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 = 0.07, 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 = {3175, 2590, 14} \[ \frac {\cos (c+d x)}{a d}+\frac {\sec (c+d x)}{a d} \]
Antiderivative was successfully verified.
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Rule 14
Rule 2590
Rule 3175
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 {\operatorname {Subst}\left (\int \frac {1-x^2}{x^2} \, dx,x,\cos (c+d x)\right )}{a d}\\ &=-\frac {\operatorname {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 \[ \frac {\frac {\cos (c+d x)}{d}+\frac {\sec (c+d x)}{d}}{a} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 25, normalized size = 0.93 \[ \frac {\cos \left (d x + c\right )^{2} + 1}{a d \cos \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 29, normalized size = 1.07 \[ \frac {\cos \left (d x + c\right )}{a d} + \frac {1}{a d \cos \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.30, size = 23, normalized size = 0.85 \[ \frac {\cos \left (d x +c \right )+\frac {1}{\cos \left (d x +c \right )}}{d a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 27, normalized size = 1.00 \[ \frac {\frac {\cos \left (d x + c\right )}{a} + \frac {1}{a \cos \left (d x + c\right )}}{d} \]
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 \[ \frac {{\cos \left (c+d\,x\right )}^2+1}{a\,d\,\cos \left (c+d\,x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 6.93, size = 36, normalized size = 1.33 \[ \begin {cases} - \frac {4}{a d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - a d} & \text {for}\: d \neq 0 \\\frac {x \sin ^{3}{\relax (c )}}{- a \sin ^{2}{\relax (c )} + a} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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