Integrand size = 43, antiderivative size = 243 \[ \int \sec ^3(c+d x) (a+a \sec (c+d x))^{3/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {2 a^2 (429 A+374 B+336 C) \tan (c+d x)}{495 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a^2 (99 A+110 B+84 C) \sec ^3(c+d x) \tan (c+d x)}{693 d \sqrt {a+a \sec (c+d x)}}-\frac {4 a (429 A+374 B+336 C) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{3465 d}+\frac {2 a (11 B+3 C) \sec ^3(c+d x) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{99 d}+\frac {2 (429 A+374 B+336 C) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{1155 d}+\frac {2 C \sec ^3(c+d x) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{11 d} \]
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Time = 0.86 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.140, Rules used = {4173, 4103, 4101, 3885, 4086, 3877} \[ \int \sec ^3(c+d x) (a+a \sec (c+d x))^{3/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {2 a^2 (99 A+110 B+84 C) \tan (c+d x) \sec ^3(c+d x)}{693 d \sqrt {a \sec (c+d x)+a}}+\frac {2 a^2 (429 A+374 B+336 C) \tan (c+d x)}{495 d \sqrt {a \sec (c+d x)+a}}+\frac {2 (429 A+374 B+336 C) \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{1155 d}-\frac {4 a (429 A+374 B+336 C) \tan (c+d x) \sqrt {a \sec (c+d x)+a}}{3465 d}+\frac {2 a (11 B+3 C) \tan (c+d x) \sec ^3(c+d x) \sqrt {a \sec (c+d x)+a}}{99 d}+\frac {2 C \tan (c+d x) \sec ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{11 d} \]
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Rule 3877
Rule 3885
Rule 4086
Rule 4101
Rule 4103
Rule 4173
Rubi steps \begin{align*} \text {integral}& = \frac {2 C \sec ^3(c+d x) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{11 d}+\frac {2 \int \sec ^3(c+d x) (a+a \sec (c+d x))^{3/2} \left (\frac {1}{2} a (11 A+6 C)+\frac {1}{2} a (11 B+3 C) \sec (c+d x)\right ) \, dx}{11 a} \\ & = \frac {2 a (11 B+3 C) \sec ^3(c+d x) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{99 d}+\frac {2 C \sec ^3(c+d x) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{11 d}+\frac {4 \int \sec ^3(c+d x) \sqrt {a+a \sec (c+d x)} \left (\frac {3}{4} a^2 (33 A+22 B+24 C)+\frac {1}{4} a^2 (99 A+110 B+84 C) \sec (c+d x)\right ) \, dx}{99 a} \\ & = \frac {2 a^2 (99 A+110 B+84 C) \sec ^3(c+d x) \tan (c+d x)}{693 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a (11 B+3 C) \sec ^3(c+d x) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{99 d}+\frac {2 C \sec ^3(c+d x) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{11 d}+\frac {1}{231} (a (429 A+374 B+336 C)) \int \sec ^3(c+d x) \sqrt {a+a \sec (c+d x)} \, dx \\ & = \frac {2 a^2 (99 A+110 B+84 C) \sec ^3(c+d x) \tan (c+d x)}{693 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a (11 B+3 C) \sec ^3(c+d x) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{99 d}+\frac {2 (429 A+374 B+336 C) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{1155 d}+\frac {2 C \sec ^3(c+d x) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{11 d}+\frac {(2 (429 A+374 B+336 C)) \int \sec (c+d x) \left (\frac {3 a}{2}-a \sec (c+d x)\right ) \sqrt {a+a \sec (c+d x)} \, dx}{1155} \\ & = \frac {2 a^2 (99 A+110 B+84 C) \sec ^3(c+d x) \tan (c+d x)}{693 d \sqrt {a+a \sec (c+d x)}}-\frac {4 a (429 A+374 B+336 C) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{3465 d}+\frac {2 a (11 B+3 C) \sec ^3(c+d x) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{99 d}+\frac {2 (429 A+374 B+336 C) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{1155 d}+\frac {2 C \sec ^3(c+d x) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{11 d}+\frac {1}{495} (a (429 A+374 B+336 C)) \int \sec (c+d x) \sqrt {a+a \sec (c+d x)} \, dx \\ & = \frac {2 a^2 (429 A+374 B+336 C) \tan (c+d x)}{495 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a^2 (99 A+110 B+84 C) \sec ^3(c+d x) \tan (c+d x)}{693 d \sqrt {a+a \sec (c+d x)}}-\frac {4 a (429 A+374 B+336 C) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{3465 d}+\frac {2 a (11 B+3 C) \sec ^3(c+d x) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{99 d}+\frac {2 (429 A+374 B+336 C) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{1155 d}+\frac {2 C \sec ^3(c+d x) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{11 d} \\ \end{align*}
Time = 2.44 (sec) , antiderivative size = 183, normalized size of antiderivative = 0.75 \[ \int \sec ^3(c+d x) (a+a \sec (c+d x))^{3/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {a^2 (3564 A+4114 B+4956 C+(12441 A+12386 B+12684 C) \cos (c+d x)+(4422 A+4862 B+4368 C) \cos (2 (c+d x))+5577 A \cos (3 (c+d x))+4862 B \cos (3 (c+d x))+4368 C \cos (3 (c+d x))+858 A \cos (4 (c+d x))+748 B \cos (4 (c+d x))+672 C \cos (4 (c+d x))+858 A \cos (5 (c+d x))+748 B \cos (5 (c+d x))+672 C \cos (5 (c+d x))) \sec ^5(c+d x) \tan (c+d x)}{6930 d \sqrt {a (1+\sec (c+d x))}} \]
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Time = 1.02 (sec) , antiderivative size = 197, normalized size of antiderivative = 0.81
method | result | size |
default | \(\frac {2 a \left (3432 A \cos \left (d x +c \right )^{5}+2992 B \cos \left (d x +c \right )^{5}+2688 C \cos \left (d x +c \right )^{5}+1716 A \cos \left (d x +c \right )^{4}+1496 B \cos \left (d x +c \right )^{4}+1344 C \cos \left (d x +c \right )^{4}+1287 A \cos \left (d x +c \right )^{3}+1122 B \cos \left (d x +c \right )^{3}+1008 C \cos \left (d x +c \right )^{3}+495 A \cos \left (d x +c \right )^{2}+935 B \cos \left (d x +c \right )^{2}+840 C \cos \left (d x +c \right )^{2}+385 B \cos \left (d x +c \right )+735 C \cos \left (d x +c \right )+315 C \right ) \sqrt {a \left (1+\sec \left (d x +c \right )\right )}\, \tan \left (d x +c \right ) \sec \left (d x +c \right )^{4}}{3465 d \left (\cos \left (d x +c \right )+1\right )}\) | \(197\) |
parts | \(\frac {2 A a \left (104 \cos \left (d x +c \right )^{3}+52 \cos \left (d x +c \right )^{2}+39 \cos \left (d x +c \right )+15\right ) \sqrt {a \left (1+\sec \left (d x +c \right )\right )}\, \tan \left (d x +c \right ) \sec \left (d x +c \right )^{2}}{105 d \left (\cos \left (d x +c \right )+1\right )}+\frac {2 B a \left (272 \cos \left (d x +c \right )^{4}+136 \cos \left (d x +c \right )^{3}+102 \cos \left (d x +c \right )^{2}+85 \cos \left (d x +c \right )+35\right ) \sqrt {a \left (1+\sec \left (d x +c \right )\right )}\, \tan \left (d x +c \right ) \sec \left (d x +c \right )^{3}}{315 d \left (\cos \left (d x +c \right )+1\right )}+\frac {2 C a \left (128 \cos \left (d x +c \right )^{5}+64 \cos \left (d x +c \right )^{4}+48 \cos \left (d x +c \right )^{3}+40 \cos \left (d x +c \right )^{2}+35 \cos \left (d x +c \right )+15\right ) \sqrt {a \left (1+\sec \left (d x +c \right )\right )}\, \tan \left (d x +c \right ) \sec \left (d x +c \right )^{4}}{165 d \left (\cos \left (d x +c \right )+1\right )}\) | \(251\) |
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Time = 0.26 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.65 \[ \int \sec ^3(c+d x) (a+a \sec (c+d x))^{3/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {2 \, {\left (8 \, {\left (429 \, A + 374 \, B + 336 \, C\right )} a \cos \left (d x + c\right )^{5} + 4 \, {\left (429 \, A + 374 \, B + 336 \, C\right )} a \cos \left (d x + c\right )^{4} + 3 \, {\left (429 \, A + 374 \, B + 336 \, C\right )} a \cos \left (d x + c\right )^{3} + 5 \, {\left (99 \, A + 187 \, B + 168 \, C\right )} a \cos \left (d x + c\right )^{2} + 35 \, {\left (11 \, B + 21 \, C\right )} a \cos \left (d x + c\right ) + 315 \, C a\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{3465 \, {\left (d \cos \left (d x + c\right )^{6} + d \cos \left (d x + c\right )^{5}\right )}} \]
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\[ \int \sec ^3(c+d x) (a+a \sec (c+d x))^{3/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int \left (a \left (\sec {\left (c + d x \right )} + 1\right )\right )^{\frac {3}{2}} \left (A + B \sec {\left (c + d x \right )} + C \sec ^{2}{\left (c + d x \right )}\right ) \sec ^{3}{\left (c + d x \right )}\, dx \]
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Timed out. \[ \int \sec ^3(c+d x) (a+a \sec (c+d x))^{3/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \]
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\[ \int \sec ^3(c+d x) (a+a \sec (c+d x))^{3/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (a \sec \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \sec \left (d x + c\right )^{3} \,d x } \]
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Time = 28.85 (sec) , antiderivative size = 852, normalized size of antiderivative = 3.51 \[ \int \sec ^3(c+d x) (a+a \sec (c+d x))^{3/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Too large to display} \]
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