Optimal. Leaf size=150 \[ \frac {1}{2} a^4 (13 A+12 B) x+\frac {a^4 (4 A+B) \tanh ^{-1}(\sin (c+d x))}{d}+\frac {5 a^4 (A+2 B) \sin (c+d x)}{2 d}-\frac {(3 A-B) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{3 d}-\frac {(3 A-8 B) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{6 d}+\frac {a A (a+a \cos (c+d x))^3 \tan (c+d x)}{d} \]
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Rubi [A]
time = 0.30, antiderivative size = 150, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {3054, 3055,
3047, 3102, 2814, 3855} \begin {gather*} \frac {5 a^4 (A+2 B) \sin (c+d x)}{2 d}+\frac {a^4 (4 A+B) \tanh ^{-1}(\sin (c+d x))}{d}-\frac {(3 A-8 B) \sin (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{6 d}+\frac {1}{2} a^4 x (13 A+12 B)-\frac {(3 A-B) \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{3 d}+\frac {a A \tan (c+d x) (a \cos (c+d x)+a)^3}{d} \end {gather*}
Antiderivative was successfully verified.
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Rule 2814
Rule 3047
Rule 3054
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 ^2(c+d x) \, dx &=\frac {a A (a+a \cos (c+d x))^3 \tan (c+d x)}{d}+\int (a+a \cos (c+d x))^3 (a (4 A+B)-a (3 A-B) \cos (c+d x)) \sec (c+d x) \, dx\\ &=-\frac {(3 A-B) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{3 d}+\frac {a A (a+a \cos (c+d x))^3 \tan (c+d x)}{d}+\frac {1}{3} \int (a+a \cos (c+d x))^2 \left (3 a^2 (4 A+B)-a^2 (3 A-8 B) \cos (c+d x)\right ) \sec (c+d x) \, dx\\ &=-\frac {(3 A-B) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{3 d}-\frac {(3 A-8 B) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{6 d}+\frac {a A (a+a \cos (c+d x))^3 \tan (c+d x)}{d}+\frac {1}{6} \int (a+a \cos (c+d x)) \left (6 a^3 (4 A+B)+15 a^3 (A+2 B) \cos (c+d x)\right ) \sec (c+d x) \, dx\\ &=-\frac {(3 A-B) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{3 d}-\frac {(3 A-8 B) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{6 d}+\frac {a A (a+a \cos (c+d x))^3 \tan (c+d x)}{d}+\frac {1}{6} \int \left (6 a^4 (4 A+B)+\left (6 a^4 (4 A+B)+15 a^4 (A+2 B)\right ) \cos (c+d x)+15 a^4 (A+2 B) \cos ^2(c+d x)\right ) \sec (c+d x) \, dx\\ &=\frac {5 a^4 (A+2 B) \sin (c+d x)}{2 d}-\frac {(3 A-B) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{3 d}-\frac {(3 A-8 B) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{6 d}+\frac {a A (a+a \cos (c+d x))^3 \tan (c+d x)}{d}+\frac {1}{6} \int \left (6 a^4 (4 A+B)+3 a^4 (13 A+12 B) \cos (c+d x)\right ) \sec (c+d x) \, dx\\ &=\frac {1}{2} a^4 (13 A+12 B) x+\frac {5 a^4 (A+2 B) \sin (c+d x)}{2 d}-\frac {(3 A-B) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{3 d}-\frac {(3 A-8 B) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{6 d}+\frac {a A (a+a \cos (c+d x))^3 \tan (c+d x)}{d}+\left (a^4 (4 A+B)\right ) \int \sec (c+d x) \, dx\\ &=\frac {1}{2} a^4 (13 A+12 B) x+\frac {a^4 (4 A+B) \tanh ^{-1}(\sin (c+d x))}{d}+\frac {5 a^4 (A+2 B) \sin (c+d x)}{2 d}-\frac {(3 A-B) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{3 d}-\frac {(3 A-8 B) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{6 d}+\frac {a A (a+a \cos (c+d x))^3 \tan (c+d x)}{d}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(312\) vs. \(2(150)=300\).
time = 1.68, size = 312, normalized size = 2.08 \begin {gather*} \frac {1}{192} a^4 (1+\cos (c+d x))^4 \sec ^8\left (\frac {1}{2} (c+d x)\right ) \left (78 A x+72 B x-\frac {12 (4 A+B) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}{d}+\frac {12 (4 A+B) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )}{d}+\frac {3 (16 A+27 B) \cos (d x) \sin (c)}{d}+\frac {3 (A+4 B) \cos (2 d x) \sin (2 c)}{d}+\frac {B \cos (3 d x) \sin (3 c)}{d}+\frac {3 (16 A+27 B) \cos (c) \sin (d x)}{d}+\frac {3 (A+4 B) \cos (2 c) \sin (2 d x)}{d}+\frac {B \cos (3 c) \sin (3 d x)}{d}+\frac {12 A \sin \left (\frac {d x}{2}\right )}{d \left (\cos \left (\frac {c}{2}\right )-\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}+\frac {12 A \sin \left (\frac {d x}{2}\right )}{d \left (\cos \left (\frac {c}{2}\right )+\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.27, size = 179, normalized size = 1.19
method | result | size |
derivativedivides | \(\frac {A \,a^{4} \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+\frac {a^{4} B \left (\cos ^{2}\left (d x +c \right )+2\right ) \sin \left (d x +c \right )}{3}+4 A \,a^{4} \sin \left (d x +c \right )+4 a^{4} B \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+6 A \,a^{4} \left (d x +c \right )+6 a^{4} B \sin \left (d x +c \right )+4 A \,a^{4} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+4 a^{4} B \left (d x +c \right )+A \,a^{4} \tan \left (d x +c \right )+a^{4} B \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}\) | \(179\) |
default | \(\frac {A \,a^{4} \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+\frac {a^{4} B \left (\cos ^{2}\left (d x +c \right )+2\right ) \sin \left (d x +c \right )}{3}+4 A \,a^{4} \sin \left (d x +c \right )+4 a^{4} B \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+6 A \,a^{4} \left (d x +c \right )+6 a^{4} B \sin \left (d x +c \right )+4 A \,a^{4} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+4 a^{4} B \left (d x +c \right )+A \,a^{4} \tan \left (d x +c \right )+a^{4} B \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}\) | \(179\) |
risch | \(\frac {13 a^{4} x A}{2}+6 a^{4} B x +\frac {i {\mathrm e}^{-2 i \left (d x +c \right )} A \,a^{4}}{8 d}+\frac {i {\mathrm e}^{-2 i \left (d x +c \right )} a^{4} B}{2 d}+\frac {2 i A \,a^{4}}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}-\frac {27 i {\mathrm e}^{i \left (d x +c \right )} a^{4} B}{8 d}-\frac {i {\mathrm e}^{2 i \left (d x +c \right )} a^{4} B}{2 d}-\frac {i {\mathrm e}^{2 i \left (d x +c \right )} A \,a^{4}}{8 d}+\frac {2 i {\mathrm e}^{-i \left (d x +c \right )} A \,a^{4}}{d}-\frac {2 i {\mathrm e}^{i \left (d x +c \right )} A \,a^{4}}{d}+\frac {27 i {\mathrm e}^{-i \left (d x +c \right )} a^{4} B}{8 d}-\frac {4 A \,a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{d}-\frac {a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) B}{d}+\frac {4 A \,a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{d}+\frac {a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) B}{d}+\frac {a^{4} B \sin \left (3 d x +3 c \right )}{12 d}\) | \(296\) |
norman | \(\frac {\left (-\frac {13}{2} A \,a^{4}-6 a^{4} B \right ) x +\left (-\frac {65}{2} A \,a^{4}-30 a^{4} B \right ) x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {13}{2} A \,a^{4}+6 a^{4} B \right ) x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {65}{2} A \,a^{4}+30 a^{4} B \right ) x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-26 A \,a^{4}-24 a^{4} B \right ) x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (26 A \,a^{4}+24 a^{4} B \right ) x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {5 a^{4} \left (A +2 B \right ) \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {2 a^{4} \left (3 A -50 B \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {a^{4} \left (11 A +18 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}-\frac {5 a^{4} \left (21 A +26 B \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}+\frac {a^{4} \left (39 A +106 B \right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {2 a^{4} \left (51 A +26 B \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {a^{4} \left (4 A +B \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}-\frac {a^{4} \left (4 A +B \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}\) | \(380\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 187, normalized size = 1.25 \begin {gather*} \frac {3 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{4} + 72 \, {\left (d x + c\right )} A a^{4} - 4 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a^{4} + 12 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{4} + 48 \, {\left (d x + c\right )} B a^{4} + 24 \, A a^{4} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 6 \, B a^{4} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 48 \, A a^{4} \sin \left (d x + c\right ) + 72 \, B a^{4} \sin \left (d x + c\right ) + 12 \, A a^{4} \tan \left (d x + c\right )}{12 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 150, normalized size = 1.00 \begin {gather*} \frac {3 \, {\left (13 \, A + 12 \, B\right )} a^{4} d x \cos \left (d x + c\right ) + 3 \, {\left (4 \, A + B\right )} a^{4} \cos \left (d x + c\right ) \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, {\left (4 \, A + B\right )} a^{4} \cos \left (d x + c\right ) \log \left (-\sin \left (d x + c\right ) + 1\right ) + {\left (2 \, B a^{4} \cos \left (d x + c\right )^{3} + 3 \, {\left (A + 4 \, B\right )} a^{4} \cos \left (d x + c\right )^{2} + 8 \, {\left (3 \, A + 5 \, B\right )} a^{4} \cos \left (d x + c\right ) + 6 \, A a^{4}\right )} \sin \left (d x + c\right )}{6 \, d \cos \left (d x + c\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.48, size = 226, normalized size = 1.51 \begin {gather*} -\frac {\frac {12 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1} - 3 \, {\left (13 \, A a^{4} + 12 \, B a^{4}\right )} {\left (d x + c\right )} - 6 \, {\left (4 \, A a^{4} + B a^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) + 6 \, {\left (4 \, A a^{4} + B a^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (21 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 30 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 48 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 76 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 27 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 54 \, 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 )}^{3}}}{6 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.42, size = 242, normalized size = 1.61 \begin {gather*} \frac {4\,A\,a^4\,\sin \left (c+d\,x\right )}{d}+\frac {20\,B\,a^4\,\sin \left (c+d\,x\right )}{3\,d}+\frac {13\,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 )}{d}+\frac {8\,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 )}{d}+\frac {12\,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 )}{d}+\frac {2\,B\,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 )}{d}+\frac {A\,a^4\,\sin \left (c+d\,x\right )}{d\,\cos \left (c+d\,x\right )}+\frac {B\,a^4\,{\cos \left (c+d\,x\right )}^2\,\sin \left (c+d\,x\right )}{3\,d}+\frac {A\,a^4\,\cos \left (c+d\,x\right )\,\sin \left (c+d\,x\right )}{2\,d}+\frac {2\,B\,a^4\,\cos \left (c+d\,x\right )\,\sin \left (c+d\,x\right )}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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