Optimal. Leaf size=79 \[ \frac {1}{2} a x \left (a^2+6 b^2\right )+\frac {5 a^2 b \sin (c+d x)}{2 d}+\frac {a^2 \sin (c+d x) \cos (c+d x) (a+b \sec (c+d x))}{2 d}+\frac {b^3 \tanh ^{-1}(\sin (c+d x))}{d} \]
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Rubi [A] time = 0.12, antiderivative size = 79, 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 = {3841, 4047, 8, 4045, 3770} \[ \frac {1}{2} a x \left (a^2+6 b^2\right )+\frac {5 a^2 b \sin (c+d x)}{2 d}+\frac {a^2 \sin (c+d x) \cos (c+d x) (a+b \sec (c+d x))}{2 d}+\frac {b^3 \tanh ^{-1}(\sin (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 8
Rule 3770
Rule 3841
Rule 4045
Rule 4047
Rubi steps
\begin {align*} \int \cos ^2(c+d x) (a+b \sec (c+d x))^3 \, dx &=\frac {a^2 \cos (c+d x) (a+b \sec (c+d x)) \sin (c+d x)}{2 d}+\frac {1}{2} \int \cos (c+d x) \left (5 a^2 b+a \left (a^2+6 b^2\right ) \sec (c+d x)+2 b^3 \sec ^2(c+d x)\right ) \, dx\\ &=\frac {a^2 \cos (c+d x) (a+b \sec (c+d x)) \sin (c+d x)}{2 d}+\frac {1}{2} \int \cos (c+d x) \left (5 a^2 b+2 b^3 \sec ^2(c+d x)\right ) \, dx+\frac {1}{2} \left (a \left (a^2+6 b^2\right )\right ) \int 1 \, dx\\ &=\frac {1}{2} a \left (a^2+6 b^2\right ) x+\frac {5 a^2 b \sin (c+d x)}{2 d}+\frac {a^2 \cos (c+d x) (a+b \sec (c+d x)) \sin (c+d x)}{2 d}+b^3 \int \sec (c+d x) \, dx\\ &=\frac {1}{2} a \left (a^2+6 b^2\right ) x+\frac {b^3 \tanh ^{-1}(\sin (c+d x))}{d}+\frac {5 a^2 b \sin (c+d x)}{2 d}+\frac {a^2 \cos (c+d x) (a+b \sec (c+d x)) \sin (c+d x)}{2 d}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 105, normalized size = 1.33 \[ \frac {a^3 \sin (2 (c+d x))+2 a \left (a^2+6 b^2\right ) (c+d x)+12 a^2 b \sin (c+d x)-4 b^3 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+4 b^3 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )}{4 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 72, normalized size = 0.91 \[ \frac {b^{3} \log \left (\sin \left (d x + c\right ) + 1\right ) - b^{3} \log \left (-\sin \left (d x + c\right ) + 1\right ) + {\left (a^{3} + 6 \, a b^{2}\right )} d x + {\left (a^{3} \cos \left (d x + c\right ) + 6 \, a^{2} b\right )} \sin \left (d x + c\right )}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.25, size = 137, normalized size = 1.73 \[ \frac {2 \, b^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 2 \, b^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + {\left (a^{3} + 6 \, a b^{2}\right )} {\left (d x + c\right )} - \frac {2 \, {\left (a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 6 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 6 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{2}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.50, size = 90, normalized size = 1.14 \[ \frac {a^{3} \cos \left (d x +c \right ) \sin \left (d x +c \right )}{2 d}+\frac {a^{3} x}{2}+\frac {a^{3} c}{2 d}+\frac {3 a^{2} b \sin \left (d x +c \right )}{d}+3 b^{2} a x +\frac {3 a \,b^{2} c}{d}+\frac {b^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.38, size = 76, normalized size = 0.96 \[ \frac {{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} a^{3} + 12 \, {\left (d x + c\right )} a b^{2} + 2 \, b^{3} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 12 \, a^{2} b \sin \left (d x + c\right )}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.03, size = 123, normalized size = 1.56 \[ \frac {a^3\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {2\,b^3\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {a^3\,\sin \left (2\,c+2\,d\,x\right )}{4\,d}+\frac {3\,a^2\,b\,\sin \left (c+d\,x\right )}{d}+\frac {6\,a\,b^2\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d} \]
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 \[ \int \left (a + b \sec {\left (c + d x \right )}\right )^{3} \cos ^{2}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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