Optimal. Leaf size=295 \[ \frac {2 a \left (7 a^2 A+27 a b B+22 A b^2\right ) \sin (c+d x)}{45 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 a^2 (9 a B+13 A b) \sin (c+d x)}{63 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (5 a^3 B+15 a^2 A b+21 a b^2 B+7 A b^3\right ) \sin (c+d x)}{21 d \sqrt {\sec (c+d x)}}+\frac {2 \left (5 a^3 B+15 a^2 A b+21 a b^2 B+7 A b^3\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d}+\frac {2 \left (7 a^3 A+27 a^2 b B+27 a A b^2+15 b^3 B\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {2 a A \sin (c+d x) (a+b \sec (c+d x))^2}{9 d \sec ^{\frac {7}{2}}(c+d x)} \]
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Rubi [A] time = 0.54, antiderivative size = 295, normalized size of antiderivative = 1.00, 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 = {4025, 4074, 4047, 3769, 3771, 2641, 4045, 2639} \[ \frac {2 a \left (7 a^2 A+27 a b B+22 A b^2\right ) \sin (c+d x)}{45 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (15 a^2 A b+5 a^3 B+21 a b^2 B+7 A b^3\right ) \sin (c+d x)}{21 d \sqrt {\sec (c+d x)}}+\frac {2 \left (15 a^2 A b+5 a^3 B+21 a b^2 B+7 A b^3\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d}+\frac {2 \left (7 a^3 A+27 a^2 b B+27 a A b^2+15 b^3 B\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {2 a^2 (9 a B+13 A b) \sin (c+d x)}{63 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 a A \sin (c+d x) (a+b \sec (c+d x))^2}{9 d \sec ^{\frac {7}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Rule 2639
Rule 2641
Rule 3769
Rule 3771
Rule 4025
Rule 4045
Rule 4047
Rule 4074
Rubi steps
\begin {align*} \int \frac {(a+b \sec (c+d x))^3 (A+B \sec (c+d x))}{\sec ^{\frac {9}{2}}(c+d x)} \, dx &=\frac {2 a A (a+b \sec (c+d x))^2 \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}-\frac {2}{9} \int \frac {(a+b \sec (c+d x)) \left (-\frac {1}{2} a (13 A b+9 a B)-\frac {1}{2} \left (7 a^2 A+9 A b^2+18 a b B\right ) \sec (c+d x)-\frac {3}{2} b (a A+3 b B) \sec ^2(c+d x)\right )}{\sec ^{\frac {7}{2}}(c+d x)} \, dx\\ &=\frac {2 a^2 (13 A b+9 a B) \sin (c+d x)}{63 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 a A (a+b \sec (c+d x))^2 \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {4}{63} \int \frac {\frac {7}{4} a \left (7 a^2 A+22 A b^2+27 a b B\right )+\frac {9}{4} \left (15 a^2 A b+7 A b^3+5 a^3 B+21 a b^2 B\right ) \sec (c+d x)+\frac {21}{4} b^2 (a A+3 b B) \sec ^2(c+d x)}{\sec ^{\frac {5}{2}}(c+d x)} \, dx\\ &=\frac {2 a^2 (13 A b+9 a B) \sin (c+d x)}{63 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 a A (a+b \sec (c+d x))^2 \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {4}{63} \int \frac {\frac {7}{4} a \left (7 a^2 A+22 A b^2+27 a b B\right )+\frac {21}{4} b^2 (a A+3 b B) \sec ^2(c+d x)}{\sec ^{\frac {5}{2}}(c+d x)} \, dx+\frac {1}{7} \left (15 a^2 A b+7 A b^3+5 a^3 B+21 a b^2 B\right ) \int \frac {1}{\sec ^{\frac {3}{2}}(c+d x)} \, dx\\ &=\frac {2 a^2 (13 A b+9 a B) \sin (c+d x)}{63 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 a \left (7 a^2 A+22 A b^2+27 a b B\right ) \sin (c+d x)}{45 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (15 a^2 A b+7 A b^3+5 a^3 B+21 a b^2 B\right ) \sin (c+d x)}{21 d \sqrt {\sec (c+d x)}}+\frac {2 a A (a+b \sec (c+d x))^2 \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {1}{21} \left (15 a^2 A b+7 A b^3+5 a^3 B+21 a b^2 B\right ) \int \sqrt {\sec (c+d x)} \, dx+\frac {1}{15} \left (7 a^3 A+27 a A b^2+27 a^2 b B+15 b^3 B\right ) \int \frac {1}{\sqrt {\sec (c+d x)}} \, dx\\ &=\frac {2 a^2 (13 A b+9 a B) \sin (c+d x)}{63 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 a \left (7 a^2 A+22 A b^2+27 a b B\right ) \sin (c+d x)}{45 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (15 a^2 A b+7 A b^3+5 a^3 B+21 a b^2 B\right ) \sin (c+d x)}{21 d \sqrt {\sec (c+d x)}}+\frac {2 a A (a+b \sec (c+d x))^2 \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {1}{21} \left (\left (15 a^2 A b+7 A b^3+5 a^3 B+21 a b^2 B\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx+\frac {1}{15} \left (\left (7 a^3 A+27 a A b^2+27 a^2 b B+15 b^3 B\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx\\ &=\frac {2 \left (7 a^3 A+27 a A b^2+27 a^2 b B+15 b^3 B\right ) \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 {2 \left (15 a^2 A b+7 A b^3+5 a^3 B+21 a b^2 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 a^2 (13 A b+9 a B) \sin (c+d x)}{63 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 a \left (7 a^2 A+22 A b^2+27 a b B\right ) \sin (c+d x)}{45 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (15 a^2 A b+7 A b^3+5 a^3 B+21 a b^2 B\right ) \sin (c+d x)}{21 d \sqrt {\sec (c+d x)}}+\frac {2 a A (a+b \sec (c+d x))^2 \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}\\ \end {align*}
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Mathematica [A] time = 2.03, size = 219, normalized size = 0.74 \[ \frac {\sqrt {\sec (c+d x)} \left (\sin (2 (c+d x)) \left (7 a \left (43 a^2 A+108 a b B+108 A b^2\right ) \cos (c+d x)+5 \left (7 a^3 A \cos (3 (c+d x))+78 a^3 B+18 a^2 (a B+3 A b) \cos (2 (c+d x))+234 a^2 A b+252 a b^2 B+84 A b^3\right )\right )+120 \left (5 a^3 B+15 a^2 A b+21 a b^2 B+7 A b^3\right ) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )+168 \left (7 a^3 A+27 a^2 b B+27 a A b^2+15 b^3 B\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )\right )}{1260 d} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {B b^{3} \sec \left (d x + c\right )^{4} + A a^{3} + {\left (3 \, B a b^{2} + A b^{3}\right )} \sec \left (d x + c\right )^{3} + 3 \, {\left (B a^{2} b + A a b^{2}\right )} \sec \left (d x + c\right )^{2} + {\left (B a^{3} + 3 \, A a^{2} b\right )} \sec \left (d x + c\right )}{\sec \left (d x + c\right )^{\frac {9}{2}}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{3}}{\sec \left (d x + c\right )^{\frac {9}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 4.76, size = 745, normalized size = 2.53 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\left (A+\frac {B}{\cos \left (c+d\,x\right )}\right )\,{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^3}{{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{9/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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