sin \left (2 c+\frac {5 d x}{2}\right )-105 A \sin \left (3 c+\frac {5 d x}{2}\right )+160 A \sin \left (3 c+\frac {7 d x}{2}\right )-126 B \sin \left (c+\frac {3 d x}{2}\right )-42 B \sin \left (2 c+\frac {5 d x}{2}\right )-6 B \sin \left (3 c+\frac {7 d x}{2}\right )-168 C \sin \left (c+\frac {3 d x}{2}\right )-56 C \sin \left (2 c+\frac {5 d x}{2}\right )-8 C \sin \left (3 c+\frac {7 d x}{2}\right )\right )+6720 A \cos ^8\left (\frac {1}{2} (c+d x)\right ) \left (\log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )\right )\right )}{210 a^4 d (\cos (c+d x)+1)^4 (2 A+2 B \cos (c+d x)+C \cos (2 (c+d x))+C)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 248, normalized size = 1.58 \[ \frac {105 \, {\left (A \cos \left (d x + c\right )^{4} + 4 \, A \cos \left (d x + c\right )^{3} + 6 \, A \cos \left (d x + c\right )^{2} + 4 \, A \cos \left (d x + c\right ) + A\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 105 \, {\left (A \cos \left (d x + c\right )^{4} + 4 \, A \cos \left (d x + c\right )^{3} + 6 \, A \cos \left (d x + c\right )^{2} + 4 \, A \cos \left (d x + c\right ) + A\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (2 \, {\left (80 \, A - 3 \, B - 4 \, C\right )} \cos \left (d x + c\right )^{3} + {\left (535 \, A - 24 \, B - 32 \, C\right )} \cos \left (d x + c\right )^{2} + {\left (620 \, A - 39 \, B - 52 \, C\right )} \cos \left (d x + c\right ) + 260 \, A - 36 \, B - 13 \, C\right )} \sin \left (d x + c\right )}{210 \, {\left (a^{4} d \cos \left (d x + c\right )^{4} + 4 \, a^{4} d \cos \left (d x + c\right )^{3} + 6 \, a^{4} d \cos \left (d x + c\right )^{2} + 4 \, a^{4} d \cos \left (d x + c\right ) + a^{4} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.39, size = 248, normalized size = 1.58 \[ \frac {\frac {840 \, A \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right )}{a^{4}} - \frac {840 \, A \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right )}{a^{4}} - \frac {15 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 15 \, B a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 15 \, C a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 105 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 63 \, B a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 21 \, C a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 385 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 105 \, B a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 35 \, C a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 1575 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 105 \, B a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 105 \, C a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{28}}}{840 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.22, size = 277, normalized size = 1.76 \[ \frac {A \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d \,a^{4}}-\frac {11 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) A}{24 d \,a^{4}}-\frac {15 A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d \,a^{4}}+\frac {C \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d \,a^{4}}-\frac {A \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d \,a^{4}}+\frac {3 B \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{40 d \,a^{4}}-\frac {C \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{40 d \,a^{4}}-\frac {\left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) A}{56 d \,a^{4}}+\frac {B \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{56 d \,a^{4}}-\frac {C \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{56 d \,a^{4}}+\frac {B \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d \,a^{4}}+\frac {B \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d \,a^{4}}+\frac {C \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d \,a^{4}}-\frac {A \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d \,a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.37, size = 313, normalized size = 1.99 \[ -\frac {5 \, A {\left (\frac {\frac {315 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {77 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {21 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {3 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a^{4}} - \frac {168 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a^{4}} + \frac {168 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a^{4}}\right )} - \frac {C {\left (\frac {105 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {35 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {21 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {15 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}\right )}}{a^{4}} - \frac {3 \, B {\left (\frac {35 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {35 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {21 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {5 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}\right )}}{a^{4}}}{840 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.14, size = 199, normalized size = 1.27 \[ \frac {2\,A\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{a^4\,d}-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {A-B+C}{8\,a^4}+\frac {4\,A-2\,B}{8\,a^4}+\frac {4\,A+2\,B}{8\,a^4}+\frac {6\,A-2\,C}{8\,a^4}\right )}{d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {A-B+C}{24\,a^4}+\frac {4\,A-2\,B}{24\,a^4}+\frac {6\,A-2\,C}{24\,a^4}\right )}{d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (\frac {A-B+C}{40\,a^4}+\frac {4\,A-2\,B}{40\,a^4}\right )}{d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (A-B+C\right )}{56\,a^4\,d} \]
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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