Optimal. Leaf size=86 \[ \frac {a^3 \sin (c+d x)}{d}-\frac {3 a^2 b \cos (c+d x)}{d}-\frac {3 a b^2 \sin (c+d x)}{d}+\frac {3 a b^2 \tanh ^{-1}(\sin (c+d x))}{d}+\frac {b^3 \cos (c+d x)}{d}+\frac {b^3 \sec (c+d x)}{d} \]
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Rubi [A] time = 0.11, antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3090, 2637, 2638, 2592, 321, 206, 2590, 14} \[ -\frac {3 a^2 b \cos (c+d x)}{d}+\frac {a^3 \sin (c+d x)}{d}-\frac {3 a b^2 \sin (c+d x)}{d}+\frac {3 a b^2 \tanh ^{-1}(\sin (c+d x))}{d}+\frac {b^3 \cos (c+d x)}{d}+\frac {b^3 \sec (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 14
Rule 206
Rule 321
Rule 2590
Rule 2592
Rule 2637
Rule 2638
Rule 3090
Rubi steps
\begin {align*} \int \sec ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 \, dx &=\int \left (a^3 \cos (c+d x)+3 a^2 b \sin (c+d x)+3 a b^2 \sin (c+d x) \tan (c+d x)+b^3 \sin (c+d x) \tan ^2(c+d x)\right ) \, dx\\ &=a^3 \int \cos (c+d x) \, dx+\left (3 a^2 b\right ) \int \sin (c+d x) \, dx+\left (3 a b^2\right ) \int \sin (c+d x) \tan (c+d x) \, dx+b^3 \int \sin (c+d x) \tan ^2(c+d x) \, dx\\ &=-\frac {3 a^2 b \cos (c+d x)}{d}+\frac {a^3 \sin (c+d x)}{d}+\frac {\left (3 a b^2\right ) \operatorname {Subst}\left (\int \frac {x^2}{1-x^2} \, dx,x,\sin (c+d x)\right )}{d}-\frac {b^3 \operatorname {Subst}\left (\int \frac {1-x^2}{x^2} \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac {3 a^2 b \cos (c+d x)}{d}+\frac {a^3 \sin (c+d x)}{d}-\frac {3 a b^2 \sin (c+d x)}{d}+\frac {\left (3 a b^2\right ) \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sin (c+d x)\right )}{d}-\frac {b^3 \operatorname {Subst}\left (\int \left (-1+\frac {1}{x^2}\right ) \, dx,x,\cos (c+d x)\right )}{d}\\ &=\frac {3 a b^2 \tanh ^{-1}(\sin (c+d x))}{d}-\frac {3 a^2 b \cos (c+d x)}{d}+\frac {b^3 \cos (c+d x)}{d}+\frac {b^3 \sec (c+d x)}{d}+\frac {a^3 \sin (c+d x)}{d}-\frac {3 a b^2 \sin (c+d x)}{d}\\ \end {align*}
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Mathematica [A] time = 1.07, size = 131, normalized size = 1.52 \[ \frac {\sec (c+d x) \left (a^3 \sin (2 (c+d x))+\left (b^3-3 a^2 b\right ) \cos (2 (c+d x))-3 a^2 b-3 a b^2 \sin (2 (c+d x))-6 a b^2 \cos (c+d x) \left (\log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )\right )+3 b^3\right )}{2 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.65, size = 109, normalized size = 1.27 \[ \frac {3 \, a b^{2} \cos \left (d x + c\right ) \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, a b^{2} \cos \left (d x + c\right ) \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, b^{3} - 2 \, {\left (3 \, a^{2} b - b^{3}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (a^{3} - 3 \, a b^{2}\right )} \cos \left (d x + c\right ) \sin \left (d x + c\right )}{2 \, d \cos \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.29, size = 150, normalized size = 1.74 \[ \frac {3 \, a b^{2} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 3 \, a b^{2} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + \frac {2 \, {\left (a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 3 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 3 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, a^{2} b - 2 \, b^{3}\right )}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 1}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 10.96, size = 126, normalized size = 1.47 \[ \frac {a^{3} \sin \left (d x +c \right )}{d}-\frac {3 a^{2} b \cos \left (d x +c \right )}{d}-\frac {3 a \,b^{2} \sin \left (d x +c \right )}{d}+\frac {3 b^{2} a \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {b^{3} \left (\sin ^{4}\left (d x +c \right )\right )}{d \cos \left (d x +c \right )}+\frac {b^{3} \cos \left (d x +c \right ) \left (\sin ^{2}\left (d x +c \right )\right )}{d}+\frac {2 b^{3} \cos \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 84, normalized size = 0.98 \[ \frac {2 \, b^{3} {\left (\frac {1}{\cos \left (d x + c\right )} + \cos \left (d x + c\right )\right )} + 3 \, a b^{2} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right ) - 2 \, \sin \left (d x + c\right )\right )} - 6 \, a^{2} b \cos \left (d x + c\right ) + 2 \, a^{3} \sin \left (d x + c\right )}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.05, size = 116, normalized size = 1.35 \[ \frac {6\,a\,b^2\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (6\,a\,b^2-2\,a^3\right )-6\,a^2\,b-\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (6\,a\,b^2-2\,a^3\right )+4\,b^3+6\,a^2\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-1\right )} \]
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 \[ \int \left (a \cos {\left (c + d x \right )} + b \sin {\left (c + d x \right )}\right )^{3} \sec ^{2}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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