Optimal. Leaf size=75 \[ \frac{\left (2 a^2-b^2\right ) \tan (c+d x)}{3 d}+\frac{a b \sec (c+d x)}{3 d}+\frac{\sec ^3(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))}{3 d} \]
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Rubi [A] time = 0.0962427, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {2691, 2669, 3767, 8} \[ \frac{\left (2 a^2-b^2\right ) \tan (c+d x)}{3 d}+\frac{a b \sec (c+d x)}{3 d}+\frac{\sec ^3(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))}{3 d} \]
Antiderivative was successfully verified.
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Rule 2691
Rule 2669
Rule 3767
Rule 8
Rubi steps
\begin{align*} \int \sec ^4(c+d x) (a+b \sin (c+d x))^2 \, dx &=\frac{\sec ^3(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))}{3 d}-\frac{1}{3} \int \sec ^2(c+d x) \left (-2 a^2+b^2-a b \sin (c+d x)\right ) \, dx\\ &=\frac{a b \sec (c+d x)}{3 d}+\frac{\sec ^3(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))}{3 d}-\frac{1}{3} \left (-2 a^2+b^2\right ) \int \sec ^2(c+d x) \, dx\\ &=\frac{a b \sec (c+d x)}{3 d}+\frac{\sec ^3(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))}{3 d}-\frac{\left (2 a^2-b^2\right ) \operatorname{Subst}(\int 1 \, dx,x,-\tan (c+d x))}{3 d}\\ &=\frac{a b \sec (c+d x)}{3 d}+\frac{\sec ^3(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))}{3 d}+\frac{\left (2 a^2-b^2\right ) \tan (c+d x)}{3 d}\\ \end{align*}
Mathematica [A] time = 0.322364, size = 105, normalized size = 1.4 \[ \frac{\left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right ) \left (3 \left (2 a^2+b^2\right ) \sin (c+d x)+\left (2 a^2-b^2\right ) \sin (3 (c+d x))+8 a b\right )}{12 d (\sin (c+d x)-1)^2 \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.051, size = 62, normalized size = 0.8 \begin{align*}{\frac{1}{d} \left ( -{a}^{2} \left ( -{\frac{2}{3}}-{\frac{ \left ( \sec \left ( dx+c \right ) \right ) ^{2}}{3}} \right ) \tan \left ( dx+c \right ) +{\frac{2\,ab}{3\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}}}+{\frac{{b}^{2} \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{3\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.972691, size = 69, normalized size = 0.92 \begin{align*} \frac{b^{2} \tan \left (d x + c\right )^{3} +{\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} a^{2} + \frac{2 \, a b}{\cos \left (d x + c\right )^{3}}}{3 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.12719, size = 122, normalized size = 1.63 \begin{align*} \frac{2 \, a b +{\left ({\left (2 \, a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} + a^{2} + b^{2}\right )} \sin \left (d x + c\right )}{3 \, d \cos \left (d x + c\right )^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11118, size = 138, normalized size = 1.84 \begin{align*} -\frac{2 \,{\left (3 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 6 \, a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 2 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 4 \, b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 3 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 2 \, a b\right )}}{3 \,{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1\right )}^{3} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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