Optimal. Leaf size=250 \[ \frac {\left (12 a^2 B+35 a A b+16 b^2 B\right ) \tan (c+d x) (a+b \sec (c+d x))^2}{60 d}+\frac {b \left (24 a^3 B+130 a^2 A b+116 a b^2 B+45 A b^3\right ) \tan (c+d x) \sec (c+d x)}{120 d}+\frac {\left (12 a^4 B+95 a^3 A b+112 a^2 b^2 B+80 a A b^3+16 b^4 B\right ) \tan (c+d x)}{30 d}+\frac {\left (8 a^4 A+16 a^3 b B+24 a^2 A b^2+12 a b^3 B+3 A b^4\right ) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac {(4 a B+5 A b) \tan (c+d x) (a+b \sec (c+d x))^3}{20 d}+\frac {B \tan (c+d x) (a+b \sec (c+d x))^4}{5 d} \]
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Rubi [A] time = 0.52, antiderivative size = 250, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {4002, 3997, 3787, 3770, 3767, 8} \[ \frac {\left (95 a^3 A b+112 a^2 b^2 B+12 a^4 B+80 a A b^3+16 b^4 B\right ) \tan (c+d x)}{30 d}+\frac {\left (24 a^2 A b^2+8 a^4 A+16 a^3 b B+12 a b^3 B+3 A b^4\right ) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac {\left (12 a^2 B+35 a A b+16 b^2 B\right ) \tan (c+d x) (a+b \sec (c+d x))^2}{60 d}+\frac {b \left (130 a^2 A b+24 a^3 B+116 a b^2 B+45 A b^3\right ) \tan (c+d x) \sec (c+d x)}{120 d}+\frac {(4 a B+5 A b) \tan (c+d x) (a+b \sec (c+d x))^3}{20 d}+\frac {B \tan (c+d x) (a+b \sec (c+d x))^4}{5 d} \]
Antiderivative was successfully verified.
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Rule 8
Rule 3767
Rule 3770
Rule 3787
Rule 3997
Rule 4002
Rubi steps
\begin {align*} \int \sec (c+d x) (a+b \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx &=\frac {B (a+b \sec (c+d x))^4 \tan (c+d x)}{5 d}+\frac {1}{5} \int \sec (c+d x) (a+b \sec (c+d x))^3 (5 a A+4 b B+(5 A b+4 a B) \sec (c+d x)) \, dx\\ &=\frac {(5 A b+4 a B) (a+b \sec (c+d x))^3 \tan (c+d x)}{20 d}+\frac {B (a+b \sec (c+d x))^4 \tan (c+d x)}{5 d}+\frac {1}{20} \int \sec (c+d x) (a+b \sec (c+d x))^2 \left (20 a^2 A+15 A b^2+28 a b B+\left (35 a A b+12 a^2 B+16 b^2 B\right ) \sec (c+d x)\right ) \, dx\\ &=\frac {\left (35 a A b+12 a^2 B+16 b^2 B\right ) (a+b \sec (c+d x))^2 \tan (c+d x)}{60 d}+\frac {(5 A b+4 a B) (a+b \sec (c+d x))^3 \tan (c+d x)}{20 d}+\frac {B (a+b \sec (c+d x))^4 \tan (c+d x)}{5 d}+\frac {1}{60} \int \sec (c+d x) (a+b \sec (c+d x)) \left (60 a^3 A+115 a A b^2+108 a^2 b B+32 b^3 B+\left (130 a^2 A b+45 A b^3+24 a^3 B+116 a b^2 B\right ) \sec (c+d x)\right ) \, dx\\ &=\frac {b \left (130 a^2 A b+45 A b^3+24 a^3 B+116 a b^2 B\right ) \sec (c+d x) \tan (c+d x)}{120 d}+\frac {\left (35 a A b+12 a^2 B+16 b^2 B\right ) (a+b \sec (c+d x))^2 \tan (c+d x)}{60 d}+\frac {(5 A b+4 a B) (a+b \sec (c+d x))^3 \tan (c+d x)}{20 d}+\frac {B (a+b \sec (c+d x))^4 \tan (c+d x)}{5 d}+\frac {1}{120} \int \sec (c+d x) \left (15 \left (8 a^4 A+24 a^2 A b^2+3 A b^4+16 a^3 b B+12 a b^3 B\right )+4 \left (95 a^3 A b+80 a A b^3+12 a^4 B+112 a^2 b^2 B+16 b^4 B\right ) \sec (c+d x)\right ) \, dx\\ &=\frac {b \left (130 a^2 A b+45 A b^3+24 a^3 B+116 a b^2 B\right ) \sec (c+d x) \tan (c+d x)}{120 d}+\frac {\left (35 a A b+12 a^2 B+16 b^2 B\right ) (a+b \sec (c+d x))^2 \tan (c+d x)}{60 d}+\frac {(5 A b+4 a B) (a+b \sec (c+d x))^3 \tan (c+d x)}{20 d}+\frac {B (a+b \sec (c+d x))^4 \tan (c+d x)}{5 d}+\frac {1}{8} \left (8 a^4 A+24 a^2 A b^2+3 A b^4+16 a^3 b B+12 a b^3 B\right ) \int \sec (c+d x) \, dx+\frac {1}{30} \left (95 a^3 A b+80 a A b^3+12 a^4 B+112 a^2 b^2 B+16 b^4 B\right ) \int \sec ^2(c+d x) \, dx\\ &=\frac {\left (8 a^4 A+24 a^2 A b^2+3 A b^4+16 a^3 b B+12 a b^3 B\right ) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac {b \left (130 a^2 A b+45 A b^3+24 a^3 B+116 a b^2 B\right ) \sec (c+d x) \tan (c+d x)}{120 d}+\frac {\left (35 a A b+12 a^2 B+16 b^2 B\right ) (a+b \sec (c+d x))^2 \tan (c+d x)}{60 d}+\frac {(5 A b+4 a B) (a+b \sec (c+d x))^3 \tan (c+d x)}{20 d}+\frac {B (a+b \sec (c+d x))^4 \tan (c+d x)}{5 d}-\frac {\left (95 a^3 A b+80 a A b^3+12 a^4 B+112 a^2 b^2 B+16 b^4 B\right ) \operatorname {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{30 d}\\ &=\frac {\left (8 a^4 A+24 a^2 A b^2+3 A b^4+16 a^3 b B+12 a b^3 B\right ) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac {\left (95 a^3 A b+80 a A b^3+12 a^4 B+112 a^2 b^2 B+16 b^4 B\right ) \tan (c+d x)}{30 d}+\frac {b \left (130 a^2 A b+45 A b^3+24 a^3 B+116 a b^2 B\right ) \sec (c+d x) \tan (c+d x)}{120 d}+\frac {\left (35 a A b+12 a^2 B+16 b^2 B\right ) (a+b \sec (c+d x))^2 \tan (c+d x)}{60 d}+\frac {(5 A b+4 a B) (a+b \sec (c+d x))^3 \tan (c+d x)}{20 d}+\frac {B (a+b \sec (c+d x))^4 \tan (c+d x)}{5 d}\\ \end {align*}
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Mathematica [A] time = 3.94, size = 198, normalized size = 0.79 \[ \frac {15 \left (8 a^4 A+16 a^3 b B+24 a^2 A b^2+12 a b^3 B+3 A b^4\right ) \tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \left (80 b^2 \left (3 a^2 B+2 a A b+b^2 B\right ) \tan ^2(c+d x)+15 b \left (16 a^3 B+24 a^2 A b+12 a b^2 B+3 A b^3\right ) \sec (c+d x)+120 \left (a^4 B+4 a^3 A b+6 a^2 b^2 B+4 a A b^3+b^4 B\right )+30 b^3 (4 a B+A b) \sec ^3(c+d x)+24 b^4 B \tan ^4(c+d x)\right )}{120 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 281, normalized size = 1.12 \[ \frac {15 \, {\left (8 \, A a^{4} + 16 \, B a^{3} b + 24 \, A a^{2} b^{2} + 12 \, B a b^{3} + 3 \, A b^{4}\right )} \cos \left (d x + c\right )^{5} \log \left (\sin \left (d x + c\right ) + 1\right ) - 15 \, {\left (8 \, A a^{4} + 16 \, B a^{3} b + 24 \, A a^{2} b^{2} + 12 \, B a b^{3} + 3 \, A b^{4}\right )} \cos \left (d x + c\right )^{5} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (24 \, B b^{4} + 8 \, {\left (15 \, B a^{4} + 60 \, A a^{3} b + 60 \, B a^{2} b^{2} + 40 \, A a b^{3} + 8 \, B b^{4}\right )} \cos \left (d x + c\right )^{4} + 15 \, {\left (16 \, B a^{3} b + 24 \, A a^{2} b^{2} + 12 \, B a b^{3} + 3 \, A b^{4}\right )} \cos \left (d x + c\right )^{3} + 16 \, {\left (15 \, B a^{2} b^{2} + 10 \, A a b^{3} + 2 \, B b^{4}\right )} \cos \left (d x + c\right )^{2} + 30 \, {\left (4 \, B a b^{3} + A b^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{240 \, d \cos \left (d x + c\right )^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.08, size = 850, normalized size = 3.40 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.67, size = 431, normalized size = 1.72 \[ \frac {A \,a^{4} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {a^{4} B \tan \left (d x +c \right )}{d}+\frac {4 A \,a^{3} b \tan \left (d x +c \right )}{d}+\frac {2 B \,a^{3} b \sec \left (d x +c \right ) \tan \left (d x +c \right )}{d}+\frac {2 B \,a^{3} b \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {3 A \,a^{2} b^{2} \sec \left (d x +c \right ) \tan \left (d x +c \right )}{d}+\frac {3 A \,a^{2} b^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {4 a^{2} b^{2} B \tan \left (d x +c \right )}{d}+\frac {2 a^{2} b^{2} B \tan \left (d x +c \right ) \left (\sec ^{2}\left (d x +c \right )\right )}{d}+\frac {8 a A \,b^{3} \tan \left (d x +c \right )}{3 d}+\frac {4 a A \,b^{3} \tan \left (d x +c \right ) \left (\sec ^{2}\left (d x +c \right )\right )}{3 d}+\frac {B a \,b^{3} \tan \left (d x +c \right ) \left (\sec ^{3}\left (d x +c \right )\right )}{d}+\frac {3 B a \,b^{3} \sec \left (d x +c \right ) \tan \left (d x +c \right )}{2 d}+\frac {3 B a \,b^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2 d}+\frac {A \,b^{4} \tan \left (d x +c \right ) \left (\sec ^{3}\left (d x +c \right )\right )}{4 d}+\frac {3 A \,b^{4} \sec \left (d x +c \right ) \tan \left (d x +c \right )}{8 d}+\frac {3 A \,b^{4} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8 d}+\frac {8 B \,b^{4} \tan \left (d x +c \right )}{15 d}+\frac {B \,b^{4} \tan \left (d x +c \right ) \left (\sec ^{4}\left (d x +c \right )\right )}{5 d}+\frac {4 B \,b^{4} \tan \left (d x +c \right ) \left (\sec ^{2}\left (d x +c \right )\right )}{15 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.02, size = 379, normalized size = 1.52 \[ \frac {480 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} B a^{2} b^{2} + 320 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} A a b^{3} + 16 \, {\left (3 \, \tan \left (d x + c\right )^{5} + 10 \, \tan \left (d x + c\right )^{3} + 15 \, \tan \left (d x + c\right )\right )} B b^{4} - 60 \, B a b^{3} {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 15 \, A b^{4} {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 240 \, B a^{3} b {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 360 \, A a^{2} b^{2} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 240 \, A a^{4} \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right ) + 240 \, B a^{4} \tan \left (d x + c\right ) + 960 \, A a^{3} b \tan \left (d x + c\right )}{240 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.01, size = 555, normalized size = 2.22 \[ \frac {\mathrm {atanh}\left (\frac {4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (A\,a^4+2\,B\,a^3\,b+3\,A\,a^2\,b^2+\frac {3\,B\,a\,b^3}{2}+\frac {3\,A\,b^4}{8}\right )}{4\,A\,a^4+8\,B\,a^3\,b+12\,A\,a^2\,b^2+6\,B\,a\,b^3+\frac {3\,A\,b^4}{2}}\right )\,\left (2\,A\,a^4+4\,B\,a^3\,b+6\,A\,a^2\,b^2+3\,B\,a\,b^3+\frac {3\,A\,b^4}{4}\right )}{d}-\frac {\left (2\,B\,a^4-\frac {5\,A\,b^4}{4}+2\,B\,b^4-6\,A\,a^2\,b^2+12\,B\,a^2\,b^2+8\,A\,a\,b^3+8\,A\,a^3\,b-5\,B\,a\,b^3-4\,B\,a^3\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (\frac {A\,b^4}{2}-8\,B\,a^4-\frac {8\,B\,b^4}{3}+12\,A\,a^2\,b^2-32\,B\,a^2\,b^2-\frac {64\,A\,a\,b^3}{3}-32\,A\,a^3\,b+2\,B\,a\,b^3+8\,B\,a^3\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (12\,B\,a^4+48\,A\,a^3\,b+40\,B\,a^2\,b^2+\frac {80\,A\,a\,b^3}{3}+\frac {116\,B\,b^4}{15}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (-\frac {A\,b^4}{2}-8\,B\,a^4-\frac {8\,B\,b^4}{3}-12\,A\,a^2\,b^2-32\,B\,a^2\,b^2-\frac {64\,A\,a\,b^3}{3}-32\,A\,a^3\,b-2\,B\,a\,b^3-8\,B\,a^3\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (\frac {5\,A\,b^4}{4}+2\,B\,a^4+2\,B\,b^4+6\,A\,a^2\,b^2+12\,B\,a^2\,b^2+8\,A\,a\,b^3+8\,A\,a^3\,b+5\,B\,a\,b^3+4\,B\,a^3\,b\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )} \]
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 \[ \int \left (A + B \sec {\left (c + d x \right )}\right ) \left (a + b \sec {\left (c + d x \right )}\right )^{4} \sec {\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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