Optimal. Leaf size=68 \[ -\frac {b \left (a^2-b^2\right ) \sin (c+d x)}{d}+\frac {3 a^2 b \tanh ^{-1}(\sin (c+d x))}{d}+\frac {a^2 \tan (c+d x) (a+b \cos (c+d x))}{d}+3 a b^2 x \]
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Rubi [A] time = 0.12, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {2792, 3023, 2735, 3770} \[ -\frac {b \left (a^2-b^2\right ) \sin (c+d x)}{d}+\frac {3 a^2 b \tanh ^{-1}(\sin (c+d x))}{d}+\frac {a^2 \tan (c+d x) (a+b \cos (c+d x))}{d}+3 a b^2 x \]
Antiderivative was successfully verified.
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Rule 2735
Rule 2792
Rule 3023
Rule 3770
Rubi steps
\begin {align*} \int (a+b \cos (c+d x))^3 \sec ^2(c+d x) \, dx &=\frac {a^2 (a+b \cos (c+d x)) \tan (c+d x)}{d}+\int \left (3 a^2 b+3 a b^2 \cos (c+d x)-b \left (a^2-b^2\right ) \cos ^2(c+d x)\right ) \sec (c+d x) \, dx\\ &=-\frac {b \left (a^2-b^2\right ) \sin (c+d x)}{d}+\frac {a^2 (a+b \cos (c+d x)) \tan (c+d x)}{d}+\int \left (3 a^2 b+3 a b^2 \cos (c+d x)\right ) \sec (c+d x) \, dx\\ &=3 a b^2 x-\frac {b \left (a^2-b^2\right ) \sin (c+d x)}{d}+\frac {a^2 (a+b \cos (c+d x)) \tan (c+d x)}{d}+\left (3 a^2 b\right ) \int \sec (c+d x) \, dx\\ &=3 a b^2 x+\frac {3 a^2 b \tanh ^{-1}(\sin (c+d x))}{d}-\frac {b \left (a^2-b^2\right ) \sin (c+d x)}{d}+\frac {a^2 (a+b \cos (c+d x)) \tan (c+d x)}{d}\\ \end {align*}
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Mathematica [A] time = 0.35, size = 88, normalized size = 1.29 \[ \frac {a^3 \tan (c+d x)+3 a b \left (-a \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+a \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )+b c+b d x\right )+b^3 \sin (c+d x)}{d} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.08, size = 94, normalized size = 1.38 \[ \frac {6 \, a b^{2} d x \cos \left (d x + c\right ) + 3 \, a^{2} b \cos \left (d x + c\right ) \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, a^{2} b \cos \left (d x + c\right ) \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (b^{3} \cos \left (d x + c\right ) + a^{3}\right )} \sin \left (d x + c\right )}{2 \, d \cos \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.58, size = 129, normalized size = 1.90 \[ \frac {3 \, {\left (d x + c\right )} a b^{2} + 3 \, a^{2} b \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 3 \, a^{2} b \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 1}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 68, normalized size = 1.00 \[ 3 a \,b^{2} x +\frac {a^{3} \tan \left (d x +c \right )}{d}+\frac {b^{3} \sin \left (d x +c \right )}{d}+\frac {3 a^{2} b \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {3 a \,b^{2} c}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.89, size = 66, normalized size = 0.97 \[ \frac {6 \, {\left (d x + c\right )} a b^{2} + 3 \, a^{2} b {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 2 \, b^{3} \sin \left (d x + c\right ) + 2 \, a^{3} \tan \left (d x + c\right )}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.62, size = 97, normalized size = 1.43 \[ \frac {b^3\,\sin \left (c+d\,x\right )}{d}+\frac {a^3\,\sin \left (c+d\,x\right )}{d\,\cos \left (c+d\,x\right )}+\frac {6\,a\,b^2\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {6\,a^2\,b\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d} \]
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 \[ \int \left (a + b \cos {\left (c + d x \right )}\right )^{3} \sec ^{2}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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