Optimal. Leaf size=175 \[ \frac {8 (83 A-20 B) \tan (c+d x)}{105 a^4 d}-\frac {(4 A-B) \tanh ^{-1}(\sin (c+d x))}{a^4 d}-\frac {(4 A-B) \tan (c+d x)}{a^4 d (\cos (c+d x)+1)}-\frac {(88 A-25 B) \tan (c+d x)}{105 a^4 d (\cos (c+d x)+1)^2}-\frac {(12 A-5 B) \tan (c+d x)}{35 a d (a \cos (c+d x)+a)^3}-\frac {(A-B) \tan (c+d x)}{7 d (a \cos (c+d x)+a)^4} \]
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Rubi [A] time = 0.67, antiderivative size = 175, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {2978, 2748, 3767, 8, 3770} \[ \frac {8 (83 A-20 B) \tan (c+d x)}{105 a^4 d}-\frac {(4 A-B) \tanh ^{-1}(\sin (c+d x))}{a^4 d}-\frac {(4 A-B) \tan (c+d x)}{a^4 d (\cos (c+d x)+1)}-\frac {(88 A-25 B) \tan (c+d x)}{105 a^4 d (\cos (c+d x)+1)^2}-\frac {(12 A-5 B) \tan (c+d x)}{35 a d (a \cos (c+d x)+a)^3}-\frac {(A-B) \tan (c+d x)}{7 d (a \cos (c+d x)+a)^4} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2748
Rule 2978
Rule 3767
Rule 3770
Rubi steps
\begin {align*} \int \frac {(A+B \cos (c+d x)) \sec ^2(c+d x)}{(a+a \cos (c+d x))^4} \, dx &=-\frac {(A-B) \tan (c+d x)}{7 d (a+a \cos (c+d x))^4}+\frac {\int \frac {(a (8 A-B)-4 a (A-B) \cos (c+d x)) \sec ^2(c+d x)}{(a+a \cos (c+d x))^3} \, dx}{7 a^2}\\ &=-\frac {(A-B) \tan (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac {(12 A-5 B) \tan (c+d x)}{35 a d (a+a \cos (c+d x))^3}+\frac {\int \frac {\left (2 a^2 (26 A-5 B)-3 a^2 (12 A-5 B) \cos (c+d x)\right ) \sec ^2(c+d x)}{(a+a \cos (c+d x))^2} \, dx}{35 a^4}\\ &=-\frac {(88 A-25 B) \tan (c+d x)}{105 a^4 d (1+\cos (c+d x))^2}-\frac {(A-B) \tan (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac {(12 A-5 B) \tan (c+d x)}{35 a d (a+a \cos (c+d x))^3}+\frac {\int \frac {\left (a^3 (244 A-55 B)-2 a^3 (88 A-25 B) \cos (c+d x)\right ) \sec ^2(c+d x)}{a+a \cos (c+d x)} \, dx}{105 a^6}\\ &=-\frac {(88 A-25 B) \tan (c+d x)}{105 a^4 d (1+\cos (c+d x))^2}-\frac {(A-B) \tan (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac {(12 A-5 B) \tan (c+d x)}{35 a d (a+a \cos (c+d x))^3}-\frac {(4 A-B) \tan (c+d x)}{d \left (a^4+a^4 \cos (c+d x)\right )}+\frac {\int \left (8 a^4 (83 A-20 B)-105 a^4 (4 A-B) \cos (c+d x)\right ) \sec ^2(c+d x) \, dx}{105 a^8}\\ &=-\frac {(88 A-25 B) \tan (c+d x)}{105 a^4 d (1+\cos (c+d x))^2}-\frac {(A-B) \tan (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac {(12 A-5 B) \tan (c+d x)}{35 a d (a+a \cos (c+d x))^3}-\frac {(4 A-B) \tan (c+d x)}{d \left (a^4+a^4 \cos (c+d x)\right )}+\frac {(8 (83 A-20 B)) \int \sec ^2(c+d x) \, dx}{105 a^4}-\frac {(4 A-B) \int \sec (c+d x) \, dx}{a^4}\\ &=-\frac {(4 A-B) \tanh ^{-1}(\sin (c+d x))}{a^4 d}-\frac {(88 A-25 B) \tan (c+d x)}{105 a^4 d (1+\cos (c+d x))^2}-\frac {(A-B) \tan (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac {(12 A-5 B) \tan (c+d x)}{35 a d (a+a \cos (c+d x))^3}-\frac {(4 A-B) \tan (c+d x)}{d \left (a^4+a^4 \cos (c+d x)\right )}-\frac {(8 (83 A-20 B)) \operatorname {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{105 a^4 d}\\ &=-\frac {(4 A-B) \tanh ^{-1}(\sin (c+d x))}{a^4 d}+\frac {8 (83 A-20 B) \tan (c+d x)}{105 a^4 d}-\frac {(88 A-25 B) \tan (c+d x)}{105 a^4 d (1+\cos (c+d x))^2}-\frac {(A-B) \tan (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac {(12 A-5 B) \tan (c+d x)}{35 a d (a+a \cos (c+d x))^3}-\frac {(4 A-B) \tan (c+d x)}{d \left (a^4+a^4 \cos (c+d x)\right )}\\ \end {align*}
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Mathematica [B] time = 5.73, size = 595, normalized size = 3.40 \[ \frac {26880 (4 A-B) \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 )+\sec \left (\frac {c}{2}\right ) \sec (c) \cos \left (\frac {1}{2} (c+d x)\right ) \sec (c+d x) \left (-245 (44 A-17 B) \sin \left (\frac {d x}{2}\right )+7 (2684 A-635 B) \sin \left (\frac {3 d x}{2}\right )-20524 A \sin \left (c-\frac {d x}{2}\right )+14644 A \sin \left (c+\frac {d x}{2}\right )-16660 A \sin \left (2 c+\frac {d x}{2}\right )-4690 A \sin \left (c+\frac {3 d x}{2}\right )+14378 A \sin \left (2 c+\frac {3 d x}{2}\right )-9100 A \sin \left (3 c+\frac {3 d x}{2}\right )+11668 A \sin \left (c+\frac {5 d x}{2}\right )-630 A \sin \left (2 c+\frac {5 d x}{2}\right )+9358 A \sin \left (3 c+\frac {5 d x}{2}\right )-2940 A \sin \left (4 c+\frac {5 d x}{2}\right )+4228 A \sin \left (2 c+\frac {7 d x}{2}\right )+315 A \sin \left (3 c+\frac {7 d x}{2}\right )+3493 A \sin \left (4 c+\frac {7 d x}{2}\right )-420 A \sin \left (5 c+\frac {7 d x}{2}\right )+664 A \sin \left (3 c+\frac {9 d x}{2}\right )+105 A \sin \left (4 c+\frac {9 d x}{2}\right )+559 A \sin \left (5 c+\frac {9 d x}{2}\right )+4795 B \sin \left (c-\frac {d x}{2}\right )-4795 B \sin \left (c+\frac {d x}{2}\right )+4165 B \sin \left (2 c+\frac {d x}{2}\right )+2275 B \sin \left (c+\frac {3 d x}{2}\right )-4445 B \sin \left (2 c+\frac {3 d x}{2}\right )+2275 B \sin \left (3 c+\frac {3 d x}{2}\right )-2785 B \sin \left (c+\frac {5 d x}{2}\right )+735 B \sin \left (2 c+\frac {5 d x}{2}\right )-2785 B \sin \left (3 c+\frac {5 d x}{2}\right )+735 B \sin \left (4 c+\frac {5 d x}{2}\right )-1015 B \sin \left (2 c+\frac {7 d x}{2}\right )+105 B \sin \left (3 c+\frac {7 d x}{2}\right )-1015 B \sin \left (4 c+\frac {7 d x}{2}\right )+105 B \sin \left (5 c+\frac {7 d x}{2}\right )-160 B \sin \left (3 c+\frac {9 d x}{2}\right )-160 B \sin \left (5 c+\frac {9 d x}{2}\right )\right )}{1680 a^4 d (\cos (c+d x)+1)^4} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.96, size = 337, normalized size = 1.93 \[ -\frac {105 \, {\left ({\left (4 \, A - B\right )} \cos \left (d x + c\right )^{5} + 4 \, {\left (4 \, A - B\right )} \cos \left (d x + c\right )^{4} + 6 \, {\left (4 \, A - B\right )} \cos \left (d x + c\right )^{3} + 4 \, {\left (4 \, A - B\right )} \cos \left (d x + c\right )^{2} + {\left (4 \, A - B\right )} \cos \left (d x + c\right )\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 105 \, {\left ({\left (4 \, A - B\right )} \cos \left (d x + c\right )^{5} + 4 \, {\left (4 \, A - B\right )} \cos \left (d x + c\right )^{4} + 6 \, {\left (4 \, A - B\right )} \cos \left (d x + c\right )^{3} + 4 \, {\left (4 \, A - B\right )} \cos \left (d x + c\right )^{2} + {\left (4 \, A - B\right )} \cos \left (d x + c\right )\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (8 \, {\left (83 \, A - 20 \, B\right )} \cos \left (d x + c\right )^{4} + {\left (2236 \, A - 535 \, B\right )} \cos \left (d x + c\right )^{3} + 4 \, {\left (659 \, A - 155 \, B\right )} \cos \left (d x + c\right )^{2} + 4 \, {\left (296 \, A - 65 \, B\right )} \cos \left (d x + c\right ) + 105 \, A\right )} \sin \left (d x + c\right )}{210 \, {\left (a^{4} d \cos \left (d x + c\right )^{5} + 4 \, a^{4} d \cos \left (d x + c\right )^{4} + 6 \, a^{4} d \cos \left (d x + c\right )^{3} + 4 \, a^{4} d \cos \left (d x + c\right )^{2} + a^{4} d \cos \left (d x + c\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.41, size = 224, normalized size = 1.28 \[ -\frac {\frac {840 \, {\left (4 \, A - B\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right )}{a^{4}} - \frac {840 \, {\left (4 \, A - B\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right )}{a^{4}} + \frac {1680 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )} a^{4}} - \frac {15 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 15 \, B a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 147 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 105 \, B a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 805 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 385 \, B a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 5145 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1575 \, B a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{28}}}{840 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.16, size = 285, normalized size = 1.63 \[ \frac {\left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) A}{56 d \,a^{4}}-\frac {B \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{56 d \,a^{4}}+\frac {7 A \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{40 d \,a^{4}}-\frac {B \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d \,a^{4}}+\frac {23 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) A}{24 d \,a^{4}}-\frac {11 B \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d \,a^{4}}+\frac {49 A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d \,a^{4}}-\frac {15 B \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d \,a^{4}}+\frac {4 A \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d \,a^{4}}-\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) B}{d \,a^{4}}-\frac {A}{d \,a^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {4 A \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d \,a^{4}}+\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) B}{d \,a^{4}}-\frac {A}{d \,a^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.65, size = 326, normalized size = 1.86 \[ \frac {A {\left (\frac {1680 \, \sin \left (d x + c\right )}{{\left (a^{4} - \frac {a^{4} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )} {\left (\cos \left (d x + c\right ) + 1\right )}} + \frac {\frac {5145 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {805 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {147 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {15 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a^{4}} - \frac {3360 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a^{4}} + \frac {3360 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a^{4}}\right )} - 5 \, B {\left (\frac {\frac {315 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {77 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {21 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {3 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a^{4}} - \frac {168 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a^{4}} + \frac {168 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a^{4}}\right )}}{840 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.28, size = 236, normalized size = 1.35 \[ \frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {A-B}{8\,a^4}+\frac {5\,A-3\,B}{12\,a^4}+\frac {10\,A-2\,B}{24\,a^4}\right )}{d}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (\frac {A-B}{20\,a^4}+\frac {5\,A-3\,B}{40\,a^4}\right )}{d}+\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {A-B}{2\,a^4}+\frac {3\,\left (5\,A-3\,B\right )}{8\,a^4}+\frac {10\,A-2\,B}{4\,a^4}+\frac {10\,A+2\,B}{8\,a^4}\right )}{d}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (A-B\right )}{56\,a^4\,d}-\frac {2\,A\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left (a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-a^4\right )}-\frac {2\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (4\,A-B\right )}{a^4\,d} \]
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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