Integrand size = 35, antiderivative size = 422 \[ \int \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^{5/2} (A+B \sec (c+d x)) \, dx=\frac {\left (48 a^3 A+66 a A b^2+59 a^2 b B+16 b^3 B\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right ) \sqrt {\sec (c+d x)}}{24 d \sqrt {a+b \sec (c+d x)}}+\frac {\left (30 a^2 A b+8 A b^3+5 a^3 B+20 a b^2 B\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right ) \sqrt {\sec (c+d x)}}{8 d \sqrt {a+b \sec (c+d x)}}-\frac {\left (54 a A b+33 a^2 B+16 b^2 B\right ) E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right ) \sqrt {a+b \sec (c+d x)}}{24 d \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \sqrt {\sec (c+d x)}}+\frac {\left (54 a A b+33 a^2 B+16 b^2 B\right ) \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{24 d}+\frac {b (2 A b+3 a B) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{4 d}+\frac {b B \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{3 d} \]
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Time = 2.03 (sec) , antiderivative size = 422, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {4111, 4181, 4187, 4193, 3944, 2886, 2884, 4120, 3941, 2734, 2732, 3943, 2742, 2740} \[ \int \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^{5/2} (A+B \sec (c+d x)) \, dx=\frac {\left (33 a^2 B+54 a A b+16 b^2 B\right ) \sin (c+d x) \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}{24 d}-\frac {\left (33 a^2 B+54 a A b+16 b^2 B\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{24 d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}+\frac {\left (48 a^3 A+59 a^2 b B+66 a A b^2+16 b^3 B\right ) \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{24 d \sqrt {a+b \sec (c+d x)}}+\frac {\left (5 a^3 B+30 a^2 A b+20 a b^2 B+8 A b^3\right ) \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{8 d \sqrt {a+b \sec (c+d x)}}+\frac {b (3 a B+2 A b) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{4 d}+\frac {b B \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}{3 d} \]
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Rule 2732
Rule 2734
Rule 2740
Rule 2742
Rule 2884
Rule 2886
Rule 3941
Rule 3943
Rule 3944
Rule 4111
Rule 4120
Rule 4181
Rule 4187
Rule 4193
Rubi steps \begin{align*} \text {integral}& = \frac {b B \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{3 d}+\frac {1}{3} \int \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)} \left (\frac {1}{2} a (6 a A+b B)+\left (6 a A b+3 a^2 B+2 b^2 B\right ) \sec (c+d x)+\frac {3}{2} b (2 A b+3 a B) \sec ^2(c+d x)\right ) \, dx \\ & = \frac {b (2 A b+3 a B) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{4 d}+\frac {b B \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{3 d}+\frac {1}{6} \int \frac {\sqrt {\sec (c+d x)} \left (\frac {1}{4} a \left (24 a^2 A+6 A b^2+13 a b B\right )+\frac {1}{2} \left (36 a^2 A b+6 A b^3+12 a^3 B+19 a b^2 B\right ) \sec (c+d x)+\frac {1}{4} b \left (54 a A b+33 a^2 B+16 b^2 B\right ) \sec ^2(c+d x)\right )}{\sqrt {a+b \sec (c+d x)}} \, dx \\ & = \frac {\left (54 a A b+33 a^2 B+16 b^2 B\right ) \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{24 d}+\frac {b (2 A b+3 a B) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{4 d}+\frac {b B \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{3 d}+\frac {\int \frac {-\frac {1}{8} a b \left (54 a A b+33 a^2 B+16 b^2 B\right )+\frac {1}{4} a b \left (24 a^2 A+6 A b^2+13 a b B\right ) \sec (c+d x)+\frac {3}{8} b \left (30 a^2 A b+8 A b^3+5 a^3 B+20 a b^2 B\right ) \sec ^2(c+d x)}{\sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}} \, dx}{6 b} \\ & = \frac {\left (54 a A b+33 a^2 B+16 b^2 B\right ) \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{24 d}+\frac {b (2 A b+3 a B) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{4 d}+\frac {b B \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{3 d}+\frac {\int \frac {-\frac {1}{8} a b \left (54 a A b+33 a^2 B+16 b^2 B\right )+\frac {1}{4} a b \left (24 a^2 A+6 A b^2+13 a b B\right ) \sec (c+d x)}{\sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}} \, dx}{6 b}+\frac {1}{16} \left (30 a^2 A b+8 A b^3+5 a^3 B+20 a b^2 B\right ) \int \frac {\sec ^{\frac {3}{2}}(c+d x)}{\sqrt {a+b \sec (c+d x)}} \, dx \\ & = \frac {\left (54 a A b+33 a^2 B+16 b^2 B\right ) \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{24 d}+\frac {b (2 A b+3 a B) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{4 d}+\frac {b B \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{3 d}+\frac {1}{48} \left (-54 a A b-33 a^2 B-16 b^2 B\right ) \int \frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {\sec (c+d x)}} \, dx+\frac {1}{48} \left (48 a^3 A+66 a A b^2+59 a^2 b B+16 b^3 B\right ) \int \frac {\sqrt {\sec (c+d x)}}{\sqrt {a+b \sec (c+d x)}} \, dx+\frac {\left (\left (30 a^2 A b+8 A b^3+5 a^3 B+20 a b^2 B\right ) \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}{16 \sqrt {a+b \sec (c+d x)}} \\ & = \frac {\left (54 a A b+33 a^2 B+16 b^2 B\right ) \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{24 d}+\frac {b (2 A b+3 a B) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{4 d}+\frac {b B \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{3 d}+\frac {\left (\left (48 a^3 A+66 a A b^2+59 a^2 b B+16 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}{48 \sqrt {a+b \sec (c+d x)}}+\frac {\left (\left (30 a^2 A b+8 A b^3+5 a^3 B+20 a b^2 B\right ) \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}{16 \sqrt {a+b \sec (c+d x)}}+\frac {\left (\left (-54 a A b-33 a^2 B-16 b^2 B\right ) \sqrt {a+b \sec (c+d x)}\right ) \int \sqrt {b+a \cos (c+d x)} \, dx}{48 \sqrt {b+a \cos (c+d x)} \sqrt {\sec (c+d x)}} \\ & = \frac {\left (30 a^2 A b+8 A b^3+5 a^3 B+20 a b^2 B\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 {\sec (c+d x)}}{8 d \sqrt {a+b \sec (c+d x)}}+\frac {\left (54 a A b+33 a^2 B+16 b^2 B\right ) \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{24 d}+\frac {b (2 A b+3 a B) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{4 d}+\frac {b B \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{3 d}+\frac {\left (\left (48 a^3 A+66 a A b^2+59 a^2 b B+16 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}{48 \sqrt {a+b \sec (c+d x)}}+\frac {\left (\left (-54 a A b-33 a^2 B-16 b^2 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}{48 \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \sqrt {\sec (c+d x)}} \\ & = \frac {\left (48 a^3 A+66 a A b^2+59 a^2 b B+16 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)}}{24 d \sqrt {a+b \sec (c+d x)}}+\frac {\left (30 a^2 A b+8 A b^3+5 a^3 B+20 a b^2 B\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 {\sec (c+d x)}}{8 d \sqrt {a+b \sec (c+d x)}}-\frac {\left (54 a A b+33 a^2 B+16 b^2 B\right ) E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right ) \sqrt {a+b \sec (c+d x)}}{24 d \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \sqrt {\sec (c+d x)}}+\frac {\left (54 a A b+33 a^2 B+16 b^2 B\right ) \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{24 d}+\frac {b (2 A b+3 a B) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{4 d}+\frac {b B \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{3 d} \\ \end{align*}
Result contains complex when optimal does not.
Time = 8.12 (sec) , antiderivative size = 678, normalized size of antiderivative = 1.61 \[ \int \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^{5/2} (A+B \sec (c+d x)) \, dx=\frac {(a+b \sec (c+d x))^{5/2} \left (\frac {2 \left (96 a^3 A+24 a A b^2+52 a^2 b B\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{\sqrt {b+a \cos (c+d x)}}+\frac {2 \left (126 a^2 A b+48 A b^3-3 a^3 B+104 a b^2 B\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{\sqrt {b+a \cos (c+d x)}}+\frac {2 i \left (-54 a^2 A b-33 a^3 B-16 a b^2 B\right ) \sqrt {\frac {a-a \cos (c+d x)}{a+b}} \sqrt {\frac {a+a \cos (c+d x)}{a-b}} \cos (2 (c+d x)) \left (-2 b (a+b) E\left (i \text {arcsinh}\left (\sqrt {\frac {1}{a-b}} \sqrt {b+a \cos (c+d x)}\right )|\frac {-a+b}{a+b}\right )+a \left (2 b \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {1}{a-b}} \sqrt {b+a \cos (c+d x)}\right ),\frac {-a+b}{a+b}\right )+a \operatorname {EllipticPi}\left (1-\frac {a}{b},i \text {arcsinh}\left (\sqrt {\frac {1}{a-b}} \sqrt {b+a \cos (c+d x)}\right ),\frac {-a+b}{a+b}\right )\right )\right ) \sin (c+d x)}{\sqrt {\frac {1}{a-b}} b \sqrt {1-\cos ^2(c+d x)} \sqrt {\frac {a^2-a^2 \cos ^2(c+d x)}{a^2}} \left (-a^2+2 b^2-4 b (b+a \cos (c+d x))+2 (b+a \cos (c+d x))^2\right )}\right )}{96 d (b+a \cos (c+d x))^{5/2} \sec ^{\frac {5}{2}}(c+d x)}+\frac {(a+b \sec (c+d x))^{5/2} \left (\frac {1}{12} \sec ^2(c+d x) \left (6 A b^2 \sin (c+d x)+13 a b B \sin (c+d x)\right )+\frac {1}{24} \sec (c+d x) \left (54 a A b \sin (c+d x)+33 a^2 B \sin (c+d x)+16 b^2 B \sin (c+d x)\right )+\frac {1}{3} b^2 B \sec ^2(c+d x) \tan (c+d x)\right )}{d (b+a \cos (c+d x))^2 \sec ^{\frac {5}{2}}(c+d x)} \]
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Result contains complex when optimal does not.
Time = 29.30 (sec) , antiderivative size = 5606, normalized size of antiderivative = 13.28
method | result | size |
default | \(\text {Expression too large to display}\) | \(5606\) |
parts | \(\text {Expression too large to display}\) | \(5641\) |
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Timed out. \[ \int \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^{5/2} (A+B \sec (c+d x)) \, dx=\text {Timed out} \]
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Timed out. \[ \int \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^{5/2} (A+B \sec (c+d x)) \, dx=\text {Timed out} \]
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\[ \int \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^{5/2} (A+B \sec (c+d x)) \, dx=\int { {\left (B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \sqrt {\sec \left (d x + c\right )} \,d x } \]
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\[ \int \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^{5/2} (A+B \sec (c+d x)) \, dx=\int { {\left (B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \sqrt {\sec \left (d x + c\right )} \,d x } \]
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Timed out. \[ \int \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^{5/2} (A+B \sec (c+d x)) \, dx=\int \left (A+\frac {B}{\cos \left (c+d\,x\right )}\right )\,{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^{5/2}\,\sqrt {\frac {1}{\cos \left (c+d\,x\right )}} \,d x \]
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