Integrand size = 23, antiderivative size = 234 \[ \int \frac {(a+b \cos (c+d x))^3}{\sec ^{\frac {3}{2}}(c+d x)} \, dx=\frac {2 b \left (27 a^2+7 b^2\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{15 d}+\frac {2 a \left (7 a^2+15 b^2\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{21 d}+\frac {40 a b^2 \sin (c+d x)}{63 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 b \left (27 a^2+7 b^2\right ) \sin (c+d x)}{45 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 a \left (7 a^2+15 b^2\right ) \sin (c+d x)}{21 d \sqrt {\sec (c+d x)}}+\frac {2 b^2 (b+a \sec (c+d x)) \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)} \]
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Time = 0.32 (sec) , antiderivative size = 234, 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 = {3317, 3926, 4132, 3854, 3856, 2719, 4130, 2720} \[ \int \frac {(a+b \cos (c+d x))^3}{\sec ^{\frac {3}{2}}(c+d x)} \, dx=\frac {2 b \left (27 a^2+7 b^2\right ) \sin (c+d x)}{45 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 a \left (7 a^2+15 b^2\right ) \sin (c+d x)}{21 d \sqrt {\sec (c+d x)}}+\frac {2 a \left (7 a^2+15 b^2\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{21 d}+\frac {2 b \left (27 a^2+7 b^2\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {2 b^2 \sin (c+d x) (a \sec (c+d x)+b)}{9 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {40 a b^2 \sin (c+d x)}{63 d \sec ^{\frac {5}{2}}(c+d x)} \]
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Rule 2719
Rule 2720
Rule 3317
Rule 3854
Rule 3856
Rule 3926
Rule 4130
Rule 4132
Rubi steps \begin{align*} \text {integral}& = \int \frac {(b+a \sec (c+d x))^3}{\sec ^{\frac {9}{2}}(c+d x)} \, dx \\ & = \frac {2 b^2 (b+a \sec (c+d x)) \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2}{9} \int \frac {10 a b^2+\frac {1}{2} b \left (27 a^2+7 b^2\right ) \sec (c+d x)+\frac {1}{2} a \left (9 a^2+5 b^2\right ) \sec ^2(c+d x)}{\sec ^{\frac {7}{2}}(c+d x)} \, dx \\ & = \frac {2 b^2 (b+a \sec (c+d x)) \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2}{9} \int \frac {10 a b^2+\frac {1}{2} a \left (9 a^2+5 b^2\right ) \sec ^2(c+d x)}{\sec ^{\frac {7}{2}}(c+d x)} \, dx+\frac {1}{9} \left (b \left (27 a^2+7 b^2\right )\right ) \int \frac {1}{\sec ^{\frac {5}{2}}(c+d x)} \, dx \\ & = \frac {40 a b^2 \sin (c+d x)}{63 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 b \left (27 a^2+7 b^2\right ) \sin (c+d x)}{45 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 b^2 (b+a \sec (c+d x)) \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {1}{15} \left (b \left (27 a^2+7 b^2\right )\right ) \int \frac {1}{\sqrt {\sec (c+d x)}} \, dx+\frac {1}{7} \left (a \left (7 a^2+15 b^2\right )\right ) \int \frac {1}{\sec ^{\frac {3}{2}}(c+d x)} \, dx \\ & = \frac {40 a b^2 \sin (c+d x)}{63 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 b \left (27 a^2+7 b^2\right ) \sin (c+d x)}{45 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 a \left (7 a^2+15 b^2\right ) \sin (c+d x)}{21 d \sqrt {\sec (c+d x)}}+\frac {2 b^2 (b+a \sec (c+d x)) \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {1}{21} \left (a \left (7 a^2+15 b^2\right )\right ) \int \sqrt {\sec (c+d x)} \, dx+\frac {1}{15} \left (b \left (27 a^2+7 b^2\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx \\ & = \frac {2 b \left (27 a^2+7 b^2\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 {40 a b^2 \sin (c+d x)}{63 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 b \left (27 a^2+7 b^2\right ) \sin (c+d x)}{45 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 a \left (7 a^2+15 b^2\right ) \sin (c+d x)}{21 d \sqrt {\sec (c+d x)}}+\frac {2 b^2 (b+a \sec (c+d x)) \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {1}{21} \left (a \left (7 a^2+15 b^2\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx \\ & = \frac {2 b \left (27 a^2+7 b^2\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 {2 a \left (7 a^2+15 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 {40 a b^2 \sin (c+d x)}{63 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 b \left (27 a^2+7 b^2\right ) \sin (c+d x)}{45 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 a \left (7 a^2+15 b^2\right ) \sin (c+d x)}{21 d \sqrt {\sec (c+d x)}}+\frac {2 b^2 (b+a \sec (c+d x)) \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)} \\ \end{align*}
Time = 1.53 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.68 \[ \int \frac {(a+b \cos (c+d x))^3}{\sec ^{\frac {3}{2}}(c+d x)} \, dx=\frac {\sqrt {\sec (c+d x)} \left (168 b \left (27 a^2+7 b^2\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+120 a \left (7 a^2+15 b^2\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )+\left (7 b \left (108 a^2+43 b^2\right ) \cos (c+d x)+5 \left (84 a^3+234 a b^2+54 a b^2 \cos (2 (c+d x))+7 b^3 \cos (3 (c+d x))\right )\right ) \sin (2 (c+d x))\right )}{1260 d} \]
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Time = 13.29 (sec) , antiderivative size = 470, normalized size of antiderivative = 2.01
method | result | size |
default | \(-\frac {2 \sqrt {\left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \left (-1120 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{3}+\left (2160 a \,b^{2}+2240 b^{3}\right ) \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (-1512 a^{2} b -3240 a \,b^{2}-2072 b^{3}\right ) \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (420 a^{3}+1512 a^{2} b +2520 a \,b^{2}+952 b^{3}\right ) \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (-210 a^{3}-378 a^{2} b -720 a \,b^{2}-168 b^{3}\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+105 a^{3} \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+225 a \,b^{2} \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-567 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, E\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) a^{2} b -147 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, E\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) b^{3}\right )}{315 \sqrt {-2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, d}\) | \(470\) |
parts | \(\text {Expression too large to display}\) | \(804\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.12 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.02 \[ \int \frac {(a+b \cos (c+d x))^3}{\sec ^{\frac {3}{2}}(c+d x)} \, dx=-\frac {15 \, \sqrt {2} {\left (7 i \, a^{3} + 15 i \, a b^{2}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 15 \, \sqrt {2} {\left (-7 i \, a^{3} - 15 i \, a b^{2}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 21 \, \sqrt {2} {\left (-27 i \, a^{2} b - 7 i \, b^{3}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 21 \, \sqrt {2} {\left (27 i \, a^{2} b + 7 i \, b^{3}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - \frac {2 \, {\left (35 \, b^{3} \cos \left (d x + c\right )^{4} + 135 \, a b^{2} \cos \left (d x + c\right )^{3} + 7 \, {\left (27 \, a^{2} b + 7 \, b^{3}\right )} \cos \left (d x + c\right )^{2} + 15 \, {\left (7 \, a^{3} + 15 \, a b^{2}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}}{315 \, d} \]
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\[ \int \frac {(a+b \cos (c+d x))^3}{\sec ^{\frac {3}{2}}(c+d x)} \, dx=\int \frac {\left (a + b \cos {\left (c + d x \right )}\right )^{3}}{\sec ^{\frac {3}{2}}{\left (c + d x \right )}}\, dx \]
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\[ \int \frac {(a+b \cos (c+d x))^3}{\sec ^{\frac {3}{2}}(c+d x)} \, dx=\int { \frac {{\left (b \cos \left (d x + c\right ) + a\right )}^{3}}{\sec \left (d x + c\right )^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {(a+b \cos (c+d x))^3}{\sec ^{\frac {3}{2}}(c+d x)} \, dx=\int { \frac {{\left (b \cos \left (d x + c\right ) + a\right )}^{3}}{\sec \left (d x + c\right )^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(a+b \cos (c+d x))^3}{\sec ^{\frac {3}{2}}(c+d x)} \, dx=\int \frac {{\left (a+b\,\cos \left (c+d\,x\right )\right )}^3}{{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{3/2}} \,d x \]
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