Optimal. Leaf size=200 \[ a^4 A x+\frac {b^2 \left (26 a^2 B+32 a A b+9 b^2 B\right ) \tan (c+d x) \sec (c+d x)}{24 d}+\frac {b \left (19 a^3 B+34 a^2 A b+16 a b^2 B+4 A b^3\right ) \tan (c+d x)}{6 d}+\frac {\left (8 a^4 B+32 a^3 A b+24 a^2 b^2 B+16 a A b^3+3 b^4 B\right ) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac {b (7 a B+4 A b) \tan (c+d x) (a+b \sec (c+d x))^2}{12 d}+\frac {b B \tan (c+d x) (a+b \sec (c+d x))^3}{4 d} \]
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Rubi [A] time = 0.33, antiderivative size = 200, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {3918, 4056, 4048, 3770, 3767, 8} \[ \frac {b \left (34 a^2 A b+19 a^3 B+16 a b^2 B+4 A b^3\right ) \tan (c+d x)}{6 d}+\frac {\left (32 a^3 A b+24 a^2 b^2 B+8 a^4 B+16 a A b^3+3 b^4 B\right ) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac {b^2 \left (26 a^2 B+32 a A b+9 b^2 B\right ) \tan (c+d x) \sec (c+d x)}{24 d}+a^4 A x+\frac {b (7 a B+4 A b) \tan (c+d x) (a+b \sec (c+d x))^2}{12 d}+\frac {b B \tan (c+d x) (a+b \sec (c+d x))^3}{4 d} \]
Antiderivative was successfully verified.
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Rule 8
Rule 3767
Rule 3770
Rule 3918
Rule 4048
Rule 4056
Rubi steps
\begin {align*} \int (a+b \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx &=\frac {b B (a+b \sec (c+d x))^3 \tan (c+d x)}{4 d}+\frac {1}{4} \int (a+b \sec (c+d x))^2 \left (4 a^2 A+\left (8 a A b+4 a^2 B+3 b^2 B\right ) \sec (c+d x)+b (4 A b+7 a B) \sec ^2(c+d x)\right ) \, dx\\ &=\frac {b (4 A b+7 a B) (a+b \sec (c+d x))^2 \tan (c+d x)}{12 d}+\frac {b B (a+b \sec (c+d x))^3 \tan (c+d x)}{4 d}+\frac {1}{12} \int (a+b \sec (c+d x)) \left (12 a^3 A+\left (36 a^2 A b+8 A b^3+12 a^3 B+23 a b^2 B\right ) \sec (c+d x)+b \left (32 a A b+26 a^2 B+9 b^2 B\right ) \sec ^2(c+d x)\right ) \, dx\\ &=\frac {b^2 \left (32 a A b+26 a^2 B+9 b^2 B\right ) \sec (c+d x) \tan (c+d x)}{24 d}+\frac {b (4 A b+7 a B) (a+b \sec (c+d x))^2 \tan (c+d x)}{12 d}+\frac {b B (a+b \sec (c+d x))^3 \tan (c+d x)}{4 d}+\frac {1}{24} \int \left (24 a^4 A+3 \left (32 a^3 A b+16 a A b^3+8 a^4 B+24 a^2 b^2 B+3 b^4 B\right ) \sec (c+d x)+4 b \left (34 a^2 A b+4 A b^3+19 a^3 B+16 a b^2 B\right ) \sec ^2(c+d x)\right ) \, dx\\ &=a^4 A x+\frac {b^2 \left (32 a A b+26 a^2 B+9 b^2 B\right ) \sec (c+d x) \tan (c+d x)}{24 d}+\frac {b (4 A b+7 a B) (a+b \sec (c+d x))^2 \tan (c+d x)}{12 d}+\frac {b B (a+b \sec (c+d x))^3 \tan (c+d x)}{4 d}+\frac {1}{6} \left (b \left (34 a^2 A b+4 A b^3+19 a^3 B+16 a b^2 B\right )\right ) \int \sec ^2(c+d x) \, dx+\frac {1}{8} \left (32 a^3 A b+16 a A b^3+8 a^4 B+24 a^2 b^2 B+3 b^4 B\right ) \int \sec (c+d x) \, dx\\ &=a^4 A x+\frac {\left (32 a^3 A b+16 a A b^3+8 a^4 B+24 a^2 b^2 B+3 b^4 B\right ) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac {b^2 \left (32 a A b+26 a^2 B+9 b^2 B\right ) \sec (c+d x) \tan (c+d x)}{24 d}+\frac {b (4 A b+7 a B) (a+b \sec (c+d x))^2 \tan (c+d x)}{12 d}+\frac {b B (a+b \sec (c+d x))^3 \tan (c+d x)}{4 d}-\frac {\left (b \left (34 a^2 A b+4 A b^3+19 a^3 B+16 a b^2 B\right )\right ) \operatorname {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{6 d}\\ &=a^4 A x+\frac {\left (32 a^3 A b+16 a A b^3+8 a^4 B+24 a^2 b^2 B+3 b^4 B\right ) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac {b \left (34 a^2 A b+4 A b^3+19 a^3 B+16 a b^2 B\right ) \tan (c+d x)}{6 d}+\frac {b^2 \left (32 a A b+26 a^2 B+9 b^2 B\right ) \sec (c+d x) \tan (c+d x)}{24 d}+\frac {b (4 A b+7 a B) (a+b \sec (c+d x))^2 \tan (c+d x)}{12 d}+\frac {b B (a+b \sec (c+d x))^3 \tan (c+d x)}{4 d}\\ \end {align*}
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Mathematica [A] time = 1.03, size = 160, normalized size = 0.80 \[ \frac {24 a^4 A d x+3 b \tan (c+d x) \left (b \left (24 a^2 B+16 a A b+3 b^2 B\right ) \sec (c+d x)+8 \left (4 a^3 B+6 a^2 A b+4 a b^2 B+A b^3\right )+2 b^3 B \sec ^3(c+d x)\right )+3 \left (8 a^4 B+32 a^3 A b+24 a^2 b^2 B+16 a A b^3+3 b^4 B\right ) \tanh ^{-1}(\sin (c+d x))+8 b^3 (4 a B+A b) \tan ^3(c+d x)}{24 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 250, normalized size = 1.25 \[ \frac {48 \, A a^{4} d x \cos \left (d x + c\right )^{4} + 3 \, {\left (8 \, B a^{4} + 32 \, A a^{3} b + 24 \, B a^{2} b^{2} + 16 \, A a b^{3} + 3 \, B b^{4}\right )} \cos \left (d x + c\right )^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, {\left (8 \, B a^{4} + 32 \, A a^{3} b + 24 \, B a^{2} b^{2} + 16 \, A a b^{3} + 3 \, B b^{4}\right )} \cos \left (d x + c\right )^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (6 \, B b^{4} + 16 \, {\left (6 \, B a^{3} b + 9 \, A a^{2} b^{2} + 4 \, B a b^{3} + A b^{4}\right )} \cos \left (d x + c\right )^{3} + 3 \, {\left (24 \, B a^{2} b^{2} + 16 \, A a b^{3} + 3 \, B b^{4}\right )} \cos \left (d x + c\right )^{2} + 8 \, {\left (4 \, B a b^{3} + A b^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{48 \, d \cos \left (d x + c\right )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 2.39, size = 635, normalized size = 3.18 \[ \frac {24 \, {\left (d x + c\right )} A a^{4} + 3 \, {\left (8 \, B a^{4} + 32 \, A a^{3} b + 24 \, B a^{2} b^{2} + 16 \, A a b^{3} + 3 \, B b^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 3 \, {\left (8 \, B a^{4} + 32 \, A a^{3} b + 24 \, B a^{2} b^{2} + 16 \, A a b^{3} + 3 \, B b^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (96 \, B a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 144 \, A a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 72 \, B a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 48 \, A a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 96 \, B a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 24 \, A b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 15 \, B b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 288 \, B a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 432 \, A a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 72 \, B a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 48 \, A a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 160 \, B a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 40 \, A b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 9 \, B b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 288 \, B a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 432 \, A a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 72 \, B a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 48 \, A a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 160 \, B a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 40 \, A b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 9 \, B b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 96 \, B a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 144 \, A a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 72 \, B a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 48 \, A a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 96 \, B a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 24 \, A b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 15 \, B b^{4} \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 = 1.38, size = 338, normalized size = 1.69 \[ A \,a^{4} x +\frac {A \,a^{4} c}{d}+\frac {a^{4} B \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {4 A \,a^{3} b \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {4 B \,a^{3} b \tan \left (d x +c \right )}{d}+\frac {6 A \,a^{2} b^{2} \tan \left (d x +c \right )}{d}+\frac {3 a^{2} b^{2} B \sec \left (d x +c \right ) \tan \left (d x +c \right )}{d}+\frac {3 a^{2} b^{2} B \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {2 a A \,b^{3} \sec \left (d x +c \right ) \tan \left (d x +c \right )}{d}+\frac {2 a A \,b^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {8 B a \,b^{3} \tan \left (d x +c \right )}{3 d}+\frac {4 B a \,b^{3} \tan \left (d x +c \right ) \left (\sec ^{2}\left (d x +c \right )\right )}{3 d}+\frac {2 A \,b^{4} \tan \left (d x +c \right )}{3 d}+\frac {A \,b^{4} \tan \left (d x +c \right ) \left (\sec ^{2}\left (d x +c \right )\right )}{3 d}+\frac {B \,b^{4} \tan \left (d x +c \right ) \left (\sec ^{3}\left (d x +c \right )\right )}{4 d}+\frac {3 B \,b^{4} \sec \left (d x +c \right ) \tan \left (d x +c \right )}{8 d}+\frac {3 B \,b^{4} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.84, size = 303, normalized size = 1.52 \[ \frac {48 \, {\left (d x + c\right )} A a^{4} + 64 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} B a b^{3} + 16 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} A b^{4} - 3 \, B b^{4} {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 72 \, B a^{2} b^{2} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 48 \, A a b^{3} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 48 \, B a^{4} \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right ) + 192 \, A a^{3} b \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right ) + 192 \, B a^{3} b \tan \left (d x + c\right ) + 288 \, A a^{2} b^{2} \tan \left (d x + c\right )}{48 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.01, size = 1969, normalized size = 9.84 \[ \text {result too large to display} \]
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 \[ \int \left (A + B \sec {\left (c + d x \right )}\right ) \left (a + b \sec {\left (c + d x \right )}\right )^{4}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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