Optimal. Leaf size=64 \[ -\frac{3 a^2 b \log (\cos (c+d x))}{d}-\frac{a^3 \cos (c+d x)}{d}+\frac{3 a b^2 \sec (c+d x)}{d}+\frac{b^3 \sec ^2(c+d x)}{2 d} \]
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Rubi [A] time = 0.101173, antiderivative size = 64, 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} \[ -\frac{3 a^2 b \log (\cos (c+d x))}{d}-\frac{a^3 \cos (c+d x)}{d}+\frac{3 a b^2 \sec (c+d x)}{d}+\frac{b^3 \sec ^2(c+d x)}{2 d} \]
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))^3 \sin (c+d x) \, dx &=-\int (-b-a \cos (c+d x))^3 \sec ^2(c+d x) \tan (c+d x) \, dx\\ &=\frac{\operatorname{Subst}\left (\int \frac{a^3 (-b+x)^3}{x^3} \, dx,x,-a \cos (c+d x)\right )}{a d}\\ &=\frac{a^2 \operatorname{Subst}\left (\int \frac{(-b+x)^3}{x^3} \, dx,x,-a \cos (c+d x)\right )}{d}\\ &=\frac{a^2 \operatorname{Subst}\left (\int \left (1-\frac{b^3}{x^3}+\frac{3 b^2}{x^2}-\frac{3 b}{x}\right ) \, dx,x,-a \cos (c+d x)\right )}{d}\\ &=-\frac{a^3 \cos (c+d x)}{d}-\frac{3 a^2 b \log (\cos (c+d x))}{d}+\frac{3 a b^2 \sec (c+d x)}{d}+\frac{b^3 \sec ^2(c+d x)}{2 d}\\ \end{align*}
Mathematica [A] time = 0.107135, size = 56, normalized size = 0.88 \[ \frac{b \left (-6 a^2 \log (\cos (c+d x))+6 a b \sec (c+d x)+b^2 \sec ^2(c+d x)\right )-2 a^3 \cos (c+d x)}{2 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.022, size = 65, normalized size = 1. \begin{align*}{\frac{{b}^{3} \left ( \sec \left ( dx+c \right ) \right ) ^{2}}{2\,d}}+3\,{\frac{a{b}^{2}\sec \left ( dx+c \right ) }{d}}+3\,{\frac{{a}^{2}b\ln \left ( \sec \left ( dx+c \right ) \right ) }{d}}-{\frac{{a}^{3}}{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 = 0.961768, size = 77, normalized size = 1.2 \begin{align*} -\frac{2 \, a^{3} \cos \left (d x + c\right ) + 6 \, a^{2} b \log \left (\cos \left (d x + c\right )\right ) - \frac{6 \, a b^{2}}{\cos \left (d x + c\right )} - \frac{b^{3}}{\cos \left (d x + c\right )^{2}}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.82262, size = 163, normalized size = 2.55 \begin{align*} -\frac{2 \, a^{3} \cos \left (d x + c\right )^{3} + 6 \, a^{2} b \cos \left (d x + c\right )^{2} \log \left (-\cos \left (d x + c\right )\right ) - 6 \, a b^{2} \cos \left (d x + c\right ) - b^{3}}{2 \, d \cos \left (d x + c\right )^{2}} \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 )^{3} \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 = 1.37943, size = 89, normalized size = 1.39 \begin{align*} -\frac{a^{3} \cos \left (d x + c\right )}{d} - \frac{3 \, a^{2} b \log \left (\frac{{\left | \cos \left (d x + c\right ) \right |}}{{\left | d \right |}}\right )}{d} + \frac{6 \, a b^{2} \cos \left (d x + c\right ) + b^{3}}{2 \, d \cos \left (d x + c\right )^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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