Optimal. Leaf size=48 \[ \frac {a^2 \sec ^2(c+d x)}{2 d}+\frac {2 a^2 \sec (c+d x)}{d}-\frac {a^2 \log (\cos (c+d x))}{d} \]
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Rubi [A] time = 0.04, antiderivative size = 48, 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 = {3879, 43} \[ \frac {a^2 \sec ^2(c+d x)}{2 d}+\frac {2 a^2 \sec (c+d x)}{d}-\frac {a^2 \log (\cos (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 43
Rule 3879
Rubi steps
\begin {align*} \int (a+a \sec (c+d x))^2 \tan (c+d x) \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {(a+a x)^2}{x^3} \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac {\operatorname {Subst}\left (\int \left (\frac {a^2}{x^3}+\frac {2 a^2}{x^2}+\frac {a^2}{x}\right ) \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac {a^2 \log (\cos (c+d x))}{d}+\frac {2 a^2 \sec (c+d x)}{d}+\frac {a^2 \sec ^2(c+d x)}{2 d}\\ \end {align*}
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Mathematica [A] time = 0.15, size = 51, normalized size = 1.06 \[ -\frac {a^2 \sec ^2(c+d x) (-4 \cos (c+d x)+\cos (2 (c+d x)) \log (\cos (c+d x))+\log (\cos (c+d x))-1)}{2 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.62, size = 52, normalized size = 1.08 \[ -\frac {2 \, a^{2} \cos \left (d x + c\right )^{2} \log \left (-\cos \left (d x + c\right )\right ) - 4 \, a^{2} \cos \left (d x + c\right ) - a^{2}}{2 \, d \cos \left (d x + c\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.39, size = 142, normalized size = 2.96 \[ \frac {2 \, a^{2} \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right ) - 2 \, a^{2} \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right ) + \frac {11 \, a^{2} + \frac {10 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {3 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}}{{\left (\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1\right )}^{2}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.30, size = 46, normalized size = 0.96 \[ \frac {a^{2} \left (\sec ^{2}\left (d x +c \right )\right )}{2 d}+\frac {2 a^{2} \sec \left (d x +c \right )}{d}+\frac {a^{2} \ln \left (\sec \left (d x +c \right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 43, normalized size = 0.90 \[ -\frac {2 \, a^{2} \log \left (\cos \left (d x + c\right )\right ) - \frac {4 \, a^{2} \cos \left (d x + c\right ) + a^{2}}{\cos \left (d x + c\right )^{2}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.21, size = 76, normalized size = 1.58 \[ \frac {2\,a^2\,\mathrm {atanh}\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\right )}{d}-\frac {2\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-4\,a^2}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.48, size = 60, normalized size = 1.25 \[ \begin {cases} \frac {a^{2} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac {a^{2} \sec ^{2}{\left (c + d x \right )}}{2 d} + \frac {2 a^{2} \sec {\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \left (a \sec {\relax (c )} + a\right )^{2} \tan {\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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