Optimal. Leaf size=151 \[ \frac {1}{8} a^4 (48 A+35 B) x+\frac {a^4 A \tanh ^{-1}(\sin (c+d x))}{d}+\frac {5 a^4 (8 A+7 B) \sin (c+d x)}{8 d}+\frac {a B (a+a \cos (c+d x))^3 \sin (c+d x)}{4 d}+\frac {(4 A+7 B) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{12 d}+\frac {(32 A+35 B) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{24 d} \]
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Rubi [A]
time = 0.27, antiderivative size = 151, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 5, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {3055, 3047,
3102, 2814, 3855} \begin {gather*} \frac {5 a^4 (8 A+7 B) \sin (c+d x)}{8 d}+\frac {(32 A+35 B) \sin (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{24 d}+\frac {1}{8} a^4 x (48 A+35 B)+\frac {a^4 A \tanh ^{-1}(\sin (c+d x))}{d}+\frac {(4 A+7 B) \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{12 d}+\frac {a B \sin (c+d x) (a \cos (c+d x)+a)^3}{4 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 2814
Rule 3047
Rule 3055
Rule 3102
Rule 3855
Rubi steps
\begin {align*} \int (a+a \cos (c+d x))^4 (A+B \cos (c+d x)) \sec (c+d x) \, dx &=\frac {a B (a+a \cos (c+d x))^3 \sin (c+d x)}{4 d}+\frac {1}{4} \int (a+a \cos (c+d x))^3 (4 a A+a (4 A+7 B) \cos (c+d x)) \sec (c+d x) \, dx\\ &=\frac {a B (a+a \cos (c+d x))^3 \sin (c+d x)}{4 d}+\frac {(4 A+7 B) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{12 d}+\frac {1}{12} \int (a+a \cos (c+d x))^2 \left (12 a^2 A+a^2 (32 A+35 B) \cos (c+d x)\right ) \sec (c+d x) \, dx\\ &=\frac {a B (a+a \cos (c+d x))^3 \sin (c+d x)}{4 d}+\frac {(4 A+7 B) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{12 d}+\frac {(32 A+35 B) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{24 d}+\frac {1}{24} \int (a+a \cos (c+d x)) \left (24 a^3 A+15 a^3 (8 A+7 B) \cos (c+d x)\right ) \sec (c+d x) \, dx\\ &=\frac {a B (a+a \cos (c+d x))^3 \sin (c+d x)}{4 d}+\frac {(4 A+7 B) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{12 d}+\frac {(32 A+35 B) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{24 d}+\frac {1}{24} \int \left (24 a^4 A+\left (24 a^4 A+15 a^4 (8 A+7 B)\right ) \cos (c+d x)+15 a^4 (8 A+7 B) \cos ^2(c+d x)\right ) \sec (c+d x) \, dx\\ &=\frac {5 a^4 (8 A+7 B) \sin (c+d x)}{8 d}+\frac {a B (a+a \cos (c+d x))^3 \sin (c+d x)}{4 d}+\frac {(4 A+7 B) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{12 d}+\frac {(32 A+35 B) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{24 d}+\frac {1}{24} \int \left (24 a^4 A+3 a^4 (48 A+35 B) \cos (c+d x)\right ) \sec (c+d x) \, dx\\ &=\frac {1}{8} a^4 (48 A+35 B) x+\frac {5 a^4 (8 A+7 B) \sin (c+d x)}{8 d}+\frac {a B (a+a \cos (c+d x))^3 \sin (c+d x)}{4 d}+\frac {(4 A+7 B) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{12 d}+\frac {(32 A+35 B) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{24 d}+\left (a^4 A\right ) \int \sec (c+d x) \, dx\\ &=\frac {1}{8} a^4 (48 A+35 B) x+\frac {a^4 A \tanh ^{-1}(\sin (c+d x))}{d}+\frac {5 a^4 (8 A+7 B) \sin (c+d x)}{8 d}+\frac {a B (a+a \cos (c+d x))^3 \sin (c+d x)}{4 d}+\frac {(4 A+7 B) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{12 d}+\frac {(32 A+35 B) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{24 d}\\ \end {align*}
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Mathematica [A]
time = 0.42, size = 138, normalized size = 0.91 \begin {gather*} \frac {a^4 \left (576 A d x+420 B d x-96 A \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+96 A \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+24 (27 A+28 B) \sin (c+d x)+24 (4 A+7 B) \sin (2 (c+d x))+8 A \sin (3 (c+d x))+32 B \sin (3 (c+d x))+3 B \sin (4 (c+d x))\right )}{96 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.22, size = 208, normalized size = 1.38
method | result | size |
derivativedivides | \(\frac {\frac {A \,a^{4} \left (\cos ^{2}\left (d x +c \right )+2\right ) \sin \left (d x +c \right )}{3}+a^{4} B \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+4 A \,a^{4} \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+\frac {4 a^{4} B \left (\cos ^{2}\left (d x +c \right )+2\right ) \sin \left (d x +c \right )}{3}+6 A \,a^{4} \sin \left (d x +c \right )+6 a^{4} B \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+4 A \,a^{4} \left (d x +c \right )+4 a^{4} B \sin \left (d x +c \right )+A \,a^{4} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+a^{4} B \left (d x +c \right )}{d}\) | \(208\) |
default | \(\frac {\frac {A \,a^{4} \left (\cos ^{2}\left (d x +c \right )+2\right ) \sin \left (d x +c \right )}{3}+a^{4} B \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+4 A \,a^{4} \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+\frac {4 a^{4} B \left (\cos ^{2}\left (d x +c \right )+2\right ) \sin \left (d x +c \right )}{3}+6 A \,a^{4} \sin \left (d x +c \right )+6 a^{4} B \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+4 A \,a^{4} \left (d x +c \right )+4 a^{4} B \sin \left (d x +c \right )+A \,a^{4} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+a^{4} B \left (d x +c \right )}{d}\) | \(208\) |
risch | \(6 a^{4} x A +\frac {35 a^{4} B x}{8}-\frac {27 i {\mathrm e}^{i \left (d x +c \right )} A \,a^{4}}{8 d}-\frac {7 i {\mathrm e}^{i \left (d x +c \right )} a^{4} B}{2 d}+\frac {27 i {\mathrm e}^{-i \left (d x +c \right )} A \,a^{4}}{8 d}+\frac {7 i {\mathrm e}^{-i \left (d x +c \right )} a^{4} B}{2 d}+\frac {A \,a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{d}-\frac {A \,a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{d}+\frac {a^{4} B \sin \left (4 d x +4 c \right )}{32 d}+\frac {\sin \left (3 d x +3 c \right ) A \,a^{4}}{12 d}+\frac {a^{4} B \sin \left (3 d x +3 c \right )}{3 d}+\frac {\sin \left (2 d x +2 c \right ) A \,a^{4}}{d}+\frac {7 a^{4} B \sin \left (2 d x +2 c \right )}{4 d}\) | \(224\) |
norman | \(\frac {\left (6 A \,a^{4}+\frac {35}{8} a^{4} B \right ) x +\left (6 A \,a^{4}+\frac {35}{8} a^{4} B \right ) x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (30 A \,a^{4}+\frac {175}{8} a^{4} B \right ) x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (30 A \,a^{4}+\frac {175}{8} a^{4} B \right ) x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (60 A \,a^{4}+\frac {175}{4} a^{4} B \right ) x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (60 A \,a^{4}+\frac {175}{4} a^{4} B \right ) x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {5 a^{4} \left (8 A +7 B \right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}+\frac {4 a^{4} \left (59 A +56 B \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}+\frac {a^{4} \left (272 A +245 B \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 d}+\frac {a^{4} \left (368 A +395 B \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 d}+\frac {3 a^{4} \left (31 B +24 A \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}+\frac {A \,a^{4} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}-\frac {A \,a^{4} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}\) | \(333\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 198, normalized size = 1.31 \begin {gather*} -\frac {32 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a^{4} - 96 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{4} - 384 \, {\left (d x + c\right )} A a^{4} + 128 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a^{4} - 3 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{4} - 144 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{4} - 96 \, {\left (d x + c\right )} B a^{4} - 96 \, A a^{4} \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right ) - 576 \, A a^{4} \sin \left (d x + c\right ) - 384 \, B a^{4} \sin \left (d x + c\right )}{96 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 118, normalized size = 0.78 \begin {gather*} \frac {3 \, {\left (48 \, A + 35 \, B\right )} a^{4} d x + 12 \, A a^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) - 12 \, A a^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) + {\left (6 \, B a^{4} \cos \left (d x + c\right )^{3} + 8 \, {\left (A + 4 \, B\right )} a^{4} \cos \left (d x + c\right )^{2} + 3 \, {\left (16 \, A + 27 \, B\right )} a^{4} \cos \left (d x + c\right ) + 160 \, {\left (A + B\right )} a^{4}\right )} \sin \left (d x + c\right )}{24 \, d} \end {gather*}
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 \begin {gather*} a^{4} \left (\int A \sec {\left (c + d x \right )}\, dx + \int 4 A \cos {\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int 6 A \cos ^{2}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int 4 A \cos ^{3}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int A \cos ^{4}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int B \cos {\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int 4 B \cos ^{2}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int 6 B \cos ^{3}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int 4 B \cos ^{4}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int B \cos ^{5}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.48, size = 214, normalized size = 1.42 \begin {gather*} \frac {24 \, A a^{4} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 24 \, A a^{4} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + 3 \, {\left (48 \, A a^{4} + 35 \, B a^{4}\right )} {\left (d x + c\right )} + \frac {2 \, {\left (120 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 105 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 424 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 385 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 520 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 511 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 216 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 279 \, B a^{4} \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 )}^{4}}}{24 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.67, size = 188, normalized size = 1.25 \begin {gather*} \frac {144\,A\,a^4\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )+24\,A\,a^4\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )+105\,B\,a^4\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )+12\,A\,a^4\,\sin \left (2\,c+2\,d\,x\right )+A\,a^4\,\sin \left (3\,c+3\,d\,x\right )+21\,B\,a^4\,\sin \left (2\,c+2\,d\,x\right )+4\,B\,a^4\,\sin \left (3\,c+3\,d\,x\right )+\frac {3\,B\,a^4\,\sin \left (4\,c+4\,d\,x\right )}{8}+81\,A\,a^4\,\sin \left (c+d\,x\right )+84\,B\,a^4\,\sin \left (c+d\,x\right )}{12\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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