Optimal. Leaf size=234 \[ \frac {2 a^2 (7 A+9 B) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{35 d}+\frac {4 a^2 (7 A+6 B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{21 d}+\frac {4 a^2 (4 A+3 B) \sin (c+d x) \sqrt {\sec (c+d x)}}{5 d}+\frac {4 a^2 (7 A+6 B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d}-\frac {4 a^2 (4 A+3 B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {2 B \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \left (a^2 \sec (c+d x)+a^2\right )}{7 d} \]
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Rubi [A] time = 0.34, antiderivative size = 234, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.212, Rules used = {4018, 3997, 3787, 3768, 3771, 2639, 2641} \[ \frac {2 a^2 (7 A+9 B) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{35 d}+\frac {4 a^2 (7 A+6 B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{21 d}+\frac {4 a^2 (4 A+3 B) \sin (c+d x) \sqrt {\sec (c+d x)}}{5 d}+\frac {4 a^2 (7 A+6 B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d}-\frac {4 a^2 (4 A+3 B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {2 B \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \left (a^2 \sec (c+d x)+a^2\right )}{7 d} \]
Antiderivative was successfully verified.
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Rule 2639
Rule 2641
Rule 3768
Rule 3771
Rule 3787
Rule 3997
Rule 4018
Rubi steps
\begin {align*} \int \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx &=\frac {2 B \sec ^{\frac {5}{2}}(c+d x) \left (a^2+a^2 \sec (c+d x)\right ) \sin (c+d x)}{7 d}+\frac {2}{7} \int \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x)) \left (\frac {1}{2} a (7 A+3 B)+\frac {1}{2} a (7 A+9 B) \sec (c+d x)\right ) \, dx\\ &=\frac {2 a^2 (7 A+9 B) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{35 d}+\frac {2 B \sec ^{\frac {5}{2}}(c+d x) \left (a^2+a^2 \sec (c+d x)\right ) \sin (c+d x)}{7 d}+\frac {4}{35} \int \sec ^{\frac {3}{2}}(c+d x) \left (\frac {7}{2} a^2 (4 A+3 B)+\frac {5}{2} a^2 (7 A+6 B) \sec (c+d x)\right ) \, dx\\ &=\frac {2 a^2 (7 A+9 B) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{35 d}+\frac {2 B \sec ^{\frac {5}{2}}(c+d x) \left (a^2+a^2 \sec (c+d x)\right ) \sin (c+d x)}{7 d}+\frac {1}{5} \left (2 a^2 (4 A+3 B)\right ) \int \sec ^{\frac {3}{2}}(c+d x) \, dx+\frac {1}{7} \left (2 a^2 (7 A+6 B)\right ) \int \sec ^{\frac {5}{2}}(c+d x) \, dx\\ &=\frac {4 a^2 (4 A+3 B) \sqrt {\sec (c+d x)} \sin (c+d x)}{5 d}+\frac {4 a^2 (7 A+6 B) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac {2 a^2 (7 A+9 B) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{35 d}+\frac {2 B \sec ^{\frac {5}{2}}(c+d x) \left (a^2+a^2 \sec (c+d x)\right ) \sin (c+d x)}{7 d}-\frac {1}{5} \left (2 a^2 (4 A+3 B)\right ) \int \frac {1}{\sqrt {\sec (c+d x)}} \, dx+\frac {1}{21} \left (2 a^2 (7 A+6 B)\right ) \int \sqrt {\sec (c+d x)} \, dx\\ &=\frac {4 a^2 (4 A+3 B) \sqrt {\sec (c+d x)} \sin (c+d x)}{5 d}+\frac {4 a^2 (7 A+6 B) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac {2 a^2 (7 A+9 B) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{35 d}+\frac {2 B \sec ^{\frac {5}{2}}(c+d x) \left (a^2+a^2 \sec (c+d x)\right ) \sin (c+d x)}{7 d}-\frac {1}{5} \left (2 a^2 (4 A+3 B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx+\frac {1}{21} \left (2 a^2 (7 A+6 B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx\\ &=-\frac {4 a^2 (4 A+3 B) \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 {4 a^2 (7 A+6 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^2 (4 A+3 B) \sqrt {\sec (c+d x)} \sin (c+d x)}{5 d}+\frac {4 a^2 (7 A+6 B) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac {2 a^2 (7 A+9 B) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{35 d}+\frac {2 B \sec ^{\frac {5}{2}}(c+d x) \left (a^2+a^2 \sec (c+d x)\right ) \sin (c+d x)}{7 d}\\ \end {align*}
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Mathematica [C] time = 4.67, size = 463, normalized size = 1.98 \[ \frac {a^2 \csc (c) e^{-i d x} \cos ^3(c+d x) \sec ^4\left (\frac {1}{2} (c+d x)\right ) (\sec (c+d x)+1)^2 (A+B \sec (c+d x)) \left (7 \sqrt {2} \left (-1+e^{2 i c}\right ) (4 A+3 B) e^{2 i d x} \sqrt {\frac {e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt {1+e^{2 i (c+d x)}} \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};-e^{2 i (c+d x)}\right )-\frac {\left (-1+e^{2 i c}\right ) e^{-i (c-d x)} \sqrt {\sec (c+d x)} \left (5 i (7 A+6 B) \left (1+e^{2 i (c+d x)}\right )^3 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )+7 A \left (9 e^{i (c+d x)}-5 e^{2 i (c+d x)}+36 e^{3 i (c+d x)}+5 e^{4 i (c+d x)}+39 e^{5 i (c+d x)}+5 e^{6 i (c+d x)}+12 e^{7 i (c+d x)}-5\right )+3 B \left (7 e^{i (c+d x)}-20 e^{2 i (c+d x)}+63 e^{3 i (c+d x)}+20 e^{4 i (c+d x)}+77 e^{5 i (c+d x)}+10 e^{6 i (c+d x)}+21 e^{7 i (c+d x)}-10\right )\right )}{\left (1+e^{2 i (c+d x)}\right )^3}\right )}{210 d (A \cos (c+d x)+B)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (B a^{2} \sec \left (d x + c\right )^{4} + {\left (A + 2 \, B\right )} a^{2} \sec \left (d x + c\right )^{3} + {\left (2 \, A + B\right )} a^{2} \sec \left (d x + c\right )^{2} + A a^{2} \sec \left (d x + c\right )\right )} \sqrt {\sec \left (d x + c\right )}, x\right ) \]
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 \[ \int {\left (B \sec \left (d x + c\right ) + A\right )} {\left (a \sec \left (d x + c\right ) + a\right )}^{2} \sec \left (d x + c\right )^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 15.56, size = 852, normalized size = 3.64 \[ \text {result too large to display} \]
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 \[ \int \left (A+\frac {B}{\cos \left (c+d\,x\right )}\right )\,{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^2\,{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{3/2} \,d x \]
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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