Integrand size = 35, antiderivative size = 472 \[ \int \frac {A+B \sec (c+d x)}{\sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}} \, dx=\frac {2 \left (a^4 A+16 a^2 A b^2-16 A b^4-9 a^3 b B+8 a b^3 B\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right ) \sqrt {\sec (c+d x)}}{3 a^4 \left (a^2-b^2\right ) d \sqrt {a+b \sec (c+d x)}}-\frac {2 \left (8 a^4 A b-28 a^2 A b^3+16 A b^5-3 a^5 B+15 a^3 b^2 B-8 a b^4 B\right ) E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right ) \sqrt {a+b \sec (c+d x)}}{3 a^4 \left (a^2-b^2\right )^2 d \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \sqrt {\sec (c+d x)}}+\frac {2 b (A b-a B) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^{3/2}}+\frac {2 b \left (10 a^2 A b-6 A b^3-7 a^3 B+3 a b^2 B\right ) \sin (c+d x)}{3 a^2 \left (a^2-b^2\right )^2 d \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (a^4 A-13 a^2 A b^2+8 A b^4+8 a^3 b B-4 a b^3 B\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{3 a^3 \left (a^2-b^2\right )^2 d \sqrt {\sec (c+d x)}} \]
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Time = 1.55 (sec) , antiderivative size = 472, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {4115, 4185, 4189, 4120, 3941, 2734, 2732, 3943, 2742, 2740} \[ \int \frac {A+B \sec (c+d x)}{\sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}} \, dx=\frac {2 b (A b-a B) \sin (c+d x)}{3 a d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^{3/2}}+\frac {2 b \left (-7 a^3 B+10 a^2 A b+3 a b^2 B-6 A b^3\right ) \sin (c+d x)}{3 a^2 d \left (a^2-b^2\right )^2 \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (a^4 A+8 a^3 b B-13 a^2 A b^2-4 a b^3 B+8 A b^4\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{3 a^3 d \left (a^2-b^2\right )^2 \sqrt {\sec (c+d x)}}+\frac {2 \left (a^4 A-9 a^3 b B+16 a^2 A b^2+8 a b^3 B-16 A b^4\right ) \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{3 a^4 d \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}-\frac {2 \left (-3 a^5 B+8 a^4 A b+15 a^3 b^2 B-28 a^2 A b^3-8 a b^4 B+16 A b^5\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{3 a^4 d \left (a^2-b^2\right )^2 \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}} \]
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Rule 2732
Rule 2734
Rule 2740
Rule 2742
Rule 3941
Rule 3943
Rule 4115
Rule 4120
Rule 4185
Rule 4189
Rubi steps \begin{align*} \text {integral}& = \frac {2 b (A b-a B) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^{3/2}}-\frac {2 \int \frac {-\frac {3}{2} \left (a^2 A-2 A b^2+a b B\right )+\frac {3}{2} a (A b-a B) \sec (c+d x)-2 b (A b-a B) \sec ^2(c+d x)}{\sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}} \, dx}{3 a \left (a^2-b^2\right )} \\ & = \frac {2 b (A b-a B) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^{3/2}}+\frac {2 b \left (10 a^2 A b-6 A b^3-7 a^3 B+3 a b^2 B\right ) \sin (c+d x)}{3 a^2 \left (a^2-b^2\right )^2 d \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}+\frac {4 \int \frac {\frac {3}{4} \left (a^4 A-13 a^2 A b^2+8 A b^4+8 a^3 b B-4 a b^3 B\right )-\frac {1}{4} a \left (6 a^2 A b-2 A b^3-3 a^3 B-a b^2 B\right ) \sec (c+d x)+\frac {1}{2} b \left (10 a^2 A b-6 A b^3-7 a^3 B+3 a b^2 B\right ) \sec ^2(c+d x)}{\sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}} \, dx}{3 a^2 \left (a^2-b^2\right )^2} \\ & = \frac {2 b (A b-a B) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^{3/2}}+\frac {2 b \left (10 a^2 A b-6 A b^3-7 a^3 B+3 a b^2 B\right ) \sin (c+d x)}{3 a^2 \left (a^2-b^2\right )^2 d \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (a^4 A-13 a^2 A b^2+8 A b^4+8 a^3 b B-4 a b^3 B\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{3 a^3 \left (a^2-b^2\right )^2 d \sqrt {\sec (c+d x)}}-\frac {8 \int \frac {\frac {3}{8} \left (8 a^4 A b-28 a^2 A b^3+16 A b^5-3 a^5 B+15 a^3 b^2 B-8 a b^4 B\right )-\frac {3}{8} a \left (a^4 A+7 a^2 A b^2-4 A b^4-6 a^3 b B+2 a b^3 B\right ) \sec (c+d x)}{\sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}} \, dx}{9 a^3 \left (a^2-b^2\right )^2} \\ & = \frac {2 b (A b-a B) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^{3/2}}+\frac {2 b \left (10 a^2 A b-6 A b^3-7 a^3 B+3 a b^2 B\right ) \sin (c+d x)}{3 a^2 \left (a^2-b^2\right )^2 d \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (a^4 A-13 a^2 A b^2+8 A b^4+8 a^3 b B-4 a b^3 B\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{3 a^3 \left (a^2-b^2\right )^2 d \sqrt {\sec (c+d x)}}+\frac {\left (a^4 A+16 a^2 A b^2-16 A b^4-9 a^3 b B+8 a b^3 B\right ) \int \frac {\sqrt {\sec (c+d x)}}{\sqrt {a+b \sec (c+d x)}} \, dx}{3 a^4 \left (a^2-b^2\right )}-\frac {\left (8 a^4 A b-28 a^2 A b^3+16 A b^5-3 a^5 B+15 a^3 b^2 B-8 a b^4 B\right ) \int \frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {\sec (c+d x)}} \, dx}{3 a^4 \left (a^2-b^2\right )^2} \\ & = \frac {2 b (A b-a B) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^{3/2}}+\frac {2 b \left (10 a^2 A b-6 A b^3-7 a^3 B+3 a b^2 B\right ) \sin (c+d x)}{3 a^2 \left (a^2-b^2\right )^2 d \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (a^4 A-13 a^2 A b^2+8 A b^4+8 a^3 b B-4 a b^3 B\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{3 a^3 \left (a^2-b^2\right )^2 d \sqrt {\sec (c+d x)}}+\frac {\left (\left (a^4 A+16 a^2 A b^2-16 A b^4-9 a^3 b B+8 a b^3 B\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}{3 a^4 \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}-\frac {\left (\left (8 a^4 A b-28 a^2 A b^3+16 A b^5-3 a^5 B+15 a^3 b^2 B-8 a b^4 B\right ) \sqrt {a+b \sec (c+d x)}\right ) \int \sqrt {b+a \cos (c+d x)} \, dx}{3 a^4 \left (a^2-b^2\right )^2 \sqrt {b+a \cos (c+d x)} \sqrt {\sec (c+d x)}} \\ & = \frac {2 b (A b-a B) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^{3/2}}+\frac {2 b \left (10 a^2 A b-6 A b^3-7 a^3 B+3 a b^2 B\right ) \sin (c+d x)}{3 a^2 \left (a^2-b^2\right )^2 d \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (a^4 A-13 a^2 A b^2+8 A b^4+8 a^3 b B-4 a b^3 B\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{3 a^3 \left (a^2-b^2\right )^2 d \sqrt {\sec (c+d x)}}+\frac {\left (\left (a^4 A+16 a^2 A b^2-16 A b^4-9 a^3 b B+8 a b^3 B\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}{3 a^4 \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}-\frac {\left (\left (8 a^4 A b-28 a^2 A b^3+16 A b^5-3 a^5 B+15 a^3 b^2 B-8 a b^4 B\right ) \sqrt {a+b \sec (c+d x)}\right ) \int \sqrt {\frac {b}{a+b}+\frac {a \cos (c+d x)}{a+b}} \, dx}{3 a^4 \left (a^2-b^2\right )^2 \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \sqrt {\sec (c+d x)}} \\ & = \frac {2 \left (a^4 A+16 a^2 A b^2-16 A b^4-9 a^3 b B+8 a b^3 B\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 a^4 \left (a^2-b^2\right ) d \sqrt {a+b \sec (c+d x)}}-\frac {2 \left (8 a^4 A b-28 a^2 A b^3+16 A b^5-3 a^5 B+15 a^3 b^2 B-8 a b^4 B\right ) E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right ) \sqrt {a+b \sec (c+d x)}}{3 a^4 \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 b (A b-a B) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^{3/2}}+\frac {2 b \left (10 a^2 A b-6 A b^3-7 a^3 B+3 a b^2 B\right ) \sin (c+d x)}{3 a^2 \left (a^2-b^2\right )^2 d \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (a^4 A-13 a^2 A b^2+8 A b^4+8 a^3 b B-4 a b^3 B\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{3 a^3 \left (a^2-b^2\right )^2 d \sqrt {\sec (c+d x)}} \\ \end{align*}
Time = 4.53 (sec) , antiderivative size = 353, normalized size of antiderivative = 0.75 \[ \int \frac {A+B \sec (c+d x)}{\sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}} \, dx=\frac {2 (b+a \cos (c+d x)) \sec ^{\frac {5}{2}}(c+d x) \left (\frac {\left (\frac {b+a \cos (c+d x)}{a+b}\right )^{3/2} \left (a^2 \left (a^4 A+7 a^2 A b^2-4 A b^4-6 a^3 b B+2 a b^3 B\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )+\left (-8 a^4 A b+28 a^2 A b^3-16 A b^5+3 a^5 B-15 a^3 b^2 B+8 a b^4 B\right ) \left ((a+b) E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )-b \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )\right )\right )}{(a-b)^2 (a+b)}+\frac {a \left (a^6 A-25 a^2 A b^4+16 A b^6+16 a^3 b^3 B-8 a b^5 B+2 a b \left (2 a^4 A-16 a^2 A b^2+10 A b^4+9 a^3 b B-5 a b^3 B\right ) \cos (c+d x)+A \left (a^3-a b^2\right )^2 \cos (2 (c+d x))\right ) \sin (c+d x)}{2 \left (a^2-b^2\right )^2}\right )}{3 a^4 d (a+b \sec (c+d x))^{5/2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(8329\) vs. \(2(496)=992\).
Time = 22.82 (sec) , antiderivative size = 8330, normalized size of antiderivative = 17.65
method | result | size |
default | \(\text {Expression too large to display}\) | \(8330\) |
parts | \(\text {Expression too large to display}\) | \(8568\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.25 (sec) , antiderivative size = 1343, normalized size of antiderivative = 2.85 \[ \int \frac {A+B \sec (c+d x)}{\sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {A+B \sec (c+d x)}{\sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {A+B \sec (c+d x)}{\sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}} \, dx=\int { \frac {B \sec \left (d x + c\right ) + A}{{\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \sec \left (d x + c\right )^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {A+B \sec (c+d x)}{\sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}} \, dx=\int { \frac {B \sec \left (d x + c\right ) + A}{{\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \sec \left (d x + c\right )^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {A+B \sec (c+d x)}{\sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}} \, dx=\int \frac {A+\frac {B}{\cos \left (c+d\,x\right )}}{{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^{5/2}\,{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{3/2}} \,d x \]
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