Optimal. Leaf size=145 \[ \frac {(5 a B+4 b C) \tan ^3(c+d x)}{15 d}+\frac {(5 a B+4 b C) \tan (c+d x)}{5 d}+\frac {3 (a C+b B) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac {(a C+b B) \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac {3 (a C+b B) \tan (c+d x) \sec (c+d x)}{8 d}+\frac {b C \tan (c+d x) \sec ^4(c+d x)}{5 d} \]
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Rubi [A] time = 0.20, antiderivative size = 145, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {4072, 3997, 3787, 3767, 3768, 3770} \[ \frac {(5 a B+4 b C) \tan ^3(c+d x)}{15 d}+\frac {(5 a B+4 b C) \tan (c+d x)}{5 d}+\frac {3 (a C+b B) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac {(a C+b B) \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac {3 (a C+b B) \tan (c+d x) \sec (c+d x)}{8 d}+\frac {b C \tan (c+d x) \sec ^4(c+d x)}{5 d} \]
Antiderivative was successfully verified.
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Rule 3767
Rule 3768
Rule 3770
Rule 3787
Rule 3997
Rule 4072
Rubi steps
\begin {align*} \int \sec ^3(c+d x) (a+b \sec (c+d x)) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx &=\int \sec ^4(c+d x) (a+b \sec (c+d x)) (B+C \sec (c+d x)) \, dx\\ &=\frac {b C \sec ^4(c+d x) \tan (c+d x)}{5 d}+\frac {1}{5} \int \sec ^4(c+d x) (5 a B+4 b C+5 (b B+a C) \sec (c+d x)) \, dx\\ &=\frac {b C \sec ^4(c+d x) \tan (c+d x)}{5 d}+(b B+a C) \int \sec ^5(c+d x) \, dx+\frac {1}{5} (5 a B+4 b C) \int \sec ^4(c+d x) \, dx\\ &=\frac {(b B+a C) \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac {b C \sec ^4(c+d x) \tan (c+d x)}{5 d}+\frac {1}{4} (3 (b B+a C)) \int \sec ^3(c+d x) \, dx-\frac {(5 a B+4 b C) \operatorname {Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (c+d x)\right )}{5 d}\\ &=\frac {(5 a B+4 b C) \tan (c+d x)}{5 d}+\frac {3 (b B+a C) \sec (c+d x) \tan (c+d x)}{8 d}+\frac {(b B+a C) \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac {b C \sec ^4(c+d x) \tan (c+d x)}{5 d}+\frac {(5 a B+4 b C) \tan ^3(c+d x)}{15 d}+\frac {1}{8} (3 (b B+a C)) \int \sec (c+d x) \, dx\\ &=\frac {3 (b B+a C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac {(5 a B+4 b C) \tan (c+d x)}{5 d}+\frac {3 (b B+a C) \sec (c+d x) \tan (c+d x)}{8 d}+\frac {(b B+a C) \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac {b C \sec ^4(c+d x) \tan (c+d x)}{5 d}+\frac {(5 a B+4 b C) \tan ^3(c+d x)}{15 d}\\ \end {align*}
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Mathematica [A] time = 1.07, size = 106, normalized size = 0.73 \[ \frac {45 (a C+b B) \tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \left (8 \left (5 (a B+2 b C) \tan ^2(c+d x)+15 (a B+b C)+3 b C \tan ^4(c+d x)\right )+30 (a C+b B) \sec ^3(c+d x)+45 (a C+b B) \sec (c+d x)\right )}{120 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.98, size = 151, normalized size = 1.04 \[ \frac {45 \, {\left (C a + B b\right )} \cos \left (d x + c\right )^{5} \log \left (\sin \left (d x + c\right ) + 1\right ) - 45 \, {\left (C a + B b\right )} \cos \left (d x + c\right )^{5} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (16 \, {\left (5 \, B a + 4 \, C b\right )} \cos \left (d x + c\right )^{4} + 45 \, {\left (C a + B b\right )} \cos \left (d x + c\right )^{3} + 8 \, {\left (5 \, B a + 4 \, C b\right )} \cos \left (d x + c\right )^{2} + 24 \, C b + 30 \, {\left (C a + B b\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{240 \, d \cos \left (d x + c\right )^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.27, size = 330, normalized size = 2.28 \[ \frac {45 \, {\left (C a + B b\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 45 \, {\left (C a + B b\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (120 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 75 \, C a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 75 \, B b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 120 \, C b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 320 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 30 \, C a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 30 \, B b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 160 \, C b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 400 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 464 \, C b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 320 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 30 \, C a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 30 \, B b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 160 \, C b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 120 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 75 \, C a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 75 \, B b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 120 \, C b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{5}}}{120 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.26, size = 213, normalized size = 1.47 \[ \frac {2 a B \tan \left (d x +c \right )}{3 d}+\frac {a B \tan \left (d x +c \right ) \left (\sec ^{2}\left (d x +c \right )\right )}{3 d}+\frac {a C \left (\sec ^{3}\left (d x +c \right )\right ) \tan \left (d x +c \right )}{4 d}+\frac {3 a C \sec \left (d x +c \right ) \tan \left (d x +c \right )}{8 d}+\frac {3 a C \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8 d}+\frac {b B \left (\sec ^{3}\left (d x +c \right )\right ) \tan \left (d x +c \right )}{4 d}+\frac {3 b B \sec \left (d x +c \right ) \tan \left (d x +c \right )}{8 d}+\frac {3 B b \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8 d}+\frac {8 b C \tan \left (d x +c \right )}{15 d}+\frac {b C \left (\sec ^{4}\left (d x +c \right )\right ) \tan \left (d x +c \right )}{5 d}+\frac {4 b C \left (\sec ^{2}\left (d x +c \right )\right ) \tan \left (d x +c \right )}{15 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.35, size = 200, normalized size = 1.38 \[ \frac {80 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} B a + 16 \, {\left (3 \, \tan \left (d x + c\right )^{5} + 10 \, \tan \left (d x + c\right )^{3} + 15 \, \tan \left (d x + c\right )\right )} C b - 15 \, C a {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 15 \, B b {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )}}{240 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 7.95, size = 234, normalized size = 1.61 \[ \frac {\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (\frac {3\,B\,b}{4}+\frac {3\,C\,a}{4}\right )}{d}-\frac {\left (2\,B\,a-\frac {5\,B\,b}{4}-\frac {5\,C\,a}{4}+2\,C\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (\frac {B\,b}{2}-\frac {16\,B\,a}{3}+\frac {C\,a}{2}-\frac {8\,C\,b}{3}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {20\,B\,a}{3}+\frac {116\,C\,b}{15}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (-\frac {16\,B\,a}{3}-\frac {B\,b}{2}-\frac {C\,a}{2}-\frac {8\,C\,b}{3}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (2\,B\,a+\frac {5\,B\,b}{4}+\frac {5\,C\,a}{4}+2\,C\,b\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )} \]
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 \[ \int \left (B + C \sec {\left (c + d x \right )}\right ) \left (a + b \sec {\left (c + d x \right )}\right ) \sec ^{4}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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