Optimal. Leaf size=249 \[ \frac {(A-13 C) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{6 d \left (a^3 \sec (c+d x)+a^3\right )}-\frac {(A-49 C) \sin (c+d x) \sqrt {\sec (c+d x)}}{10 a^3 d}+\frac {(A-13 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{6 a^3 d}+\frac {(A-49 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{10 a^3 d}-\frac {(A+C) \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x)}{5 d (a \sec (c+d x)+a)^3}+\frac {2 (A-4 C) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{15 a d (a \sec (c+d x)+a)^2} \]
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Rubi [A] time = 0.52, antiderivative size = 249, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4085, 4019, 3787, 3771, 2641, 3768, 2639} \[ \frac {(A-13 C) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{6 d \left (a^3 \sec (c+d x)+a^3\right )}-\frac {(A-49 C) \sin (c+d x) \sqrt {\sec (c+d x)}}{10 a^3 d}+\frac {(A-13 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{6 a^3 d}+\frac {(A-49 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{10 a^3 d}-\frac {(A+C) \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x)}{5 d (a \sec (c+d x)+a)^3}+\frac {2 (A-4 C) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{15 a d (a \sec (c+d x)+a)^2} \]
Antiderivative was successfully verified.
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Rule 2639
Rule 2641
Rule 3768
Rule 3771
Rule 3787
Rule 4019
Rule 4085
Rubi steps
\begin {align*} \int \frac {\sec ^{\frac {5}{2}}(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^3} \, dx &=-\frac {(A+C) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac {\int \frac {\sec ^{\frac {5}{2}}(c+d x) \left (-\frac {5}{2} a (A-C)-\frac {1}{2} a (A+11 C) \sec (c+d x)\right )}{(a+a \sec (c+d x))^2} \, dx}{5 a^2}\\ &=-\frac {(A+C) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}+\frac {2 (A-4 C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{15 a d (a+a \sec (c+d x))^2}-\frac {\int \frac {\sec ^{\frac {3}{2}}(c+d x) \left (-3 a^2 (A-4 C)-\frac {1}{2} a^2 (A+41 C) \sec (c+d x)\right )}{a+a \sec (c+d x)} \, dx}{15 a^4}\\ &=-\frac {(A+C) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}+\frac {2 (A-4 C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{15 a d (a+a \sec (c+d x))^2}+\frac {(A-13 C) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{6 d \left (a^3+a^3 \sec (c+d x)\right )}-\frac {\int \sqrt {\sec (c+d x)} \left (-\frac {5}{4} a^3 (A-13 C)+\frac {3}{4} a^3 (A-49 C) \sec (c+d x)\right ) \, dx}{15 a^6}\\ &=-\frac {(A+C) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}+\frac {2 (A-4 C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{15 a d (a+a \sec (c+d x))^2}+\frac {(A-13 C) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{6 d \left (a^3+a^3 \sec (c+d x)\right )}-\frac {(A-49 C) \int \sec ^{\frac {3}{2}}(c+d x) \, dx}{20 a^3}+\frac {(A-13 C) \int \sqrt {\sec (c+d x)} \, dx}{12 a^3}\\ &=-\frac {(A-49 C) \sqrt {\sec (c+d x)} \sin (c+d x)}{10 a^3 d}-\frac {(A+C) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}+\frac {2 (A-4 C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{15 a d (a+a \sec (c+d x))^2}+\frac {(A-13 C) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{6 d \left (a^3+a^3 \sec (c+d x)\right )}+\frac {(A-49 C) \int \frac {1}{\sqrt {\sec (c+d x)}} \, dx}{20 a^3}+\frac {\left ((A-13 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{12 a^3}\\ &=\frac {(A-13 C) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{6 a^3 d}-\frac {(A-49 C) \sqrt {\sec (c+d x)} \sin (c+d x)}{10 a^3 d}-\frac {(A+C) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}+\frac {2 (A-4 C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{15 a d (a+a \sec (c+d x))^2}+\frac {(A-13 C) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{6 d \left (a^3+a^3 \sec (c+d x)\right )}+\frac {\left ((A-49 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx}{20 a^3}\\ &=\frac {(A-49 C) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{10 a^3 d}+\frac {(A-13 C) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{6 a^3 d}-\frac {(A-49 C) \sqrt {\sec (c+d x)} \sin (c+d x)}{10 a^3 d}-\frac {(A+C) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}+\frac {2 (A-4 C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{15 a d (a+a \sec (c+d x))^2}+\frac {(A-13 C) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{6 d \left (a^3+a^3 \sec (c+d x)\right )}\\ \end {align*}
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Mathematica [C] time = 7.20, size = 953, normalized size = 3.83 \[ -\frac {2 \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)}} \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 ) \sec (c+d x) \left (C \sec ^2(c+d x)+A\right ) \cos ^6\left (\frac {c}{2}+\frac {d x}{2}\right )}{15 d (\cos (2 c+2 d x) A+A+2 C) (\sec (c+d x) a+a)^3}+\frac {98 \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)}} \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 ) \sec (c+d x) \left (C \sec ^2(c+d x)+A\right ) \cos ^6\left (\frac {c}{2}+\frac {d x}{2}\right )}{15 d (\cos (2 c+2 d x) A+A+2 C) (\sec (c+d x) a+a)^3}+\frac {4 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 ) \sec ^{\frac {3}{2}}(c+d x) \left (C \sec ^2(c+d x)+A\right ) \sin (c) \cos ^6\left (\frac {c}{2}+\frac {d x}{2}\right )}{3 d (\cos (2 c+2 d x) A+A+2 C) (\sec (c+d x) a+a)^3}-\frac {52 C \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 ) \sec ^{\frac {3}{2}}(c+d x) \left (C \sec ^2(c+d x)+A\right ) \sin (c) \cos ^6\left (\frac {c}{2}+\frac {d x}{2}\right )}{3 d (\cos (2 c+2 d x) A+A+2 C) (\sec (c+d x) a+a)^3}+\frac {\sec ^{\frac {3}{2}}(c+d x) \left (C \sec ^2(c+d x)+A\right ) \left (-\frac {4 \sec \left (\frac {c}{2}\right ) \left (A \sin \left (\frac {d x}{2}\right )+C \sin \left (\frac {d x}{2}\right )\right ) \sec ^5\left (\frac {c}{2}+\frac {d x}{2}\right )}{5 d}-\frac {4 (A+C) \tan \left (\frac {c}{2}\right ) \sec ^4\left (\frac {c}{2}+\frac {d x}{2}\right )}{5 d}+\frac {16 \sec \left (\frac {c}{2}\right ) \left (A \sin \left (\frac {d x}{2}\right )-4 C \sin \left (\frac {d x}{2}\right )\right ) \sec ^3\left (\frac {c}{2}+\frac {d x}{2}\right )}{15 d}+\frac {16 (A-4 C) \tan \left (\frac {c}{2}\right ) \sec ^2\left (\frac {c}{2}+\frac {d x}{2}\right )}{15 d}+\frac {8 \sec \left (\frac {c}{2}\right ) \left (A \sin \left (\frac {d x}{2}\right )-13 C \sin \left (\frac {d x}{2}\right )\right ) \sec \left (\frac {c}{2}+\frac {d x}{2}\right )}{3 d}-\frac {4 (A-49 C) \cos (d x) \csc \left (\frac {c}{2}\right ) \sec \left (\frac {c}{2}\right )}{5 d}-\frac {8 (13 C-A) \tan \left (\frac {c}{2}\right )}{3 d}\right ) \cos ^6\left (\frac {c}{2}+\frac {d x}{2}\right )}{(\cos (2 c+2 d x) A+A+2 C) (\sec (c+d x) a+a)^3} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (C \sec \left (d x + c\right )^{4} + A \sec \left (d x + c\right )^{2}\right )} \sqrt {\sec \left (d x + c\right )}}{a^{3} \sec \left (d x + c\right )^{3} + 3 \, a^{3} \sec \left (d x + c\right )^{2} + 3 \, a^{3} \sec \left (d x + c\right ) + a^{3}}, x\right ) \]
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 \[ \int \frac {{\left (C \sec \left (d x + c\right )^{2} + A\right )} \sec \left (d x + c\right )^{\frac {5}{2}}}{{\left (a \sec \left (d x + c\right ) + a\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 6.26, size = 679, normalized size = 2.73 \[ \text {result too large to display} \]
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 \[ \int \frac {\left (A+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right )\,{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{5/2}}{{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^3} \,d x \]
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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