Integrand size = 23, antiderivative size = 194 \[ \int \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3 \, dx=\frac {2 b \left (27 a^2+7 b^2\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {2 a \left (7 a^2+15 b^2\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{21 d}+\frac {2 a \left (7 a^2+15 b^2\right ) \sqrt {\cos (c+d x)} \sin (c+d x)}{21 d}+\frac {2 b \left (27 a^2+7 b^2\right ) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac {40 a b^2 \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{63 d}+\frac {2 b^2 \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x)) \sin (c+d x)}{9 d} \]
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Time = 0.26 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {2872, 3102, 2827, 2715, 2720, 2719} \[ \int \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3 \, dx=\frac {2 a \left (7 a^2+15 b^2\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{21 d}+\frac {2 b \left (27 a^2+7 b^2\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {2 b \left (27 a^2+7 b^2\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{45 d}+\frac {2 a \left (7 a^2+15 b^2\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{21 d}+\frac {2 b^2 \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))}{9 d}+\frac {40 a b^2 \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{63 d} \]
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Rule 2715
Rule 2719
Rule 2720
Rule 2827
Rule 2872
Rule 3102
Rubi steps \begin{align*} \text {integral}& = \frac {2 b^2 \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x)) \sin (c+d x)}{9 d}+\frac {2}{9} \int \cos ^{\frac {3}{2}}(c+d x) \left (\frac {1}{2} a \left (9 a^2+5 b^2\right )+\frac {1}{2} b \left (27 a^2+7 b^2\right ) \cos (c+d x)+10 a b^2 \cos ^2(c+d x)\right ) \, dx \\ & = \frac {40 a b^2 \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{63 d}+\frac {2 b^2 \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x)) \sin (c+d x)}{9 d}+\frac {4}{63} \int \cos ^{\frac {3}{2}}(c+d x) \left (\frac {9}{4} a \left (7 a^2+15 b^2\right )+\frac {7}{4} b \left (27 a^2+7 b^2\right ) \cos (c+d x)\right ) \, dx \\ & = \frac {40 a b^2 \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{63 d}+\frac {2 b^2 \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x)) \sin (c+d x)}{9 d}+\frac {1}{9} \left (b \left (27 a^2+7 b^2\right )\right ) \int \cos ^{\frac {5}{2}}(c+d x) \, dx+\frac {1}{7} \left (a \left (7 a^2+15 b^2\right )\right ) \int \cos ^{\frac {3}{2}}(c+d x) \, dx \\ & = \frac {2 a \left (7 a^2+15 b^2\right ) \sqrt {\cos (c+d x)} \sin (c+d x)}{21 d}+\frac {2 b \left (27 a^2+7 b^2\right ) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac {40 a b^2 \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{63 d}+\frac {2 b^2 \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x)) \sin (c+d x)}{9 d}+\frac {1}{15} \left (b \left (27 a^2+7 b^2\right )\right ) \int \sqrt {\cos (c+d x)} \, dx+\frac {1}{21} \left (a \left (7 a^2+15 b^2\right )\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx \\ & = \frac {2 b \left (27 a^2+7 b^2\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {2 a \left (7 a^2+15 b^2\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{21 d}+\frac {2 a \left (7 a^2+15 b^2\right ) \sqrt {\cos (c+d x)} \sin (c+d x)}{21 d}+\frac {2 b \left (27 a^2+7 b^2\right ) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac {40 a b^2 \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{63 d}+\frac {2 b^2 \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x)) \sin (c+d x)}{9 d} \\ \end{align*}
Time = 1.00 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.71 \[ \int \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3 \, dx=\frac {84 \left (27 a^2 b+7 b^3\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+60 \left (7 a^3+15 a b^2\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )+\sqrt {\cos (c+d x)} \left (7 b \left (108 a^2+43 b^2\right ) \cos (c+d x)+5 \left (84 a^3+234 a b^2+54 a b^2 \cos (2 (c+d x))+7 b^3 \cos (3 (c+d x))\right )\right ) \sin (c+d x)}{630 d} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(469\) vs. \(2(226)=452\).
Time = 13.53 (sec) , antiderivative size = 470, normalized size of antiderivative = 2.42
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 = 227, normalized size of antiderivative = 1.17 \[ \int \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3 \, dx=\frac {2 \, {\left (35 \, b^{3} \cos \left (d x + c\right )^{3} + 135 \, a b^{2} \cos \left (d x + c\right )^{2} + 105 \, a^{3} + 225 \, a b^{2} + 7 \, {\left (27 \, a^{2} b + 7 \, b^{3}\right )} \cos \left (d x + c\right )\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) - 15 \, \sqrt {2} {\left (7 i \, a^{3} + 15 i \, a b^{2}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) - 15 \, \sqrt {2} {\left (-7 i \, a^{3} - 15 i \, a b^{2}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) - 21 \, \sqrt {2} {\left (-27 i \, a^{2} b - 7 i \, b^{3}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) - 21 \, \sqrt {2} {\left (27 i \, a^{2} b + 7 i \, b^{3}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right )}{315 \, d} \]
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Timed out. \[ \int \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3 \, dx=\text {Timed out} \]
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\[ \int \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3 \, dx=\int { {\left (b \cos \left (d x + c\right ) + a\right )}^{3} \cos \left (d x + c\right )^{\frac {3}{2}} \,d x } \]
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\[ \int \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3 \, dx=\int { {\left (b \cos \left (d x + c\right ) + a\right )}^{3} \cos \left (d x + c\right )^{\frac {3}{2}} \,d x } \]
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Time = 14.77 (sec) , antiderivative size = 178, normalized size of antiderivative = 0.92 \[ \int \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3 \, dx=\frac {2\,a^3\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )}{3\,d}+\frac {2\,a^3\,\sqrt {\cos \left (c+d\,x\right )}\,\sin \left (c+d\,x\right )}{3\,d}-\frac {2\,b^3\,{\cos \left (c+d\,x\right )}^{11/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {11}{4};\ \frac {15}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{11\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {6\,a^2\,b\,{\cos \left (c+d\,x\right )}^{7/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {7}{4};\ \frac {11}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{7\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {2\,a\,b^2\,{\cos \left (c+d\,x\right )}^{9/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {9}{4};\ \frac {13}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{3\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}} \]
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