Optimal. Leaf size=145 \[ \frac {5 a^3 b \sin (c+d x) \cos ^2(c+d x)}{6 d}+\frac {4 a b \left (2 a^2+3 b^2\right ) \sin (c+d x)}{3 d}+\frac {a^2 \left (3 a^2+22 b^2\right ) \sin (c+d x) \cos (c+d x)}{8 d}+\frac {a^2 \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^2}{4 d}+\frac {1}{8} x \left (3 a^4+24 a^2 b^2+8 b^4\right ) \]
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Rubi [A] time = 0.31, antiderivative size = 145, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3841, 4074, 4047, 2637, 4045, 8} \[ \frac {4 a b \left (2 a^2+3 b^2\right ) \sin (c+d x)}{3 d}+\frac {a^2 \left (3 a^2+22 b^2\right ) \sin (c+d x) \cos (c+d x)}{8 d}+\frac {1}{8} x \left (24 a^2 b^2+3 a^4+8 b^4\right )+\frac {5 a^3 b \sin (c+d x) \cos ^2(c+d x)}{6 d}+\frac {a^2 \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^2}{4 d} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2637
Rule 3841
Rule 4045
Rule 4047
Rule 4074
Rubi steps
\begin {align*} \int \cos ^4(c+d x) (a+b \sec (c+d x))^4 \, dx &=\frac {a^2 \cos ^3(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{4 d}+\frac {1}{4} \int \cos ^3(c+d x) (a+b \sec (c+d x)) \left (10 a^2 b+3 a \left (a^2+4 b^2\right ) \sec (c+d x)+b \left (a^2+4 b^2\right ) \sec ^2(c+d x)\right ) \, dx\\ &=\frac {5 a^3 b \cos ^2(c+d x) \sin (c+d x)}{6 d}+\frac {a^2 \cos ^3(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{4 d}-\frac {1}{12} \int \cos ^2(c+d x) \left (-3 a^2 \left (3 a^2+22 b^2\right )-16 a b \left (2 a^2+3 b^2\right ) \sec (c+d x)-3 b^2 \left (a^2+4 b^2\right ) \sec ^2(c+d x)\right ) \, dx\\ &=\frac {5 a^3 b \cos ^2(c+d x) \sin (c+d x)}{6 d}+\frac {a^2 \cos ^3(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{4 d}-\frac {1}{12} \int \cos ^2(c+d x) \left (-3 a^2 \left (3 a^2+22 b^2\right )-3 b^2 \left (a^2+4 b^2\right ) \sec ^2(c+d x)\right ) \, dx+\frac {1}{3} \left (4 a b \left (2 a^2+3 b^2\right )\right ) \int \cos (c+d x) \, dx\\ &=\frac {4 a b \left (2 a^2+3 b^2\right ) \sin (c+d x)}{3 d}+\frac {a^2 \left (3 a^2+22 b^2\right ) \cos (c+d x) \sin (c+d x)}{8 d}+\frac {5 a^3 b \cos ^2(c+d x) \sin (c+d x)}{6 d}+\frac {a^2 \cos ^3(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{4 d}-\frac {1}{8} \left (-3 a^4-24 a^2 b^2-8 b^4\right ) \int 1 \, dx\\ &=\frac {1}{8} \left (3 a^4+24 a^2 b^2+8 b^4\right ) x+\frac {4 a b \left (2 a^2+3 b^2\right ) \sin (c+d x)}{3 d}+\frac {a^2 \left (3 a^2+22 b^2\right ) \cos (c+d x) \sin (c+d x)}{8 d}+\frac {5 a^3 b \cos ^2(c+d x) \sin (c+d x)}{6 d}+\frac {a^2 \cos ^3(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{4 d}\\ \end {align*}
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Mathematica [A] time = 0.24, size = 104, normalized size = 0.72 \[ \frac {3 a^4 \sin (4 (c+d x))+32 a^3 b \sin (3 (c+d x))+24 a^2 \left (a^2+6 b^2\right ) \sin (2 (c+d x))+96 a b \left (3 a^2+4 b^2\right ) \sin (c+d x)+12 \left (3 a^4+24 a^2 b^2+8 b^4\right ) (c+d x)}{96 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 96, normalized size = 0.66 \[ \frac {3 \, {\left (3 \, a^{4} + 24 \, a^{2} b^{2} + 8 \, b^{4}\right )} d x + {\left (6 \, a^{4} \cos \left (d x + c\right )^{3} + 32 \, a^{3} b \cos \left (d x + c\right )^{2} + 64 \, a^{3} b + 96 \, a b^{3} + 9 \, {\left (a^{4} + 8 \, a^{2} b^{2}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{24 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.24, size = 318, normalized size = 2.19 \[ \frac {3 \, {\left (3 \, a^{4} + 24 \, a^{2} b^{2} + 8 \, b^{4}\right )} {\left (d x + c\right )} - \frac {2 \, {\left (15 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 96 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 72 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 96 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 9 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 160 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 72 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 288 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 9 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 160 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 72 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 288 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 15 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 96 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 72 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 96 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{4}}}{24 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.95, size = 116, normalized size = 0.80 \[ \frac {a^{4} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {4 a^{3} b \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+6 a^{2} b^{2} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+4 a \,b^{3} \sin \left (d x +c \right )+b^{4} \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.46, size = 109, normalized size = 0.75 \[ \frac {3 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a^{4} - 128 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} a^{3} b + 144 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} a^{2} b^{2} + 96 \, {\left (d x + c\right )} b^{4} + 384 \, a b^{3} \sin \left (d x + c\right )}{96 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.88, size = 123, normalized size = 0.85 \[ \frac {3\,a^4\,x}{8}+b^4\,x+3\,a^2\,b^2\,x+\frac {a^4\,\sin \left (2\,c+2\,d\,x\right )}{4\,d}+\frac {a^4\,\sin \left (4\,c+4\,d\,x\right )}{32\,d}+\frac {a^3\,b\,\sin \left (3\,c+3\,d\,x\right )}{3\,d}+\frac {3\,a^2\,b^2\,\sin \left (2\,c+2\,d\,x\right )}{2\,d}+\frac {4\,a\,b^3\,\sin \left (c+d\,x\right )}{d}+\frac {3\,a^3\,b\,\sin \left (c+d\,x\right )}{d} \]
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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