Integrand size = 23, antiderivative size = 287 \[ \int \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^4 \, dx=-\frac {2 \left (15 a^4+54 a^2 b^2+7 b^4\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{15 d}+\frac {8 a b \left (7 a^2+5 b^2\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{21 d}+\frac {2 \left (15 a^4+54 a^2 b^2+7 b^4\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{15 d}+\frac {8 a b \left (7 a^2+5 b^2\right ) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac {14 b^2 \left (7 a^2+b^2\right ) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac {44 a b^3 \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{63 d}+\frac {2 b^2 \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{9 d} \]
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Time = 0.48 (sec) , antiderivative size = 287, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {3927, 4161, 4132, 3853, 3856, 2720, 4131, 2719} \[ \int \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^4 \, dx=\frac {14 b^2 \left (7 a^2+b^2\right ) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{45 d}+\frac {8 a b \left (7 a^2+5 b^2\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{21 d}+\frac {8 a b \left (7 a^2+5 b^2\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{21 d}+\frac {2 \left (15 a^4+54 a^2 b^2+7 b^4\right ) \sin (c+d x) \sqrt {\sec (c+d x)}}{15 d}-\frac {2 \left (15 a^4+54 a^2 b^2+7 b^4\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {44 a b^3 \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x)}{63 d}+\frac {2 b^2 \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^2}{9 d} \]
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Rule 2719
Rule 2720
Rule 3853
Rule 3856
Rule 3927
Rule 4131
Rule 4132
Rule 4161
Rubi steps \begin{align*} \text {integral}& = \frac {2 b^2 \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{9 d}+\frac {2}{9} \int \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x)) \left (\frac {3}{2} a \left (3 a^2+b^2\right )+\frac {1}{2} b \left (27 a^2+7 b^2\right ) \sec (c+d x)+11 a b^2 \sec ^2(c+d x)\right ) \, dx \\ & = \frac {44 a b^3 \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{63 d}+\frac {2 b^2 \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{9 d}+\frac {4}{63} \int \sec ^{\frac {3}{2}}(c+d x) \left (\frac {21}{4} a^2 \left (3 a^2+b^2\right )+9 a b \left (7 a^2+5 b^2\right ) \sec (c+d x)+\frac {49}{4} b^2 \left (7 a^2+b^2\right ) \sec ^2(c+d x)\right ) \, dx \\ & = \frac {44 a b^3 \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{63 d}+\frac {2 b^2 \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{9 d}+\frac {4}{63} \int \sec ^{\frac {3}{2}}(c+d x) \left (\frac {21}{4} a^2 \left (3 a^2+b^2\right )+\frac {49}{4} b^2 \left (7 a^2+b^2\right ) \sec ^2(c+d x)\right ) \, dx+\frac {1}{7} \left (4 a b \left (7 a^2+5 b^2\right )\right ) \int \sec ^{\frac {5}{2}}(c+d x) \, dx \\ & = \frac {8 a b \left (7 a^2+5 b^2\right ) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac {14 b^2 \left (7 a^2+b^2\right ) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac {44 a b^3 \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{63 d}+\frac {2 b^2 \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{9 d}+\frac {1}{21} \left (4 a b \left (7 a^2+5 b^2\right )\right ) \int \sqrt {\sec (c+d x)} \, dx+\frac {1}{15} \left (15 a^4+54 a^2 b^2+7 b^4\right ) \int \sec ^{\frac {3}{2}}(c+d x) \, dx \\ & = \frac {2 \left (15 a^4+54 a^2 b^2+7 b^4\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{15 d}+\frac {8 a b \left (7 a^2+5 b^2\right ) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac {14 b^2 \left (7 a^2+b^2\right ) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac {44 a b^3 \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{63 d}+\frac {2 b^2 \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{9 d}+\frac {1}{15} \left (-15 a^4-54 a^2 b^2-7 b^4\right ) \int \frac {1}{\sqrt {\sec (c+d x)}} \, dx+\frac {1}{21} \left (4 a b \left (7 a^2+5 b^2\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx \\ & = \frac {8 a b \left (7 a^2+5 b^2\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{21 d}+\frac {2 \left (15 a^4+54 a^2 b^2+7 b^4\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{15 d}+\frac {8 a b \left (7 a^2+5 b^2\right ) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac {14 b^2 \left (7 a^2+b^2\right ) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac {44 a b^3 \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{63 d}+\frac {2 b^2 \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{9 d}+\frac {1}{15} \left (\left (-15 a^4-54 a^2 b^2-7 b^4\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx \\ & = -\frac {2 \left (15 a^4+54 a^2 b^2+7 b^4\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{15 d}+\frac {8 a b \left (7 a^2+5 b^2\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{21 d}+\frac {2 \left (15 a^4+54 a^2 b^2+7 b^4\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{15 d}+\frac {8 a b \left (7 a^2+5 b^2\right ) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac {14 b^2 \left (7 a^2+b^2\right ) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac {44 a b^3 \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{63 d}+\frac {2 b^2 \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{9 d} \\ \end{align*}
Time = 2.61 (sec) , antiderivative size = 256, normalized size of antiderivative = 0.89 \[ \int \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^4 \, dx=-\frac {2 (a+b \sec (c+d x))^4 \left (21 \left (15 a^4+54 a^2 b^2+7 b^4\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )-60 a b \left (7 a^2+5 b^2\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )-315 a^4 \sin (c+d x)-1134 a^2 b^2 \sin (c+d x)-147 b^4 \sin (c+d x)-420 a^3 b \tan (c+d x)-300 a b^3 \tan (c+d x)-378 a^2 b^2 \sec (c+d x) \tan (c+d x)-49 b^4 \sec (c+d x) \tan (c+d x)-180 a b^3 \sec ^2(c+d x) \tan (c+d x)-35 b^4 \sec ^3(c+d x) \tan (c+d x)\right )}{315 d (b+a \cos (c+d x))^4 \sec ^{\frac {7}{2}}(c+d x)} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(1146\) vs. \(2(307)=614\).
Time = 64.92 (sec) , antiderivative size = 1147, normalized size of antiderivative = 4.00
method | result | size |
default | \(\text {Expression too large to display}\) | \(1147\) |
parts | \(\text {Expression too large to display}\) | \(1396\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.12 (sec) , antiderivative size = 318, normalized size of antiderivative = 1.11 \[ \int \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^4 \, dx=-\frac {60 \, \sqrt {2} {\left (7 i \, a^{3} b + 5 i \, a b^{3}\right )} \cos \left (d x + c\right )^{4} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 60 \, \sqrt {2} {\left (-7 i \, a^{3} b - 5 i \, a b^{3}\right )} \cos \left (d x + c\right )^{4} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 21 \, \sqrt {2} {\left (15 i \, a^{4} + 54 i \, a^{2} b^{2} + 7 i \, b^{4}\right )} \cos \left (d x + c\right )^{4} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 21 \, \sqrt {2} {\left (-15 i \, a^{4} - 54 i \, a^{2} b^{2} - 7 i \, b^{4}\right )} \cos \left (d x + c\right )^{4} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - \frac {2 \, {\left (180 \, a b^{3} \cos \left (d x + c\right ) + 21 \, {\left (15 \, a^{4} + 54 \, a^{2} b^{2} + 7 \, b^{4}\right )} \cos \left (d x + c\right )^{4} + 35 \, b^{4} + 60 \, {\left (7 \, a^{3} b + 5 \, a b^{3}\right )} \cos \left (d x + c\right )^{3} + 7 \, {\left (54 \, a^{2} b^{2} + 7 \, b^{4}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}}{315 \, d \cos \left (d x + c\right )^{4}} \]
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Timed out. \[ \int \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^4 \, dx=\text {Timed out} \]
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Timed out. \[ \int \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^4 \, dx=\text {Timed out} \]
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\[ \int \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^4 \, dx=\int { {\left (b \sec \left (d x + c\right ) + a\right )}^{4} \sec \left (d x + c\right )^{\frac {3}{2}} \,d x } \]
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Timed out. \[ \int \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^4 \, dx=\int {\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^4\,{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{3/2} \,d x \]
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