Optimal. Leaf size=42 \[ \frac {2 \tan (c+d x)}{3 a d}-\frac {\sec (c+d x)}{3 d (a \sin (c+d x)+a)} \]
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Rubi [A] time = 0.05, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2672, 3767, 8} \[ \frac {2 \tan (c+d x)}{3 a d}-\frac {\sec (c+d x)}{3 d (a \sin (c+d x)+a)} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2672
Rule 3767
Rubi steps
\begin {align*} \int \frac {\sec ^2(c+d x)}{a+a \sin (c+d x)} \, dx &=-\frac {\sec (c+d x)}{3 d (a+a \sin (c+d x))}+\frac {2 \int \sec ^2(c+d x) \, dx}{3 a}\\ &=-\frac {\sec (c+d x)}{3 d (a+a \sin (c+d x))}-\frac {2 \operatorname {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{3 a d}\\ &=-\frac {\sec (c+d x)}{3 d (a+a \sin (c+d x))}+\frac {2 \tan (c+d x)}{3 a d}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 45, normalized size = 1.07 \[ \frac {2 \tan (c+d x)-\cos (2 (c+d x)) \sec (c+d x)}{3 a d (\sin (c+d x)+1)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.67, size = 49, normalized size = 1.17 \[ -\frac {2 \, \cos \left (d x + c\right )^{2} - 2 \, \sin \left (d x + c\right ) - 1}{3 \, {\left (a d \cos \left (d x + c\right ) \sin \left (d x + c\right ) + a d \cos \left (d x + c\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.65, size = 67, normalized size = 1.60 \[ -\frac {\frac {3}{a {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}} + \frac {9 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 12 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 7}{a {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}^{3}}}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.15, size = 70, normalized size = 1.67 \[ \frac {-\frac {1}{2 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {2}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {1}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {3}{2 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}}{d a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.61, size = 129, normalized size = 3.07 \[ \frac {2 \, {\left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {3 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {3 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - 1\right )}}{3 \, {\left (a + \frac {2 \, a \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {2 \, a \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {a \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}\right )} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.56, size = 71, normalized size = 1.69 \[ -\frac {2\,\left (3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-1\right )}{3\,a\,d\,\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-1\right )\,{\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )}^3} \]
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 \[ \frac {\int \frac {\sec ^{2}{\left (c + d x \right )}}{\sin {\left (c + d x \right )} + 1}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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