Optimal. Leaf size=27 \[ \frac {\cos (c+d x)}{a d}+\frac {\sec (c+d x)}{a d} \]
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Rubi [A]
time = 0.05, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {3254, 2670, 14}
\begin {gather*} \frac {\cos (c+d x)}{a d}+\frac {\sec (c+d x)}{a d} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 2670
Rule 3254
Rubi steps
\begin {align*} \int \frac {\sin ^3(c+d x)}{a-a \sin ^2(c+d x)} \, dx &=\frac {\int \sin (c+d x) \tan ^2(c+d x) \, dx}{a}\\ &=-\frac {\text {Subst}\left (\int \frac {1-x^2}{x^2} \, dx,x,\cos (c+d x)\right )}{a d}\\ &=-\frac {\text {Subst}\left (\int \left (-1+\frac {1}{x^2}\right ) \, dx,x,\cos (c+d x)\right )}{a d}\\ &=\frac {\cos (c+d x)}{a d}+\frac {\sec (c+d x)}{a d}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 25, normalized size = 0.93 \begin {gather*} \frac {\frac {\cos (c+d x)}{d}+\frac {\sec (c+d x)}{d}}{a} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.18, size = 23, normalized size = 0.85
method | result | size |
derivativedivides | \(\frac {\cos \left (d x +c \right )+\frac {1}{\cos \left (d x +c \right )}}{d a}\) | \(23\) |
default | \(\frac {\cos \left (d x +c \right )+\frac {1}{\cos \left (d x +c \right )}}{d a}\) | \(23\) |
risch | \(\frac {{\mathrm e}^{i \left (d x +c \right )}}{2 a d}+\frac {{\mathrm e}^{-i \left (d x +c \right )}}{2 a d}+\frac {2 \,{\mathrm e}^{i \left (d x +c \right )}}{d a \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}\) | \(66\) |
norman | \(\frac {-\frac {4 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d}-\frac {4}{a d}-\frac {8 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}\) | \(79\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 27, normalized size = 1.00 \begin {gather*} \frac {\frac {\cos \left (d x + c\right )}{a} + \frac {1}{a \cos \left (d x + c\right )}}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 25, normalized size = 0.93 \begin {gather*} \frac {\cos \left (d x + c\right )^{2} + 1}{a d \cos \left (d x + c\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.68, size = 36, normalized size = 1.33 \begin {gather*} \begin {cases} - \frac {4}{a d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - a d} & \text {for}\: d \neq 0 \\\frac {x \sin ^{3}{\left (c \right )}}{- a \sin ^{2}{\left (c \right )} + a} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.54, size = 29, normalized size = 1.07 \begin {gather*} \frac {\cos \left (d x + c\right )}{a d} + \frac {1}{a d \cos \left (d x + c\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 25, normalized size = 0.93 \begin {gather*} \frac {{\cos \left (c+d\,x\right )}^2+1}{a\,d\,\cos \left (c+d\,x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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