Optimal. Leaf size=236 \[ \frac {2 a^2 (5 a B+9 A b) \sin (c+d x)}{15 d \sqrt {\sec (c+d x)}}+\frac {2 \left (a^3 B+3 a^2 A b+9 a b^2 B+3 A b^3\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{3 d}+\frac {2 \left (3 a^3 A+15 a^2 b B+15 a A b^2-5 b^3 B\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}-\frac {2 b^2 (a A-5 b B) \sin (c+d x) \sqrt {\sec (c+d x)}}{5 d}+\frac {2 a A \sin (c+d x) (a+b \sec (c+d x))^2}{5 d \sec ^{\frac {3}{2}}(c+d x)} \]
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Rubi [A] time = 0.46, antiderivative size = 236, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.212, Rules used = {4025, 4074, 4047, 3771, 2641, 4046, 2639} \[ \frac {2 \left (3 a^2 A b+a^3 B+9 a b^2 B+3 A b^3\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{3 d}+\frac {2 \left (3 a^3 A+15 a^2 b B+15 a A b^2-5 b^3 B\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {2 a^2 (5 a B+9 A b) \sin (c+d x)}{15 d \sqrt {\sec (c+d x)}}-\frac {2 b^2 (a A-5 b B) \sin (c+d x) \sqrt {\sec (c+d x)}}{5 d}+\frac {2 a A \sin (c+d x) (a+b \sec (c+d x))^2}{5 d \sec ^{\frac {3}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Rule 2639
Rule 2641
Rule 3771
Rule 4025
Rule 4046
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 {5}{2}}(c+d x)} \, dx &=\frac {2 a A (a+b \sec (c+d x))^2 \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x)}-\frac {2}{5} \int \frac {(a+b \sec (c+d x)) \left (-\frac {1}{2} a (9 A b+5 a B)-\frac {1}{2} \left (3 a^2 A+5 A b^2+10 a b B\right ) \sec (c+d x)+\frac {1}{2} b (a A-5 b B) \sec ^2(c+d x)\right )}{\sec ^{\frac {3}{2}}(c+d x)} \, dx\\ &=\frac {2 a^2 (9 A b+5 a B) \sin (c+d x)}{15 d \sqrt {\sec (c+d x)}}+\frac {2 a A (a+b \sec (c+d x))^2 \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {4}{15} \int \frac {\frac {3}{4} a \left (3 a^2 A+14 A b^2+15 a b B\right )+\frac {5}{4} \left (3 a^2 A b+3 A b^3+a^3 B+9 a b^2 B\right ) \sec (c+d x)-\frac {3}{4} b^2 (a A-5 b B) \sec ^2(c+d x)}{\sqrt {\sec (c+d x)}} \, dx\\ &=\frac {2 a^2 (9 A b+5 a B) \sin (c+d x)}{15 d \sqrt {\sec (c+d x)}}+\frac {2 a A (a+b \sec (c+d x))^2 \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {4}{15} \int \frac {\frac {3}{4} a \left (3 a^2 A+14 A b^2+15 a b B\right )-\frac {3}{4} b^2 (a A-5 b B) \sec ^2(c+d x)}{\sqrt {\sec (c+d x)}} \, dx+\frac {1}{3} \left (3 a^2 A b+3 A b^3+a^3 B+9 a b^2 B\right ) \int \sqrt {\sec (c+d x)} \, dx\\ &=\frac {2 a^2 (9 A b+5 a B) \sin (c+d x)}{15 d \sqrt {\sec (c+d x)}}-\frac {2 b^2 (a A-5 b B) \sqrt {\sec (c+d x)} \sin (c+d x)}{5 d}+\frac {2 a A (a+b \sec (c+d x))^2 \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {1}{5} \left (3 a^3 A+15 a A b^2+15 a^2 b B-5 b^3 B\right ) \int \frac {1}{\sqrt {\sec (c+d x)}} \, dx+\frac {1}{3} \left (\left (3 a^2 A b+3 A b^3+a^3 B+9 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 {2 \left (3 a^2 A b+3 A b^3+a^3 B+9 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)}}{3 d}+\frac {2 a^2 (9 A b+5 a B) \sin (c+d x)}{15 d \sqrt {\sec (c+d x)}}-\frac {2 b^2 (a A-5 b B) \sqrt {\sec (c+d x)} \sin (c+d x)}{5 d}+\frac {2 a A (a+b \sec (c+d x))^2 \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {1}{5} \left (\left (3 a^3 A+15 a A b^2+15 a^2 b B-5 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 (3 a^3 A+15 a A b^2+15 a^2 b B-5 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)}}{5 d}+\frac {2 \left (3 a^2 A b+3 A b^3+a^3 B+9 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)}}{3 d}+\frac {2 a^2 (9 A b+5 a B) \sin (c+d x)}{15 d \sqrt {\sec (c+d x)}}-\frac {2 b^2 (a A-5 b B) \sqrt {\sec (c+d x)} \sin (c+d x)}{5 d}+\frac {2 a A (a+b \sec (c+d x))^2 \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x)}\\ \end {align*}
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Mathematica [A] time = 1.69, size = 172, normalized size = 0.73 \[ \frac {\sqrt {\sec (c+d x)} \left (2 \sin (c+d x) \left (3 \left (a^3 A \cos (2 (c+d x))+a^3 A+10 b^3 B\right )+10 a^2 (a B+3 A b) \cos (c+d x)\right )+20 \left (a^3 B+3 a^2 A b+9 a b^2 B+3 A b^3\right ) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )+12 \left (3 a^3 A+15 a^2 b B+15 a A b^2-5 b^3 B\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )\right )}{30 d} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {B b^{3} \sec \left (d x + c\right )^{4} + A a^{3} + {\left (3 \, B a b^{2} + A b^{3}\right )} \sec \left (d x + c\right )^{3} + 3 \, {\left (B a^{2} b + A a b^{2}\right )} \sec \left (d x + c\right )^{2} + {\left (B a^{3} + 3 \, A a^{2} b\right )} \sec \left (d x + c\right )}{\sec \left (d x + c\right )^{\frac {5}{2}}}, x\right ) \]
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 \[ \int \frac {{\left (B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{3}}{\sec \left (d x + c\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 5.21, size = 867, normalized size = 3.67 \[ \text {result too large to display} \]
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 \[ \int \frac {\left (A+\frac {B}{\cos \left (c+d\,x\right )}\right )\,{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^3}{{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (A + B \sec {\left (c + d x \right )}\right ) \left (a + b \sec {\left (c + d x \right )}\right )^{3}}{\sec ^{\frac {5}{2}}{\left (c + d x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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