Optimal. Leaf size=108 \[ \frac {3 a^3 b \tan (c+d x)}{d}-\frac {b^2 \left (a^2-2 b^2\right ) \sin (c+d x)}{2 d}+\frac {a^2 \left (a^2+12 b^2\right ) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac {a^2 \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^2}{2 d}+4 a b^3 x \]
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Rubi [A] time = 0.25, antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {2792, 3031, 3023, 2735, 3770} \[ -\frac {b^2 \left (a^2-2 b^2\right ) \sin (c+d x)}{2 d}+\frac {a^2 \left (a^2+12 b^2\right ) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac {3 a^3 b \tan (c+d x)}{d}+\frac {a^2 \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^2}{2 d}+4 a b^3 x \]
Antiderivative was successfully verified.
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Rule 2735
Rule 2792
Rule 3023
Rule 3031
Rule 3770
Rubi steps
\begin {align*} \int (a+b \cos (c+d x))^4 \sec ^3(c+d x) \, dx &=\frac {a^2 (a+b \cos (c+d x))^2 \sec (c+d x) \tan (c+d x)}{2 d}+\frac {1}{2} \int (a+b \cos (c+d x)) \left (6 a^2 b+a \left (a^2+6 b^2\right ) \cos (c+d x)-b \left (a^2-2 b^2\right ) \cos ^2(c+d x)\right ) \sec ^2(c+d x) \, dx\\ &=\frac {3 a^3 b \tan (c+d x)}{d}+\frac {a^2 (a+b \cos (c+d x))^2 \sec (c+d x) \tan (c+d x)}{2 d}-\frac {1}{2} \int \left (-a^2 \left (a^2+12 b^2\right )-8 a b^3 \cos (c+d x)+b^2 \left (a^2-2 b^2\right ) \cos ^2(c+d x)\right ) \sec (c+d x) \, dx\\ &=-\frac {b^2 \left (a^2-2 b^2\right ) \sin (c+d x)}{2 d}+\frac {3 a^3 b \tan (c+d x)}{d}+\frac {a^2 (a+b \cos (c+d x))^2 \sec (c+d x) \tan (c+d x)}{2 d}-\frac {1}{2} \int \left (-a^2 \left (a^2+12 b^2\right )-8 a b^3 \cos (c+d x)\right ) \sec (c+d x) \, dx\\ &=4 a b^3 x-\frac {b^2 \left (a^2-2 b^2\right ) \sin (c+d x)}{2 d}+\frac {3 a^3 b \tan (c+d x)}{d}+\frac {a^2 (a+b \cos (c+d x))^2 \sec (c+d x) \tan (c+d x)}{2 d}+\frac {1}{2} \left (a^2 \left (a^2+12 b^2\right )\right ) \int \sec (c+d x) \, dx\\ &=4 a b^3 x+\frac {a^2 \left (a^2+12 b^2\right ) \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac {b^2 \left (a^2-2 b^2\right ) \sin (c+d x)}{2 d}+\frac {3 a^3 b \tan (c+d x)}{d}+\frac {a^2 (a+b \cos (c+d x))^2 \sec (c+d x) \tan (c+d x)}{2 d}\\ \end {align*}
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Mathematica [A] time = 2.44, size = 174, normalized size = 1.61 \[ \frac {16 a^3 b \tan (c+d x)+a \left (\frac {a^3}{\left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}-\frac {a^3}{\left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^2}-2 a \left (a^2+12 b^2\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+2 a \left (a^2+12 b^2\right ) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )+16 b^3 c+16 b^3 d x\right )+4 b^4 \sin (c+d x)}{4 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.75, size = 130, normalized size = 1.20 \[ \frac {16 \, a b^{3} d x \cos \left (d x + c\right )^{2} + {\left (a^{4} + 12 \, a^{2} b^{2}\right )} \cos \left (d x + c\right )^{2} \log \left (\sin \left (d x + c\right ) + 1\right ) - {\left (a^{4} + 12 \, a^{2} b^{2}\right )} \cos \left (d x + c\right )^{2} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (2 \, b^{4} \cos \left (d x + c\right )^{2} + 8 \, a^{3} b \cos \left (d x + c\right ) + a^{4}\right )} \sin \left (d x + c\right )}{4 \, d \cos \left (d x + c\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.86, size = 177, normalized size = 1.64 \[ \frac {8 \, {\left (d x + c\right )} a b^{3} + \frac {4 \, b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1} + {\left (a^{4} + 12 \, a^{2} b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - {\left (a^{4} + 12 \, a^{2} b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + \frac {2 \, {\left (a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 8 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 8 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{2}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.10, size = 114, normalized size = 1.06 \[ \frac {a^{4} \sec \left (d x +c \right ) \tan \left (d x +c \right )}{2 d}+\frac {a^{4} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2 d}+\frac {4 a^{3} b \tan \left (d x +c \right )}{d}+\frac {6 a^{2} b^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+4 a \,b^{3} x +\frac {4 a \,b^{3} c}{d}+\frac {b^{4} \sin \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.86, size = 115, normalized size = 1.06 \[ \frac {16 \, {\left (d x + c\right )} a b^{3} - a^{4} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 12 \, a^{2} b^{2} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 4 \, b^{4} \sin \left (d x + c\right ) + 16 \, a^{3} b \tan \left (d x + c\right )}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.72, size = 152, normalized size = 1.41 \[ \frac {b^4\,\sin \left (c+d\,x\right )}{d}+\frac {a^4\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {a^4\,\sin \left (c+d\,x\right )}{2\,d\,{\cos \left (c+d\,x\right )}^2}+\frac {12\,a^2\,b^2\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {8\,a\,b^3\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {4\,a^3\,b\,\sin \left (c+d\,x\right )}{d\,\cos \left (c+d\,x\right )} \]
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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