Optimal. Leaf size=34 \[ -\frac {a^2 \sin (c+d x)}{d}-\frac {2 a^2 \log (1-\sin (c+d x))}{d} \]
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Rubi [A] time = 0.04, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2667, 43} \[ -\frac {a^2 \sin (c+d x)}{d}-\frac {2 a^2 \log (1-\sin (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 43
Rule 2667
Rubi steps
\begin {align*} \int \sec (c+d x) (a+a \sin (c+d x))^2 \, dx &=\frac {a \operatorname {Subst}\left (\int \frac {a+x}{a-x} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {a \operatorname {Subst}\left (\int \left (-1+\frac {2 a}{a-x}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=-\frac {2 a^2 \log (1-\sin (c+d x))}{d}-\frac {a^2 \sin (c+d x)}{d}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 29, normalized size = 0.85 \[ \frac {a^2 (-\sin (c+d x)-2 \log (1-\sin (c+d x)))}{d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.60, size = 32, normalized size = 0.94 \[ -\frac {2 \, a^{2} \log \left (-\sin \left (d x + c\right ) + 1\right ) + a^{2} \sin \left (d x + c\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.87, size = 91, normalized size = 2.68 \[ \frac {2 \, {\left (a^{2} \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right ) - 2 \, a^{2} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a^{2}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1}\right )}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.14, size = 53, normalized size = 1.56 \[ -\frac {a^{2} \sin \left (d x +c \right )}{d}+\frac {2 a^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}-\frac {2 a^{2} \ln \left (\cos \left (d x +c \right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.39, size = 30, normalized size = 0.88 \[ -\frac {2 \, a^{2} \log \left (\sin \left (d x + c\right ) - 1\right ) + a^{2} \sin \left (d x + c\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 26, normalized size = 0.76 \[ -\frac {a^2\,\left (2\,\ln \left (\sin \left (c+d\,x\right )-1\right )+\sin \left (c+d\,x\right )\right )}{d} \]
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 \[ a^{2} \left (\int 2 \sin {\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int \sin ^{2}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int \sec {\left (c + d x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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