Optimal. Leaf size=244 \[ \frac{4 a^3 (24 A+23 B) \sin (c+d x)}{105 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 (9 A+13 B) \sin (c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{63 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{4 a^3 (13 A+11 B) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{4 a^3 (13 A+11 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d}+\frac{4 a^3 (21 A+17 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 d}+\frac{2 a B \sin (c+d x) (a \sec (c+d x)+a)^2}{9 d \sec ^{\frac{7}{2}}(c+d x)} \]
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Rubi [A] time = 0.535339, antiderivative size = 244, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.242, Rules used = {2960, 4017, 3996, 3787, 3769, 3771, 2641, 2639} \[ \frac{4 a^3 (24 A+23 B) \sin (c+d x)}{105 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 (9 A+13 B) \sin (c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{63 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{4 a^3 (13 A+11 B) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{4 a^3 (13 A+11 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d}+\frac{4 a^3 (21 A+17 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 d}+\frac{2 a B \sin (c+d x) (a \sec (c+d x)+a)^2}{9 d \sec ^{\frac{7}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Rule 2960
Rule 4017
Rule 3996
Rule 3787
Rule 3769
Rule 3771
Rule 2641
Rule 2639
Rubi steps
\begin{align*} \int \frac{(a+a \cos (c+d x))^3 (A+B \cos (c+d x))}{\sqrt{\sec (c+d x)}} \, dx &=\int \frac{(a+a \sec (c+d x))^3 (B+A \sec (c+d x))}{\sec ^{\frac{9}{2}}(c+d x)} \, dx\\ &=\frac{2 a B (a+a \sec (c+d x))^2 \sin (c+d x)}{9 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2}{9} \int \frac{(a+a \sec (c+d x))^2 \left (\frac{1}{2} a (9 A+13 B)+\frac{3}{2} a (3 A+B) \sec (c+d x)\right )}{\sec ^{\frac{7}{2}}(c+d x)} \, dx\\ &=\frac{2 a B (a+a \sec (c+d x))^2 \sin (c+d x)}{9 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2 (9 A+13 B) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{63 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{4}{63} \int \frac{(a+a \sec (c+d x)) \left (\frac{3}{2} a^2 (24 A+23 B)+\frac{15}{2} a^2 (3 A+2 B) \sec (c+d x)\right )}{\sec ^{\frac{5}{2}}(c+d x)} \, dx\\ &=\frac{4 a^3 (24 A+23 B) \sin (c+d x)}{105 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 a B (a+a \sec (c+d x))^2 \sin (c+d x)}{9 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2 (9 A+13 B) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{63 d \sec ^{\frac{5}{2}}(c+d x)}-\frac{8}{315} \int \frac{-\frac{45}{4} a^3 (13 A+11 B)-\frac{21}{4} a^3 (21 A+17 B) \sec (c+d x)}{\sec ^{\frac{3}{2}}(c+d x)} \, dx\\ &=\frac{4 a^3 (24 A+23 B) \sin (c+d x)}{105 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 a B (a+a \sec (c+d x))^2 \sin (c+d x)}{9 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2 (9 A+13 B) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{63 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{1}{7} \left (2 a^3 (13 A+11 B)\right ) \int \frac{1}{\sec ^{\frac{3}{2}}(c+d x)} \, dx+\frac{1}{15} \left (2 a^3 (21 A+17 B)\right ) \int \frac{1}{\sqrt{\sec (c+d x)}} \, dx\\ &=\frac{4 a^3 (24 A+23 B) \sin (c+d x)}{105 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{4 a^3 (13 A+11 B) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{2 a B (a+a \sec (c+d x))^2 \sin (c+d x)}{9 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2 (9 A+13 B) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{63 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{1}{21} \left (2 a^3 (13 A+11 B)\right ) \int \sqrt{\sec (c+d x)} \, dx+\frac{1}{15} \left (2 a^3 (21 A+17 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx\\ &=\frac{4 a^3 (21 A+17 B) \sqrt{\cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{15 d}+\frac{4 a^3 (24 A+23 B) \sin (c+d x)}{105 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{4 a^3 (13 A+11 B) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{2 a B (a+a \sec (c+d x))^2 \sin (c+d x)}{9 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2 (9 A+13 B) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{63 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{1}{21} \left (2 a^3 (13 A+11 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{4 a^3 (21 A+17 B) \sqrt{\cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{15 d}+\frac{4 a^3 (13 A+11 B) \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{4 a^3 (24 A+23 B) \sin (c+d x)}{105 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{4 a^3 (13 A+11 B) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{2 a B (a+a \sec (c+d x))^2 \sin (c+d x)}{9 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2 (9 A+13 B) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{63 d \sec ^{\frac{5}{2}}(c+d x)}\\ \end{align*}
Mathematica [C] time = 2.76805, size = 196, normalized size = 0.8 \[ \frac{a^3 \sqrt{\sec (c+d x)} \left (-112 i (21 A+17 B) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right )+\cos (c+d x) (30 (107 A+97 B) \sin (c+d x)+14 (54 A+73 B) \sin (2 (c+d x))+90 A \sin (3 (c+d x))+7056 i A+270 B \sin (3 (c+d x))+35 B \sin (4 (c+d x))+5712 i B)+240 (13 A+11 B) \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )\right )}{1260 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 3.016, size = 413, normalized size = 1.7 \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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (B \cos \left (d x + c\right ) + A\right )}{\left (a \cos \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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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{B a^{3} \cos \left (d x + c\right )^{4} +{\left (A + 3 \, B\right )} a^{3} \cos \left (d x + c\right )^{3} + 3 \,{\left (A + B\right )} a^{3} \cos \left (d x + c\right )^{2} +{\left (3 \, A + B\right )} a^{3} \cos \left (d x + c\right ) + A a^{3}}{\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 \frac{{\left (B \cos \left (d x + c\right ) + A\right )}{\left (a \cos \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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