Optimal. Leaf size=42 \[ -\frac{a^2 \cos (c+d x)}{d}-\frac{2 a b \log (\cos (c+d x))}{d}+\frac{b^2 \sec (c+d x)}{d} \]
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Rubi [A] time = 0.0775606, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {3872, 2833, 12, 43} \[ -\frac{a^2 \cos (c+d x)}{d}-\frac{2 a b \log (\cos (c+d x))}{d}+\frac{b^2 \sec (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 3872
Rule 2833
Rule 12
Rule 43
Rubi steps
\begin{align*} \int (a+b \sec (c+d x))^2 \sin (c+d x) \, dx &=\int (-b-a \cos (c+d x))^2 \sec (c+d x) \tan (c+d x) \, dx\\ &=\frac{\operatorname{Subst}\left (\int \frac{a^2 (-b+x)^2}{x^2} \, dx,x,-a \cos (c+d x)\right )}{a d}\\ &=\frac{a \operatorname{Subst}\left (\int \frac{(-b+x)^2}{x^2} \, dx,x,-a \cos (c+d x)\right )}{d}\\ &=\frac{a \operatorname{Subst}\left (\int \left (1+\frac{b^2}{x^2}-\frac{2 b}{x}\right ) \, dx,x,-a \cos (c+d x)\right )}{d}\\ &=-\frac{a^2 \cos (c+d x)}{d}-\frac{2 a b \log (\cos (c+d x))}{d}+\frac{b^2 \sec (c+d x)}{d}\\ \end{align*}
Mathematica [A] time = 0.0565242, size = 37, normalized size = 0.88 \[ \frac{b (b \sec (c+d x)-2 a \log (\cos (c+d x)))-a^2 \cos (c+d x)}{d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.022, size = 45, normalized size = 1.1 \begin{align*}{\frac{{b}^{2}\sec \left ( dx+c \right ) }{d}}+2\,{\frac{ab\ln \left ( \sec \left ( dx+c \right ) \right ) }{d}}-{\frac{{a}^{2}}{d\sec \left ( dx+c \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.940664, size = 54, normalized size = 1.29 \begin{align*} -\frac{a^{2} \cos \left (d x + c\right ) + 2 \, a b \log \left (\cos \left (d x + c\right )\right ) - \frac{b^{2}}{\cos \left (d x + c\right )}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.75211, size = 116, normalized size = 2.76 \begin{align*} -\frac{a^{2} \cos \left (d x + c\right )^{2} + 2 \, a b \cos \left (d x + c\right ) \log \left (-\cos \left (d x + c\right )\right ) - b^{2}}{d \cos \left (d x + c\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a + b \sec{\left (c + d x \right )}\right )^{2} \sin{\left (c + d x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.28743, size = 68, normalized size = 1.62 \begin{align*} -\frac{a^{2} \cos \left (d x + c\right )}{d} - \frac{2 \, a b \log \left (\frac{{\left | \cos \left (d x + c\right ) \right |}}{{\left | d \right |}}\right )}{d} + \frac{b^{2}}{d \cos \left (d x + c\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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