Integrand size = 45, antiderivative size = 334 \[ \int \frac {(a+a \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {13}{2}}(c+d x)} \, dx=\frac {2 a^3 (2224 A+2522 B+2717 C) \sin (c+d x)}{9009 d \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {2 a^3 (8368 A+9230 B+10439 C) \sin (c+d x)}{15015 d \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {8 a^3 (8368 A+9230 B+10439 C) \sin (c+d x)}{45045 d \sqrt {\sec (c+d x)} \sqrt {a+a \sec (c+d x)}}+\frac {16 a^3 (8368 A+9230 B+10439 C) \sqrt {\sec (c+d x)} \sin (c+d x)}{45045 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a^2 (136 A+182 B+143 C) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{1287 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 a (5 A+13 B) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{143 d \sec ^{\frac {9}{2}}(c+d x)}+\frac {2 A (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{13 d \sec ^{\frac {11}{2}}(c+d x)} \]
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Time = 1.16 (sec) , antiderivative size = 334, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {4171, 4102, 4100, 3890, 3889} \[ \int \frac {(a+a \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {13}{2}}(c+d x)} \, dx=\frac {2 a^3 (8368 A+9230 B+10439 C) \sin (c+d x)}{15015 d \sec ^{\frac {3}{2}}(c+d x) \sqrt {a \sec (c+d x)+a}}+\frac {2 a^3 (2224 A+2522 B+2717 C) \sin (c+d x)}{9009 d \sec ^{\frac {5}{2}}(c+d x) \sqrt {a \sec (c+d x)+a}}+\frac {16 a^3 (8368 A+9230 B+10439 C) \sin (c+d x) \sqrt {\sec (c+d x)}}{45045 d \sqrt {a \sec (c+d x)+a}}+\frac {8 a^3 (8368 A+9230 B+10439 C) \sin (c+d x)}{45045 d \sqrt {\sec (c+d x)} \sqrt {a \sec (c+d x)+a}}+\frac {2 a^2 (136 A+182 B+143 C) \sin (c+d x) \sqrt {a \sec (c+d x)+a}}{1287 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 a (5 A+13 B) \sin (c+d x) (a \sec (c+d x)+a)^{3/2}}{143 d \sec ^{\frac {9}{2}}(c+d x)}+\frac {2 A \sin (c+d x) (a \sec (c+d x)+a)^{5/2}}{13 d \sec ^{\frac {11}{2}}(c+d x)} \]
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Rule 3889
Rule 3890
Rule 4100
Rule 4102
Rule 4171
Rubi steps \begin{align*} \text {integral}& = \frac {2 A (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{13 d \sec ^{\frac {11}{2}}(c+d x)}+\frac {2 \int \frac {(a+a \sec (c+d x))^{5/2} \left (\frac {1}{2} a (5 A+13 B)+\frac {1}{2} a (6 A+13 C) \sec (c+d x)\right )}{\sec ^{\frac {11}{2}}(c+d x)} \, dx}{13 a} \\ & = \frac {2 a (5 A+13 B) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{143 d \sec ^{\frac {9}{2}}(c+d x)}+\frac {2 A (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{13 d \sec ^{\frac {11}{2}}(c+d x)}+\frac {4 \int \frac {(a+a \sec (c+d x))^{3/2} \left (\frac {1}{4} a^2 (136 A+182 B+143 C)+\frac {1}{4} a^2 (96 A+78 B+143 C) \sec (c+d x)\right )}{\sec ^{\frac {9}{2}}(c+d x)} \, dx}{143 a} \\ & = \frac {2 a^2 (136 A+182 B+143 C) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{1287 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 a (5 A+13 B) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{143 d \sec ^{\frac {9}{2}}(c+d x)}+\frac {2 A (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{13 d \sec ^{\frac {11}{2}}(c+d x)}+\frac {8 \int \frac {\sqrt {a+a \sec (c+d x)} \left (\frac {1}{8} a^3 (2224 A+2522 B+2717 C)+\frac {3}{8} a^3 (560 A+598 B+715 C) \sec (c+d x)\right )}{\sec ^{\frac {7}{2}}(c+d x)} \, dx}{1287 a} \\ & = \frac {2 a^3 (2224 A+2522 B+2717 C) \sin (c+d x)}{9009 d \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {2 a^2 (136 A+182 B+143 C) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{1287 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 a (5 A+13 B) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{143 d \sec ^{\frac {9}{2}}(c+d x)}+\frac {2 A (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{13 d \sec ^{\frac {11}{2}}(c+d x)}+\frac {\left (a^2 (8368 A+9230 B+10439 C)\right ) \int \frac {\sqrt {a+a \sec (c+d x)}}{\sec ^{\frac {5}{2}}(c+d x)} \, dx}{3003} \\ & = \frac {2 a^3 (2224 A+2522 B+2717 C) \sin (c+d x)}{9009 d \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {2 a^3 (8368 A+9230 B+10439 C) \sin (c+d x)}{15015 d \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {2 a^2 (136 A+182 B+143 C) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{1287 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 a (5 A+13 B) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{143 d \sec ^{\frac {9}{2}}(c+d x)}+\frac {2 A (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{13 d \sec ^{\frac {11}{2}}(c+d x)}+\frac {\left (4 a^2 (8368 A+9230 B+10439 C)\right ) \int \frac {\sqrt {a+a \sec (c+d x)}}{\sec ^{\frac {3}{2}}(c+d x)} \, dx}{15015} \\ & = \frac {2 a^3 (2224 A+2522 B+2717 C) \sin (c+d x)}{9009 d \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {2 a^3 (8368 A+9230 B+10439 C) \sin (c+d x)}{15015 d \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {8 a^3 (8368 A+9230 B+10439 C) \sin (c+d x)}{45045 d \sqrt {\sec (c+d x)} \sqrt {a+a \sec (c+d x)}}+\frac {2 a^2 (136 A+182 B+143 C) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{1287 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 a (5 A+13 B) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{143 d \sec ^{\frac {9}{2}}(c+d x)}+\frac {2 A (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{13 d \sec ^{\frac {11}{2}}(c+d x)}+\frac {\left (8 a^2 (8368 A+9230 B+10439 C)\right ) \int \frac {\sqrt {a+a \sec (c+d x)}}{\sqrt {\sec (c+d x)}} \, dx}{45045} \\ & = \frac {2 a^3 (2224 A+2522 B+2717 C) \sin (c+d x)}{9009 d \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {2 a^3 (8368 A+9230 B+10439 C) \sin (c+d x)}{15015 d \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {8 a^3 (8368 A+9230 B+10439 C) \sin (c+d x)}{45045 d \sqrt {\sec (c+d x)} \sqrt {a+a \sec (c+d x)}}+\frac {16 a^3 (8368 A+9230 B+10439 C) \sqrt {\sec (c+d x)} \sin (c+d x)}{45045 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a^2 (136 A+182 B+143 C) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{1287 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 a (5 A+13 B) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{143 d \sec ^{\frac {9}{2}}(c+d x)}+\frac {2 A (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{13 d \sec ^{\frac {11}{2}}(c+d x)} \\ \end{align*}
Time = 5.98 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.48 \[ \int \frac {(a+a \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {13}{2}}(c+d x)} \, dx=\frac {2 a^3 \left (3465 A+315 (38 A+13 B) \sec (c+d x)+35 (523 A+416 B+143 C) \sec ^2(c+d x)+5 (4184 A+4615 B+3718 C) \sec ^3(c+d x)+3 (8368 A+9230 B+10439 C) \sec ^4(c+d x)+4 (8368 A+9230 B+10439 C) \sec ^5(c+d x)+8 (8368 A+9230 B+10439 C) \sec ^6(c+d x)\right ) \sin (c+d x)}{45045 d \sec ^{\frac {11}{2}}(c+d x) \sqrt {a (1+\sec (c+d x))}} \]
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Time = 1.55 (sec) , antiderivative size = 214, normalized size of antiderivative = 0.64
method | result | size |
default | \(\frac {2 a^{2} \left (3465 A \cos \left (d x +c \right )^{6}+11970 A \cos \left (d x +c \right )^{5}+4095 B \cos \left (d x +c \right )^{5}+18305 A \cos \left (d x +c \right )^{4}+14560 B \cos \left (d x +c \right )^{4}+5005 C \cos \left (d x +c \right )^{4}+20920 A \cos \left (d x +c \right )^{3}+23075 B \cos \left (d x +c \right )^{3}+18590 C \cos \left (d x +c \right )^{3}+25104 A \cos \left (d x +c \right )^{2}+27690 B \cos \left (d x +c \right )^{2}+31317 C \cos \left (d x +c \right )^{2}+33472 A \cos \left (d x +c \right )+36920 B \cos \left (d x +c \right )+41756 C \cos \left (d x +c \right )+66944 A +73840 B +83512 C \right ) \sqrt {a \left (1+\sec \left (d x +c \right )\right )}\, \tan \left (d x +c \right )}{45045 d \left (\cos \left (d x +c \right )+1\right ) \sec \left (d x +c \right )^{\frac {3}{2}}}\) | \(214\) |
parts | \(\frac {2 A \,a^{2} \left (3465 \cos \left (d x +c \right )^{6}+11970 \cos \left (d x +c \right )^{5}+18305 \cos \left (d x +c \right )^{4}+20920 \cos \left (d x +c \right )^{3}+25104 \cos \left (d x +c \right )^{2}+33472 \cos \left (d x +c \right )+66944\right ) \sqrt {a \left (1+\sec \left (d x +c \right )\right )}\, \tan \left (d x +c \right )}{45045 d \left (\cos \left (d x +c \right )+1\right ) \sec \left (d x +c \right )^{\frac {3}{2}}}+\frac {2 B \,a^{2} \left (63 \cos \left (d x +c \right )^{5}+224 \cos \left (d x +c \right )^{4}+355 \cos \left (d x +c \right )^{3}+426 \cos \left (d x +c \right )^{2}+568 \cos \left (d x +c \right )+1136\right ) \sqrt {a \left (1+\sec \left (d x +c \right )\right )}\, \tan \left (d x +c \right )}{693 d \left (\cos \left (d x +c \right )+1\right ) \sec \left (d x +c \right )^{\frac {3}{2}}}+\frac {2 C \,a^{2} \left (35 \cos \left (d x +c \right )^{4}+130 \cos \left (d x +c \right )^{3}+219 \cos \left (d x +c \right )^{2}+292 \cos \left (d x +c \right )+584\right ) \sqrt {a \left (1+\sec \left (d x +c \right )\right )}\, \tan \left (d x +c \right )}{315 d \left (\cos \left (d x +c \right )+1\right ) \sec \left (d x +c \right )^{\frac {3}{2}}}\) | \(287\) |
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Time = 0.27 (sec) , antiderivative size = 197, normalized size of antiderivative = 0.59 \[ \int \frac {(a+a \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {13}{2}}(c+d x)} \, dx=\frac {2 \, {\left (3465 \, A a^{2} \cos \left (d x + c\right )^{7} + 315 \, {\left (38 \, A + 13 \, B\right )} a^{2} \cos \left (d x + c\right )^{6} + 35 \, {\left (523 \, A + 416 \, B + 143 \, C\right )} a^{2} \cos \left (d x + c\right )^{5} + 5 \, {\left (4184 \, A + 4615 \, B + 3718 \, C\right )} a^{2} \cos \left (d x + c\right )^{4} + 3 \, {\left (8368 \, A + 9230 \, B + 10439 \, C\right )} a^{2} \cos \left (d x + c\right )^{3} + 4 \, {\left (8368 \, A + 9230 \, B + 10439 \, C\right )} a^{2} \cos \left (d x + c\right )^{2} + 8 \, {\left (8368 \, A + 9230 \, B + 10439 \, C\right )} a^{2} \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{45045 \, {\left (d \cos \left (d x + c\right ) + d\right )} \sqrt {\cos \left (d x + c\right )}} \]
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Timed out. \[ \int \frac {(a+a \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {13}{2}}(c+d x)} \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1562 vs. \(2 (292) = 584\).
Time = 0.57 (sec) , antiderivative size = 1562, normalized size of antiderivative = 4.68 \[ \int \frac {(a+a \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {13}{2}}(c+d x)} \, dx=\text {Too large to display} \]
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\[ \int \frac {(a+a \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {13}{2}}(c+d x)} \, dx=\int { \frac {{\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (a \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}}}{\sec \left (d x + c\right )^{\frac {13}{2}}} \,d x } \]
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Time = 25.60 (sec) , antiderivative size = 458, normalized size of antiderivative = 1.37 \[ \int \frac {(a+a \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {13}{2}}(c+d x)} \, dx=\frac {\sqrt {a-\frac {a}{2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1}}\,\left (2\,{\sin \left (\frac {13\,c}{4}+\frac {13\,d\,x}{4}\right )}^2+\sin \left (\frac {13\,c}{2}+\frac {13\,d\,x}{2}\right )\,1{}\mathrm {i}-1\right )\,\left (\frac {A\,a^2\,\sin \left (\frac {13\,c}{2}+\frac {13\,d\,x}{2}\right )\,\left (-2\,{\sin \left (\frac {13\,c}{4}+\frac {13\,d\,x}{4}\right )}^2+\sin \left (\frac {13\,c}{2}+\frac {13\,d\,x}{2}\right )\,1{}\mathrm {i}+1\right )}{208\,d}+\frac {a^2\,\sin \left (\frac {9\,c}{2}+\frac {9\,d\,x}{2}\right )\,\left (-2\,{\sin \left (\frac {13\,c}{4}+\frac {13\,d\,x}{4}\right )}^2+\sin \left (\frac {13\,c}{2}+\frac {13\,d\,x}{2}\right )\,1{}\mathrm {i}+1\right )\,\left (7\,A+5\,B+2\,C\right )}{72\,d}+\frac {a^2\,\sin \left (\frac {7\,c}{2}+\frac {7\,d\,x}{2}\right )\,\left (-2\,{\sin \left (\frac {13\,c}{4}+\frac {13\,d\,x}{4}\right )}^2+\sin \left (\frac {13\,c}{2}+\frac {13\,d\,x}{2}\right )\,1{}\mathrm {i}+1\right )\,\left (15\,A+13\,B+10\,C\right )}{56\,d}+\frac {a^2\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (-2\,{\sin \left (\frac {13\,c}{4}+\frac {13\,d\,x}{4}\right )}^2+\sin \left (\frac {13\,c}{2}+\frac {13\,d\,x}{2}\right )\,1{}\mathrm {i}+1\right )\,\left (21\,A+23\,B+26\,C\right )}{4\,d}+\frac {a^2\,\sin \left (\frac {5\,c}{2}+\frac {5\,d\,x}{2}\right )\,\left (-2\,{\sin \left (\frac {13\,c}{4}+\frac {13\,d\,x}{4}\right )}^2+\sin \left (\frac {13\,c}{2}+\frac {13\,d\,x}{2}\right )\,1{}\mathrm {i}+1\right )\,\left (51\,A+50\,B+48\,C\right )}{80\,d}+\frac {a^2\,\sin \left (\frac {3\,c}{2}+\frac {3\,d\,x}{2}\right )\,\left (-2\,{\sin \left (\frac {13\,c}{4}+\frac {13\,d\,x}{4}\right )}^2+\sin \left (\frac {13\,c}{2}+\frac {13\,d\,x}{2}\right )\,1{}\mathrm {i}+1\right )\,\left (71\,A+76\,B+80\,C\right )}{48\,d}+\frac {a^2\,\sin \left (\frac {11\,c}{2}+\frac {11\,d\,x}{2}\right )\,\left (5\,A+2\,B\right )\,\left (-2\,{\sin \left (\frac {13\,c}{4}+\frac {13\,d\,x}{4}\right )}^2+\sin \left (\frac {13\,c}{2}+\frac {13\,d\,x}{2}\right )\,1{}\mathrm {i}+1\right )}{176\,d}\right )}{2\,\sqrt {-\frac {1}{2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1}}\,\left (2\,{\sin \left (\frac {c}{4}+\frac {d\,x}{4}\right )}^2-1\right )} \]
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