Optimal. Leaf size=232 \[ -\frac {32 (5 A+54 C) \tan (c+d x)}{105 a^4 d}+\frac {(2 A+21 C) \tanh ^{-1}(\sin (c+d x))}{2 a^4 d}-\frac {(10 A+129 C) \tan (c+d x) \sec ^3(c+d x)}{105 a^4 d (\sec (c+d x)+1)^2}-\frac {16 (5 A+54 C) \tan (c+d x) \sec ^2(c+d x)}{105 a^4 d (\sec (c+d x)+1)}+\frac {(2 A+21 C) \tan (c+d x) \sec (c+d x)}{2 a^4 d}-\frac {(A+C) \tan (c+d x) \sec ^5(c+d x)}{7 d (a \sec (c+d x)+a)^4}-\frac {2 C \tan (c+d x) \sec ^4(c+d x)}{5 a d (a \sec (c+d x)+a)^3} \]
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Rubi [A] time = 0.65, antiderivative size = 232, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.212, Rules used = {4085, 4019, 3787, 3767, 8, 3768, 3770} \[ -\frac {32 (5 A+54 C) \tan (c+d x)}{105 a^4 d}+\frac {(2 A+21 C) \tanh ^{-1}(\sin (c+d x))}{2 a^4 d}-\frac {(10 A+129 C) \tan (c+d x) \sec ^3(c+d x)}{105 a^4 d (\sec (c+d x)+1)^2}-\frac {16 (5 A+54 C) \tan (c+d x) \sec ^2(c+d x)}{105 a^4 d (\sec (c+d x)+1)}+\frac {(2 A+21 C) \tan (c+d x) \sec (c+d x)}{2 a^4 d}-\frac {(A+C) \tan (c+d x) \sec ^5(c+d x)}{7 d (a \sec (c+d x)+a)^4}-\frac {2 C \tan (c+d x) \sec ^4(c+d x)}{5 a d (a \sec (c+d x)+a)^3} \]
Antiderivative was successfully verified.
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Rule 8
Rule 3767
Rule 3768
Rule 3770
Rule 3787
Rule 4019
Rule 4085
Rubi steps
\begin {align*} \int \frac {\sec ^5(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^4} \, dx &=-\frac {(A+C) \sec ^5(c+d x) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}-\frac {\int \frac {\sec ^5(c+d x) (-a (2 A-5 C)-a (2 A+9 C) \sec (c+d x))}{(a+a \sec (c+d x))^3} \, dx}{7 a^2}\\ &=-\frac {(A+C) \sec ^5(c+d x) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}-\frac {2 C \sec ^4(c+d x) \tan (c+d x)}{5 a d (a+a \sec (c+d x))^3}-\frac {\int \frac {\sec ^4(c+d x) \left (56 a^2 C-a^2 (10 A+73 C) \sec (c+d x)\right )}{(a+a \sec (c+d x))^2} \, dx}{35 a^4}\\ &=-\frac {(10 A+129 C) \sec ^3(c+d x) \tan (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}-\frac {(A+C) \sec ^5(c+d x) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}-\frac {2 C \sec ^4(c+d x) \tan (c+d x)}{5 a d (a+a \sec (c+d x))^3}-\frac {\int \frac {\sec ^3(c+d x) \left (3 a^3 (10 A+129 C)-a^3 (50 A+477 C) \sec (c+d x)\right )}{a+a \sec (c+d x)} \, dx}{105 a^6}\\ &=-\frac {(10 A+129 C) \sec ^3(c+d x) \tan (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}-\frac {(A+C) \sec ^5(c+d x) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}-\frac {2 C \sec ^4(c+d x) \tan (c+d x)}{5 a d (a+a \sec (c+d x))^3}-\frac {16 (5 A+54 C) \sec ^2(c+d x) \tan (c+d x)}{105 d \left (a^4+a^4 \sec (c+d x)\right )}-\frac {\int \sec ^2(c+d x) \left (32 a^4 (5 A+54 C)-105 a^4 (2 A+21 C) \sec (c+d x)\right ) \, dx}{105 a^8}\\ &=-\frac {(10 A+129 C) \sec ^3(c+d x) \tan (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}-\frac {(A+C) \sec ^5(c+d x) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}-\frac {2 C \sec ^4(c+d x) \tan (c+d x)}{5 a d (a+a \sec (c+d x))^3}-\frac {16 (5 A+54 C) \sec ^2(c+d x) \tan (c+d x)}{105 d \left (a^4+a^4 \sec (c+d x)\right )}+\frac {(2 A+21 C) \int \sec ^3(c+d x) \, dx}{a^4}-\frac {(32 (5 A+54 C)) \int \sec ^2(c+d x) \, dx}{105 a^4}\\ &=\frac {(2 A+21 C) \sec (c+d x) \tan (c+d x)}{2 a^4 d}-\frac {(10 A+129 C) \sec ^3(c+d x) \tan (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}-\frac {(A+C) \sec ^5(c+d x) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}-\frac {2 C \sec ^4(c+d x) \tan (c+d x)}{5 a d (a+a \sec (c+d x))^3}-\frac {16 (5 A+54 C) \sec ^2(c+d x) \tan (c+d x)}{105 d \left (a^4+a^4 \sec (c+d x)\right )}+\frac {(2 A+21 C) \int \sec (c+d x) \, dx}{2 a^4}+\frac {(32 (5 A+54 C)) \operatorname {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{105 a^4 d}\\ &=\frac {(2 A+21 C) \tanh ^{-1}(\sin (c+d x))}{2 a^4 d}-\frac {32 (5 A+54 C) \tan (c+d x)}{105 a^4 d}+\frac {(2 A+21 C) \sec (c+d x) \tan (c+d x)}{2 a^4 d}-\frac {(10 A+129 C) \sec ^3(c+d x) \tan (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}-\frac {(A+C) \sec ^5(c+d x) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}-\frac {2 C \sec ^4(c+d x) \tan (c+d x)}{5 a d (a+a \sec (c+d x))^3}-\frac {16 (5 A+54 C) \sec ^2(c+d x) \tan (c+d x)}{105 d \left (a^4+a^4 \sec (c+d x)\right )}\\ \end {align*}
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Mathematica [B] time = 4.40, size = 746, normalized size = 3.22 \[ -\frac {\cos \left (\frac {1}{2} (c+d x)\right ) \sec ^2(c+d x) \left (A+C \sec ^2(c+d x)\right ) \left (\sec \left (\frac {c}{2}\right ) \sec (c) \sec ^2(c+d x) \left (-17220 A \sin \left (c-\frac {d x}{2}\right )+17220 A \sin \left (c+\frac {d x}{2}\right )-14140 A \sin \left (2 c+\frac {d x}{2}\right )-9800 A \sin \left (c+\frac {3 d x}{2}\right )+15160 A \sin \left (2 c+\frac {3 d x}{2}\right )-9800 A \sin \left (3 c+\frac {3 d x}{2}\right )+10920 A \sin \left (c+\frac {5 d x}{2}\right )-4760 A \sin \left (2 c+\frac {5 d x}{2}\right )+10920 A \sin \left (3 c+\frac {5 d x}{2}\right )-4760 A \sin \left (4 c+\frac {5 d x}{2}\right )+5890 A \sin \left (2 c+\frac {7 d x}{2}\right )-1470 A \sin \left (3 c+\frac {7 d x}{2}\right )+5890 A \sin \left (4 c+\frac {7 d x}{2}\right )-1470 A \sin \left (5 c+\frac {7 d x}{2}\right )+2030 A \sin \left (3 c+\frac {9 d x}{2}\right )-210 A \sin \left (4 c+\frac {9 d x}{2}\right )+2030 A \sin \left (5 c+\frac {9 d x}{2}\right )-210 A \sin \left (6 c+\frac {9 d x}{2}\right )+320 A \sin \left (4 c+\frac {11 d x}{2}\right )+320 A \sin \left (6 c+\frac {11 d x}{2}\right )-14 (1010 A+5229 C) \sin \left (\frac {d x}{2}\right )+4 (3790 A+41667 C) \sin \left (\frac {3 d x}{2}\right )-183162 C \sin \left (c-\frac {d x}{2}\right )+100842 C \sin \left (c+\frac {d x}{2}\right )-155526 C \sin \left (2 c+\frac {d x}{2}\right )-37380 C \sin \left (c+\frac {3 d x}{2}\right )+101148 C \sin \left (2 c+\frac {3 d x}{2}\right )-102900 C \sin \left (3 c+\frac {3 d x}{2}\right )+119364 C \sin \left (c+\frac {5 d x}{2}\right )-8820 C \sin \left (2 c+\frac {5 d x}{2}\right )+78204 C \sin \left (3 c+\frac {5 d x}{2}\right )-49980 C \sin \left (4 c+\frac {5 d x}{2}\right )+64053 C \sin \left (2 c+\frac {7 d x}{2}\right )+3885 C \sin \left (3 c+\frac {7 d x}{2}\right )+44733 C \sin \left (4 c+\frac {7 d x}{2}\right )-15435 C \sin \left (5 c+\frac {7 d x}{2}\right )+21987 C \sin \left (3 c+\frac {9 d x}{2}\right )+3675 C \sin \left (4 c+\frac {9 d x}{2}\right )+16107 C \sin \left (5 c+\frac {9 d x}{2}\right )-2205 C \sin \left (6 c+\frac {9 d x}{2}\right )+3456 C \sin \left (4 c+\frac {11 d x}{2}\right )+840 C \sin \left (5 c+\frac {11 d x}{2}\right )+2616 C \sin \left (6 c+\frac {11 d x}{2}\right )\right )+53760 (2 A+21 C) \cos ^7\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 )}{3360 a^4 d (\sec (c+d x)+1)^4 (A \cos (2 (c+d x))+A+2 C)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 354, normalized size = 1.53 \[ \frac {105 \, {\left ({\left (2 \, A + 21 \, C\right )} \cos \left (d x + c\right )^{6} + 4 \, {\left (2 \, A + 21 \, C\right )} \cos \left (d x + c\right )^{5} + 6 \, {\left (2 \, A + 21 \, C\right )} \cos \left (d x + c\right )^{4} + 4 \, {\left (2 \, A + 21 \, C\right )} \cos \left (d x + c\right )^{3} + {\left (2 \, A + 21 \, C\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 105 \, {\left ({\left (2 \, A + 21 \, C\right )} \cos \left (d x + c\right )^{6} + 4 \, {\left (2 \, A + 21 \, C\right )} \cos \left (d x + c\right )^{5} + 6 \, {\left (2 \, A + 21 \, C\right )} \cos \left (d x + c\right )^{4} + 4 \, {\left (2 \, A + 21 \, C\right )} \cos \left (d x + c\right )^{3} + {\left (2 \, A + 21 \, C\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (64 \, {\left (5 \, A + 54 \, C\right )} \cos \left (d x + c\right )^{5} + {\left (1070 \, A + 11619 \, C\right )} \cos \left (d x + c\right )^{4} + 4 \, {\left (310 \, A + 3411 \, C\right )} \cos \left (d x + c\right )^{3} + 4 \, {\left (130 \, A + 1509 \, C\right )} \cos \left (d x + c\right )^{2} + 420 \, C \cos \left (d x + c\right ) - 105 \, C\right )} \sin \left (d x + c\right )}{420 \, {\left (a^{4} d \cos \left (d x + c\right )^{6} + 4 \, a^{4} d \cos \left (d x + c\right )^{5} + 6 \, a^{4} d \cos \left (d x + c\right )^{4} + 4 \, a^{4} d \cos \left (d x + c\right )^{3} + a^{4} d \cos \left (d x + c\right )^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.51, size = 241, normalized size = 1.04 \[ \frac {\frac {420 \, {\left (2 \, A + 21 \, C\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right )}{a^{4}} - \frac {420 \, {\left (2 \, A + 21 \, C\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right )}{a^{4}} + \frac {840 \, {\left (9 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 7 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{2} a^{4}} - \frac {15 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 15 \, C a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 105 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 189 \, C a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 385 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 1365 \, C a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 1575 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 11655 \, C a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{28}}}{840 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.67, size = 329, normalized size = 1.42 \[ -\frac {\left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) A}{56 d \,a^{4}}-\frac {C \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{56 d \,a^{4}}-\frac {A \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d \,a^{4}}-\frac {9 C \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{40 d \,a^{4}}-\frac {11 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) A}{24 d \,a^{4}}-\frac {13 C \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d \,a^{4}}-\frac {15 A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d \,a^{4}}-\frac {111 C \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d \,a^{4}}-\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) A}{d \,a^{4}}-\frac {21 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) C}{2 d \,a^{4}}+\frac {C}{2 d \,a^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {9 C}{2 d \,a^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) A}{d \,a^{4}}+\frac {21 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) C}{2 d \,a^{4}}-\frac {C}{2 d \,a^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {9 C}{2 d \,a^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.48, size = 372, normalized size = 1.60 \[ -\frac {3 \, C {\left (\frac {280 \, {\left (\frac {7 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {9 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}\right )}}{a^{4} - \frac {2 \, a^{4} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {a^{4} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}} + \frac {\frac {3885 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {455 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {63 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {5 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a^{4}} - \frac {2940 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a^{4}} + \frac {2940 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a^{4}}\right )} + 5 \, A {\left (\frac {\frac {315 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {77 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {21 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {3 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a^{4}} - \frac {168 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a^{4}} + \frac {168 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a^{4}}\right )}}{840 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.64, size = 256, normalized size = 1.10 \[ \frac {2\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (A+\frac {21\,C}{2}\right )}{a^4\,d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (\frac {3\,\left (A+C\right )}{40\,a^4}+\frac {2\,A+6\,C}{40\,a^4}\right )}{d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {A+C}{4\,a^4}-\frac {A-15\,C}{24\,a^4}+\frac {2\,A+6\,C}{8\,a^4}\right )}{d}-\frac {7\,C\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-9\,C\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{d\,\left (a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-2\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+a^4\right )}-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {5\,\left (A+C\right )}{4\,a^4}-\frac {3\,\left (A-15\,C\right )}{8\,a^4}+\frac {3\,\left (2\,A+6\,C\right )}{4\,a^4}-\frac {4\,A-20\,C}{8\,a^4}\right )}{d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (A+C\right )}{56\,a^4\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {A \sec ^{5}{\left (c + d x \right )}}{\sec ^{4}{\left (c + d x \right )} + 4 \sec ^{3}{\left (c + d x \right )} + 6 \sec ^{2}{\left (c + d x \right )} + 4 \sec {\left (c + d x \right )} + 1}\, dx + \int \frac {C \sec ^{7}{\left (c + d x \right )}}{\sec ^{4}{\left (c + d x \right )} + 4 \sec ^{3}{\left (c + d x \right )} + 6 \sec ^{2}{\left (c + d x \right )} + 4 \sec {\left (c + d x \right )} + 1}\, dx}{a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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