Integrand size = 42, antiderivative size = 204 \[ \int \frac {(a+b \cos (c+d x))^3 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {9}{2}}(c+d x)} \, dx=-\frac {2 \left (3 a^3 B+15 a b^2 B+15 a^2 b C-5 b^3 C\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {2 \left (3 a^2 b B+3 b^3 B+a^3 C+9 a b^2 C\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 d}+\frac {2 a^2 (9 b B+5 a C) \sin (c+d x)}{15 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 a \left (3 a^2 B+14 b^2 B+15 a b C\right ) \sin (c+d x)}{5 d \sqrt {\cos (c+d x)}}+\frac {2 a B (a+b \cos (c+d x))^2 \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x)} \]
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Time = 0.61 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3108, 3068, 3110, 3100, 2827, 2720, 2719} \[ \int \frac {(a+b \cos (c+d x))^3 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {9}{2}}(c+d x)} \, dx=\frac {2 a \left (3 a^2 B+15 a b C+14 b^2 B\right ) \sin (c+d x)}{5 d \sqrt {\cos (c+d x)}}+\frac {2 a^2 (5 a C+9 b B) \sin (c+d x)}{15 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (a^3 C+3 a^2 b B+9 a b^2 C+3 b^3 B\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 d}-\frac {2 \left (3 a^3 B+15 a^2 b C+15 a b^2 B-5 b^3 C\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {2 a B \sin (c+d x) (a+b \cos (c+d x))^2}{5 d \cos ^{\frac {5}{2}}(c+d x)} \]
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Rule 2719
Rule 2720
Rule 2827
Rule 3068
Rule 3100
Rule 3108
Rule 3110
Rubi steps \begin{align*} \text {integral}& = \int \frac {(a+b \cos (c+d x))^3 (B+C \cos (c+d x))}{\cos ^{\frac {7}{2}}(c+d x)} \, dx \\ & = \frac {2 a B (a+b \cos (c+d x))^2 \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2}{5} \int \frac {(a+b \cos (c+d x)) \left (\frac {1}{2} a (9 b B+5 a C)+\frac {1}{2} \left (3 a^2 B+5 b^2 B+10 a b C\right ) \cos (c+d x)-\frac {1}{2} b (a B-5 b C) \cos ^2(c+d x)\right )}{\cos ^{\frac {5}{2}}(c+d x)} \, dx \\ & = \frac {2 a^2 (9 b B+5 a C) \sin (c+d x)}{15 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 a B (a+b \cos (c+d x))^2 \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x)}-\frac {4}{15} \int \frac {-\frac {3}{4} a \left (3 a^2 B+14 b^2 B+15 a b C\right )-\frac {5}{4} \left (3 a^2 b B+3 b^3 B+a^3 C+9 a b^2 C\right ) \cos (c+d x)+\frac {3}{4} b^2 (a B-5 b C) \cos ^2(c+d x)}{\cos ^{\frac {3}{2}}(c+d x)} \, dx \\ & = \frac {2 a^2 (9 b B+5 a C) \sin (c+d x)}{15 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 a \left (3 a^2 B+14 b^2 B+15 a b C\right ) \sin (c+d x)}{5 d \sqrt {\cos (c+d x)}}+\frac {2 a B (a+b \cos (c+d x))^2 \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x)}-\frac {8}{15} \int \frac {-\frac {5}{8} \left (3 a^2 b B+3 b^3 B+a^3 C+9 a b^2 C\right )+\frac {3}{8} \left (3 a^3 B+15 a b^2 B+15 a^2 b C-5 b^3 C\right ) \cos (c+d x)}{\sqrt {\cos (c+d x)}} \, dx \\ & = \frac {2 a^2 (9 b B+5 a C) \sin (c+d x)}{15 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 a \left (3 a^2 B+14 b^2 B+15 a b C\right ) \sin (c+d x)}{5 d \sqrt {\cos (c+d x)}}+\frac {2 a B (a+b \cos (c+d x))^2 \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x)}-\frac {1}{3} \left (-3 a^2 b B-3 b^3 B-a^3 C-9 a b^2 C\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx-\frac {1}{5} \left (3 a^3 B+15 a b^2 B+15 a^2 b C-5 b^3 C\right ) \int \sqrt {\cos (c+d x)} \, dx \\ & = -\frac {2 \left (3 a^3 B+15 a b^2 B+15 a^2 b C-5 b^3 C\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {2 \left (3 a^2 b B+3 b^3 B+a^3 C+9 a b^2 C\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 d}+\frac {2 a^2 (9 b B+5 a C) \sin (c+d x)}{15 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 a \left (3 a^2 B+14 b^2 B+15 a b C\right ) \sin (c+d x)}{5 d \sqrt {\cos (c+d x)}}+\frac {2 a B (a+b \cos (c+d x))^2 \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x)} \\ \end{align*}
Time = 4.30 (sec) , antiderivative size = 176, normalized size of antiderivative = 0.86 \[ \int \frac {(a+b \cos (c+d x))^3 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {9}{2}}(c+d x)} \, dx=\frac {-6 \left (3 a^3 B+15 a b^2 B+15 a^2 b C-5 b^3 C\right ) \cos ^{\frac {3}{2}}(c+d x) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+10 \left (3 a^2 b B+3 b^3 B+a^3 C+9 a b^2 C\right ) \cos ^{\frac {3}{2}}(c+d x) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )+10 a^2 (3 b B+a C) \sin (c+d x)+9 a \left (a^2 B+5 b^2 B+5 a b C\right ) \sin (2 (c+d x))+6 a^3 B \tan (c+d x)}{15 d \cos ^{\frac {3}{2}}(c+d x)} \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 10.31 (sec) , antiderivative size = 950, normalized size of antiderivative = 4.66
method | result | size |
parts | \(\text {Expression too large to display}\) | \(950\) |
default | \(\text {Expression too large to display}\) | \(970\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.14 (sec) , antiderivative size = 326, normalized size of antiderivative = 1.60 \[ \int \frac {(a+b \cos (c+d x))^3 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {9}{2}}(c+d x)} \, dx=-\frac {5 \, \sqrt {2} {\left (i \, C a^{3} + 3 i \, B a^{2} b + 9 i \, C a b^{2} + 3 i \, B b^{3}\right )} \cos \left (d x + c\right )^{3} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 5 \, \sqrt {2} {\left (-i \, C a^{3} - 3 i \, B a^{2} b - 9 i \, C a b^{2} - 3 i \, B b^{3}\right )} \cos \left (d x + c\right )^{3} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 3 \, \sqrt {2} {\left (3 i \, B a^{3} + 15 i \, C a^{2} b + 15 i \, B a b^{2} - 5 i \, C b^{3}\right )} \cos \left (d x + c\right )^{3} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 3 \, \sqrt {2} {\left (-3 i \, B a^{3} - 15 i \, C a^{2} b - 15 i \, B a b^{2} + 5 i \, C b^{3}\right )} \cos \left (d x + c\right )^{3} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - 2 \, {\left (3 \, B a^{3} + 9 \, {\left (B a^{3} + 5 \, C a^{2} b + 5 \, B a b^{2}\right )} \cos \left (d x + c\right )^{2} + 5 \, {\left (C a^{3} + 3 \, B a^{2} b\right )} \cos \left (d x + c\right )\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )}{15 \, d \cos \left (d x + c\right )^{3}} \]
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Timed out. \[ \int \frac {(a+b \cos (c+d x))^3 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {9}{2}}(c+d x)} \, dx=\text {Timed out} \]
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\[ \int \frac {(a+b \cos (c+d x))^3 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {9}{2}}(c+d x)} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right )\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{3}}{\cos \left (d x + c\right )^{\frac {9}{2}}} \,d x } \]
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\[ \int \frac {(a+b \cos (c+d x))^3 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {9}{2}}(c+d x)} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right )\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{3}}{\cos \left (d x + c\right )^{\frac {9}{2}}} \,d x } \]
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Time = 5.57 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.43 \[ \int \frac {(a+b \cos (c+d x))^3 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {9}{2}}(c+d x)} \, dx=\frac {2\,\left (C\,\mathrm {E}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )\,b^3+3\,C\,a\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )\,b^2\right )}{d}+\frac {2\,B\,b^3\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )}{d}+\frac {2\,B\,a^3\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (-\frac {5}{4},\frac {1}{2};\ -\frac {1}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{5\,d\,{\cos \left (c+d\,x\right )}^{5/2}\,\sqrt {{\sin \left (c+d\,x\right )}^2}}+\frac {2\,C\,a^3\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (-\frac {3}{4},\frac {1}{2};\ \frac {1}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{3\,d\,{\cos \left (c+d\,x\right )}^{3/2}\,\sqrt {{\sin \left (c+d\,x\right )}^2}}+\frac {6\,B\,a\,b^2\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{4},\frac {1}{2};\ \frac {3}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{d\,\sqrt {\cos \left (c+d\,x\right )}\,\sqrt {{\sin \left (c+d\,x\right )}^2}}+\frac {2\,B\,a^2\,b\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (-\frac {3}{4},\frac {1}{2};\ \frac {1}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{d\,{\cos \left (c+d\,x\right )}^{3/2}\,\sqrt {{\sin \left (c+d\,x\right )}^2}}+\frac {6\,C\,a^2\,b\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{4},\frac {1}{2};\ \frac {3}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{d\,\sqrt {\cos \left (c+d\,x\right )}\,\sqrt {{\sin \left (c+d\,x\right )}^2}} \]
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