Optimal. Leaf size=111 \[ \frac {5 a^3 (B+C) \tan (c+d x)}{2 d}+\frac {a^3 (7 B+5 C) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac {(3 B+5 C) \tan (c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{6 d}+a^3 B x+\frac {a C \tan (c+d x) (a \sec (c+d x)+a)^2}{3 d} \]
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Rubi [A] time = 0.20, antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {4072, 3917, 3914, 3767, 8, 3770} \[ \frac {5 a^3 (B+C) \tan (c+d x)}{2 d}+\frac {a^3 (7 B+5 C) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac {(3 B+5 C) \tan (c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{6 d}+a^3 B x+\frac {a C \tan (c+d x) (a \sec (c+d x)+a)^2}{3 d} \]
Antiderivative was successfully verified.
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Rule 8
Rule 3767
Rule 3770
Rule 3914
Rule 3917
Rule 4072
Rubi steps
\begin {align*} \int \cos (c+d x) (a+a \sec (c+d x))^3 \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx &=\int (a+a \sec (c+d x))^3 (B+C \sec (c+d x)) \, dx\\ &=\frac {a C (a+a \sec (c+d x))^2 \tan (c+d x)}{3 d}+\frac {1}{3} \int (a+a \sec (c+d x))^2 (3 a B+a (3 B+5 C) \sec (c+d x)) \, dx\\ &=\frac {a C (a+a \sec (c+d x))^2 \tan (c+d x)}{3 d}+\frac {(3 B+5 C) \left (a^3+a^3 \sec (c+d x)\right ) \tan (c+d x)}{6 d}+\frac {1}{6} \int (a+a \sec (c+d x)) \left (6 a^2 B+15 a^2 (B+C) \sec (c+d x)\right ) \, dx\\ &=a^3 B x+\frac {a C (a+a \sec (c+d x))^2 \tan (c+d x)}{3 d}+\frac {(3 B+5 C) \left (a^3+a^3 \sec (c+d x)\right ) \tan (c+d x)}{6 d}+\frac {1}{2} \left (5 a^3 (B+C)\right ) \int \sec ^2(c+d x) \, dx+\frac {1}{2} \left (a^3 (7 B+5 C)\right ) \int \sec (c+d x) \, dx\\ &=a^3 B x+\frac {a^3 (7 B+5 C) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac {a C (a+a \sec (c+d x))^2 \tan (c+d x)}{3 d}+\frac {(3 B+5 C) \left (a^3+a^3 \sec (c+d x)\right ) \tan (c+d x)}{6 d}-\frac {\left (5 a^3 (B+C)\right ) \operatorname {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{2 d}\\ &=a^3 B x+\frac {a^3 (7 B+5 C) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac {5 a^3 (B+C) \tan (c+d x)}{2 d}+\frac {a C (a+a \sec (c+d x))^2 \tan (c+d x)}{3 d}+\frac {(3 B+5 C) \left (a^3+a^3 \sec (c+d x)\right ) \tan (c+d x)}{6 d}\\ \end {align*}
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Mathematica [B] time = 6.45, size = 772, normalized size = 6.95 \[ a^3 \left (\frac {(\cos (c+d x)+1)^3 \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) \left (9 B \sin \left (\frac {d x}{2}\right )+11 C \sin \left (\frac {d x}{2}\right )\right )}{24 d \left (\cos \left (\frac {c}{2}\right )-\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )-\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right )}+\frac {(\cos (c+d x)+1)^3 \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) \left (9 B \sin \left (\frac {d x}{2}\right )+11 C \sin \left (\frac {d x}{2}\right )\right )}{24 d \left (\sin \left (\frac {c}{2}\right )+\cos \left (\frac {c}{2}\right )\right ) \left (\sin \left (\frac {c}{2}+\frac {d x}{2}\right )+\cos \left (\frac {c}{2}+\frac {d x}{2}\right )\right )}+\frac {(\cos (c+d x)+1)^3 \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) \left (-3 B \sin \left (\frac {c}{2}\right )+3 B \cos \left (\frac {c}{2}\right )-8 C \sin \left (\frac {c}{2}\right )+10 C \cos \left (\frac {c}{2}\right )\right )}{96 d \left (\cos \left (\frac {c}{2}\right )-\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )-\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right )^2}+\frac {(\cos (c+d x)+1)^3 \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) \left (-3 B \sin \left (\frac {c}{2}\right )-3 B \cos \left (\frac {c}{2}\right )-8 C \sin \left (\frac {c}{2}\right )-10 C \cos \left (\frac {c}{2}\right )\right )}{96 d \left (\sin \left (\frac {c}{2}\right )+\cos \left (\frac {c}{2}\right )\right ) \left (\sin \left (\frac {c}{2}+\frac {d x}{2}\right )+\cos \left (\frac {c}{2}+\frac {d x}{2}\right )\right )^2}+\frac {(-7 B-5 C) (\cos (c+d x)+1)^3 \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) \log \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )-\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right )}{16 d}+\frac {(7 B+5 C) (\cos (c+d x)+1)^3 \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) \log \left (\sin \left (\frac {c}{2}+\frac {d x}{2}\right )+\cos \left (\frac {c}{2}+\frac {d x}{2}\right )\right )}{16 d}+\frac {1}{8} B x (\cos (c+d x)+1)^3 \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right )+\frac {C \sin \left (\frac {d x}{2}\right ) (\cos (c+d x)+1)^3 \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right )}{48 d \left (\cos \left (\frac {c}{2}\right )-\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )-\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right )^3}+\frac {C \sin \left (\frac {d x}{2}\right ) (\cos (c+d x)+1)^3 \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right )}{48 d \left (\sin \left (\frac {c}{2}\right )+\cos \left (\frac {c}{2}\right )\right ) \left (\sin \left (\frac {c}{2}+\frac {d x}{2}\right )+\cos \left (\frac {c}{2}+\frac {d x}{2}\right )\right )^3}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 141, normalized size = 1.27 \[ \frac {12 \, B a^{3} d x \cos \left (d x + c\right )^{3} + 3 \, {\left (7 \, B + 5 \, C\right )} a^{3} \cos \left (d x + c\right )^{3} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, {\left (7 \, B + 5 \, C\right )} a^{3} \cos \left (d x + c\right )^{3} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (2 \, {\left (9 \, B + 11 \, C\right )} a^{3} \cos \left (d x + c\right )^{2} + 3 \, {\left (B + 3 \, C\right )} a^{3} \cos \left (d x + c\right ) + 2 \, C a^{3}\right )} \sin \left (d x + c\right )}{12 \, d \cos \left (d x + c\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.34, size = 189, normalized size = 1.70 \[ \frac {6 \, {\left (d x + c\right )} B a^{3} + 3 \, {\left (7 \, B a^{3} + 5 \, C a^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 3 \, {\left (7 \, B a^{3} + 5 \, C a^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (15 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 15 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 36 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 40 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 21 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 33 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{3}}}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.53, size = 158, normalized size = 1.42 \[ a^{3} B x +\frac {a^{3} B c}{d}+\frac {5 C \,a^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2 d}+\frac {7 a^{3} B \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2 d}+\frac {11 a^{3} C \tan \left (d x +c \right )}{3 d}+\frac {3 a^{3} B \tan \left (d x +c \right )}{d}+\frac {3 C \,a^{3} \sec \left (d x +c \right ) \tan \left (d x +c \right )}{2 d}+\frac {a^{3} B \sec \left (d x +c \right ) \tan \left (d x +c \right )}{2 d}+\frac {C \,a^{3} \tan \left (d x +c \right ) \left (\sec ^{2}\left (d x +c \right )\right )}{3 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.45, size = 212, normalized size = 1.91 \[ \frac {12 \, {\left (d x + c\right )} B a^{3} + 4 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} C a^{3} - 3 \, B a^{3} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 9 \, C a^{3} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 18 \, B a^{3} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 6 \, C a^{3} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 36 \, B a^{3} \tan \left (d x + c\right ) + 36 \, C a^{3} \tan \left (d x + c\right )}{12 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.01, size = 209, normalized size = 1.88 \[ \frac {2\,B\,a^3\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {7\,B\,a^3\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {5\,C\,a^3\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {3\,B\,a^3\,\sin \left (c+d\,x\right )}{d\,\cos \left (c+d\,x\right )}+\frac {B\,a^3\,\sin \left (c+d\,x\right )}{2\,d\,{\cos \left (c+d\,x\right )}^2}+\frac {11\,C\,a^3\,\sin \left (c+d\,x\right )}{3\,d\,\cos \left (c+d\,x\right )}+\frac {3\,C\,a^3\,\sin \left (c+d\,x\right )}{2\,d\,{\cos \left (c+d\,x\right )}^2}+\frac {C\,a^3\,\sin \left (c+d\,x\right )}{3\,d\,{\cos \left (c+d\,x\right )}^3} \]
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 \[ a^{3} \left (\int B \cos {\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int 3 B \cos {\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int 3 B \cos {\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}\, dx + \int B \cos {\left (c + d x \right )} \sec ^{4}{\left (c + d x \right )}\, dx + \int C \cos {\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int 3 C \cos {\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}\, dx + \int 3 C \cos {\left (c + d x \right )} \sec ^{4}{\left (c + d x \right )}\, dx + \int C \cos {\left (c + d x \right )} \sec ^{5}{\left (c + d x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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