Optimal. Leaf size=295 \[ \frac{2 \left (21 a^3 A+21 a^2 b B+21 a A b^2+5 b^3 B\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right )}{21 d}+\frac{2 b \left (18 a^2 B+21 a A b+5 b^2 B\right ) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{21 d}+\frac{2 \left (15 a^2 A b+5 a^3 B+9 a b^2 B+3 A b^3\right ) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}-\frac{2 \left (15 a^2 A b+5 a^3 B+9 a b^2 B+3 A b^3\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d}+\frac{2 b^2 (11 a B+7 A b) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{35 d}+\frac{2 b B \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^2}{7 d} \]
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Rubi [A] time = 0.504427, antiderivative size = 295, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.242, Rules used = {4026, 4076, 4047, 3768, 3771, 2639, 4046, 2641} \[ \frac{2 b \left (18 a^2 B+21 a A b+5 b^2 B\right ) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{21 d}+\frac{2 \left (15 a^2 A b+5 a^3 B+9 a b^2 B+3 A b^3\right ) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}+\frac{2 \left (21 a^3 A+21 a^2 b B+21 a A b^2+5 b^3 B\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d}-\frac{2 \left (15 a^2 A b+5 a^3 B+9 a b^2 B+3 A b^3\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d}+\frac{2 b^2 (11 a B+7 A b) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{35 d}+\frac{2 b B \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^2}{7 d} \]
Antiderivative was successfully verified.
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Rule 4026
Rule 4076
Rule 4047
Rule 3768
Rule 3771
Rule 2639
Rule 4046
Rule 2641
Rubi steps
\begin{align*} \int \sqrt{\sec (c+d x)} (a+b \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx &=\frac{2 b B \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{7 d}+\frac{2}{7} \int \sqrt{\sec (c+d x)} (a+b \sec (c+d x)) \left (\frac{1}{2} a (7 a A+b B)+\frac{1}{2} \left (5 b^2 B+7 a (2 A b+a B)\right ) \sec (c+d x)+\frac{1}{2} b (7 A b+11 a B) \sec ^2(c+d x)\right ) \, dx\\ &=\frac{2 b^2 (7 A b+11 a B) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{35 d}+\frac{2 b B \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{7 d}+\frac{4}{35} \int \sqrt{\sec (c+d x)} \left (\frac{5}{4} a^2 (7 a A+b B)+\frac{7}{4} \left (15 a^2 A b+3 A b^3+5 a^3 B+9 a b^2 B\right ) \sec (c+d x)+\frac{5}{4} b \left (21 a A b+18 a^2 B+5 b^2 B\right ) \sec ^2(c+d x)\right ) \, dx\\ &=\frac{2 b^2 (7 A b+11 a B) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{35 d}+\frac{2 b B \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{7 d}+\frac{4}{35} \int \sqrt{\sec (c+d x)} \left (\frac{5}{4} a^2 (7 a A+b B)+\frac{5}{4} b \left (21 a A b+18 a^2 B+5 b^2 B\right ) \sec ^2(c+d x)\right ) \, dx+\frac{1}{5} \left (15 a^2 A b+3 A b^3+5 a^3 B+9 a b^2 B\right ) \int \sec ^{\frac{3}{2}}(c+d x) \, dx\\ &=\frac{2 \left (15 a^2 A b+3 A b^3+5 a^3 B+9 a b^2 B\right ) \sqrt{\sec (c+d x)} \sin (c+d x)}{5 d}+\frac{2 b \left (21 a A b+18 a^2 B+5 b^2 B\right ) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac{2 b^2 (7 A b+11 a B) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{35 d}+\frac{2 b B \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{7 d}+\frac{1}{5} \left (-15 a^2 A b-3 A b^3-5 a^3 B-9 a b^2 B\right ) \int \frac{1}{\sqrt{\sec (c+d x)}} \, dx+\frac{1}{21} \left (21 a^3 A+21 a A b^2+21 a^2 b B+5 b^3 B\right ) \int \sqrt{\sec (c+d x)} \, dx\\ &=\frac{2 \left (15 a^2 A b+3 A b^3+5 a^3 B+9 a b^2 B\right ) \sqrt{\sec (c+d x)} \sin (c+d x)}{5 d}+\frac{2 b \left (21 a A b+18 a^2 B+5 b^2 B\right ) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac{2 b^2 (7 A b+11 a B) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{35 d}+\frac{2 b B \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{7 d}+\frac{1}{5} \left (\left (-15 a^2 A b-3 A b^3-5 a^3 B-9 a b^2 B\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx+\frac{1}{21} \left (\left (21 a^3 A+21 a A b^2+21 a^2 b B+5 b^3 B\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx\\ &=-\frac{2 \left (15 a^2 A b+3 A b^3+5 a^3 B+9 a b^2 B\right ) \sqrt{\cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{5 d}+\frac{2 \left (21 a^3 A+21 a A b^2+21 a^2 b B+5 b^3 B\right ) \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{21 d}+\frac{2 \left (15 a^2 A b+3 A b^3+5 a^3 B+9 a b^2 B\right ) \sqrt{\sec (c+d x)} \sin (c+d x)}{5 d}+\frac{2 b \left (21 a A b+18 a^2 B+5 b^2 B\right ) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac{2 b^2 (7 A b+11 a B) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{35 d}+\frac{2 b B \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{7 d}\\ \end{align*}
Mathematica [A] time = 3.88042, size = 225, normalized size = 0.76 \[ \frac{2 \sqrt{\sec (c+d x)} \left (5 \left (21 a^3 A+21 a^2 b B+21 a A b^2+5 b^3 B\right ) \sqrt{\cos (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right )+21 \left (15 a^2 A b+5 a^3 B+9 a b^2 B+3 A b^3\right ) \sin (c+d x)+5 b \left (21 a^2 B+21 a A b+5 b^2 B\right ) \tan (c+d x)-21 \left (15 a^2 A b+5 a^3 B+9 a b^2 B+3 A b^3\right ) \sqrt{\cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )+21 b^2 (3 a B+A b) \tan (c+d x) \sec (c+d x)+15 b^3 B \tan (c+d x) \sec ^2(c+d x)\right )}{105 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 8.33, size = 944, normalized size = 3.2 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [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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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (B b^{3} \sec \left (d x + c\right )^{4} + A a^{3} +{\left (3 \, B a b^{2} + A b^{3}\right )} \sec \left (d x + c\right )^{3} + 3 \,{\left (B a^{2} b + A a b^{2}\right )} \sec \left (d x + c\right )^{2} +{\left (B a^{3} + 3 \, A a^{2} b\right )} \sec \left (d x + c\right )\right )} \sqrt{\sec \left (d x + c\right )}, x\right ) \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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (B \sec \left (d x + c\right ) + A\right )}{\left (b \sec \left (d x + c\right ) + a\right )}^{3} \sqrt{\sec \left (d x + c\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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