Optimal. Leaf size=158 \[ a^5 x+\frac {a b^2 \left (5 a^2-4 b^2\right ) \tan (c+d x)}{2 d}+\frac {b^3 \left (2 a^2-3 b^2\right ) \tan (c+d x) \sec (c+d x)}{8 d}+\frac {b \left (24 a^4-8 a^2 b^2-3 b^4\right ) \tanh ^{-1}(\sin (c+d x))}{8 d}-\frac {a b^2 \tan (c+d x) (a+b \sec (c+d x))^2}{4 d}-\frac {b^2 \tan (c+d x) (a+b \sec (c+d x))^3}{4 d} \]
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Rubi [A] time = 0.29, antiderivative size = 158, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.233, Rules used = {4042, 3918, 4056, 4048, 3770, 3767, 8} \[ \frac {a b^2 \left (5 a^2-4 b^2\right ) \tan (c+d x)}{2 d}+\frac {b \left (-8 a^2 b^2+24 a^4-3 b^4\right ) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac {b^3 \left (2 a^2-3 b^2\right ) \tan (c+d x) \sec (c+d x)}{8 d}+a^5 x-\frac {a b^2 \tan (c+d x) (a+b \sec (c+d x))^2}{4 d}-\frac {b^2 \tan (c+d x) (a+b \sec (c+d x))^3}{4 d} \]
Antiderivative was successfully verified.
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Rule 8
Rule 3767
Rule 3770
Rule 3918
Rule 4042
Rule 4048
Rule 4056
Rubi steps
\begin {align*} \int (a+b \sec (c+d x))^3 \left (a^2-b^2 \sec ^2(c+d x)\right ) \, dx &=-\int (-a+b \sec (c+d x)) (a+b \sec (c+d x))^4 \, dx\\ &=-\frac {b^2 (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^3-b \left (4 a^2-3 b^2\right ) \sec (c+d x)+3 a b^2 \sec ^2(c+d x)\right ) \, dx\\ &=-\frac {a b^2 (a+b \sec (c+d x))^2 \tan (c+d x)}{4 d}-\frac {b^2 (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^4-3 a b \left (8 a^2-5 b^2\right ) \sec (c+d x)-3 b^2 \left (2 a^2-3 b^2\right ) \sec ^2(c+d x)\right ) \, dx\\ &=\frac {b^3 \left (2 a^2-3 b^2\right ) \sec (c+d x) \tan (c+d x)}{8 d}-\frac {a b^2 (a+b \sec (c+d x))^2 \tan (c+d x)}{4 d}-\frac {b^2 (a+b \sec (c+d x))^3 \tan (c+d x)}{4 d}-\frac {1}{24} \int \left (-24 a^5-3 b \left (24 a^4-8 a^2 b^2-3 b^4\right ) \sec (c+d x)-12 a b^2 \left (5 a^2-4 b^2\right ) \sec ^2(c+d x)\right ) \, dx\\ &=a^5 x+\frac {b^3 \left (2 a^2-3 b^2\right ) \sec (c+d x) \tan (c+d x)}{8 d}-\frac {a b^2 (a+b \sec (c+d x))^2 \tan (c+d x)}{4 d}-\frac {b^2 (a+b \sec (c+d x))^3 \tan (c+d x)}{4 d}+\frac {1}{2} \left (a b^2 \left (5 a^2-4 b^2\right )\right ) \int \sec ^2(c+d x) \, dx+\frac {1}{8} \left (b \left (24 a^4-8 a^2 b^2-3 b^4\right )\right ) \int \sec (c+d x) \, dx\\ &=a^5 x+\frac {b \left (24 a^4-8 a^2 b^2-3 b^4\right ) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac {b^3 \left (2 a^2-3 b^2\right ) \sec (c+d x) \tan (c+d x)}{8 d}-\frac {a b^2 (a+b \sec (c+d x))^2 \tan (c+d x)}{4 d}-\frac {b^2 (a+b \sec (c+d x))^3 \tan (c+d x)}{4 d}-\frac {\left (a b^2 \left (5 a^2-4 b^2\right )\right ) \operatorname {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{2 d}\\ &=a^5 x+\frac {b \left (24 a^4-8 a^2 b^2-3 b^4\right ) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac {a b^2 \left (5 a^2-4 b^2\right ) \tan (c+d x)}{2 d}+\frac {b^3 \left (2 a^2-3 b^2\right ) \sec (c+d x) \tan (c+d x)}{8 d}-\frac {a b^2 (a+b \sec (c+d x))^2 \tan (c+d x)}{4 d}-\frac {b^2 (a+b \sec (c+d x))^3 \tan (c+d x)}{4 d}\\ \end {align*}
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Mathematica [B] time = 6.41, size = 1299, normalized size = 8.22 \[ \text {result too large to display} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.50, size = 181, normalized size = 1.15 \[ \frac {16 \, a^{5} d x \cos \left (d x + c\right )^{4} + {\left (24 \, a^{4} b - 8 \, a^{2} b^{3} - 3 \, b^{5}\right )} \cos \left (d x + c\right )^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) - {\left (24 \, a^{4} b - 8 \, a^{2} b^{3} - 3 \, b^{5}\right )} \cos \left (d x + c\right )^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (8 \, a b^{4} \cos \left (d x + c\right ) + 2 \, b^{5} - 16 \, {\left (a^{3} b^{2} - a b^{4}\right )} \cos \left (d x + c\right )^{3} + {\left (8 \, a^{2} b^{3} + 3 \, b^{5}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{16 \, d \cos \left (d x + c\right )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.31, size = 379, normalized size = 2.40 \[ \frac {8 \, {\left (d x + c\right )} a^{5} + {\left (24 \, a^{4} b - 8 \, a^{2} b^{3} - 3 \, b^{5}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - {\left (24 \, a^{4} b - 8 \, a^{2} b^{3} - 3 \, b^{5}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (16 \, a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 8 \, a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 24 \, a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 5 \, b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 48 \, a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 8 \, a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 40 \, a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 3 \, b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 48 \, a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 8 \, a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 40 \, a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 3 \, b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 16 \, a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 8 \, a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 24 \, a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 5 \, b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{4}}}{8 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.53, size = 205, normalized size = 1.30 \[ a^{5} x +\frac {a^{5} c}{d}+\frac {2 b^{2} a^{3} \tan \left (d x +c \right )}{d}+\frac {3 b \,a^{4} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}-\frac {a^{2} b^{3} \sec \left (d x +c \right ) \tan \left (d x +c \right )}{d}-\frac {a^{2} b^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}-\frac {2 a \,b^{4} \tan \left (d x +c \right )}{d}-\frac {a \,b^{4} \tan \left (d x +c \right ) \left (\sec ^{2}\left (d x +c \right )\right )}{d}-\frac {b^{5} \tan \left (d x +c \right ) \left (\sec ^{3}\left (d x +c \right )\right )}{4 d}-\frac {3 b^{5} \sec \left (d x +c \right ) \tan \left (d x +c \right )}{8 d}-\frac {3 b^{5} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 192, normalized size = 1.22 \[ \frac {16 \, {\left (d x + c\right )} a^{5} - 16 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} a b^{4} + b^{5} {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 8 \, a^{2} b^{3} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 48 \, a^{4} b \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right ) + 32 \, a^{3} b^{2} \tan \left (d x + c\right )}{16 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.07, size = 274, normalized size = 1.73 \[ \frac {2\,a^5\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}-\frac {3\,b^5\,\sin \left (c+d\,x\right )}{8\,d\,{\cos \left (c+d\,x\right )}^2}-\frac {b^5\,\sin \left (c+d\,x\right )}{4\,d\,{\cos \left (c+d\,x\right )}^4}-\frac {2\,a\,b^4\,\sin \left (c+d\,x\right )}{d\,\cos \left (c+d\,x\right )}-\frac {a\,b^4\,\sin \left (c+d\,x\right )}{d\,{\cos \left (c+d\,x\right )}^3}+\frac {2\,a^3\,b^2\,\sin \left (c+d\,x\right )}{d\,\cos \left (c+d\,x\right )}-\frac {a^2\,b^3\,\sin \left (c+d\,x\right )}{d\,{\cos \left (c+d\,x\right )}^2}+\frac {b^5\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,1{}\mathrm {i}}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,3{}\mathrm {i}}{4\,d}+\frac {a^2\,b^3\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,1{}\mathrm {i}}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,2{}\mathrm {i}}{d}-\frac {a^4\,b\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,1{}\mathrm {i}}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,6{}\mathrm {i}}{d} \]
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 \[ \int \left (a - b \sec {\left (c + d x \right )}\right ) \left (a + b \sec {\left (c + d x \right )}\right )^{4}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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