Optimal. Leaf size=129 \[ \frac{\left (6 a^2-b^2\right ) \tan ^5(c+d x)}{35 d}+\frac{2 \left (6 a^2-b^2\right ) \tan ^3(c+d x)}{21 d}+\frac{\left (6 a^2-b^2\right ) \tan (c+d x)}{7 d}+\frac{a b \sec ^5(c+d x)}{7 d}+\frac{\sec ^7(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))}{7 d} \]
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Rubi [A] time = 0.122862, antiderivative size = 129, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {2691, 2669, 3767} \[ \frac{\left (6 a^2-b^2\right ) \tan ^5(c+d x)}{35 d}+\frac{2 \left (6 a^2-b^2\right ) \tan ^3(c+d x)}{21 d}+\frac{\left (6 a^2-b^2\right ) \tan (c+d x)}{7 d}+\frac{a b \sec ^5(c+d x)}{7 d}+\frac{\sec ^7(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))}{7 d} \]
Antiderivative was successfully verified.
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Rule 2691
Rule 2669
Rule 3767
Rubi steps
\begin{align*} \int \sec ^8(c+d x) (a+b \sin (c+d x))^2 \, dx &=\frac{\sec ^7(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))}{7 d}-\frac{1}{7} \int \sec ^6(c+d x) \left (-6 a^2+b^2-5 a b \sin (c+d x)\right ) \, dx\\ &=\frac{a b \sec ^5(c+d x)}{7 d}+\frac{\sec ^7(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))}{7 d}-\frac{1}{7} \left (-6 a^2+b^2\right ) \int \sec ^6(c+d x) \, dx\\ &=\frac{a b \sec ^5(c+d x)}{7 d}+\frac{\sec ^7(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))}{7 d}-\frac{\left (6 a^2-b^2\right ) \operatorname{Subst}\left (\int \left (1+2 x^2+x^4\right ) \, dx,x,-\tan (c+d x)\right )}{7 d}\\ &=\frac{a b \sec ^5(c+d x)}{7 d}+\frac{\sec ^7(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))}{7 d}+\frac{\left (6 a^2-b^2\right ) \tan (c+d x)}{7 d}+\frac{2 \left (6 a^2-b^2\right ) \tan ^3(c+d x)}{21 d}+\frac{\left (6 a^2-b^2\right ) \tan ^5(c+d x)}{35 d}\\ \end{align*}
Mathematica [A] time = 0.835627, size = 110, normalized size = 0.85 \[ \frac{\sec ^7(c+d x) \left (105 \left (2 a^2+b^2\right ) \sin (c+d x)+21 \left (6 a^2-b^2\right ) \sin (3 (c+d x))+42 a^2 \sin (5 (c+d x))+6 a^2 \sin (7 (c+d x))+240 a b-7 b^2 \sin (5 (c+d x))-b^2 \sin (7 (c+d x))\right )}{840 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.062, size = 120, normalized size = 0.9 \begin{align*}{\frac{1}{d} \left ( -{a}^{2} \left ( -{\frac{16}{35}}-{\frac{ \left ( \sec \left ( dx+c \right ) \right ) ^{6}}{7}}-{\frac{6\, \left ( \sec \left ( dx+c \right ) \right ) ^{4}}{35}}-{\frac{8\, \left ( \sec \left ( dx+c \right ) \right ) ^{2}}{35}} \right ) \tan \left ( dx+c \right ) +{\frac{2\,ab}{7\, \left ( \cos \left ( dx+c \right ) \right ) ^{7}}}+{b}^{2} \left ({\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{7\, \left ( \cos \left ( dx+c \right ) \right ) ^{7}}}+{\frac{4\, \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{35\, \left ( \cos \left ( dx+c \right ) \right ) ^{5}}}+{\frac{8\, \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{105\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.971905, size = 131, normalized size = 1.02 \begin{align*} \frac{3 \,{\left (5 \, \tan \left (d x + c\right )^{7} + 21 \, \tan \left (d x + c\right )^{5} + 35 \, \tan \left (d x + c\right )^{3} + 35 \, \tan \left (d x + c\right )\right )} a^{2} +{\left (15 \, \tan \left (d x + c\right )^{7} + 42 \, \tan \left (d x + c\right )^{5} + 35 \, \tan \left (d x + c\right )^{3}\right )} b^{2} + \frac{30 \, a b}{\cos \left (d x + c\right )^{7}}}{105 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.20787, size = 225, normalized size = 1.74 \begin{align*} \frac{30 \, a b +{\left (8 \,{\left (6 \, a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{6} + 4 \,{\left (6 \, a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{4} + 3 \,{\left (6 \, a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} + 15 \, a^{2} + 15 \, b^{2}\right )} \sin \left (d x + c\right )}{105 \, d \cos \left (d x + c\right )^{7}} \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 [B] time = 1.12001, size = 351, normalized size = 2.72 \begin{align*} -\frac{2 \,{\left (105 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{13} + 210 \, a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{12} - 210 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{11} + 140 \, b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{11} + 903 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} + 112 \, b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} + 1050 \, a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{8} - 636 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 456 \, b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 903 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 112 \, b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 630 \, a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 210 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 140 \, b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 105 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 30 \, a b\right )}}{105 \,{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1\right )}^{7} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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