Optimal. Leaf size=107 \[ \frac {a^4 \tanh ^{-1}(\sin (c+d x))}{d}+\frac {b^2 \left (17 a^2+2 b^2\right ) \sin (c+d x)}{3 d}+2 a b x \left (2 a^2+b^2\right )+\frac {4 a b^3 \sin (c+d x) \cos (c+d x)}{3 d}+\frac {b^2 \sin (c+d x) (a+b \cos (c+d x))^2}{3 d} \]
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Rubi [A] time = 0.23, antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {2793, 3033, 3023, 2735, 3770} \[ \frac {b^2 \left (17 a^2+2 b^2\right ) \sin (c+d x)}{3 d}+2 a b x \left (2 a^2+b^2\right )+\frac {a^4 \tanh ^{-1}(\sin (c+d x))}{d}+\frac {4 a b^3 \sin (c+d x) \cos (c+d x)}{3 d}+\frac {b^2 \sin (c+d x) (a+b \cos (c+d x))^2}{3 d} \]
Antiderivative was successfully verified.
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Rule 2735
Rule 2793
Rule 3023
Rule 3033
Rule 3770
Rubi steps
\begin {align*} \int (a+b \cos (c+d x))^4 \sec (c+d x) \, dx &=\frac {b^2 (a+b \cos (c+d x))^2 \sin (c+d x)}{3 d}+\frac {1}{3} \int (a+b \cos (c+d x)) \left (3 a^3+b \left (9 a^2+2 b^2\right ) \cos (c+d x)+8 a b^2 \cos ^2(c+d x)\right ) \sec (c+d x) \, dx\\ &=\frac {4 a b^3 \cos (c+d x) \sin (c+d x)}{3 d}+\frac {b^2 (a+b \cos (c+d x))^2 \sin (c+d x)}{3 d}+\frac {1}{6} \int \left (6 a^4+12 a b \left (2 a^2+b^2\right ) \cos (c+d x)+2 b^2 \left (17 a^2+2 b^2\right ) \cos ^2(c+d x)\right ) \sec (c+d x) \, dx\\ &=\frac {b^2 \left (17 a^2+2 b^2\right ) \sin (c+d x)}{3 d}+\frac {4 a b^3 \cos (c+d x) \sin (c+d x)}{3 d}+\frac {b^2 (a+b \cos (c+d x))^2 \sin (c+d x)}{3 d}+\frac {1}{6} \int \left (6 a^4+12 a b \left (2 a^2+b^2\right ) \cos (c+d x)\right ) \sec (c+d x) \, dx\\ &=2 a b \left (2 a^2+b^2\right ) x+\frac {b^2 \left (17 a^2+2 b^2\right ) \sin (c+d x)}{3 d}+\frac {4 a b^3 \cos (c+d x) \sin (c+d x)}{3 d}+\frac {b^2 (a+b \cos (c+d x))^2 \sin (c+d x)}{3 d}+a^4 \int \sec (c+d x) \, dx\\ &=2 a b \left (2 a^2+b^2\right ) x+\frac {a^4 \tanh ^{-1}(\sin (c+d x))}{d}+\frac {b^2 \left (17 a^2+2 b^2\right ) \sin (c+d x)}{3 d}+\frac {4 a b^3 \cos (c+d x) \sin (c+d x)}{3 d}+\frac {b^2 (a+b \cos (c+d x))^2 \sin (c+d x)}{3 d}\\ \end {align*}
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Mathematica [A] time = 0.16, size = 128, normalized size = 1.20 \[ \frac {-12 a^4 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+12 a^4 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )+24 a b \left (2 a^2+b^2\right ) (c+d x)+9 b^2 \left (8 a^2+b^2\right ) \sin (c+d x)+12 a b^3 \sin (2 (c+d x))+b^4 \sin (3 (c+d x))}{12 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.15, size = 98, normalized size = 0.92 \[ \frac {3 \, a^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, a^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 12 \, {\left (2 \, a^{3} b + a b^{3}\right )} d x + 2 \, {\left (b^{4} \cos \left (d x + c\right )^{2} + 6 \, a b^{3} \cos \left (d x + c\right ) + 18 \, a^{2} b^{2} + 2 \, b^{4}\right )} \sin \left (d x + c\right )}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.59, size = 212, normalized size = 1.98 \[ \frac {3 \, a^{4} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 3 \, a^{4} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + 6 \, {\left (2 \, a^{3} b + a b^{3}\right )} {\left (d x + c\right )} + \frac {2 \, {\left (18 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 6 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 3 \, b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 36 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 2 \, b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 18 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{3}}}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 131, normalized size = 1.22 \[ \frac {a^{4} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+4 a^{3} b x +\frac {4 a^{3} b c}{d}+\frac {6 a^{2} b^{2} \sin \left (d x +c \right )}{d}+\frac {2 a \,b^{3} \cos \left (d x +c \right ) \sin \left (d x +c \right )}{d}+2 a \,b^{3} x +\frac {2 a \,b^{3} c}{d}+\frac {\sin \left (d x +c \right ) \left (\cos ^{2}\left (d x +c \right )\right ) b^{4}}{3 d}+\frac {2 b^{4} \sin \left (d x +c \right )}{3 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.68, size = 95, normalized size = 0.89 \[ \frac {12 \, {\left (d x + c\right )} a^{3} b + 3 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} a b^{3} - {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} b^{4} + 3 \, a^{4} \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right ) + 18 \, a^{2} b^{2} \sin \left (d x + c\right )}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.82, size = 158, normalized size = 1.48 \[ \frac {3\,b^4\,\sin \left (c+d\,x\right )}{4\,d}+\frac {2\,a^4\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {b^4\,\sin \left (3\,c+3\,d\,x\right )}{12\,d}+\frac {a\,b^3\,\sin \left (2\,c+2\,d\,x\right )}{d}+\frac {6\,a^2\,b^2\,\sin \left (c+d\,x\right )}{d}+\frac {4\,a\,b^3\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {8\,a^3\,b\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d} \]
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 \[ \int \left (a + b \cos {\left (c + d x \right )}\right )^{4} \sec {\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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