Optimal. Leaf size=343 \[ \frac {4 a^3 (175 A+195 B+221 C) \sin (c+d x)}{585 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {20 a^3 (236 A+273 B+286 C) \sin (c+d x)}{9009 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 (145 A+195 B+143 C) \sin (c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{1287 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {4 a^3 (95 A+105 B+121 C) \sin (c+d x)}{231 d \sqrt {\sec (c+d x)}}+\frac {4 a^3 (95 A+105 B+121 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{231 d}+\frac {4 a^3 (175 A+195 B+221 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{195 d}+\frac {2 (6 A+13 B) \sin (c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{143 a d \sec ^{\frac {9}{2}}(c+d x)}+\frac {2 A \sin (c+d x) (a \sec (c+d x)+a)^3}{13 d \sec ^{\frac {11}{2}}(c+d x)} \]
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Rubi [A] time = 0.72, antiderivative size = 343, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.186, Rules used = {4086, 4017, 3996, 3787, 3769, 3771, 2639, 2641} \[ \frac {4 a^3 (175 A+195 B+221 C) \sin (c+d x)}{585 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {20 a^3 (236 A+273 B+286 C) \sin (c+d x)}{9009 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 (145 A+195 B+143 C) \sin (c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{1287 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {4 a^3 (95 A+105 B+121 C) \sin (c+d x)}{231 d \sqrt {\sec (c+d x)}}+\frac {4 a^3 (95 A+105 B+121 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{231 d}+\frac {4 a^3 (175 A+195 B+221 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{195 d}+\frac {2 (6 A+13 B) \sin (c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{143 a d \sec ^{\frac {9}{2}}(c+d x)}+\frac {2 A \sin (c+d x) (a \sec (c+d x)+a)^3}{13 d \sec ^{\frac {11}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Rule 2639
Rule 2641
Rule 3769
Rule 3771
Rule 3787
Rule 3996
Rule 4017
Rule 4086
Rubi steps
\begin {align*} \int \frac {(a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {13}{2}}(c+d x)} \, dx &=\frac {2 A (a+a \sec (c+d x))^3 \sin (c+d x)}{13 d \sec ^{\frac {11}{2}}(c+d x)}+\frac {2 \int \frac {(a+a \sec (c+d x))^3 \left (\frac {1}{2} a (6 A+13 B)+\frac {1}{2} a (5 A+13 C) \sec (c+d x)\right )}{\sec ^{\frac {11}{2}}(c+d x)} \, dx}{13 a}\\ &=\frac {2 A (a+a \sec (c+d x))^3 \sin (c+d x)}{13 d \sec ^{\frac {11}{2}}(c+d x)}+\frac {2 (6 A+13 B) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{143 a d \sec ^{\frac {9}{2}}(c+d x)}+\frac {4 \int \frac {(a+a \sec (c+d x))^2 \left (\frac {1}{4} a^2 (145 A+195 B+143 C)+\frac {1}{4} a^2 (85 A+65 B+143 C) \sec (c+d x)\right )}{\sec ^{\frac {9}{2}}(c+d x)} \, dx}{143 a}\\ &=\frac {2 A (a+a \sec (c+d x))^3 \sin (c+d x)}{13 d \sec ^{\frac {11}{2}}(c+d x)}+\frac {2 (6 A+13 B) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{143 a d \sec ^{\frac {9}{2}}(c+d x)}+\frac {2 (145 A+195 B+143 C) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{1287 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {8 \int \frac {(a+a \sec (c+d x)) \left (\frac {5}{4} a^3 (236 A+273 B+286 C)+\frac {1}{4} a^3 (745 A+780 B+1001 C) \sec (c+d x)\right )}{\sec ^{\frac {7}{2}}(c+d x)} \, dx}{1287 a}\\ &=\frac {20 a^3 (236 A+273 B+286 C) \sin (c+d x)}{9009 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 A (a+a \sec (c+d x))^3 \sin (c+d x)}{13 d \sec ^{\frac {11}{2}}(c+d x)}+\frac {2 (6 A+13 B) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{143 a d \sec ^{\frac {9}{2}}(c+d x)}+\frac {2 (145 A+195 B+143 C) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{1287 d \sec ^{\frac {7}{2}}(c+d x)}-\frac {16 \int \frac {-\frac {77}{8} a^4 (175 A+195 B+221 C)-\frac {117}{8} a^4 (95 A+105 B+121 C) \sec (c+d x)}{\sec ^{\frac {5}{2}}(c+d x)} \, dx}{9009 a}\\ &=\frac {20 a^3 (236 A+273 B+286 C) \sin (c+d x)}{9009 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 A (a+a \sec (c+d x))^3 \sin (c+d x)}{13 d \sec ^{\frac {11}{2}}(c+d x)}+\frac {2 (6 A+13 B) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{143 a d \sec ^{\frac {9}{2}}(c+d x)}+\frac {2 (145 A+195 B+143 C) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{1287 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {1}{77} \left (2 a^3 (95 A+105 B+121 C)\right ) \int \frac {1}{\sec ^{\frac {3}{2}}(c+d x)} \, dx+\frac {1}{117} \left (2 a^3 (175 A+195 B+221 C)\right ) \int \frac {1}{\sec ^{\frac {5}{2}}(c+d x)} \, dx\\ &=\frac {20 a^3 (236 A+273 B+286 C) \sin (c+d x)}{9009 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {4 a^3 (175 A+195 B+221 C) \sin (c+d x)}{585 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {4 a^3 (95 A+105 B+121 C) \sin (c+d x)}{231 d \sqrt {\sec (c+d x)}}+\frac {2 A (a+a \sec (c+d x))^3 \sin (c+d x)}{13 d \sec ^{\frac {11}{2}}(c+d x)}+\frac {2 (6 A+13 B) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{143 a d \sec ^{\frac {9}{2}}(c+d x)}+\frac {2 (145 A+195 B+143 C) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{1287 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {1}{231} \left (2 a^3 (95 A+105 B+121 C)\right ) \int \sqrt {\sec (c+d x)} \, dx+\frac {1}{195} \left (2 a^3 (175 A+195 B+221 C)\right ) \int \frac {1}{\sqrt {\sec (c+d x)}} \, dx\\ &=\frac {20 a^3 (236 A+273 B+286 C) \sin (c+d x)}{9009 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {4 a^3 (175 A+195 B+221 C) \sin (c+d x)}{585 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {4 a^3 (95 A+105 B+121 C) \sin (c+d x)}{231 d \sqrt {\sec (c+d x)}}+\frac {2 A (a+a \sec (c+d x))^3 \sin (c+d x)}{13 d \sec ^{\frac {11}{2}}(c+d x)}+\frac {2 (6 A+13 B) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{143 a d \sec ^{\frac {9}{2}}(c+d x)}+\frac {2 (145 A+195 B+143 C) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{1287 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {1}{231} \left (2 a^3 (95 A+105 B+121 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx+\frac {1}{195} \left (2 a^3 (175 A+195 B+221 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx\\ &=\frac {4 a^3 (175 A+195 B+221 C) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{195 d}+\frac {4 a^3 (95 A+105 B+121 C) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{231 d}+\frac {20 a^3 (236 A+273 B+286 C) \sin (c+d x)}{9009 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {4 a^3 (175 A+195 B+221 C) \sin (c+d x)}{585 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {4 a^3 (95 A+105 B+121 C) \sin (c+d x)}{231 d \sqrt {\sec (c+d x)}}+\frac {2 A (a+a \sec (c+d x))^3 \sin (c+d x)}{13 d \sec ^{\frac {11}{2}}(c+d x)}+\frac {2 (6 A+13 B) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{143 a d \sec ^{\frac {9}{2}}(c+d x)}+\frac {2 (145 A+195 B+143 C) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{1287 d \sec ^{\frac {7}{2}}(c+d x)}\\ \end {align*}
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Mathematica [C] time = 6.16, size = 300, normalized size = 0.87 \[ \frac {a^3 e^{-i d x} \sqrt {\sec (c+d x)} (\cos (d x)+i \sin (d x)) \left (-4928 i (175 A+195 B+221 C) e^{i (c+d x)} \sqrt {1+e^{2 i (c+d x)}} \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};-e^{2 i (c+d x)}\right )+\cos (c+d x) (780 (1811 A+1953 B+2134 C) \sin (c+d x)+77 (7825 A+7800 B+7592 C) \sin (2 (c+d x))+251550 A \sin (3 (c+d x))+90860 A \sin (4 (c+d x))+24570 A \sin (5 (c+d x))+3465 A \sin (6 (c+d x))+2587200 i A+221130 B \sin (3 (c+d x))+60060 B \sin (4 (c+d x))+8190 B \sin (5 (c+d x))+2882880 i B+154440 C \sin (3 (c+d x))+20020 C \sin (4 (c+d x))+3267264 i C)+12480 (95 A+105 B+121 C) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )\right )}{720720 d} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.51, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {C a^{3} \sec \left (d x + c\right )^{5} + {\left (B + 3 \, C\right )} a^{3} \sec \left (d x + c\right )^{4} + {\left (A + 3 \, B + 3 \, C\right )} a^{3} \sec \left (d x + c\right )^{3} + {\left (3 \, A + 3 \, B + C\right )} a^{3} \sec \left (d x + c\right )^{2} + {\left (3 \, A + B\right )} a^{3} \sec \left (d x + c\right ) + A a^{3}}{\sec \left (d x + c\right )^{\frac {13}{2}}}, x\right ) \]
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 \[ \int \frac {{\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (a \sec \left (d x + c\right ) + a\right )}^{3}}{\sec \left (d x + c\right )^{\frac {13}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 5.11, size = 576, normalized size = 1.68 \[ -\frac {4 \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 )}\, a^{3} \left (-221760 A \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (1058400 A +131040 B \right ) \left (\sin ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (-2122400 A -567840 B -80080 C \right ) \left (\sin ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (2331040 A +1004640 B +314600 C \right ) \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (-1535860 A -939120 B -487916 C \right ) \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (633710 A +510510 B +386386 C \right ) \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (-121230 A -114660 B -105534 C \right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+18525 A \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}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-40425 A \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}\, \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+20475 B \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}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-45045 B \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}\, \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+23595 C \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}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-51051 C \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}\, \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )\right )}{45045 \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(-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 \[ \int \frac {{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^3\,\left (A+\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right )}{{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{13/2}} \,d x \]
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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