Optimal. Leaf size=114 \[ \frac {(a B+A b) \tan ^3(c+d x)}{3 d}+\frac {(a B+A b) \tan (c+d x)}{d}+\frac {(4 a A+3 b B) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac {(4 a A+3 b B) \tan (c+d x) \sec (c+d x)}{8 d}+\frac {b B \tan (c+d x) \sec ^3(c+d x)}{4 d} \]
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Rubi [A] time = 0.15, antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {3997, 3787, 3768, 3770, 3767} \[ \frac {(a B+A b) \tan ^3(c+d x)}{3 d}+\frac {(a B+A b) \tan (c+d x)}{d}+\frac {(4 a A+3 b B) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac {(4 a A+3 b B) \tan (c+d x) \sec (c+d x)}{8 d}+\frac {b B \tan (c+d x) \sec ^3(c+d x)}{4 d} \]
Antiderivative was successfully verified.
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Rule 3767
Rule 3768
Rule 3770
Rule 3787
Rule 3997
Rubi steps
\begin {align*} \int \sec ^3(c+d x) (a+b \sec (c+d x)) (A+B \sec (c+d x)) \, dx &=\frac {b B \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac {1}{4} \int \sec ^3(c+d x) (4 a A+3 b B+4 (A b+a B) \sec (c+d x)) \, dx\\ &=\frac {b B \sec ^3(c+d x) \tan (c+d x)}{4 d}+(A b+a B) \int \sec ^4(c+d x) \, dx+\frac {1}{4} (4 a A+3 b B) \int \sec ^3(c+d x) \, dx\\ &=\frac {(4 a A+3 b B) \sec (c+d x) \tan (c+d x)}{8 d}+\frac {b B \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac {1}{8} (4 a A+3 b B) \int \sec (c+d x) \, dx-\frac {(A b+a B) \operatorname {Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (c+d x)\right )}{d}\\ &=\frac {(4 a A+3 b B) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac {(A b+a B) \tan (c+d x)}{d}+\frac {(4 a A+3 b B) \sec (c+d x) \tan (c+d x)}{8 d}+\frac {b B \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac {(A b+a B) \tan ^3(c+d x)}{3 d}\\ \end {align*}
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Mathematica [A] time = 0.63, size = 85, normalized size = 0.75 \[ \frac {3 (4 a A+3 b B) \tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \sec (c+d x) \left (8 (a B+A b) (\cos (2 (c+d x))+2) \sec (c+d x)+12 a A+6 b B \sec ^2(c+d x)+9 b B\right )}{24 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.97, size = 136, normalized size = 1.19 \[ \frac {3 \, {\left (4 \, A a + 3 \, B b\right )} \cos \left (d x + c\right )^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, {\left (4 \, A a + 3 \, B b\right )} \cos \left (d x + c\right )^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (16 \, {\left (B a + A b\right )} \cos \left (d x + c\right )^{3} + 3 \, {\left (4 \, A a + 3 \, B b\right )} \cos \left (d x + c\right )^{2} + 6 \, B b + 8 \, {\left (B a + A b\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{48 \, d \cos \left (d x + c\right )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.30, size = 304, normalized size = 2.67 \[ \frac {3 \, {\left (4 \, A a + 3 \, B b\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 3 \, {\left (4 \, A a + 3 \, B b\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + \frac {2 \, {\left (12 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 24 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 24 \, A b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 15 \, B b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 12 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 40 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 40 \, A b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 9 \, B b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 12 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 40 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 40 \, A b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 9 \, B b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 12 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 24 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 24 \, A b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 15 \, B b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{4}}}{24 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.25, size = 171, normalized size = 1.50 \[ \frac {a A \sec \left (d x +c \right ) \tan \left (d x +c \right )}{2 d}+\frac {a A \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2 d}+\frac {2 a B \tan \left (d x +c \right )}{3 d}+\frac {a B \tan \left (d x +c \right ) \left (\sec ^{2}\left (d x +c \right )\right )}{3 d}+\frac {2 A b \tan \left (d x +c \right )}{3 d}+\frac {A b \left (\sec ^{2}\left (d x +c \right )\right ) \tan \left (d x +c \right )}{3 d}+\frac {b B \left (\sec ^{3}\left (d x +c \right )\right ) \tan \left (d x +c \right )}{4 d}+\frac {3 b B \sec \left (d x +c \right ) \tan \left (d x +c \right )}{8 d}+\frac {3 B b \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.88, size = 163, normalized size = 1.43 \[ \frac {16 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} B a + 16 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} A b - 3 \, B b {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 12 \, A a {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )}}{48 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.73, size = 194, normalized size = 1.70 \[ \frac {\left (A\,a-2\,A\,b-2\,B\,a+\frac {5\,B\,b}{4}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {10\,A\,b}{3}-A\,a+\frac {10\,B\,a}{3}+\frac {3\,B\,b}{4}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (\frac {3\,B\,b}{4}-\frac {10\,A\,b}{3}-\frac {10\,B\,a}{3}-A\,a\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (A\,a+2\,A\,b+2\,B\,a+\frac {5\,B\,b}{4}\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}+\frac {\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (A\,a+\frac {3\,B\,b}{4}\right )}{d} \]
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 \[ \int \left (A + B \sec {\left (c + d x \right )}\right ) \left (a + b \sec {\left (c + d x \right )}\right ) \sec ^{3}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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