Optimal. Leaf size=229 \[ \frac {(A-B) \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x)}{d (a \sec (c+d x)+a)}-\frac {(5 A-7 B) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 a d}+\frac {5 (A-B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 a d}-\frac {3 (5 A-7 B) \sin (c+d x) \sqrt {\sec (c+d x)}}{5 a d}+\frac {5 (A-B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{3 a d}+\frac {3 (5 A-7 B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 a d} \]
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Rubi [A] time = 0.25, antiderivative size = 229, 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 = {4019, 3787, 3768, 3771, 2641, 2639} \[ \frac {(A-B) \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x)}{d (a \sec (c+d x)+a)}-\frac {(5 A-7 B) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 a d}+\frac {5 (A-B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 a d}-\frac {3 (5 A-7 B) \sin (c+d x) \sqrt {\sec (c+d x)}}{5 a d}+\frac {5 (A-B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{3 a d}+\frac {3 (5 A-7 B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 a d} \]
Antiderivative was successfully verified.
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Rule 2639
Rule 2641
Rule 3768
Rule 3771
Rule 3787
Rule 4019
Rubi steps
\begin {align*} \int \frac {\sec ^{\frac {7}{2}}(c+d x) (A+B \sec (c+d x))}{a+a \sec (c+d x)} \, dx &=\frac {(A-B) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{d (a+a \sec (c+d x))}+\frac {\int \sec ^{\frac {5}{2}}(c+d x) \left (\frac {5}{2} a (A-B)-\frac {1}{2} a (5 A-7 B) \sec (c+d x)\right ) \, dx}{a^2}\\ &=\frac {(A-B) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{d (a+a \sec (c+d x))}-\frac {(5 A-7 B) \int \sec ^{\frac {7}{2}}(c+d x) \, dx}{2 a}+\frac {(5 (A-B)) \int \sec ^{\frac {5}{2}}(c+d x) \, dx}{2 a}\\ &=\frac {5 (A-B) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{3 a d}-\frac {(5 A-7 B) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{5 a d}+\frac {(A-B) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{d (a+a \sec (c+d x))}-\frac {(3 (5 A-7 B)) \int \sec ^{\frac {3}{2}}(c+d x) \, dx}{10 a}+\frac {(5 (A-B)) \int \sqrt {\sec (c+d x)} \, dx}{6 a}\\ &=-\frac {3 (5 A-7 B) \sqrt {\sec (c+d x)} \sin (c+d x)}{5 a d}+\frac {5 (A-B) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{3 a d}-\frac {(5 A-7 B) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{5 a d}+\frac {(A-B) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{d (a+a \sec (c+d x))}+\frac {(3 (5 A-7 B)) \int \frac {1}{\sqrt {\sec (c+d x)}} \, dx}{10 a}+\frac {\left (5 (A-B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{6 a}\\ &=\frac {5 (A-B) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{3 a d}-\frac {3 (5 A-7 B) \sqrt {\sec (c+d x)} \sin (c+d x)}{5 a d}+\frac {5 (A-B) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{3 a d}-\frac {(5 A-7 B) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{5 a d}+\frac {(A-B) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{d (a+a \sec (c+d x))}+\frac {\left (3 (5 A-7 B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx}{10 a}\\ &=\frac {3 (5 A-7 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 (A-B) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{3 a d}-\frac {3 (5 A-7 B) \sqrt {\sec (c+d x)} \sin (c+d x)}{5 a d}+\frac {5 (A-B) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{3 a d}-\frac {(5 A-7 B) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{5 a d}+\frac {(A-B) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{d (a+a \sec (c+d x))}\\ \end {align*}
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Mathematica [C] time = 7.70, size = 814, normalized size = 3.55 \[ -\frac {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)}} \csc \left (\frac {c}{2}\right ) \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 \left (\frac {c}{2}\right ) (A+B \sec (c+d x)) \cos ^2\left (\frac {c}{2}+\frac {d x}{2}\right )}{\sqrt {2} d (B+A \cos (c+d x)) (\sec (c+d x) a+a)}+\frac {7 B 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)}} \csc \left (\frac {c}{2}\right ) \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 \left (\frac {c}{2}\right ) (A+B \sec (c+d x)) \cos ^2\left (\frac {c}{2}+\frac {d x}{2}\right )}{5 \sqrt {2} d (B+A \cos (c+d x)) (\sec (c+d x) a+a)}+\frac {5 A \sqrt {\cos (c+d x)} \csc \left (\frac {c}{2}\right ) F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sec \left (\frac {c}{2}\right ) \sqrt {\sec (c+d x)} (A+B \sec (c+d x)) \sin (c) \cos ^2\left (\frac {c}{2}+\frac {d x}{2}\right )}{3 d (B+A \cos (c+d x)) (\sec (c+d x) a+a)}-\frac {5 B \sqrt {\cos (c+d x)} \csc \left (\frac {c}{2}\right ) F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sec \left (\frac {c}{2}\right ) \sqrt {\sec (c+d x)} (A+B \sec (c+d x)) \sin (c) \cos ^2\left (\frac {c}{2}+\frac {d x}{2}\right )}{3 d (B+A \cos (c+d x)) (\sec (c+d x) a+a)}+\frac {\sqrt {\sec (c+d x)} (A+B \sec (c+d x)) \left (\frac {4 B \sec (c) \sin (d x) \sec ^2(c+d x)}{5 d}+\frac {4 \sec (c) (3 B \sin (c)+5 A \sin (d x)-5 B \sin (d x)) \sec (c+d x)}{15 d}+\frac {3 (7 B-5 A) \cos (d x) \csc \left (\frac {c}{2}\right ) \sec \left (\frac {c}{2}\right )}{5 d}-\frac {(B-A) \sec \left (\frac {c}{2}\right ) \sec (c) \left (5 \sin \left (\frac {3 c}{2}\right )-\sin \left (\frac {c}{2}\right )\right )}{3 d}-\frac {2 \sec \left (\frac {c}{2}\right ) \sec \left (\frac {c}{2}+\frac {d x}{2}\right ) \left (B \sin \left (\frac {d x}{2}\right )-A \sin \left (\frac {d x}{2}\right )\right )}{d}\right ) \cos ^2\left (\frac {c}{2}+\frac {d x}{2}\right )}{(B+A \cos (c+d x)) (\sec (c+d x) a+a)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (B \sec \left (d x + c\right )^{4} + A \sec \left (d x + c\right )^{3}\right )} \sqrt {\sec \left (d x + c\right )}}{a \sec \left (d x + c\right ) + a}, x\right ) \]
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 \[ \int \frac {{\left (B \sec \left (d x + c\right ) + A\right )} \sec \left (d x + c\right )^{\frac {7}{2}}}{a \sec \left (d x + c\right ) + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 14.32, size = 806, normalized size = 3.52 \[ \text {result too large to display} \]
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 \[ \int \frac {{\left (B \sec \left (d x + c\right ) + A\right )} \sec \left (d x + c\right )^{\frac {7}{2}}}{a \sec \left (d x + c\right ) + a}\,{d x} \]
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 \[ \int \frac {\left (A+\frac {B}{\cos \left (c+d\,x\right )}\right )\,{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{7/2}}{a+\frac {a}{\cos \left (c+d\,x\right )}} \,d x \]
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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