Optimal. Leaf size=270 \[ \frac {2 a^2 (21 A+19 C) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{105 d}+\frac {4 a^2 (7 A+5 C) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{21 d}+\frac {16 a^2 (3 A+2 C) \sin (c+d x) \sqrt {\sec (c+d x)}}{15 d}+\frac {4 a^2 (7 A+5 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d}-\frac {16 a^2 (3 A+2 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {8 C \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \left (a^2 \sec (c+d x)+a^2\right )}{63 d}+\frac {2 C \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a \sec (c+d x)+a)^2}{9 d} \]
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Rubi [A] time = 0.44, antiderivative size = 270, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.229, Rules used = {4089, 4018, 3997, 3787, 3768, 3771, 2639, 2641} \[ \frac {2 a^2 (21 A+19 C) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{105 d}+\frac {4 a^2 (7 A+5 C) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{21 d}+\frac {16 a^2 (3 A+2 C) \sin (c+d x) \sqrt {\sec (c+d x)}}{15 d}+\frac {4 a^2 (7 A+5 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d}-\frac {16 a^2 (3 A+2 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {8 C \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \left (a^2 \sec (c+d x)+a^2\right )}{63 d}+\frac {2 C \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a \sec (c+d x)+a)^2}{9 d} \]
Antiderivative was successfully verified.
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Rule 2639
Rule 2641
Rule 3768
Rule 3771
Rule 3787
Rule 3997
Rule 4018
Rule 4089
Rubi steps
\begin {align*} \int \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^2 \left (A+C \sec ^2(c+d x)\right ) \, dx &=\frac {2 C \sec ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{9 d}+\frac {2 \int \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^2 \left (\frac {3}{2} a (3 A+C)+2 a C \sec (c+d x)\right ) \, dx}{9 a}\\ &=\frac {2 C \sec ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{9 d}+\frac {8 C \sec ^{\frac {5}{2}}(c+d x) \left (a^2+a^2 \sec (c+d x)\right ) \sin (c+d x)}{63 d}+\frac {4 \int \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x)) \left (\frac {3}{4} a^2 (21 A+11 C)+\frac {3}{4} a^2 (21 A+19 C) \sec (c+d x)\right ) \, dx}{63 a}\\ &=\frac {2 a^2 (21 A+19 C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac {2 C \sec ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{9 d}+\frac {8 C \sec ^{\frac {5}{2}}(c+d x) \left (a^2+a^2 \sec (c+d x)\right ) \sin (c+d x)}{63 d}+\frac {8 \int \sec ^{\frac {3}{2}}(c+d x) \left (21 a^3 (3 A+2 C)+\frac {45}{4} a^3 (7 A+5 C) \sec (c+d x)\right ) \, dx}{315 a}\\ &=\frac {2 a^2 (21 A+19 C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac {2 C \sec ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{9 d}+\frac {8 C \sec ^{\frac {5}{2}}(c+d x) \left (a^2+a^2 \sec (c+d x)\right ) \sin (c+d x)}{63 d}+\frac {1}{15} \left (8 a^2 (3 A+2 C)\right ) \int \sec ^{\frac {3}{2}}(c+d x) \, dx+\frac {1}{7} \left (2 a^2 (7 A+5 C)\right ) \int \sec ^{\frac {5}{2}}(c+d x) \, dx\\ &=\frac {16 a^2 (3 A+2 C) \sqrt {\sec (c+d x)} \sin (c+d x)}{15 d}+\frac {4 a^2 (7 A+5 C) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac {2 a^2 (21 A+19 C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac {2 C \sec ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{9 d}+\frac {8 C \sec ^{\frac {5}{2}}(c+d x) \left (a^2+a^2 \sec (c+d x)\right ) \sin (c+d x)}{63 d}-\frac {1}{15} \left (8 a^2 (3 A+2 C)\right ) \int \frac {1}{\sqrt {\sec (c+d x)}} \, dx+\frac {1}{21} \left (2 a^2 (7 A+5 C)\right ) \int \sqrt {\sec (c+d x)} \, dx\\ &=\frac {16 a^2 (3 A+2 C) \sqrt {\sec (c+d x)} \sin (c+d x)}{15 d}+\frac {4 a^2 (7 A+5 C) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac {2 a^2 (21 A+19 C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac {2 C \sec ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{9 d}+\frac {8 C \sec ^{\frac {5}{2}}(c+d x) \left (a^2+a^2 \sec (c+d x)\right ) \sin (c+d x)}{63 d}-\frac {1}{15} \left (8 a^2 (3 A+2 C) \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+5 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx\\ &=-\frac {16 a^2 (3 A+2 C) \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^2 (7 A+5 C) \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 {16 a^2 (3 A+2 C) \sqrt {\sec (c+d x)} \sin (c+d x)}{15 d}+\frac {4 a^2 (7 A+5 C) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac {2 a^2 (21 A+19 C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac {2 C \sec ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{9 d}+\frac {8 C \sec ^{\frac {5}{2}}(c+d x) \left (a^2+a^2 \sec (c+d x)\right ) \sin (c+d x)}{63 d}\\ \end {align*}
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Mathematica [C] time = 6.89, size = 821, normalized size = 3.04 \[ \frac {4 \sqrt {2} A e^{-i d x} \sqrt {\frac {e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt {1+e^{2 i (c+d x)}} \cos ^4(c+d x) \csc (c) \left (e^{2 i d x} \left (-1+e^{2 i c}\right ) \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};-e^{2 i (c+d x)}\right )-3 \sqrt {1+e^{2 i (c+d x)}}\right ) (\sec (c+d x) a+a)^2 \left (C \sec ^2(c+d x)+A\right ) \sec ^4\left (\frac {c}{2}+\frac {d x}{2}\right )}{15 d (\cos (2 c+2 d x) A+A+2 C)}+\frac {8 \sqrt {2} C e^{-i d x} \sqrt {\frac {e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt {1+e^{2 i (c+d x)}} \cos ^4(c+d x) \csc (c) \left (e^{2 i d x} \left (-1+e^{2 i c}\right ) \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};-e^{2 i (c+d x)}\right )-3 \sqrt {1+e^{2 i (c+d x)}}\right ) (\sec (c+d x) a+a)^2 \left (C \sec ^2(c+d x)+A\right ) \sec ^4\left (\frac {c}{2}+\frac {d x}{2}\right )}{45 d (\cos (2 c+2 d x) A+A+2 C)}+\frac {2 A \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) (\sec (c+d x) a+a)^2 \left (C \sec ^2(c+d x)+A\right ) \sec ^4\left (\frac {c}{2}+\frac {d x}{2}\right )}{3 d (\cos (2 c+2 d x) A+A+2 C) \sec ^{\frac {7}{2}}(c+d x)}+\frac {10 C \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) (\sec (c+d x) a+a)^2 \left (C \sec ^2(c+d x)+A\right ) \sec ^4\left (\frac {c}{2}+\frac {d x}{2}\right )}{21 d (\cos (2 c+2 d x) A+A+2 C) \sec ^{\frac {7}{2}}(c+d x)}+\frac {(\sec (c+d x) a+a)^2 \left (C \sec ^2(c+d x)+A\right ) \left (\frac {C \sec (c) \sin (d x) \sec ^4(c+d x)}{9 d}+\frac {\sec (c) (7 C \sin (c)+18 C \sin (d x)) \sec ^3(c+d x)}{63 d}+\frac {\sec (c) (90 C \sin (c)+63 A \sin (d x)+112 C \sin (d x)) \sec ^2(c+d x)}{315 d}+\frac {\sec (c) (63 A \sin (c)+112 C \sin (c)+210 A \sin (d x)+150 C \sin (d x)) \sec (c+d x)}{315 d}+\frac {8 (3 A+2 C) \cos (d x) \csc (c)}{15 d}+\frac {2 (7 A+5 C) \tan (c)}{21 d}\right ) \sec ^4\left (\frac {c}{2}+\frac {d x}{2}\right )}{(\cos (2 c+2 d x) A+A+2 C) \sec ^{\frac {7}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (C a^{2} \sec \left (d x + c\right )^{5} + 2 \, C a^{2} \sec \left (d x + c\right )^{4} + {\left (A + C\right )} a^{2} \sec \left (d x + c\right )^{3} + 2 \, A a^{2} \sec \left (d x + c\right )^{2} + A a^{2} \sec \left (d x + c\right )\right )} \sqrt {\sec \left (d x + c\right )}, x\right ) \]
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 \[ \int {\left (C \sec \left (d x + c\right )^{2} + A\right )} {\left (a \sec \left (d x + c\right ) + a\right )}^{2} \sec \left (d x + c\right )^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 19.17, size = 1168, normalized size = 4.33 \[ \text {result too large to display} \]
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 \[ \int \left (A+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right )\,{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^2\,{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{3/2} \,d x \]
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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