Integrand size = 23, antiderivative size = 124 \[ \int \frac {(a+b \cos (c+d x))^3}{\cos ^{\frac {3}{2}}(c+d x)} \, dx=-\frac {2 a \left (a^2-3 b^2\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}+\frac {2 b \left (9 a^2+b^2\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 d}-\frac {2 b \left (3 a^2-b^2\right ) \sqrt {\cos (c+d x)} \sin (c+d x)}{3 d}+\frac {2 a^2 (a+b \cos (c+d x)) \sin (c+d x)}{d \sqrt {\cos (c+d x)}} \]
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Time = 0.22 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {2871, 3102, 2827, 2720, 2719} \[ \int \frac {(a+b \cos (c+d x))^3}{\cos ^{\frac {3}{2}}(c+d x)} \, dx=\frac {2 b \left (9 a^2+b^2\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 d}-\frac {2 a \left (a^2-3 b^2\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}-\frac {2 b \left (3 a^2-b^2\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}+\frac {2 a^2 \sin (c+d x) (a+b \cos (c+d x))}{d \sqrt {\cos (c+d x)}} \]
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Rule 2719
Rule 2720
Rule 2827
Rule 2871
Rule 3102
Rubi steps \begin{align*} \text {integral}& = \frac {2 a^2 (a+b \cos (c+d x)) \sin (c+d x)}{d \sqrt {\cos (c+d x)}}+2 \int \frac {2 a^2 b-\frac {1}{2} a \left (a^2-3 b^2\right ) \cos (c+d x)-\frac {1}{2} b \left (3 a^2-b^2\right ) \cos ^2(c+d x)}{\sqrt {\cos (c+d x)}} \, dx \\ & = -\frac {2 b \left (3 a^2-b^2\right ) \sqrt {\cos (c+d x)} \sin (c+d x)}{3 d}+\frac {2 a^2 (a+b \cos (c+d x)) \sin (c+d x)}{d \sqrt {\cos (c+d x)}}+\frac {4}{3} \int \frac {\frac {1}{4} b \left (9 a^2+b^2\right )-\frac {3}{4} a \left (a^2-3 b^2\right ) \cos (c+d x)}{\sqrt {\cos (c+d x)}} \, dx \\ & = -\frac {2 b \left (3 a^2-b^2\right ) \sqrt {\cos (c+d x)} \sin (c+d x)}{3 d}+\frac {2 a^2 (a+b \cos (c+d x)) \sin (c+d x)}{d \sqrt {\cos (c+d x)}}-\left (a \left (a^2-3 b^2\right )\right ) \int \sqrt {\cos (c+d x)} \, dx+\frac {1}{3} \left (b \left (9 a^2+b^2\right )\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx \\ & = -\frac {2 a \left (a^2-3 b^2\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}+\frac {2 b \left (9 a^2+b^2\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 d}-\frac {2 b \left (3 a^2-b^2\right ) \sqrt {\cos (c+d x)} \sin (c+d x)}{3 d}+\frac {2 a^2 (a+b \cos (c+d x)) \sin (c+d x)}{d \sqrt {\cos (c+d x)}} \\ \end{align*}
Time = 0.60 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.69 \[ \int \frac {(a+b \cos (c+d x))^3}{\cos ^{\frac {3}{2}}(c+d x)} \, dx=\frac {2 \left (-3 \left (a^3-3 a b^2\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+\left (9 a^2 b+b^3\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )+\frac {\left (3 a^3+b^3 \cos (c+d x)\right ) \sin (c+d x)}{\sqrt {\cos (c+d x)}}\right )}{3 d} \]
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Time = 8.83 (sec) , antiderivative size = 303, normalized size of antiderivative = 2.44
method | result | size |
default | \(-\frac {2 \left (4 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{3}-6 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{3}-2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{3}+9 a^{2} b \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+b^{3} \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+3 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, E\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) a^{3}-9 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, E\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) a \,b^{2}\right )}{3 \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, d}\) | \(303\) |
parts | \(-\frac {2 a^{3} \left (-2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \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 )}\, \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\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}\, \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 )}\, E\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )\right )}{\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}-\frac {2 b^{3} \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 (4 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\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 )\right )}{3 \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}+\frac {6 a^{2} b \,\operatorname {am}^{-1}\left (\frac {d x}{2}+\frac {c}{2}| \sqrt {2}\right )}{d}+\frac {6 a \,b^{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 )}\, \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \left (\cos ^{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 )}{\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}\) | \(524\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.73 \[ \int \frac {(a+b \cos (c+d x))^3}{\cos ^{\frac {3}{2}}(c+d x)} \, dx=\frac {\sqrt {2} {\left (-9 i \, a^{2} b - i \, b^{3}\right )} \cos \left (d x + c\right ) {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + \sqrt {2} {\left (9 i \, a^{2} b + i \, b^{3}\right )} \cos \left (d x + c\right ) {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) - 3 \, \sqrt {2} {\left (i \, a^{3} - 3 i \, a b^{2}\right )} \cos \left (d x + c\right ) {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) - 3 \, \sqrt {2} {\left (-i \, a^{3} + 3 i \, a b^{2}\right )} \cos \left (d x + c\right ) {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) + 2 \, {\left (b^{3} \cos \left (d x + c\right ) + 3 \, a^{3}\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )}{3 \, d \cos \left (d x + c\right )} \]
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Timed out. \[ \int \frac {(a+b \cos (c+d x))^3}{\cos ^{\frac {3}{2}}(c+d x)} \, dx=\text {Timed out} \]
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\[ \int \frac {(a+b \cos (c+d x))^3}{\cos ^{\frac {3}{2}}(c+d x)} \, dx=\int { \frac {{\left (b \cos \left (d x + c\right ) + a\right )}^{3}}{\cos \left (d x + c\right )^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {(a+b \cos (c+d x))^3}{\cos ^{\frac {3}{2}}(c+d x)} \, dx=\int { \frac {{\left (b \cos \left (d x + c\right ) + a\right )}^{3}}{\cos \left (d x + c\right )^{\frac {3}{2}}} \,d x } \]
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Time = 15.14 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b \cos (c+d x))^3}{\cos ^{\frac {3}{2}}(c+d x)} \, dx=\frac {2\,b^3\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )}{3\,d}+\frac {6\,a\,b^2\,\mathrm {E}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )}{d}+\frac {6\,a^2\,b\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )}{d}+\frac {2\,b^3\,\sqrt {\cos \left (c+d\,x\right )}\,\sin \left (c+d\,x\right )}{3\,d}+\frac {2\,a^3\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{4},\frac {1}{2};\ \frac {3}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{d\,\sqrt {\cos \left (c+d\,x\right )}\,\sqrt {{\sin \left (c+d\,x\right )}^2}} \]
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