Integrand size = 25, antiderivative size = 458 \[ \int \frac {\sec ^{\frac {9}{2}}(c+d x)}{(a+b \sec (c+d x))^{5/2}} \, dx=\frac {\left (5 a^2-3 b^2\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right ) \sqrt {\sec (c+d x)}}{3 b^2 \left (a^2-b^2\right ) d \sqrt {a+b \sec (c+d x)}}-\frac {5 a \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right ) \sqrt {\sec (c+d x)}}{b^3 d \sqrt {a+b \sec (c+d x)}}-\frac {\left (15 a^4-26 a^2 b^2+3 b^4\right ) E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right ) \sqrt {a+b \sec (c+d x)}}{3 b^3 \left (a^2-b^2\right )^2 d \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \sqrt {\sec (c+d x)}}-\frac {2 a^2 \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^{3/2}}-\frac {2 a^2 \left (5 a^2-9 b^2\right ) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{3 b^2 \left (a^2-b^2\right )^2 d \sqrt {a+b \sec (c+d x)}}+\frac {\left (15 a^4-26 a^2 b^2+3 b^4\right ) \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{3 b^3 \left (a^2-b^2\right )^2 d} \]
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Time = 1.72 (sec) , antiderivative size = 458, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.560, Rules used = {3930, 4183, 4187, 4193, 3944, 2886, 2884, 4120, 3941, 2734, 2732, 3943, 2742, 2740} \[ \int \frac {\sec ^{\frac {9}{2}}(c+d x)}{(a+b \sec (c+d x))^{5/2}} \, dx=-\frac {2 a^2 \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{3/2}}-\frac {2 a^2 \left (5 a^2-9 b^2\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 b^2 d \left (a^2-b^2\right )^2 \sqrt {a+b \sec (c+d x)}}+\frac {\left (5 a^2-3 b^2\right ) \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{3 b^2 d \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}+\frac {\left (15 a^4-26 a^2 b^2+3 b^4\right ) \sin (c+d x) \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}{3 b^3 d \left (a^2-b^2\right )^2}-\frac {\left (15 a^4-26 a^2 b^2+3 b^4\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{3 b^3 d \left (a^2-b^2\right )^2 \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}-\frac {5 a \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{b^3 d \sqrt {a+b \sec (c+d x)}} \]
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Rule 2732
Rule 2734
Rule 2740
Rule 2742
Rule 2884
Rule 2886
Rule 3930
Rule 3941
Rule 3943
Rule 3944
Rule 4120
Rule 4183
Rule 4187
Rule 4193
Rubi steps \begin{align*} \text {integral}& = -\frac {2 a^2 \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^{3/2}}-\frac {2 \int \frac {\sec ^{\frac {3}{2}}(c+d x) \left (\frac {3 a^2}{2}-\frac {3}{2} a b \sec (c+d x)-\frac {1}{2} \left (5 a^2-3 b^2\right ) \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^{3/2}} \, dx}{3 b \left (a^2-b^2\right )} \\ & = -\frac {2 a^2 \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^{3/2}}-\frac {2 a^2 \left (5 a^2-9 b^2\right ) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{3 b^2 \left (a^2-b^2\right )^2 d \sqrt {a+b \sec (c+d x)}}+\frac {4 \int \frac {\sqrt {\sec (c+d x)} \left (-\frac {1}{4} a^2 \left (5 a^2-9 b^2\right )+\frac {1}{2} a b \left (a^2-3 b^2\right ) \sec (c+d x)+\frac {1}{4} \left (15 a^4-26 a^2 b^2+3 b^4\right ) \sec ^2(c+d x)\right )}{\sqrt {a+b \sec (c+d x)}} \, dx}{3 b^2 \left (a^2-b^2\right )^2} \\ & = -\frac {2 a^2 \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^{3/2}}-\frac {2 a^2 \left (5 a^2-9 b^2\right ) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{3 b^2 \left (a^2-b^2\right )^2 d \sqrt {a+b \sec (c+d x)}}+\frac {\left (15 a^4-26 a^2 b^2+3 b^4\right ) \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{3 b^3 \left (a^2-b^2\right )^2 d}+\frac {4 \int \frac {-\frac {1}{8} a \left (15 a^4-26 a^2 b^2+3 b^4\right )-\frac {1}{4} a^2 b \left (5 a^2-9 b^2\right ) \sec (c+d x)-\frac {15}{8} a \left (a^2-b^2\right )^2 \sec ^2(c+d x)}{\sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}} \, dx}{3 b^3 \left (a^2-b^2\right )^2} \\ & = -\frac {2 a^2 \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^{3/2}}-\frac {2 a^2 \left (5 a^2-9 b^2\right ) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{3 b^2 \left (a^2-b^2\right )^2 d \sqrt {a+b \sec (c+d x)}}+\frac {\left (15 a^4-26 a^2 b^2+3 b^4\right ) \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{3 b^3 \left (a^2-b^2\right )^2 d}-\frac {(5 a) \int \frac {\sec ^{\frac {3}{2}}(c+d x)}{\sqrt {a+b \sec (c+d x)}} \, dx}{2 b^3}+\frac {4 \int \frac {-\frac {1}{8} a \left (15 a^4-26 a^2 b^2+3 b^4\right )-\frac {1}{4} a^2 b \left (5 a^2-9 b^2\right ) \sec (c+d x)}{\sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}} \, dx}{3 b^3 \left (a^2-b^2\right )^2} \\ & = -\frac {2 a^2 \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^{3/2}}-\frac {2 a^2 \left (5 a^2-9 b^2\right ) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{3 b^2 \left (a^2-b^2\right )^2 d \sqrt {a+b \sec (c+d x)}}+\frac {\left (15 a^4-26 a^2 b^2+3 b^4\right ) \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{3 b^3 \left (a^2-b^2\right )^2 d}+\frac {\left (5 a^2-3 b^2\right ) \int \frac {\sqrt {\sec (c+d x)}}{\sqrt {a+b \sec (c+d x)}} \, dx}{6 b^2 \left (a^2-b^2\right )}-\frac {\left (15 a^4-26 a^2 b^2+3 b^4\right ) \int \frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {\sec (c+d x)}} \, dx}{6 b^3 \left (a^2-b^2\right )^2}-\frac {\left (5 a \sqrt {b+a \cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\sec (c+d x)}{\sqrt {b+a \cos (c+d x)}} \, dx}{2 b^3 \sqrt {a+b \sec (c+d x)}} \\ & = -\frac {2 a^2 \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^{3/2}}-\frac {2 a^2 \left (5 a^2-9 b^2\right ) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{3 b^2 \left (a^2-b^2\right )^2 d \sqrt {a+b \sec (c+d x)}}+\frac {\left (15 a^4-26 a^2 b^2+3 b^4\right ) \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{3 b^3 \left (a^2-b^2\right )^2 d}+\frac {\left (\left (5 a^2-3 b^2\right ) \sqrt {b+a \cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {b+a \cos (c+d x)}} \, dx}{6 b^2 \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}-\frac {\left (5 a \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \sqrt {\sec (c+d x)}\right ) \int \frac {\sec (c+d x)}{\sqrt {\frac {b}{a+b}+\frac {a \cos (c+d x)}{a+b}}} \, dx}{2 b^3 \sqrt {a+b \sec (c+d x)}}-\frac {\left (\left (15 a^4-26 a^2 b^2+3 b^4\right ) \sqrt {a+b \sec (c+d x)}\right ) \int \sqrt {b+a \cos (c+d x)} \, dx}{6 b^3 \left (a^2-b^2\right )^2 \sqrt {b+a \cos (c+d x)} \sqrt {\sec (c+d x)}} \\ & = -\frac {5 a \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right ) \sqrt {\sec (c+d x)}}{b^3 d \sqrt {a+b \sec (c+d x)}}-\frac {2 a^2 \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^{3/2}}-\frac {2 a^2 \left (5 a^2-9 b^2\right ) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{3 b^2 \left (a^2-b^2\right )^2 d \sqrt {a+b \sec (c+d x)}}+\frac {\left (15 a^4-26 a^2 b^2+3 b^4\right ) \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{3 b^3 \left (a^2-b^2\right )^2 d}+\frac {\left (\left (5 a^2-3 b^2\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\frac {b}{a+b}+\frac {a \cos (c+d x)}{a+b}}} \, dx}{6 b^2 \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}-\frac {\left (\left (15 a^4-26 a^2 b^2+3 b^4\right ) \sqrt {a+b \sec (c+d x)}\right ) \int \sqrt {\frac {b}{a+b}+\frac {a \cos (c+d x)}{a+b}} \, dx}{6 b^3 \left (a^2-b^2\right )^2 \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \sqrt {\sec (c+d x)}} \\ & = \frac {\left (5 a^2-3 b^2\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right ) \sqrt {\sec (c+d x)}}{3 b^2 \left (a^2-b^2\right ) d \sqrt {a+b \sec (c+d x)}}-\frac {5 a \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right ) \sqrt {\sec (c+d x)}}{b^3 d \sqrt {a+b \sec (c+d x)}}-\frac {\left (15 a^4-26 a^2 b^2+3 b^4\right ) E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right ) \sqrt {a+b \sec (c+d x)}}{3 b^3 \left (a^2-b^2\right )^2 d \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \sqrt {\sec (c+d x)}}-\frac {2 a^2 \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^{3/2}}-\frac {2 a^2 \left (5 a^2-9 b^2\right ) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{3 b^2 \left (a^2-b^2\right )^2 d \sqrt {a+b \sec (c+d x)}}+\frac {\left (15 a^4-26 a^2 b^2+3 b^4\right ) \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{3 b^3 \left (a^2-b^2\right )^2 d} \\ \end{align*}
Result contains complex when optimal does not.
Time = 5.78 (sec) , antiderivative size = 561, normalized size of antiderivative = 1.22 \[ \int \frac {\sec ^{\frac {9}{2}}(c+d x)}{(a+b \sec (c+d x))^{5/2}} \, dx=\frac {\sec ^{\frac {5}{2}}(c+d x) \left (-\frac {a (b+a \cos (c+d x))^{5/2} \left (\frac {8 a b \left (5 a^2-9 b^2\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{\sqrt {b+a \cos (c+d x)}}+\frac {2 \left (45 a^4-86 a^2 b^2+33 b^4\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{\sqrt {b+a \cos (c+d x)}}+\frac {2 i \left (15 a^4-26 a^2 b^2+3 b^4\right ) \sqrt {-\frac {a (-1+\cos (c+d x))}{a+b}} \sqrt {\frac {a (1+\cos (c+d x))}{a-b}} \csc (c+d x) \left (-2 b (a+b) E\left (i \text {arcsinh}\left (\sqrt {\frac {1}{a-b}} \sqrt {b+a \cos (c+d x)}\right )|\frac {-a+b}{a+b}\right )+a \left (2 b \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {1}{a-b}} \sqrt {b+a \cos (c+d x)}\right ),\frac {-a+b}{a+b}\right )+a \operatorname {EllipticPi}\left (1-\frac {a}{b},i \text {arcsinh}\left (\sqrt {\frac {1}{a-b}} \sqrt {b+a \cos (c+d x)}\right ),\frac {-a+b}{a+b}\right )\right )\right )}{a^2 \sqrt {\frac {1}{a-b}} b}\right )}{(a-b)^2 (a+b)^2}+\frac {2 (b+a \cos (c+d x)) \left (15 a^6-20 a^4 b^2-9 a^2 b^4+6 b^6+4 a b \left (10 a^4-17 a^2 b^2+3 b^4\right ) \cos (c+d x)+\left (15 a^6-26 a^4 b^2+3 a^2 b^4\right ) \cos (2 (c+d x))\right ) \tan (c+d x)}{\left (a^2-b^2\right )^2}\right )}{12 b^3 d (a+b \sec (c+d x))^{5/2}} \]
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Result contains complex when optimal does not.
Time = 11.48 (sec) , antiderivative size = 5987, normalized size of antiderivative = 13.07
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Timed out. \[ \int \frac {\sec ^{\frac {9}{2}}(c+d x)}{(a+b \sec (c+d x))^{5/2}} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {\sec ^{\frac {9}{2}}(c+d x)}{(a+b \sec (c+d x))^{5/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {\sec ^{\frac {9}{2}}(c+d x)}{(a+b \sec (c+d x))^{5/2}} \, dx=\int { \frac {\sec \left (d x + c\right )^{\frac {9}{2}}}{{\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]
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\[ \int \frac {\sec ^{\frac {9}{2}}(c+d x)}{(a+b \sec (c+d x))^{5/2}} \, dx=\int { \frac {\sec \left (d x + c\right )^{\frac {9}{2}}}{{\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {\sec ^{\frac {9}{2}}(c+d x)}{(a+b \sec (c+d x))^{5/2}} \, dx=\int \frac {{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{9/2}}{{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^{5/2}} \,d x \]
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