Optimal. Leaf size=230 \[ -\frac {(A-B) \sin (c+d x)}{d \sec ^{\frac {5}{2}}(c+d x) (a \sec (c+d x)+a)}-\frac {7 (A-B) \sin (c+d x)}{5 a d \sec ^{\frac {3}{2}}(c+d x)}+\frac {(9 A-7 B) \sin (c+d x)}{7 a d \sec ^{\frac {5}{2}}(c+d x)}+\frac {5 (9 A-7 B) \sin (c+d x)}{21 a d \sqrt {\sec (c+d x)}}+\frac {5 (9 A-7 B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 a d}-\frac {21 (A-B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 a d} \]
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Rubi [A] time = 0.23, antiderivative size = 230, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {4020, 3787, 3769, 3771, 2641, 2639} \[ -\frac {(A-B) \sin (c+d x)}{d \sec ^{\frac {5}{2}}(c+d x) (a \sec (c+d x)+a)}-\frac {7 (A-B) \sin (c+d x)}{5 a d \sec ^{\frac {3}{2}}(c+d x)}+\frac {(9 A-7 B) \sin (c+d x)}{7 a d \sec ^{\frac {5}{2}}(c+d x)}+\frac {5 (9 A-7 B) \sin (c+d x)}{21 a d \sqrt {\sec (c+d x)}}+\frac {5 (9 A-7 B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 a d}-\frac {21 (A-B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 a d} \]
Antiderivative was successfully verified.
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Rule 2639
Rule 2641
Rule 3769
Rule 3771
Rule 3787
Rule 4020
Rubi steps
\begin {align*} \int \frac {A+B \sec (c+d x)}{\sec ^{\frac {7}{2}}(c+d x) (a+a \sec (c+d x))} \, dx &=-\frac {(A-B) \sin (c+d x)}{d \sec ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))}+\frac {\int \frac {\frac {1}{2} a (9 A-7 B)-\frac {7}{2} a (A-B) \sec (c+d x)}{\sec ^{\frac {7}{2}}(c+d x)} \, dx}{a^2}\\ &=-\frac {(A-B) \sin (c+d x)}{d \sec ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))}+\frac {(9 A-7 B) \int \frac {1}{\sec ^{\frac {7}{2}}(c+d x)} \, dx}{2 a}-\frac {(7 (A-B)) \int \frac {1}{\sec ^{\frac {5}{2}}(c+d x)} \, dx}{2 a}\\ &=\frac {(9 A-7 B) \sin (c+d x)}{7 a d \sec ^{\frac {5}{2}}(c+d x)}-\frac {7 (A-B) \sin (c+d x)}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {(A-B) \sin (c+d x)}{d \sec ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))}+\frac {(5 (9 A-7 B)) \int \frac {1}{\sec ^{\frac {3}{2}}(c+d x)} \, dx}{14 a}-\frac {(21 (A-B)) \int \frac {1}{\sqrt {\sec (c+d x)}} \, dx}{10 a}\\ &=\frac {(9 A-7 B) \sin (c+d x)}{7 a d \sec ^{\frac {5}{2}}(c+d x)}-\frac {7 (A-B) \sin (c+d x)}{5 a d \sec ^{\frac {3}{2}}(c+d x)}+\frac {5 (9 A-7 B) \sin (c+d x)}{21 a d \sqrt {\sec (c+d x)}}-\frac {(A-B) \sin (c+d x)}{d \sec ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))}+\frac {(5 (9 A-7 B)) \int \sqrt {\sec (c+d x)} \, dx}{42 a}-\frac {\left (21 (A-B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx}{10 a}\\ &=-\frac {21 (A-B) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{5 a d}+\frac {(9 A-7 B) \sin (c+d x)}{7 a d \sec ^{\frac {5}{2}}(c+d x)}-\frac {7 (A-B) \sin (c+d x)}{5 a d \sec ^{\frac {3}{2}}(c+d x)}+\frac {5 (9 A-7 B) \sin (c+d x)}{21 a d \sqrt {\sec (c+d x)}}-\frac {(A-B) \sin (c+d x)}{d \sec ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))}+\frac {\left (5 (9 A-7 B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{42 a}\\ &=-\frac {21 (A-B) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{5 a d}+\frac {5 (9 A-7 B) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{21 a d}+\frac {(9 A-7 B) \sin (c+d x)}{7 a d \sec ^{\frac {5}{2}}(c+d x)}-\frac {7 (A-B) \sin (c+d x)}{5 a d \sec ^{\frac {3}{2}}(c+d x)}+\frac {5 (9 A-7 B) \sin (c+d x)}{21 a d \sqrt {\sec (c+d x)}}-\frac {(A-B) \sin (c+d x)}{d \sec ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))}\\ \end {align*}
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Mathematica [C] time = 4.16, size = 568, normalized size = 2.47 \[ \frac {\cos ^2\left (\frac {1}{2} (c+d x)\right ) (A+B \sec (c+d x)) \left (\sqrt {\sec (c+d x)} \left (20 (27 A-14 B) \sin (2 c) \cos (2 d x)-84 (A-B) \sin (3 c) \cos (3 d x)-2772 (A-B) \cos (c) \sin (d x)+20 (27 A-14 B) \cos (2 c) \sin (2 d x)-84 (A-B) \cos (3 c) \sin (3 d x)-840 (A-B) \sec \left (\frac {c}{2}\right ) \sin \left (\frac {d x}{2}\right ) \sec \left (\frac {1}{2} (c+d x)\right )+63 (A-B) (11 \cos (2 c)+17) \csc \left (\frac {c}{2}\right ) \sec \left (\frac {c}{2}\right ) \cos (d x)-840 (A-B) \tan \left (\frac {c}{2}\right )+30 A \sin (4 c) \cos (4 d x)+30 A \cos (4 c) \sin (4 d x)\right )+588 \sqrt {2} A \csc (c) e^{-i d x} \sqrt {\frac {e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt {1+e^{2 i (c+d x)}} \left (\left (-1+e^{2 i c}\right ) e^{2 i d x} \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};-e^{2 i (c+d x)}\right )-3 \sqrt {1+e^{2 i (c+d x)}}\right )+1800 A \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )-588 \sqrt {2} B \csc (c) e^{-i d x} \sqrt {\frac {e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt {1+e^{2 i (c+d x)}} \left (\left (-1+e^{2 i c}\right ) e^{2 i d x} \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};-e^{2 i (c+d x)}\right )-3 \sqrt {1+e^{2 i (c+d x)}}\right )-1400 B \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )\right )}{420 a d (\sec (c+d x)+1) (A \cos (c+d x)+B)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (B \sec \left (d x + c\right ) + A\right )} \sqrt {\sec \left (d x + c\right )}}{a \sec \left (d x + c\right )^{5} + a \sec \left (d x + c\right )^{4}}, x\right ) \]
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 \[ \int \frac {B \sec \left (d x + c\right ) + A}{{\left (a \sec \left (d x + c\right ) + a\right )} \sec \left (d x + c\right )^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 5.00, size = 300, normalized size = 1.30 \[ -\frac {\sqrt {\left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \left (225 A \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+441 A \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-175 B \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-441 B \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )\right )-480 A \left (\sin ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (864 A +336 B \right ) \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-888 A -392 B \right ) \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (930 A -210 B \right ) \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-321 A +161 B \right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )}{105 a \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {B \sec \left (d x + c\right ) + A}{{\left (a \sec \left (d x + c\right ) + a\right )} \sec \left (d x + c\right )^{\frac {7}{2}}}\,{d x} \]
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 \[ \int \frac {A+\frac {B}{\cos \left (c+d\,x\right )}}{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )\,{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{7/2}} \,d x \]
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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