Optimal. Leaf size=183 \[ \frac {1}{8} a^3 (15 A+20 B+28 C) x+\frac {a^3 C \tanh ^{-1}(\sin (c+d x))}{d}+\frac {5 a^3 (3 A+4 (B+C)) \sin (c+d x)}{8 d}+\frac {A \cos ^3(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{4 d}+\frac {(3 A+4 B) \cos ^2(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{12 a d}+\frac {(15 A+20 B+12 C) \cos (c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{24 d} \]
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Rubi [A]
time = 0.31, antiderivative size = 183, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 4, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.098, Rules used = {4171, 4102,
4081, 3855} \begin {gather*} \frac {5 a^3 (3 A+4 (B+C)) \sin (c+d x)}{8 d}+\frac {(15 A+20 B+12 C) \sin (c+d x) \cos (c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{24 d}+\frac {1}{8} a^3 x (15 A+20 B+28 C)+\frac {a^3 C \tanh ^{-1}(\sin (c+d x))}{d}+\frac {(3 A+4 B) \sin (c+d x) \cos ^2(c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{12 a d}+\frac {A \sin (c+d x) \cos ^3(c+d x) (a \sec (c+d x)+a)^3}{4 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 3855
Rule 4081
Rule 4102
Rule 4171
Rubi steps
\begin {align*} \int \cos ^4(c+d x) (a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx &=\frac {A \cos ^3(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{4 d}+\frac {\int \cos ^3(c+d x) (a+a \sec (c+d x))^3 (a (3 A+4 B)+4 a C \sec (c+d x)) \, dx}{4 a}\\ &=\frac {A \cos ^3(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{4 d}+\frac {(3 A+4 B) \cos ^2(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{12 a d}+\frac {\int \cos ^2(c+d x) (a+a \sec (c+d x))^2 \left (a^2 (15 A+20 B+12 C)+12 a^2 C \sec (c+d x)\right ) \, dx}{12 a}\\ &=\frac {A \cos ^3(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{4 d}+\frac {(3 A+4 B) \cos ^2(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{12 a d}+\frac {(15 A+20 B+12 C) \cos (c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{24 d}+\frac {\int \cos (c+d x) (a+a \sec (c+d x)) \left (15 a^3 (3 A+4 (B+C))+24 a^3 C \sec (c+d x)\right ) \, dx}{24 a}\\ &=\frac {5 a^3 (3 A+4 (B+C)) \sin (c+d x)}{8 d}+\frac {A \cos ^3(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{4 d}+\frac {(3 A+4 B) \cos ^2(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{12 a d}+\frac {(15 A+20 B+12 C) \cos (c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{24 d}-\frac {\int \left (-3 a^4 (15 A+20 B+28 C)-24 a^4 C \sec (c+d x)\right ) \, dx}{24 a}\\ &=\frac {1}{8} a^3 (15 A+20 B+28 C) x+\frac {5 a^3 (3 A+4 (B+C)) \sin (c+d x)}{8 d}+\frac {A \cos ^3(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{4 d}+\frac {(3 A+4 B) \cos ^2(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{12 a d}+\frac {(15 A+20 B+12 C) \cos (c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{24 d}+\left (a^3 C\right ) \int \sec (c+d x) \, dx\\ &=\frac {1}{8} a^3 (15 A+20 B+28 C) x+\frac {a^3 C \tanh ^{-1}(\sin (c+d x))}{d}+\frac {5 a^3 (3 A+4 (B+C)) \sin (c+d x)}{8 d}+\frac {A \cos ^3(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{4 d}+\frac {(3 A+4 B) \cos ^2(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{12 a d}+\frac {(15 A+20 B+12 C) \cos (c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{24 d}\\ \end {align*}
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Mathematica [A]
time = 0.45, size = 147, normalized size = 0.80 \begin {gather*} \frac {a^3 \left (180 A d x+240 B d x+336 C d x-96 C \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+96 C \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+24 (13 A+15 B+12 C) \sin (c+d x)+24 (4 A+3 B+C) \sin (2 (c+d x))+24 A \sin (3 (c+d x))+8 B \sin (3 (c+d x))+3 A \sin (4 (c+d x))\right )}{96 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.90, size = 245, normalized size = 1.34
method | result | size |
derivativedivides | \(\frac {A \,a^{3} \sin \left (d x +c \right )+a^{3} B \left (d x +c \right )+a^{3} C \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+3 A \,a^{3} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+3 a^{3} B \sin \left (d x +c \right )+3 a^{3} C \left (d x +c \right )+A \,a^{3} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )+3 a^{3} B \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+3 a^{3} C \sin \left (d x +c \right )+A \,a^{3} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {a^{3} B \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+a^{3} C \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}\) | \(245\) |
default | \(\frac {A \,a^{3} \sin \left (d x +c \right )+a^{3} B \left (d x +c \right )+a^{3} C \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+3 A \,a^{3} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+3 a^{3} B \sin \left (d x +c \right )+3 a^{3} C \left (d x +c \right )+A \,a^{3} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )+3 a^{3} B \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+3 a^{3} C \sin \left (d x +c \right )+A \,a^{3} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {a^{3} B \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+a^{3} C \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}\) | \(245\) |
risch | \(\frac {15 a^{3} A x}{8}+\frac {5 a^{3} B x}{2}+\frac {7 a^{3} x C}{2}+\frac {15 i a^{3} B \,{\mathrm e}^{-i \left (d x +c \right )}}{8 d}+\frac {13 i A \,a^{3} {\mathrm e}^{-i \left (d x +c \right )}}{8 d}+\frac {3 i {\mathrm e}^{-i \left (d x +c \right )} a^{3} C}{2 d}-\frac {13 i A \,a^{3} {\mathrm e}^{i \left (d x +c \right )}}{8 d}-\frac {15 i a^{3} B \,{\mathrm e}^{i \left (d x +c \right )}}{8 d}-\frac {3 i {\mathrm e}^{i \left (d x +c \right )} a^{3} C}{2 d}+\frac {a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) C}{d}-\frac {a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C}{d}+\frac {A \,a^{3} \sin \left (4 d x +4 c \right )}{32 d}+\frac {A \,a^{3} \sin \left (3 d x +3 c \right )}{4 d}+\frac {a^{3} B \sin \left (3 d x +3 c \right )}{12 d}+\frac {\sin \left (2 d x +2 c \right ) A \,a^{3}}{d}+\frac {3 \sin \left (2 d x +2 c \right ) a^{3} B}{4 d}+\frac {\sin \left (2 d x +2 c \right ) a^{3} C}{4 d}\) | \(287\) |
norman | \(\frac {\left (\frac {15}{8} A \,a^{3}+\frac {5}{2} a^{3} B +\frac {7}{2} a^{3} C \right ) x +\left (-\frac {15}{2} A \,a^{3}-10 a^{3} B -14 a^{3} C \right ) x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-\frac {15}{2} A \,a^{3}-10 a^{3} B -14 a^{3} C \right ) x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {15}{8} A \,a^{3}+\frac {5}{2} a^{3} B +\frac {7}{2} a^{3} C \right ) x \left (\tan ^{16}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {45}{4} A \,a^{3}+15 a^{3} B +21 a^{3} C \right ) x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {19 a^{3} \left (3 A +4 B +4 C \right ) \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}+\frac {5 a^{3} \left (3 A +4 B +4 C \right ) \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}-\frac {a^{3} \left (15 A +20 B +36 C \right ) \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 d}+\frac {a^{3} \left (49 A +44 B +28 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d}+\frac {a^{3} \left (81 A +44 B +156 C \right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 d}+\frac {a^{3} \left (111 A +404 B +276 C \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 d}-\frac {a^{3} \left (369 A +236 B +108 C \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 d}+\frac {a^{3} \left (-52 B -68 C +57 A \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{4}}+\frac {a^{3} C \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}-\frac {a^{3} C \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}\) | \(454\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 240, normalized size = 1.31 \begin {gather*} -\frac {96 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a^{3} - 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 )} A a^{3} - 72 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{3} + 32 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a^{3} - 72 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{3} - 96 \, {\left (d x + c\right )} B a^{3} - 24 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{3} - 288 \, {\left (d x + c\right )} C a^{3} - 48 \, C a^{3} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 96 \, A a^{3} \sin \left (d x + c\right ) - 288 \, B a^{3} \sin \left (d x + c\right ) - 288 \, C a^{3} \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 = 2.20, size = 131, normalized size = 0.72 \begin {gather*} \frac {3 \, {\left (15 \, A + 20 \, B + 28 \, C\right )} a^{3} d x + 12 \, C a^{3} \log \left (\sin \left (d x + c\right ) + 1\right ) - 12 \, C a^{3} \log \left (-\sin \left (d x + c\right ) + 1\right ) + {\left (6 \, A a^{3} \cos \left (d x + c\right )^{3} + 8 \, {\left (3 \, A + B\right )} a^{3} \cos \left (d x + c\right )^{2} + 3 \, {\left (15 \, A + 12 \, B + 4 \, C\right )} a^{3} \cos \left (d x + c\right ) + 8 \, {\left (9 \, A + 11 \, B + 9 \, C\right )} a^{3}\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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.56, size = 286, normalized size = 1.56 \begin {gather*} \frac {24 \, C a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 24 \, C a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + 3 \, {\left (15 \, A a^{3} + 20 \, B a^{3} + 28 \, C a^{3}\right )} {\left (d x + c\right )} + \frac {2 \, {\left (45 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 60 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 60 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 165 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 220 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 204 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 219 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 292 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 228 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 147 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 132 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 84 \, C a^{3} \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 = 3.71, size = 243, normalized size = 1.33 \begin {gather*} \frac {45\,A\,a^3\,\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 )+60\,B\,a^3\,\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 )+84\,C\,a^3\,\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\,C\,a^3\,\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 )+12\,A\,a^3\,\sin \left (2\,c+2\,d\,x\right )+3\,A\,a^3\,\sin \left (3\,c+3\,d\,x\right )+\frac {3\,A\,a^3\,\sin \left (4\,c+4\,d\,x\right )}{8}+9\,B\,a^3\,\sin \left (2\,c+2\,d\,x\right )+B\,a^3\,\sin \left (3\,c+3\,d\,x\right )+3\,C\,a^3\,\sin \left (2\,c+2\,d\,x\right )+39\,A\,a^3\,\sin \left (c+d\,x\right )+45\,B\,a^3\,\sin \left (c+d\,x\right )+36\,C\,a^3\,\sin \left (c+d\,x\right )}{12\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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