Optimal. Leaf size=30 \[ \frac {(A+C) \sin (c+d x)}{d}-\frac {A \sin ^3(c+d x)}{3 d} \]
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Rubi [A]
time = 0.03, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {4129, 3092}
\begin {gather*} \frac {(A+C) \sin (c+d x)}{d}-\frac {A \sin ^3(c+d x)}{3 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 3092
Rule 4129
Rubi steps
\begin {align*} \int \cos ^3(c+d x) \left (A+C \sec ^2(c+d x)\right ) \, dx &=\int \cos (c+d x) \left (C+A \cos ^2(c+d x)\right ) \, dx\\ &=-\frac {\text {Subst}\left (\int \left (A+C-A x^2\right ) \, dx,x,-\sin (c+d x)\right )}{d}\\ &=\frac {(A+C) \sin (c+d x)}{d}-\frac {A \sin ^3(c+d x)}{3 d}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 50, normalized size = 1.67 \begin {gather*} \frac {C \cos (d x) \sin (c)}{d}+\frac {C \cos (c) \sin (d x)}{d}+\frac {A \sin (c+d x)}{d}-\frac {A \sin ^3(c+d x)}{3 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.39, size = 33, normalized size = 1.10
method | result | size |
derivativedivides | \(\frac {\frac {A \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+C \sin \left (d x +c \right )}{d}\) | \(33\) |
default | \(\frac {\frac {A \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+C \sin \left (d x +c \right )}{d}\) | \(33\) |
risch | \(\frac {3 A \sin \left (d x +c \right )}{4 d}+\frac {C \sin \left (d x +c \right )}{d}+\frac {A \sin \left (3 d x +3 c \right )}{12 d}\) | \(40\) |
norman | \(\frac {\frac {2 \left (A -3 C \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {2 \left (A -3 C \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {2 \left (A +C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {2 \left (A +C \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}\) | \(111\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 27, normalized size = 0.90 \begin {gather*} -\frac {A \sin \left (d x + c\right )^{3} - 3 \, {\left (A + C\right )} \sin \left (d x + c\right )}{3 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.56, size = 28, normalized size = 0.93 \begin {gather*} \frac {{\left (A \cos \left (d x + c\right )^{2} + 2 \, A + 3 \, C\right )} \sin \left (d x + c\right )}{3 \, d} \end {gather*}
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 \begin {gather*} \int \left (A + C \sec ^{2}{\left (c + d x \right )}\right ) \cos ^{3}{\left (c + d x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, size = 34, normalized size = 1.13 \begin {gather*} -\frac {A \sin \left (d x + c\right )^{3} - 3 \, A \sin \left (d x + c\right ) - 3 \, C \sin \left (d x + c\right )}{3 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 28, normalized size = 0.93 \begin {gather*} -\frac {\frac {A\,{\sin \left (c+d\,x\right )}^3}{3}-\sin \left (c+d\,x\right )\,\left (A+C\right )}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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