Optimal. Leaf size=225 \[ \frac {4 (34 A+9 C) \tan ^3(c+d x)}{15 a^3 d}+\frac {4 (34 A+9 C) \tan (c+d x)}{5 a^3 d}-\frac {(23 A+6 C) \tanh ^{-1}(\sin (c+d x))}{2 a^3 d}-\frac {(23 A+6 C) \tan (c+d x) \sec (c+d x)}{2 a^3 d}-\frac {(23 A+6 C) \tan (c+d x) \sec ^2(c+d x)}{3 d \left (a^3 \cos (c+d x)+a^3\right )}-\frac {(13 A+3 C) \tan (c+d x) \sec ^2(c+d x)}{15 a d (a \cos (c+d x)+a)^2}-\frac {(A+C) \tan (c+d x) \sec ^2(c+d x)}{5 d (a \cos (c+d x)+a)^3} \]
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Rubi [A] time = 0.55, antiderivative size = 225, normalized size of antiderivative = 1.00, number of steps used = 8, 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} \[ \frac {4 (34 A+9 C) \tan ^3(c+d x)}{15 a^3 d}+\frac {4 (34 A+9 C) \tan (c+d x)}{5 a^3 d}-\frac {(23 A+6 C) \tanh ^{-1}(\sin (c+d x))}{2 a^3 d}-\frac {(23 A+6 C) \tan (c+d x) \sec (c+d x)}{2 a^3 d}-\frac {(23 A+6 C) \tan (c+d x) \sec ^2(c+d x)}{3 d \left (a^3 \cos (c+d x)+a^3\right )}-\frac {(13 A+3 C) \tan (c+d x) \sec ^2(c+d x)}{15 a d (a \cos (c+d x)+a)^2}-\frac {(A+C) \tan (c+d x) \sec ^2(c+d x)}{5 d (a \cos (c+d x)+a)^3} \]
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))^3} \, dx &=-\frac {(A+C) \sec ^2(c+d x) \tan (c+d x)}{5 d (a+a \cos (c+d x))^3}+\frac {\int \frac {(a (8 A+3 C)-5 a A \cos (c+d x)) \sec ^4(c+d x)}{(a+a \cos (c+d x))^2} \, dx}{5 a^2}\\ &=-\frac {(A+C) \sec ^2(c+d x) \tan (c+d x)}{5 d (a+a \cos (c+d x))^3}-\frac {(13 A+3 C) \sec ^2(c+d x) \tan (c+d x)}{15 a d (a+a \cos (c+d x))^2}+\frac {\int \frac {\left (9 a^2 (7 A+2 C)-4 a^2 (13 A+3 C) \cos (c+d x)\right ) \sec ^4(c+d x)}{a+a \cos (c+d x)} \, dx}{15 a^4}\\ &=-\frac {(A+C) \sec ^2(c+d x) \tan (c+d x)}{5 d (a+a \cos (c+d x))^3}-\frac {(13 A+3 C) \sec ^2(c+d x) \tan (c+d x)}{15 a d (a+a \cos (c+d x))^2}-\frac {(23 A+6 C) \sec ^2(c+d x) \tan (c+d x)}{3 d \left (a^3+a^3 \cos (c+d x)\right )}+\frac {\int \left (12 a^3 (34 A+9 C)-15 a^3 (23 A+6 C) \cos (c+d x)\right ) \sec ^4(c+d x) \, dx}{15 a^6}\\ &=-\frac {(A+C) \sec ^2(c+d x) \tan (c+d x)}{5 d (a+a \cos (c+d x))^3}-\frac {(13 A+3 C) \sec ^2(c+d x) \tan (c+d x)}{15 a d (a+a \cos (c+d x))^2}-\frac {(23 A+6 C) \sec ^2(c+d x) \tan (c+d x)}{3 d \left (a^3+a^3 \cos (c+d x)\right )}-\frac {(23 A+6 C) \int \sec ^3(c+d x) \, dx}{a^3}+\frac {(4 (34 A+9 C)) \int \sec ^4(c+d x) \, dx}{5 a^3}\\ &=-\frac {(23 A+6 C) \sec (c+d x) \tan (c+d x)}{2 a^3 d}-\frac {(A+C) \sec ^2(c+d x) \tan (c+d x)}{5 d (a+a \cos (c+d x))^3}-\frac {(13 A+3 C) \sec ^2(c+d x) \tan (c+d x)}{15 a d (a+a \cos (c+d x))^2}-\frac {(23 A+6 C) \sec ^2(c+d x) \tan (c+d x)}{3 d \left (a^3+a^3 \cos (c+d x)\right )}-\frac {(23 A+6 C) \int \sec (c+d x) \, dx}{2 a^3}-\frac {(4 (34 A+9 C)) \operatorname {Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (c+d x)\right )}{5 a^3 d}\\ &=-\frac {(23 A+6 C) \tanh ^{-1}(\sin (c+d x))}{2 a^3 d}+\frac {4 (34 A+9 C) \tan (c+d x)}{5 a^3 d}-\frac {(23 A+6 C) \sec (c+d x) \tan (c+d x)}{2 a^3 d}-\frac {(A+C) \sec ^2(c+d x) \tan (c+d x)}{5 d (a+a \cos (c+d x))^3}-\frac {(13 A+3 C) \sec ^2(c+d x) \tan (c+d x)}{15 a d (a+a \cos (c+d x))^2}-\frac {(23 A+6 C) \sec ^2(c+d x) \tan (c+d x)}{3 d \left (a^3+a^3 \cos (c+d x)\right )}+\frac {4 (34 A+9 C) \tan ^3(c+d x)}{15 a^3 d}\\ \end {align*}
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Mathematica [B] time = 6.48, size = 798, normalized size = 3.55 \[ \frac {4 (23 A+6 C) \log \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )-\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right ) \cos ^6\left (\frac {c}{2}+\frac {d x}{2}\right )}{d (\cos (c+d x) a+a)^3}-\frac {4 (23 A+6 C) \log \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )+\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right ) \cos ^6\left (\frac {c}{2}+\frac {d x}{2}\right )}{d (\cos (c+d x) a+a)^3}+\frac {\sec \left (\frac {c}{2}\right ) \sec (c) \sec ^3(c+d x) \left (-2484 A \sin \left (\frac {d x}{2}\right )-1764 C \sin \left (\frac {d x}{2}\right )+12622 A \sin \left (\frac {3 d x}{2}\right )+3372 C \sin \left (\frac {3 d x}{2}\right )-13340 A \sin \left (c-\frac {d x}{2}\right )-3480 C \sin \left (c-\frac {d x}{2}\right )+4140 A \sin \left (c+\frac {d x}{2}\right )+2100 C \sin \left (c+\frac {d x}{2}\right )-11684 A \sin \left (2 c+\frac {d x}{2}\right )-3144 C \sin \left (2 c+\frac {d x}{2}\right )-450 A \sin \left (c+\frac {3 d x}{2}\right )-960 C \sin \left (c+\frac {3 d x}{2}\right )+5022 A \sin \left (2 c+\frac {3 d x}{2}\right )+2232 C \sin \left (2 c+\frac {3 d x}{2}\right )-8050 A \sin \left (3 c+\frac {3 d x}{2}\right )-2100 C \sin \left (3 c+\frac {3 d x}{2}\right )+9230 A \sin \left (c+\frac {5 d x}{2}\right )+2460 C \sin \left (c+\frac {5 d x}{2}\right )+630 A \sin \left (2 c+\frac {5 d x}{2}\right )-390 C \sin \left (2 c+\frac {5 d x}{2}\right )+4230 A \sin \left (3 c+\frac {5 d x}{2}\right )+1710 C \sin \left (3 c+\frac {5 d x}{2}\right )-4370 A \sin \left (4 c+\frac {5 d x}{2}\right )-1140 C \sin \left (4 c+\frac {5 d x}{2}\right )+5347 A \sin \left (2 c+\frac {7 d x}{2}\right )+1422 C \sin \left (2 c+\frac {7 d x}{2}\right )+875 A \sin \left (3 c+\frac {7 d x}{2}\right )-60 C \sin \left (3 c+\frac {7 d x}{2}\right )+2747 A \sin \left (4 c+\frac {7 d x}{2}\right )+1032 C \sin \left (4 c+\frac {7 d x}{2}\right )-1725 A \sin \left (5 c+\frac {7 d x}{2}\right )-450 C \sin \left (5 c+\frac {7 d x}{2}\right )+2375 A \sin \left (3 c+\frac {9 d x}{2}\right )+630 C \sin \left (3 c+\frac {9 d x}{2}\right )+655 A \sin \left (4 c+\frac {9 d x}{2}\right )+60 C \sin \left (4 c+\frac {9 d x}{2}\right )+1375 A \sin \left (5 c+\frac {9 d x}{2}\right )+480 C \sin \left (5 c+\frac {9 d x}{2}\right )-345 A \sin \left (6 c+\frac {9 d x}{2}\right )-90 C \sin \left (6 c+\frac {9 d x}{2}\right )+544 A \sin \left (4 c+\frac {11 d x}{2}\right )+144 C \sin \left (4 c+\frac {11 d x}{2}\right )+200 A \sin \left (5 c+\frac {11 d x}{2}\right )+30 C \sin \left (5 c+\frac {11 d x}{2}\right )+344 A \sin \left (6 c+\frac {11 d x}{2}\right )+114 C \sin \left (6 c+\frac {11 d x}{2}\right )\right ) \cos \left (\frac {c}{2}+\frac {d x}{2}\right )}{960 d (\cos (c+d x) a+a)^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.04, size = 306, normalized size = 1.36 \[ -\frac {15 \, {\left ({\left (23 \, A + 6 \, C\right )} \cos \left (d x + c\right )^{6} + 3 \, {\left (23 \, A + 6 \, C\right )} \cos \left (d x + c\right )^{5} + 3 \, {\left (23 \, A + 6 \, C\right )} \cos \left (d x + c\right )^{4} + {\left (23 \, A + 6 \, C\right )} \cos \left (d x + c\right )^{3}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 15 \, {\left ({\left (23 \, A + 6 \, C\right )} \cos \left (d x + c\right )^{6} + 3 \, {\left (23 \, A + 6 \, C\right )} \cos \left (d x + c\right )^{5} + 3 \, {\left (23 \, A + 6 \, C\right )} \cos \left (d x + c\right )^{4} + {\left (23 \, A + 6 \, C\right )} \cos \left (d x + c\right )^{3}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (16 \, {\left (34 \, A + 9 \, C\right )} \cos \left (d x + c\right )^{5} + 9 \, {\left (143 \, A + 38 \, C\right )} \cos \left (d x + c\right )^{4} + {\left (869 \, A + 234 \, C\right )} \cos \left (d x + c\right )^{3} + 5 \, {\left (19 \, A + 6 \, C\right )} \cos \left (d x + c\right )^{2} - 15 \, A \cos \left (d x + c\right ) + 10 \, A\right )} \sin \left (d x + c\right )}{60 \, {\left (a^{3} d \cos \left (d x + c\right )^{6} + 3 \, a^{3} d \cos \left (d x + c\right )^{5} + 3 \, a^{3} d \cos \left (d x + c\right )^{4} + a^{3} d \cos \left (d x + c\right )^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.67, size = 261, normalized size = 1.16 \[ -\frac {\frac {30 \, {\left (23 \, A + 6 \, C\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right )}{a^{3}} - \frac {30 \, {\left (23 \, A + 6 \, C\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right )}{a^{3}} + \frac {20 \, {\left (51 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 6 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 76 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 12 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 33 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{3} a^{3}} - \frac {3 \, A a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 3 \, C a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 50 \, A a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 30 \, C a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 735 \, A a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 255 \, C a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{15}}}{60 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.24, size = 378, normalized size = 1.68 \[ \frac {A \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{20 d \,a^{3}}+\frac {C \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{20 d \,a^{3}}+\frac {5 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) A}{6 d \,a^{3}}+\frac {C \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d \,a^{3}}+\frac {49 A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d \,a^{3}}+\frac {17 C \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d \,a^{3}}-\frac {17 A}{2 d \,a^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {C}{d \,a^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {23 A \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 d \,a^{3}}+\frac {3 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) C}{d \,a^{3}}-\frac {A}{3 d \,a^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {2 A}{d \,a^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {23 A \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 d \,a^{3}}-\frac {3 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) C}{d \,a^{3}}-\frac {17 A}{2 d \,a^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {C}{d \,a^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {A}{3 d \,a^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {2 A}{d \,a^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 421, normalized size = 1.87 \[ \frac {A {\left (\frac {20 \, {\left (\frac {33 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {76 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {51 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}\right )}}{a^{3} - \frac {3 \, a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {3 \, a^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {a^{3} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}} + \frac {\frac {735 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {50 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {3 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}}{a^{3}} - \frac {690 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a^{3}} + \frac {690 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a^{3}}\right )} + 3 \, C {\left (\frac {40 \, \sin \left (d x + c\right )}{{\left (a^{3} - \frac {a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )} {\left (\cos \left (d x + c\right ) + 1\right )}} + \frac {\frac {85 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {10 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {\sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}}{a^{3}} - \frac {60 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a^{3}} + \frac {60 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a^{3}}\right )}}{60 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.92, size = 246, normalized size = 1.09 \[ \frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {A+C}{3\,a^3}+\frac {6\,A+2\,C}{12\,a^3}\right )}{d}-\frac {\left (17\,A+2\,C\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (-\frac {76\,A}{3}-4\,C\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (11\,A+2\,C\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left (a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-3\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+3\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-a^3\right )}+\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {5\,\left (A+C\right )}{2\,a^3}+\frac {6\,A+2\,C}{a^3}+\frac {15\,A-C}{4\,a^3}\right )}{d}-\frac {\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (23\,A+6\,C\right )}{a^3\,d}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (A+C\right )}{20\,a^3\,d} \]
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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