Optimal. Leaf size=257 \[ \frac {4 (454 A+83 C) \tan ^3(c+d x)}{105 a^4 d}+\frac {4 (454 A+83 C) \tan (c+d x)}{35 a^4 d}-\frac {2 (11 A+2 C) \tanh ^{-1}(\sin (c+d x))}{a^4 d}-\frac {2 (11 A+2 C) \tan (c+d x) \sec (c+d x)}{a^4 d}-\frac {4 (11 A+2 C) \tan (c+d x) \sec ^2(c+d x)}{3 a^4 d (\cos (c+d x)+1)}-\frac {(178 A+31 C) \tan (c+d x) \sec ^2(c+d x)}{105 a^4 d (\cos (c+d x)+1)^2}-\frac {2 (8 A+C) \tan (c+d x) \sec ^2(c+d x)}{35 a d (a \cos (c+d x)+a)^3}-\frac {(A+C) \tan (c+d x) \sec ^2(c+d x)}{7 d (a \cos (c+d x)+a)^4} \]
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Rubi [A] time = 0.71, antiderivative size = 257, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {3042, 2978, 2748, 3767, 3768, 3770} \[ \frac {4 (454 A+83 C) \tan ^3(c+d x)}{105 a^4 d}+\frac {4 (454 A+83 C) \tan (c+d x)}{35 a^4 d}-\frac {2 (11 A+2 C) \tanh ^{-1}(\sin (c+d x))}{a^4 d}-\frac {2 (11 A+2 C) \tan (c+d x) \sec (c+d x)}{a^4 d}-\frac {4 (11 A+2 C) \tan (c+d x) \sec ^2(c+d x)}{3 a^4 d (\cos (c+d x)+1)}-\frac {(178 A+31 C) \tan (c+d x) \sec ^2(c+d x)}{105 a^4 d (\cos (c+d x)+1)^2}-\frac {2 (8 A+C) \tan (c+d x) \sec ^2(c+d x)}{35 a d (a \cos (c+d x)+a)^3}-\frac {(A+C) \tan (c+d x) \sec ^2(c+d x)}{7 d (a \cos (c+d x)+a)^4} \]
Antiderivative was successfully verified.
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Rule 2748
Rule 2978
Rule 3042
Rule 3767
Rule 3768
Rule 3770
Rubi steps
\begin {align*} \int \frac {\left (A+C \cos ^2(c+d x)\right ) \sec ^4(c+d x)}{(a+a \cos (c+d x))^4} \, dx &=-\frac {(A+C) \sec ^2(c+d x) \tan (c+d x)}{7 d (a+a \cos (c+d x))^4}+\frac {\int \frac {(a (10 A+3 C)-a (6 A-C) \cos (c+d x)) \sec ^4(c+d x)}{(a+a \cos (c+d x))^3} \, dx}{7 a^2}\\ &=-\frac {(A+C) \sec ^2(c+d x) \tan (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac {2 (8 A+C) \sec ^2(c+d x) \tan (c+d x)}{35 a d (a+a \cos (c+d x))^3}+\frac {\int \frac {\left (7 a^2 (14 A+3 C)-10 a^2 (8 A+C) \cos (c+d x)\right ) \sec ^4(c+d x)}{(a+a \cos (c+d x))^2} \, dx}{35 a^4}\\ &=-\frac {(178 A+31 C) \sec ^2(c+d x) \tan (c+d x)}{105 a^4 d (1+\cos (c+d x))^2}-\frac {(A+C) \sec ^2(c+d x) \tan (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac {2 (8 A+C) \sec ^2(c+d x) \tan (c+d x)}{35 a d (a+a \cos (c+d x))^3}+\frac {\int \frac {\left (12 a^3 (69 A+13 C)-4 a^3 (178 A+31 C) \cos (c+d x)\right ) \sec ^4(c+d x)}{a+a \cos (c+d x)} \, dx}{105 a^6}\\ &=-\frac {(178 A+31 C) \sec ^2(c+d x) \tan (c+d x)}{105 a^4 d (1+\cos (c+d x))^2}-\frac {(A+C) \sec ^2(c+d x) \tan (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac {2 (8 A+C) \sec ^2(c+d x) \tan (c+d x)}{35 a d (a+a \cos (c+d x))^3}-\frac {4 (11 A+2 C) \sec ^2(c+d x) \tan (c+d x)}{3 d \left (a^4+a^4 \cos (c+d x)\right )}+\frac {\int \left (12 a^4 (454 A+83 C)-420 a^4 (11 A+2 C) \cos (c+d x)\right ) \sec ^4(c+d x) \, dx}{105 a^8}\\ &=-\frac {(178 A+31 C) \sec ^2(c+d x) \tan (c+d x)}{105 a^4 d (1+\cos (c+d x))^2}-\frac {(A+C) \sec ^2(c+d x) \tan (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac {2 (8 A+C) \sec ^2(c+d x) \tan (c+d x)}{35 a d (a+a \cos (c+d x))^3}-\frac {4 (11 A+2 C) \sec ^2(c+d x) \tan (c+d x)}{3 d \left (a^4+a^4 \cos (c+d x)\right )}-\frac {(4 (11 A+2 C)) \int \sec ^3(c+d x) \, dx}{a^4}+\frac {(4 (454 A+83 C)) \int \sec ^4(c+d x) \, dx}{35 a^4}\\ &=-\frac {2 (11 A+2 C) \sec (c+d x) \tan (c+d x)}{a^4 d}-\frac {(178 A+31 C) \sec ^2(c+d x) \tan (c+d x)}{105 a^4 d (1+\cos (c+d x))^2}-\frac {(A+C) \sec ^2(c+d x) \tan (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac {2 (8 A+C) \sec ^2(c+d x) \tan (c+d x)}{35 a d (a+a \cos (c+d x))^3}-\frac {4 (11 A+2 C) \sec ^2(c+d x) \tan (c+d x)}{3 d \left (a^4+a^4 \cos (c+d x)\right )}-\frac {(2 (11 A+2 C)) \int \sec (c+d x) \, dx}{a^4}-\frac {(4 (454 A+83 C)) \operatorname {Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (c+d x)\right )}{35 a^4 d}\\ &=-\frac {2 (11 A+2 C) \tanh ^{-1}(\sin (c+d x))}{a^4 d}+\frac {4 (454 A+83 C) \tan (c+d x)}{35 a^4 d}-\frac {2 (11 A+2 C) \sec (c+d x) \tan (c+d x)}{a^4 d}-\frac {(178 A+31 C) \sec ^2(c+d x) \tan (c+d x)}{105 a^4 d (1+\cos (c+d x))^2}-\frac {(A+C) \sec ^2(c+d x) \tan (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac {2 (8 A+C) \sec ^2(c+d x) \tan (c+d x)}{35 a d (a+a \cos (c+d x))^3}-\frac {4 (11 A+2 C) \sec ^2(c+d x) \tan (c+d x)}{3 d \left (a^4+a^4 \cos (c+d x)\right )}+\frac {4 (454 A+83 C) \tan ^3(c+d x)}{105 a^4 d}\\ \end {align*}
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Mathematica [A] time = 4.59, size = 361, normalized size = 1.40 \[ \frac {2 \cos \left (\frac {1}{2} (c+d x)\right ) \left (4 (412 A+139 C) \tan \left (\frac {c}{2}\right ) \cos ^5\left (\frac {1}{2} (c+d x)\right )+6 (31 A+17 C) \tan \left (\frac {c}{2}\right ) \cos ^3\left (\frac {1}{2} (c+d x)\right )+15 (A+C) \tan \left (\frac {c}{2}\right ) \cos \left (\frac {1}{2} (c+d x)\right )+15 (A+C) \sec \left (\frac {c}{2}\right ) \sin \left (\frac {d x}{2}\right )+8 (2512 A+559 C) \sec \left (\frac {c}{2}\right ) \sin \left (\frac {d x}{2}\right ) \cos ^6\left (\frac {1}{2} (c+d x)\right )+4 (412 A+139 C) \sec \left (\frac {c}{2}\right ) \sin \left (\frac {d x}{2}\right ) \cos ^4\left (\frac {1}{2} (c+d x)\right )+6 (31 A+17 C) \sec \left (\frac {c}{2}\right ) \sin \left (\frac {d x}{2}\right ) \cos ^2\left (\frac {1}{2} (c+d x)\right )+280 \cos ^7\left (\frac {1}{2} (c+d x)\right ) \left (\sec (c) \sin (d x) \sec (c+d x) \left (A \sec ^2(c+d x)-6 A \sec (c+d x)+32 A+3 C\right )+6 (11 A+2 C) \left (\log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )\right )+A \tan (c) \sec ^2(c+d x)-6 A \tan (c) \sec (c+d x)\right )\right )}{105 a^4 d (\cos (c+d x)+1)^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.76, size = 372, normalized size = 1.45 \[ -\frac {105 \, {\left ({\left (11 \, A + 2 \, C\right )} \cos \left (d x + c\right )^{7} + 4 \, {\left (11 \, A + 2 \, C\right )} \cos \left (d x + c\right )^{6} + 6 \, {\left (11 \, A + 2 \, C\right )} \cos \left (d x + c\right )^{5} + 4 \, {\left (11 \, A + 2 \, C\right )} \cos \left (d x + c\right )^{4} + {\left (11 \, A + 2 \, C\right )} \cos \left (d x + c\right )^{3}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 105 \, {\left ({\left (11 \, A + 2 \, C\right )} \cos \left (d x + c\right )^{7} + 4 \, {\left (11 \, A + 2 \, C\right )} \cos \left (d x + c\right )^{6} + 6 \, {\left (11 \, A + 2 \, C\right )} \cos \left (d x + c\right )^{5} + 4 \, {\left (11 \, A + 2 \, C\right )} \cos \left (d x + c\right )^{4} + {\left (11 \, A + 2 \, C\right )} \cos \left (d x + c\right )^{3}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - {\left (8 \, {\left (454 \, A + 83 \, C\right )} \cos \left (d x + c\right )^{6} + 2 \, {\left (6109 \, A + 1118 \, C\right )} \cos \left (d x + c\right )^{5} + 4 \, {\left (3592 \, A + 659 \, C\right )} \cos \left (d x + c\right )^{4} + 8 \, {\left (799 \, A + 148 \, C\right )} \cos \left (d x + c\right )^{3} + 35 \, {\left (14 \, A + 3 \, C\right )} \cos \left (d x + c\right )^{2} - 70 \, A \cos \left (d x + c\right ) + 35 \, A\right )} \sin \left (d x + c\right )}{105 \, {\left (a^{4} d \cos \left (d x + c\right )^{7} + 4 \, a^{4} d \cos \left (d x + c\right )^{6} + 6 \, a^{4} d \cos \left (d x + c\right )^{5} + 4 \, a^{4} d \cos \left (d x + c\right )^{4} + a^{4} d \cos \left (d x + c\right )^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.58, size = 295, normalized size = 1.15 \[ -\frac {\frac {1680 \, {\left (11 \, A + 2 \, C\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right )}{a^{4}} - \frac {1680 \, {\left (11 \, A + 2 \, C\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right )}{a^{4}} + \frac {560 \, {\left (39 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 3 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 62 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 6 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 27 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{3} a^{4}} - \frac {15 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 15 \, C a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 231 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 147 \, C a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 2065 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 805 \, C a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 21945 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 5145 \, C a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{28}}}{840 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.24, size = 418, normalized size = 1.63 \[ \frac {\left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) A}{56 d \,a^{4}}+\frac {C \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{56 d \,a^{4}}+\frac {11 A \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{40 d \,a^{4}}+\frac {7 C \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{40 d \,a^{4}}+\frac {59 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) A}{24 d \,a^{4}}+\frac {23 C \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d \,a^{4}}+\frac {209 A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d \,a^{4}}+\frac {49 C \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d \,a^{4}}-\frac {13 A}{d \,a^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {C}{d \,a^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {22 A \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d \,a^{4}}+\frac {4 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) C}{d \,a^{4}}-\frac {A}{3 d \,a^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {5 A}{2 d \,a^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {22 A \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d \,a^{4}}-\frac {4 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) C}{d \,a^{4}}-\frac {13 A}{d \,a^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {C}{d \,a^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {A}{3 d \,a^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {5 A}{2 d \,a^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.35, size = 461, normalized size = 1.79 \[ \frac {A {\left (\frac {560 \, {\left (\frac {27 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {62 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {39 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}\right )}}{a^{4} - \frac {3 \, a^{4} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {3 \, a^{4} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {a^{4} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}} + \frac {\frac {21945 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {2065 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {231 \, \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 {18480 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a^{4}} + \frac {18480 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a^{4}}\right )} + C {\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 )}}{840 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.94, size = 303, normalized size = 1.18 \[ \frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {5\,\left (A+C\right )}{2\,a^4}+\frac {21\,A+C}{2\,a^4}+\frac {5\,\left (7\,A+3\,C\right )}{4\,a^4}+\frac {35\,A-5\,C}{8\,a^4}\right )}{d}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (\frac {A+C}{10\,a^4}+\frac {7\,A+3\,C}{40\,a^4}\right )}{d}-\frac {\left (26\,A+2\,C\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (-\frac {124\,A}{3}-4\,C\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (18\,A+2\,C\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left (a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-3\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+3\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-a^4\right )}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {5\,\left (A+C\right )}{12\,a^4}+\frac {21\,A+C}{24\,a^4}+\frac {7\,A+3\,C}{6\,a^4}\right )}{d}-\frac {4\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (11\,A+2\,C\right )}{a^4\,d}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (A+C\right )}{56\,a^4\,d} \]
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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