Integrand size = 42, antiderivative size = 261 \[ \int \frac {\sec ^4(c+d x) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^{3/2}} \, dx=-\frac {(15 B-19 C) \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {2} \sqrt {a+a \sec (c+d x)}}\right )}{2 \sqrt {2} a^{3/2} d}+\frac {(B-C) \sec ^4(c+d x) \tan (c+d x)}{2 d (a+a \sec (c+d x))^{3/2}}+\frac {(651 B-799 C) \tan (c+d x)}{105 a d \sqrt {a+a \sec (c+d x)}}+\frac {(63 B-67 C) \sec ^2(c+d x) \tan (c+d x)}{70 a d \sqrt {a+a \sec (c+d x)}}-\frac {(7 B-11 C) \sec ^3(c+d x) \tan (c+d x)}{14 a d \sqrt {a+a \sec (c+d x)}}-\frac {(273 B-397 C) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{210 a^2 d} \]
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Time = 1.14 (sec) , antiderivative size = 261, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {4157, 4104, 4106, 4095, 4086, 3880, 209} \[ \int \frac {\sec ^4(c+d x) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^{3/2}} \, dx=-\frac {(15 B-19 C) \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {2} \sqrt {a \sec (c+d x)+a}}\right )}{2 \sqrt {2} a^{3/2} d}-\frac {(273 B-397 C) \tan (c+d x) \sqrt {a \sec (c+d x)+a}}{210 a^2 d}+\frac {(B-C) \tan (c+d x) \sec ^4(c+d x)}{2 d (a \sec (c+d x)+a)^{3/2}}-\frac {(7 B-11 C) \tan (c+d x) \sec ^3(c+d x)}{14 a d \sqrt {a \sec (c+d x)+a}}+\frac {(63 B-67 C) \tan (c+d x) \sec ^2(c+d x)}{70 a d \sqrt {a \sec (c+d x)+a}}+\frac {(651 B-799 C) \tan (c+d x)}{105 a d \sqrt {a \sec (c+d x)+a}} \]
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Rule 209
Rule 3880
Rule 4086
Rule 4095
Rule 4104
Rule 4106
Rule 4157
Rubi steps \begin{align*} \text {integral}& = \int \frac {\sec ^5(c+d x) (B+C \sec (c+d x))}{(a+a \sec (c+d x))^{3/2}} \, dx \\ & = \frac {(B-C) \sec ^4(c+d x) \tan (c+d x)}{2 d (a+a \sec (c+d x))^{3/2}}+\frac {\int \frac {\sec ^4(c+d x) \left (4 a (B-C)-\frac {1}{2} a (7 B-11 C) \sec (c+d x)\right )}{\sqrt {a+a \sec (c+d x)}} \, dx}{2 a^2} \\ & = \frac {(B-C) \sec ^4(c+d x) \tan (c+d x)}{2 d (a+a \sec (c+d x))^{3/2}}-\frac {(7 B-11 C) \sec ^3(c+d x) \tan (c+d x)}{14 a d \sqrt {a+a \sec (c+d x)}}+\frac {\int \frac {\sec ^3(c+d x) \left (-\frac {3}{2} a^2 (7 B-11 C)+\frac {1}{4} a^2 (63 B-67 C) \sec (c+d x)\right )}{\sqrt {a+a \sec (c+d x)}} \, dx}{7 a^3} \\ & = \frac {(B-C) \sec ^4(c+d x) \tan (c+d x)}{2 d (a+a \sec (c+d x))^{3/2}}+\frac {(63 B-67 C) \sec ^2(c+d x) \tan (c+d x)}{70 a d \sqrt {a+a \sec (c+d x)}}-\frac {(7 B-11 C) \sec ^3(c+d x) \tan (c+d x)}{14 a d \sqrt {a+a \sec (c+d x)}}+\frac {2 \int \frac {\sec ^2(c+d x) \left (\frac {1}{2} a^3 (63 B-67 C)-\frac {1}{8} a^3 (273 B-397 C) \sec (c+d x)\right )}{\sqrt {a+a \sec (c+d x)}} \, dx}{35 a^4} \\ & = \frac {(B-C) \sec ^4(c+d x) \tan (c+d x)}{2 d (a+a \sec (c+d x))^{3/2}}+\frac {(63 B-67 C) \sec ^2(c+d x) \tan (c+d x)}{70 a d \sqrt {a+a \sec (c+d x)}}-\frac {(7 B-11 C) \sec ^3(c+d x) \tan (c+d x)}{14 a d \sqrt {a+a \sec (c+d x)}}-\frac {(273 B-397 C) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{210 a^2 d}+\frac {4 \int \frac {\sec (c+d x) \left (-\frac {1}{16} a^4 (273 B-397 C)+\frac {1}{8} a^4 (651 B-799 C) \sec (c+d x)\right )}{\sqrt {a+a \sec (c+d x)}} \, dx}{105 a^5} \\ & = \frac {(B-C) \sec ^4(c+d x) \tan (c+d x)}{2 d (a+a \sec (c+d x))^{3/2}}+\frac {(651 B-799 C) \tan (c+d x)}{105 a d \sqrt {a+a \sec (c+d x)}}+\frac {(63 B-67 C) \sec ^2(c+d x) \tan (c+d x)}{70 a d \sqrt {a+a \sec (c+d x)}}-\frac {(7 B-11 C) \sec ^3(c+d x) \tan (c+d x)}{14 a d \sqrt {a+a \sec (c+d x)}}-\frac {(273 B-397 C) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{210 a^2 d}-\frac {(15 B-19 C) \int \frac {\sec (c+d x)}{\sqrt {a+a \sec (c+d x)}} \, dx}{4 a} \\ & = \frac {(B-C) \sec ^4(c+d x) \tan (c+d x)}{2 d (a+a \sec (c+d x))^{3/2}}+\frac {(651 B-799 C) \tan (c+d x)}{105 a d \sqrt {a+a \sec (c+d x)}}+\frac {(63 B-67 C) \sec ^2(c+d x) \tan (c+d x)}{70 a d \sqrt {a+a \sec (c+d x)}}-\frac {(7 B-11 C) \sec ^3(c+d x) \tan (c+d x)}{14 a d \sqrt {a+a \sec (c+d x)}}-\frac {(273 B-397 C) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{210 a^2 d}+\frac {(15 B-19 C) \text {Subst}\left (\int \frac {1}{2 a+x^2} \, dx,x,-\frac {a \tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{2 a d} \\ & = -\frac {(15 B-19 C) \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {2} \sqrt {a+a \sec (c+d x)}}\right )}{2 \sqrt {2} a^{3/2} d}+\frac {(B-C) \sec ^4(c+d x) \tan (c+d x)}{2 d (a+a \sec (c+d x))^{3/2}}+\frac {(651 B-799 C) \tan (c+d x)}{105 a d \sqrt {a+a \sec (c+d x)}}+\frac {(63 B-67 C) \sec ^2(c+d x) \tan (c+d x)}{70 a d \sqrt {a+a \sec (c+d x)}}-\frac {(7 B-11 C) \sec ^3(c+d x) \tan (c+d x)}{14 a d \sqrt {a+a \sec (c+d x)}}-\frac {(273 B-397 C) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{210 a^2 d} \\ \end{align*}
Time = 0.90 (sec) , antiderivative size = 204, normalized size of antiderivative = 0.78 \[ \int \frac {\sec ^4(c+d x) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^{3/2}} \, dx=\frac {\left (-210 \sqrt {2} (15 B-19 C) \text {arctanh}\left (\frac {\sqrt {1-\sec (c+d x)}}{\sqrt {2}}\right ) \cos ^2\left (\frac {1}{2} (c+d x)\right ) \sec (c+d x)+\frac {1}{4} (2751 B-2339 C+24 (217 B-213 C) \cos (c+d x)+60 (63 B-67 C) \cos (2 (c+d x))+1512 B \cos (3 (c+d x))-1608 C \cos (3 (c+d x))+1029 B \cos (4 (c+d x))-1201 C \cos (4 (c+d x))) \sqrt {1-\sec (c+d x)} \sec ^4(c+d x)\right ) \tan (c+d x)}{420 d \sqrt {1-\sec (c+d x)} (a (1+\sec (c+d x)))^{3/2}} \]
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Time = 1.04 (sec) , antiderivative size = 423, normalized size of antiderivative = 1.62
method | result | size |
default | \(-\frac {\sqrt {-\frac {2 a}{\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-1}}\, \left (105 B \left (1-\cos \left (d x +c \right )\right )^{9} \csc \left (d x +c \right )^{9}-105 C \left (1-\cos \left (d x +c \right )\right )^{9} \csc \left (d x +c \right )^{9}+1575 B \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )+\sqrt {\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-1}\right ) \left (\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-1\right )^{\frac {7}{2}}-1995 C \left (\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-1\right )^{\frac {7}{2}} \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )+\sqrt {\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-1}\right )-2772 B \left (1-\cos \left (d x +c \right )\right )^{7} \csc \left (d x +c \right )^{7}+3508 C \left (1-\cos \left (d x +c \right )\right )^{7} \csc \left (d x +c \right )^{7}+6342 B \left (1-\cos \left (d x +c \right )\right )^{5} \csc \left (d x +c \right )^{5}-7238 C \left (1-\cos \left (d x +c \right )\right )^{5} \csc \left (d x +c \right )^{5}-5460 B \left (1-\cos \left (d x +c \right )\right )^{3} \csc \left (d x +c \right )^{3}+6580 C \left (1-\cos \left (d x +c \right )\right )^{3} \csc \left (d x +c \right )^{3}+1785 B \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )-1785 C \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )\right )}{420 a^{2} d \left (\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-1\right )^{3}}\) | \(423\) |
parts | \(-\frac {B \sqrt {-\frac {2 a}{\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-1}}\, \left (5 \left (1-\cos \left (d x +c \right )\right )^{7} \csc \left (d x +c \right )^{7}+75 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )+\sqrt {\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-1}\right ) \left (\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-1\right )^{\frac {5}{2}}-127 \left (1-\cos \left (d x +c \right )\right )^{5} \csc \left (d x +c \right )^{5}+175 \left (1-\cos \left (d x +c \right )\right )^{3} \csc \left (d x +c \right )^{3}+85 \cot \left (d x +c \right )-85 \csc \left (d x +c \right )\right )}{20 d \,a^{2} \left (\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-1\right )^{2}}+\frac {C \sqrt {-\frac {2 a}{\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-1}}\, \left (105 \left (1-\cos \left (d x +c \right )\right )^{9} \csc \left (d x +c \right )^{9}+1995 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )+\sqrt {\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-1}\right ) \left (\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-1\right )^{\frac {7}{2}}-3508 \left (1-\cos \left (d x +c \right )\right )^{7} \csc \left (d x +c \right )^{7}+7238 \left (1-\cos \left (d x +c \right )\right )^{5} \csc \left (d x +c \right )^{5}-6580 \left (1-\cos \left (d x +c \right )\right )^{3} \csc \left (d x +c \right )^{3}-1785 \cot \left (d x +c \right )+1785 \csc \left (d x +c \right )\right )}{420 d \,a^{2} \left (\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-1\right )^{3}}\) | \(454\) |
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Time = 0.33 (sec) , antiderivative size = 539, normalized size of antiderivative = 2.07 \[ \int \frac {\sec ^4(c+d x) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^{3/2}} \, dx=\left [\frac {105 \, \sqrt {2} {\left ({\left (15 \, B - 19 \, C\right )} \cos \left (d x + c\right )^{5} + 2 \, {\left (15 \, B - 19 \, C\right )} \cos \left (d x + c\right )^{4} + {\left (15 \, B - 19 \, C\right )} \cos \left (d x + c\right )^{3}\right )} \sqrt {-a} \log \left (\frac {2 \, \sqrt {2} \sqrt {-a} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right ) \sin \left (d x + c\right ) + 3 \, a \cos \left (d x + c\right )^{2} + 2 \, a \cos \left (d x + c\right ) - a}{\cos \left (d x + c\right )^{2} + 2 \, \cos \left (d x + c\right ) + 1}\right ) + 4 \, {\left ({\left (1029 \, B - 1201 \, C\right )} \cos \left (d x + c\right )^{4} + 12 \, {\left (63 \, B - 67 \, C\right )} \cos \left (d x + c\right )^{3} - 28 \, {\left (3 \, B - 7 \, C\right )} \cos \left (d x + c\right )^{2} + 12 \, {\left (7 \, B - 3 \, C\right )} \cos \left (d x + c\right ) + 60 \, C\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{840 \, {\left (a^{2} d \cos \left (d x + c\right )^{5} + 2 \, a^{2} d \cos \left (d x + c\right )^{4} + a^{2} d \cos \left (d x + c\right )^{3}\right )}}, \frac {105 \, \sqrt {2} {\left ({\left (15 \, B - 19 \, C\right )} \cos \left (d x + c\right )^{5} + 2 \, {\left (15 \, B - 19 \, C\right )} \cos \left (d x + c\right )^{4} + {\left (15 \, B - 19 \, C\right )} \cos \left (d x + c\right )^{3}\right )} \sqrt {a} \arctan \left (\frac {\sqrt {2} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )}{\sqrt {a} \sin \left (d x + c\right )}\right ) + 2 \, {\left ({\left (1029 \, B - 1201 \, C\right )} \cos \left (d x + c\right )^{4} + 12 \, {\left (63 \, B - 67 \, C\right )} \cos \left (d x + c\right )^{3} - 28 \, {\left (3 \, B - 7 \, C\right )} \cos \left (d x + c\right )^{2} + 12 \, {\left (7 \, B - 3 \, C\right )} \cos \left (d x + c\right ) + 60 \, C\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{420 \, {\left (a^{2} d \cos \left (d x + c\right )^{5} + 2 \, a^{2} d \cos \left (d x + c\right )^{4} + a^{2} d \cos \left (d x + c\right )^{3}\right )}}\right ] \]
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\[ \int \frac {\sec ^4(c+d x) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^{3/2}} \, dx=\int \frac {\left (B + C \sec {\left (c + d x \right )}\right ) \sec ^{5}{\left (c + d x \right )}}{\left (a \left (\sec {\left (c + d x \right )} + 1\right )\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {\sec ^4(c+d x) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^{3/2}} \, dx=\int { \frac {{\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right )\right )} \sec \left (d x + c\right )^{4}}{{\left (a \sec \left (d x + c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
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Time = 1.78 (sec) , antiderivative size = 362, normalized size of antiderivative = 1.39 \[ \int \frac {\sec ^4(c+d x) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^{3/2}} \, dx=\frac {\frac {105 \, {\left (15 \, \sqrt {2} B - 19 \, \sqrt {2} C\right )} \log \left ({\left | -\sqrt {-a} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a} \right |}\right )}{\sqrt {-a} a \mathrm {sgn}\left (\cos \left (d x + c\right )\right )} - \frac {{\left ({\left ({\left ({\left (\frac {105 \, {\left (\sqrt {2} B a^{5} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) - \sqrt {2} C a^{5} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}}{a^{3}} - \frac {4 \, {\left (693 \, \sqrt {2} B a^{5} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) - 877 \, \sqrt {2} C a^{5} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )\right )}}{a^{3}}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + \frac {14 \, {\left (453 \, \sqrt {2} B a^{5} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) - 517 \, \sqrt {2} C a^{5} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )\right )}}{a^{3}}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - \frac {140 \, {\left (39 \, \sqrt {2} B a^{5} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) - 47 \, \sqrt {2} C a^{5} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )\right )}}{a^{3}}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + \frac {1785 \, {\left (\sqrt {2} B a^{5} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) - \sqrt {2} C a^{5} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )\right )}}{a^{3}}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{{\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a\right )}^{3} \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}}}{420 \, d} \]
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Timed out. \[ \int \frac {\sec ^4(c+d x) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^{3/2}} \, dx=\int \frac {\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}}{{\cos \left (c+d\,x\right )}^4\,{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^{3/2}} \,d x \]
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