Optimal. Leaf size=236 \[ -\frac {b \left (17 a^2-b^2\right ) \sin (c+d x)}{2 d}+\frac {3 b \left (2 a^2-b^2\right ) \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac {\left (4 a^2-b^2\right ) \sin (c+d x) (a \cos (c+d x)+b)^3}{4 b^2 d}-\frac {\left (6 a^2-b^2\right ) \sin (c+d x) (a \cos (c+d x)+b)^2}{4 b d}-\frac {a \left (21 a^2-2 b^2\right ) \sin (c+d x) \cos (c+d x)}{8 d}+\frac {3}{8} a x \left (a^2-12 b^2\right )+\frac {a \tan (c+d x) (a \cos (c+d x)+b)^4}{b^2 d}+\frac {\tan (c+d x) \sec (c+d x) (a \cos (c+d x)+b)^4}{2 b d} \]
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Rubi [A] time = 0.75, antiderivative size = 236, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3872, 2893, 3049, 3033, 3023, 2735, 3770} \[ -\frac {b \left (17 a^2-b^2\right ) \sin (c+d x)}{2 d}+\frac {3 b \left (2 a^2-b^2\right ) \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac {\left (4 a^2-b^2\right ) \sin (c+d x) (a \cos (c+d x)+b)^3}{4 b^2 d}-\frac {\left (6 a^2-b^2\right ) \sin (c+d x) (a \cos (c+d x)+b)^2}{4 b d}-\frac {a \left (21 a^2-2 b^2\right ) \sin (c+d x) \cos (c+d x)}{8 d}+\frac {3}{8} a x \left (a^2-12 b^2\right )+\frac {a \tan (c+d x) (a \cos (c+d x)+b)^4}{b^2 d}+\frac {\tan (c+d x) \sec (c+d x) (a \cos (c+d x)+b)^4}{2 b d} \]
Antiderivative was successfully verified.
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Rule 2735
Rule 2893
Rule 3023
Rule 3033
Rule 3049
Rule 3770
Rule 3872
Rubi steps
\begin {align*} \int (a+b \sec (c+d x))^3 \sin ^4(c+d x) \, dx &=-\int (-b-a \cos (c+d x))^3 \sin (c+d x) \tan ^3(c+d x) \, dx\\ &=\frac {a (b+a \cos (c+d x))^4 \tan (c+d x)}{b^2 d}+\frac {(b+a \cos (c+d x))^4 \sec (c+d x) \tan (c+d x)}{2 b d}+\frac {\int (-b-a \cos (c+d x))^3 \left (-3 \left (2 a^2-b^2\right )+3 a b \cos (c+d x)+2 \left (4 a^2-b^2\right ) \cos ^2(c+d x)\right ) \sec (c+d x) \, dx}{2 b^2}\\ &=-\frac {\left (4 a^2-b^2\right ) (b+a \cos (c+d x))^3 \sin (c+d x)}{4 b^2 d}+\frac {a (b+a \cos (c+d x))^4 \tan (c+d x)}{b^2 d}+\frac {(b+a \cos (c+d x))^4 \sec (c+d x) \tan (c+d x)}{2 b d}+\frac {\int (-b-a \cos (c+d x))^2 \left (12 b \left (2 a^2-b^2\right )-18 a b^2 \cos (c+d x)-6 b \left (6 a^2-b^2\right ) \cos ^2(c+d x)\right ) \sec (c+d x) \, dx}{8 b^2}\\ &=-\frac {\left (6 a^2-b^2\right ) (b+a \cos (c+d x))^2 \sin (c+d x)}{4 b d}-\frac {\left (4 a^2-b^2\right ) (b+a \cos (c+d x))^3 \sin (c+d x)}{4 b^2 d}+\frac {a (b+a \cos (c+d x))^4 \tan (c+d x)}{b^2 d}+\frac {(b+a \cos (c+d x))^4 \sec (c+d x) \tan (c+d x)}{2 b d}+\frac {\int (-b-a \cos (c+d x)) \left (-36 b^2 \left (2 a^2-b^2\right )+78 a b^3 \cos (c+d x)+6 b^2 \left (21 a^2-2 b^2\right ) \cos ^2(c+d x)\right ) \sec (c+d x) \, dx}{24 b^2}\\ &=-\frac {a \left (21 a^2-2 b^2\right ) \cos (c+d x) \sin (c+d x)}{8 d}-\frac {\left (6 a^2-b^2\right ) (b+a \cos (c+d x))^2 \sin (c+d x)}{4 b d}-\frac {\left (4 a^2-b^2\right ) (b+a \cos (c+d x))^3 \sin (c+d x)}{4 b^2 d}+\frac {a (b+a \cos (c+d x))^4 \tan (c+d x)}{b^2 d}+\frac {(b+a \cos (c+d x))^4 \sec (c+d x) \tan (c+d x)}{2 b d}+\frac {\int \left (72 b^3 \left (2 a^2-b^2\right )+18 a b^2 \left (a^2-12 b^2\right ) \cos (c+d x)-24 b^3 \left (17 a^2-b^2\right ) \cos ^2(c+d x)\right ) \sec (c+d x) \, dx}{48 b^2}\\ &=-\frac {b \left (17 a^2-b^2\right ) \sin (c+d x)}{2 d}-\frac {a \left (21 a^2-2 b^2\right ) \cos (c+d x) \sin (c+d x)}{8 d}-\frac {\left (6 a^2-b^2\right ) (b+a \cos (c+d x))^2 \sin (c+d x)}{4 b d}-\frac {\left (4 a^2-b^2\right ) (b+a \cos (c+d x))^3 \sin (c+d x)}{4 b^2 d}+\frac {a (b+a \cos (c+d x))^4 \tan (c+d x)}{b^2 d}+\frac {(b+a \cos (c+d x))^4 \sec (c+d x) \tan (c+d x)}{2 b d}+\frac {\int \left (72 b^3 \left (2 a^2-b^2\right )+18 a b^2 \left (a^2-12 b^2\right ) \cos (c+d x)\right ) \sec (c+d x) \, dx}{48 b^2}\\ &=\frac {3}{8} a \left (a^2-12 b^2\right ) x-\frac {b \left (17 a^2-b^2\right ) \sin (c+d x)}{2 d}-\frac {a \left (21 a^2-2 b^2\right ) \cos (c+d x) \sin (c+d x)}{8 d}-\frac {\left (6 a^2-b^2\right ) (b+a \cos (c+d x))^2 \sin (c+d x)}{4 b d}-\frac {\left (4 a^2-b^2\right ) (b+a \cos (c+d x))^3 \sin (c+d x)}{4 b^2 d}+\frac {a (b+a \cos (c+d x))^4 \tan (c+d x)}{b^2 d}+\frac {(b+a \cos (c+d x))^4 \sec (c+d x) \tan (c+d x)}{2 b d}+\frac {1}{2} \left (3 b \left (2 a^2-b^2\right )\right ) \int \sec (c+d x) \, dx\\ &=\frac {3}{8} a \left (a^2-12 b^2\right ) x+\frac {3 b \left (2 a^2-b^2\right ) \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac {b \left (17 a^2-b^2\right ) \sin (c+d x)}{2 d}-\frac {a \left (21 a^2-2 b^2\right ) \cos (c+d x) \sin (c+d x)}{8 d}-\frac {\left (6 a^2-b^2\right ) (b+a \cos (c+d x))^2 \sin (c+d x)}{4 b d}-\frac {\left (4 a^2-b^2\right ) (b+a \cos (c+d x))^3 \sin (c+d x)}{4 b^2 d}+\frac {a (b+a \cos (c+d x))^4 \tan (c+d x)}{b^2 d}+\frac {(b+a \cos (c+d x))^4 \sec (c+d x) \tan (c+d x)}{2 b d}\\ \end {align*}
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Mathematica [B] time = 6.19, size = 696, normalized size = 2.95 \[ \frac {a^3 \sin (4 (c+d x)) \cos ^3(c+d x) (a+b \sec (c+d x))^3}{32 d (a \cos (c+d x)+b)^3}+\frac {3 \left (b^3-2 a^2 b\right ) \cos ^3(c+d x) (a+b \sec (c+d x))^3 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}{2 d (a \cos (c+d x)+b)^3}-\frac {3 \left (b^3-2 a^2 b\right ) \cos ^3(c+d x) (a+b \sec (c+d x))^3 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )}{2 d (a \cos (c+d x)+b)^3}+\frac {3 a \left (a^2-12 b^2\right ) (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^3}{8 d (a \cos (c+d x)+b)^3}+\frac {b \left (4 b^2-15 a^2\right ) \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^3}{4 d (a \cos (c+d x)+b)^3}-\frac {a \left (a^2-3 b^2\right ) \sin (2 (c+d x)) \cos ^3(c+d x) (a+b \sec (c+d x))^3}{4 d (a \cos (c+d x)+b)^3}+\frac {a^2 b \sin (3 (c+d x)) \cos ^3(c+d x) (a+b \sec (c+d x))^3}{4 d (a \cos (c+d x)+b)^3}+\frac {b^3 \cos ^3(c+d x) (a+b \sec (c+d x))^3}{4 d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^2 (a \cos (c+d x)+b)^3}-\frac {b^3 \cos ^3(c+d x) (a+b \sec (c+d x))^3}{4 d \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^2 (a \cos (c+d x)+b)^3}+\frac {3 a b^2 \sin \left (\frac {1}{2} (c+d x)\right ) \cos ^3(c+d x) (a+b \sec (c+d x))^3}{d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right ) (a \cos (c+d x)+b)^3}+\frac {3 a b^2 \sin \left (\frac {1}{2} (c+d x)\right ) \cos ^3(c+d x) (a+b \sec (c+d x))^3}{d \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right ) (a \cos (c+d x)+b)^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.57, size = 196, normalized size = 0.83 \[ \frac {3 \, {\left (a^{3} - 12 \, a b^{2}\right )} d x \cos \left (d x + c\right )^{2} + 6 \, {\left (2 \, a^{2} b - b^{3}\right )} \cos \left (d x + c\right )^{2} \log \left (\sin \left (d x + c\right ) + 1\right ) - 6 \, {\left (2 \, a^{2} b - b^{3}\right )} \cos \left (d x + c\right )^{2} \log \left (-\sin \left (d x + c\right ) + 1\right ) + {\left (2 \, a^{3} \cos \left (d x + c\right )^{5} + 8 \, a^{2} b \cos \left (d x + c\right )^{4} + 24 \, a b^{2} \cos \left (d x + c\right ) - {\left (5 \, a^{3} - 12 \, a b^{2}\right )} \cos \left (d x + c\right )^{3} + 4 \, b^{3} - 8 \, {\left (4 \, a^{2} b - b^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{8 \, d \cos \left (d x + c\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.40, size = 431, normalized size = 1.83 \[ \frac {3 \, {\left (a^{3} - 12 \, a b^{2}\right )} {\left (d x + c\right )} + 12 \, {\left (2 \, a^{2} b - b^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 12 \, {\left (2 \, a^{2} b - b^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {8 \, {\left (6 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 6 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - b^{3} \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 )}^{2}} + \frac {2 \, {\left (3 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 24 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 12 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 8 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 11 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 104 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 12 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 24 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 11 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 104 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 12 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 24 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 3 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 24 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 12 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 8 \, b^{3} \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 = 0.62, size = 276, normalized size = 1.17 \[ -\frac {a^{3} \cos \left (d x +c \right ) \left (\sin ^{3}\left (d x +c \right )\right )}{4 d}-\frac {3 a^{3} \cos \left (d x +c \right ) \sin \left (d x +c \right )}{8 d}+\frac {3 a^{3} x}{8}+\frac {3 a^{3} c}{8 d}-\frac {a^{2} b \left (\sin ^{3}\left (d x +c \right )\right )}{d}-\frac {3 a^{2} b \sin \left (d x +c \right )}{d}+\frac {3 a^{2} b \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {3 b^{2} a \left (\sin ^{5}\left (d x +c \right )\right )}{d \cos \left (d x +c \right )}+\frac {3 b^{2} a \cos \left (d x +c \right ) \left (\sin ^{3}\left (d x +c \right )\right )}{d}+\frac {9 a \,b^{2} \cos \left (d x +c \right ) \sin \left (d x +c \right )}{2 d}-\frac {9 a \,b^{2} x}{2}-\frac {9 a \,b^{2} c}{2 d}+\frac {b^{3} \left (\sin ^{5}\left (d x +c \right )\right )}{2 d \cos \left (d x +c \right )^{2}}+\frac {b^{3} \left (\sin ^{3}\left (d x +c \right )\right )}{2 d}+\frac {3 b^{3} \sin \left (d x +c \right )}{2 d}-\frac {3 b^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.73, size = 183, normalized size = 0.78 \[ \frac {{\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) - 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a^{3} - 16 \, {\left (2 \, \sin \left (d x + c\right )^{3} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right ) + 6 \, \sin \left (d x + c\right )\right )} a^{2} b - 48 \, {\left (3 \, d x + 3 \, c - \frac {\tan \left (d x + c\right )}{\tan \left (d x + c\right )^{2} + 1} - 2 \, \tan \left (d x + c\right )\right )} a b^{2} - 8 \, b^{3} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} + 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, \log \left (\sin \left (d x + c\right ) - 1\right ) - 4 \, \sin \left (d x + c\right )\right )}}{32 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.47, size = 281, normalized size = 1.19 \[ \frac {b^3\,\sin \left (c+d\,x\right )}{d}+\frac {3\,a^3\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{4\,d}-\frac {3\,b^3\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {a^3\,{\cos \left (c+d\,x\right )}^3\,\sin \left (c+d\,x\right )}{4\,d}+\frac {b^3\,\sin \left (c+d\,x\right )}{2\,d\,{\cos \left (c+d\,x\right )}^2}-\frac {5\,a^3\,\cos \left (c+d\,x\right )\,\sin \left (c+d\,x\right )}{8\,d}-\frac {4\,a^2\,b\,\sin \left (c+d\,x\right )}{d}-\frac {9\,a\,b^2\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {6\,a^2\,b\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {3\,a\,b^2\,\cos \left (c+d\,x\right )\,\sin \left (c+d\,x\right )}{2\,d}+\frac {3\,a\,b^2\,\sin \left (c+d\,x\right )}{d\,\cos \left (c+d\,x\right )}+\frac {a^2\,b\,{\cos \left (c+d\,x\right )}^2\,\sin \left (c+d\,x\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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