Integrand size = 23, antiderivative size = 121 \[ \int \frac {(a+a \cos (c+d x))^4}{\sqrt {\cos (c+d x)}} \, dx=\frac {64 a^4 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {136 a^4 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{21 d}+\frac {94 a^4 \sqrt {\cos (c+d x)} \sin (c+d x)}{21 d}+\frac {8 a^4 \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{5 d}+\frac {2 a^4 \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{7 d} \]
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Time = 0.18 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {2836, 2720, 2719, 2715} \[ \int \frac {(a+a \cos (c+d x))^4}{\sqrt {\cos (c+d x)}} \, dx=\frac {136 a^4 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{21 d}+\frac {64 a^4 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {2 a^4 \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 d}+\frac {8 a^4 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}+\frac {94 a^4 \sin (c+d x) \sqrt {\cos (c+d x)}}{21 d} \]
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Rule 2715
Rule 2719
Rule 2720
Rule 2836
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {a^4}{\sqrt {\cos (c+d x)}}+4 a^4 \sqrt {\cos (c+d x)}+6 a^4 \cos ^{\frac {3}{2}}(c+d x)+4 a^4 \cos ^{\frac {5}{2}}(c+d x)+a^4 \cos ^{\frac {7}{2}}(c+d x)\right ) \, dx \\ & = a^4 \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx+a^4 \int \cos ^{\frac {7}{2}}(c+d x) \, dx+\left (4 a^4\right ) \int \sqrt {\cos (c+d x)} \, dx+\left (4 a^4\right ) \int \cos ^{\frac {5}{2}}(c+d x) \, dx+\left (6 a^4\right ) \int \cos ^{\frac {3}{2}}(c+d x) \, dx \\ & = \frac {8 a^4 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}+\frac {2 a^4 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{d}+\frac {4 a^4 \sqrt {\cos (c+d x)} \sin (c+d x)}{d}+\frac {8 a^4 \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{5 d}+\frac {2 a^4 \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{7 d}+\frac {1}{7} \left (5 a^4\right ) \int \cos ^{\frac {3}{2}}(c+d x) \, dx+\left (2 a^4\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx+\frac {1}{5} \left (12 a^4\right ) \int \sqrt {\cos (c+d x)} \, dx \\ & = \frac {64 a^4 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {6 a^4 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{d}+\frac {94 a^4 \sqrt {\cos (c+d x)} \sin (c+d x)}{21 d}+\frac {8 a^4 \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{5 d}+\frac {2 a^4 \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{7 d}+\frac {1}{21} \left (5 a^4\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx \\ & = \frac {64 a^4 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {136 a^4 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{21 d}+\frac {94 a^4 \sqrt {\cos (c+d x)} \sin (c+d x)}{21 d}+\frac {8 a^4 \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{5 d}+\frac {2 a^4 \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{7 d} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 5.78 (sec) , antiderivative size = 245, normalized size of antiderivative = 2.02 \[ \int \frac {(a+a \cos (c+d x))^4}{\sqrt {\cos (c+d x)}} \, dx=\frac {a^4 (1+\cos (c+d x))^4 \sec ^8\left (\frac {1}{2} (c+d x)\right ) \left (\frac {672 (3 \cos (c-d x-\arctan (\tan (c)))+\cos (c+d x+\arctan (\tan (c)))) \csc (c) \sec (c)}{\sqrt {\sec ^2(c)}}-1360 \cos (c+d x) \sqrt {\cos ^2(d x-\arctan (\cot (c)))} \sqrt {\csc ^2(c)} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};\sin ^2(d x-\arctan (\cot (c)))\right ) \sec (d x-\arctan (\cot (c))) \sin (c)+\cos (c+d x) (-2688 \cot (c)+955 \sin (c+d x)+168 \sin (2 (c+d x))+15 \sin (3 (c+d x)))-1344 \cos (c) \csc (d x+\arctan (\tan (c))) \, _2F_1\left (-\frac {1}{2},-\frac {1}{4};\frac {3}{4};\cos ^2(d x+\arctan (\tan (c)))\right ) \sqrt {\sec ^2(c)} \sqrt {\sin ^2(d x+\arctan (\tan (c)))}\right )}{3360 d \sqrt {\cos (c+d x)}} \]
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Time = 9.15 (sec) , antiderivative size = 272, normalized size of antiderivative = 2.25
method | result | size |
default | \(-\frac {8 \sqrt {\left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, a^{4} \left (60 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-258 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+448 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-167 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+85 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-168 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, E\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )\right )}{105 \sqrt {-2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, d}\) | \(272\) |
parts | \(\text {Expression too large to display}\) | \(743\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.34 \[ \int \frac {(a+a \cos (c+d x))^4}{\sqrt {\cos (c+d x)}} \, dx=-\frac {2 \, {\left (170 i \, \sqrt {2} a^{4} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) - 170 i \, \sqrt {2} a^{4} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) - 336 i \, \sqrt {2} a^{4} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 336 i \, \sqrt {2} a^{4} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - {\left (15 \, a^{4} \cos \left (d x + c\right )^{2} + 84 \, a^{4} \cos \left (d x + c\right ) + 235 \, a^{4}\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )\right )}}{105 \, d} \]
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Timed out. \[ \int \frac {(a+a \cos (c+d x))^4}{\sqrt {\cos (c+d x)}} \, dx=\text {Timed out} \]
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\[ \int \frac {(a+a \cos (c+d x))^4}{\sqrt {\cos (c+d x)}} \, dx=\int { \frac {{\left (a \cos \left (d x + c\right ) + a\right )}^{4}}{\sqrt {\cos \left (d x + c\right )}} \,d x } \]
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\[ \int \frac {(a+a \cos (c+d x))^4}{\sqrt {\cos (c+d x)}} \, dx=\int { \frac {{\left (a \cos \left (d x + c\right ) + a\right )}^{4}}{\sqrt {\cos \left (d x + c\right )}} \,d x } \]
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Time = 15.01 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.21 \[ \int \frac {(a+a \cos (c+d x))^4}{\sqrt {\cos (c+d x)}} \, dx=\frac {2\,\left (4\,a^4\,\mathrm {E}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )+3\,a^4\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )+2\,a^4\,\sqrt {\cos \left (c+d\,x\right )}\,\sin \left (c+d\,x\right )\right )}{d}-\frac {8\,a^4\,{\cos \left (c+d\,x\right )}^{7/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {7}{4};\ \frac {11}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{7\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {2\,a^4\,{\cos \left (c+d\,x\right )}^{9/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {9}{4};\ \frac {13}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{9\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}} \]
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