Integrand size = 35, antiderivative size = 279 \[ \int \frac {(a+a \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac {13}{2}}(c+d x)} \, dx=-\frac {4 a^3 (5 A+7 C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {4 a^3 (105 A+143 C) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{231 d}+\frac {8 a^3 (35 A+44 C) \sin (c+d x)}{385 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {4 a^3 (105 A+143 C) \sin (c+d x)}{231 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {4 a^3 (5 A+7 C) \sin (c+d x)}{5 d \sqrt {\cos (c+d x)}}+\frac {2 A (a+a \cos (c+d x))^3 \sin (c+d x)}{11 d \cos ^{\frac {11}{2}}(c+d x)}+\frac {4 A \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{33 a d \cos ^{\frac {9}{2}}(c+d x)}+\frac {2 (35 A+33 C) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{231 d \cos ^{\frac {7}{2}}(c+d x)} \]
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Time = 0.78 (sec) , antiderivative size = 279, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.229, Rules used = {3123, 3054, 3047, 3100, 2827, 2716, 2720, 2719} \[ \int \frac {(a+a \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac {13}{2}}(c+d x)} \, dx=\frac {4 a^3 (105 A+143 C) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{231 d}-\frac {4 a^3 (5 A+7 C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {4 a^3 (105 A+143 C) \sin (c+d x)}{231 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {8 a^3 (35 A+44 C) \sin (c+d x)}{385 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 (35 A+33 C) \sin (c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{231 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {4 a^3 (5 A+7 C) \sin (c+d x)}{5 d \sqrt {\cos (c+d x)}}+\frac {4 A \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{33 a d \cos ^{\frac {9}{2}}(c+d x)}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^3}{11 d \cos ^{\frac {11}{2}}(c+d x)} \]
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Rule 2716
Rule 2719
Rule 2720
Rule 2827
Rule 3047
Rule 3054
Rule 3100
Rule 3123
Rubi steps \begin{align*} \text {integral}& = \frac {2 A (a+a \cos (c+d x))^3 \sin (c+d x)}{11 d \cos ^{\frac {11}{2}}(c+d x)}+\frac {2 \int \frac {(a+a \cos (c+d x))^3 \left (3 a A+\frac {1}{2} a (3 A+11 C) \cos (c+d x)\right )}{\cos ^{\frac {11}{2}}(c+d x)} \, dx}{11 a} \\ & = \frac {2 A (a+a \cos (c+d x))^3 \sin (c+d x)}{11 d \cos ^{\frac {11}{2}}(c+d x)}+\frac {4 A \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{33 a d \cos ^{\frac {9}{2}}(c+d x)}+\frac {4 \int \frac {(a+a \cos (c+d x))^2 \left (\frac {3}{4} a^2 (35 A+33 C)+\frac {9}{4} a^2 (5 A+11 C) \cos (c+d x)\right )}{\cos ^{\frac {9}{2}}(c+d x)} \, dx}{99 a} \\ & = \frac {2 A (a+a \cos (c+d x))^3 \sin (c+d x)}{11 d \cos ^{\frac {11}{2}}(c+d x)}+\frac {4 A \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{33 a d \cos ^{\frac {9}{2}}(c+d x)}+\frac {2 (35 A+33 C) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{231 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {8 \int \frac {(a+a \cos (c+d x)) \left (\frac {9}{2} a^3 (35 A+44 C)+\frac {45}{4} a^3 (7 A+11 C) \cos (c+d x)\right )}{\cos ^{\frac {7}{2}}(c+d x)} \, dx}{693 a} \\ & = \frac {2 A (a+a \cos (c+d x))^3 \sin (c+d x)}{11 d \cos ^{\frac {11}{2}}(c+d x)}+\frac {4 A \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{33 a d \cos ^{\frac {9}{2}}(c+d x)}+\frac {2 (35 A+33 C) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{231 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {8 \int \frac {\frac {9}{2} a^4 (35 A+44 C)+\left (\frac {45}{4} a^4 (7 A+11 C)+\frac {9}{2} a^4 (35 A+44 C)\right ) \cos (c+d x)+\frac {45}{4} a^4 (7 A+11 C) \cos ^2(c+d x)}{\cos ^{\frac {7}{2}}(c+d x)} \, dx}{693 a} \\ & = \frac {8 a^3 (35 A+44 C) \sin (c+d x)}{385 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 A (a+a \cos (c+d x))^3 \sin (c+d x)}{11 d \cos ^{\frac {11}{2}}(c+d x)}+\frac {4 A \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{33 a d \cos ^{\frac {9}{2}}(c+d x)}+\frac {2 (35 A+33 C) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{231 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {16 \int \frac {\frac {45}{8} a^4 (105 A+143 C)+\frac {693}{8} a^4 (5 A+7 C) \cos (c+d x)}{\cos ^{\frac {5}{2}}(c+d x)} \, dx}{3465 a} \\ & = \frac {8 a^3 (35 A+44 C) \sin (c+d x)}{385 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 A (a+a \cos (c+d x))^3 \sin (c+d x)}{11 d \cos ^{\frac {11}{2}}(c+d x)}+\frac {4 A \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{33 a d \cos ^{\frac {9}{2}}(c+d x)}+\frac {2 (35 A+33 C) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{231 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {1}{5} \left (2 a^3 (5 A+7 C)\right ) \int \frac {1}{\cos ^{\frac {3}{2}}(c+d x)} \, dx+\frac {1}{77} \left (2 a^3 (105 A+143 C)\right ) \int \frac {1}{\cos ^{\frac {5}{2}}(c+d x)} \, dx \\ & = \frac {8 a^3 (35 A+44 C) \sin (c+d x)}{385 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {4 a^3 (105 A+143 C) \sin (c+d x)}{231 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {4 a^3 (5 A+7 C) \sin (c+d x)}{5 d \sqrt {\cos (c+d x)}}+\frac {2 A (a+a \cos (c+d x))^3 \sin (c+d x)}{11 d \cos ^{\frac {11}{2}}(c+d x)}+\frac {4 A \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{33 a d \cos ^{\frac {9}{2}}(c+d x)}+\frac {2 (35 A+33 C) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{231 d \cos ^{\frac {7}{2}}(c+d x)}-\frac {1}{5} \left (2 a^3 (5 A+7 C)\right ) \int \sqrt {\cos (c+d x)} \, dx+\frac {1}{231} \left (2 a^3 (105 A+143 C)\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx \\ & = -\frac {4 a^3 (5 A+7 C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {4 a^3 (105 A+143 C) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{231 d}+\frac {8 a^3 (35 A+44 C) \sin (c+d x)}{385 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {4 a^3 (105 A+143 C) \sin (c+d x)}{231 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {4 a^3 (5 A+7 C) \sin (c+d x)}{5 d \sqrt {\cos (c+d x)}}+\frac {2 A (a+a \cos (c+d x))^3 \sin (c+d x)}{11 d \cos ^{\frac {11}{2}}(c+d x)}+\frac {4 A \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{33 a d \cos ^{\frac {9}{2}}(c+d x)}+\frac {2 (35 A+33 C) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{231 d \cos ^{\frac {7}{2}}(c+d x)} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 12.87 (sec) , antiderivative size = 997, normalized size of antiderivative = 3.57 \[ \int \frac {(a+a \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac {13}{2}}(c+d x)} \, dx=\sqrt {\cos (c+d x)} (a+a \cos (c+d x))^3 \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) \left (\frac {(5 A+7 C) \csc (c) \sec (c)}{10 d}+\frac {A \sec (c) \sec ^6(c+d x) \sin (d x)}{44 d}+\frac {\sec (c) \sec ^5(c+d x) (3 A \sin (c)+11 A \sin (d x))}{132 d}+\frac {\sec (c) \sec ^4(c+d x) (77 A \sin (c)+126 A \sin (d x)+33 C \sin (d x))}{924 d}+\frac {\sec (c) \sec ^3(c+d x) (630 A \sin (c)+165 C \sin (c)+770 A \sin (d x)+693 C \sin (d x))}{4620 d}+\frac {\sec (c) \sec ^2(c+d x) (770 A \sin (c)+693 C \sin (c)+1050 A \sin (d x)+1430 C \sin (d x))}{4620 d}+\frac {\sec (c) \sec (c+d x) (525 A \sin (c)+715 C \sin (c)+1155 A \sin (d x)+1617 C \sin (d x))}{2310 d}\right )-\frac {5 A (a+a \cos (c+d x))^3 \csc (c) \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};\sin ^2(d x-\arctan (\cot (c)))\right ) \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) \sec (d x-\arctan (\cot (c))) \sqrt {1-\sin (d x-\arctan (\cot (c)))} \sqrt {-\sqrt {1+\cot ^2(c)} \sin (c) \sin (d x-\arctan (\cot (c)))} \sqrt {1+\sin (d x-\arctan (\cot (c)))}}{22 d \sqrt {1+\cot ^2(c)}}-\frac {13 C (a+a \cos (c+d x))^3 \csc (c) \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};\sin ^2(d x-\arctan (\cot (c)))\right ) \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) \sec (d x-\arctan (\cot (c))) \sqrt {1-\sin (d x-\arctan (\cot (c)))} \sqrt {-\sqrt {1+\cot ^2(c)} \sin (c) \sin (d x-\arctan (\cot (c)))} \sqrt {1+\sin (d x-\arctan (\cot (c)))}}{42 d \sqrt {1+\cot ^2(c)}}+\frac {A (a+a \cos (c+d x))^3 \csc (c) \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) \left (\frac {\, _2F_1\left (-\frac {1}{2},-\frac {1}{4};\frac {3}{4};\cos ^2(d x+\arctan (\tan (c)))\right ) \sin (d x+\arctan (\tan (c))) \tan (c)}{\sqrt {1-\cos (d x+\arctan (\tan (c)))} \sqrt {1+\cos (d x+\arctan (\tan (c)))} \sqrt {\cos (c) \cos (d x+\arctan (\tan (c))) \sqrt {1+\tan ^2(c)}} \sqrt {1+\tan ^2(c)}}-\frac {\frac {\sin (d x+\arctan (\tan (c))) \tan (c)}{\sqrt {1+\tan ^2(c)}}+\frac {2 \cos ^2(c) \cos (d x+\arctan (\tan (c))) \sqrt {1+\tan ^2(c)}}{\cos ^2(c)+\sin ^2(c)}}{\sqrt {\cos (c) \cos (d x+\arctan (\tan (c))) \sqrt {1+\tan ^2(c)}}}\right )}{4 d}+\frac {7 C (a+a \cos (c+d x))^3 \csc (c) \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) \left (\frac {\, _2F_1\left (-\frac {1}{2},-\frac {1}{4};\frac {3}{4};\cos ^2(d x+\arctan (\tan (c)))\right ) \sin (d x+\arctan (\tan (c))) \tan (c)}{\sqrt {1-\cos (d x+\arctan (\tan (c)))} \sqrt {1+\cos (d x+\arctan (\tan (c)))} \sqrt {\cos (c) \cos (d x+\arctan (\tan (c))) \sqrt {1+\tan ^2(c)}} \sqrt {1+\tan ^2(c)}}-\frac {\frac {\sin (d x+\arctan (\tan (c))) \tan (c)}{\sqrt {1+\tan ^2(c)}}+\frac {2 \cos ^2(c) \cos (d x+\arctan (\tan (c))) \sqrt {1+\tan ^2(c)}}{\cos ^2(c)+\sin ^2(c)}}{\sqrt {\cos (c) \cos (d x+\arctan (\tan (c))) \sqrt {1+\tan ^2(c)}}}\right )}{20 d} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(1380\) vs. \(2(303)=606\).
Time = 33.62 (sec) , antiderivative size = 1381, normalized size of antiderivative = 4.95
method | result | size |
default | \(\text {Expression too large to display}\) | \(1381\) |
parts | \(\text {Expression too large to display}\) | \(1711\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.12 (sec) , antiderivative size = 299, normalized size of antiderivative = 1.07 \[ \int \frac {(a+a \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac {13}{2}}(c+d x)} \, dx=-\frac {2 \, {\left (5 i \, \sqrt {2} {\left (105 \, A + 143 \, C\right )} a^{3} \cos \left (d x + c\right )^{6} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) - 5 i \, \sqrt {2} {\left (105 \, A + 143 \, C\right )} a^{3} \cos \left (d x + c\right )^{6} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 231 i \, \sqrt {2} {\left (5 \, A + 7 \, C\right )} a^{3} \cos \left (d x + c\right )^{6} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) - 231 i \, \sqrt {2} {\left (5 \, A + 7 \, C\right )} a^{3} \cos \left (d x + c\right )^{6} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - {\left (462 \, {\left (5 \, A + 7 \, C\right )} a^{3} \cos \left (d x + c\right )^{5} + 10 \, {\left (105 \, A + 143 \, C\right )} a^{3} \cos \left (d x + c\right )^{4} + 77 \, {\left (10 \, A + 9 \, C\right )} a^{3} \cos \left (d x + c\right )^{3} + 15 \, {\left (42 \, A + 11 \, C\right )} a^{3} \cos \left (d x + c\right )^{2} + 385 \, A a^{3} \cos \left (d x + c\right ) + 105 \, A a^{3}\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )\right )}}{1155 \, d \cos \left (d x + c\right )^{6}} \]
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Timed out. \[ \int \frac {(a+a \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac {13}{2}}(c+d x)} \, dx=\text {Timed out} \]
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\[ \int \frac {(a+a \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac {13}{2}}(c+d x)} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + A\right )} {\left (a \cos \left (d x + c\right ) + a\right )}^{3}}{\cos \left (d x + c\right )^{\frac {13}{2}}} \,d x } \]
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\[ \int \frac {(a+a \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac {13}{2}}(c+d x)} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + A\right )} {\left (a \cos \left (d x + c\right ) + a\right )}^{3}}{\cos \left (d x + c\right )^{\frac {13}{2}}} \,d x } \]
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Time = 4.21 (sec) , antiderivative size = 621, normalized size of antiderivative = 2.23 \[ \int \frac {(a+a \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac {13}{2}}(c+d x)} \, dx=\frac {8\,{{}}_2{\mathrm {F}}_1\left (-\frac {3}{4},\frac {1}{2};\ \frac {5}{4};\ {\cos \left (c+d\,x\right )}^2\right )\,\left (\frac {42\,A\,a^3\,\sin \left (c+d\,x\right )}{{\cos \left (c+d\,x\right )}^{3/2}\,\sqrt {1-{\cos \left (c+d\,x\right )}^2}}+\frac {7\,A\,a^3\,\sin \left (c+d\,x\right )}{{\cos \left (c+d\,x\right )}^{7/2}\,\sqrt {1-{\cos \left (c+d\,x\right )}^2}}+\frac {11\,C\,a^3\,\sin \left (c+d\,x\right )}{{\cos \left (c+d\,x\right )}^{3/2}\,\sqrt {1-{\cos \left (c+d\,x\right )}^2}}\right )}{231\,d}-\frac {8\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{4},\frac {1}{2};\ \frac {7}{4};\ {\cos \left (c+d\,x\right )}^2\right )\,\left (\frac {10\,A\,a^3\,\sin \left (c+d\,x\right )}{\sqrt {\cos \left (c+d\,x\right )}\,\sqrt {1-{\cos \left (c+d\,x\right )}^2}}+\frac {5\,A\,a^3\,\sin \left (c+d\,x\right )}{{\cos \left (c+d\,x\right )}^{5/2}\,\sqrt {1-{\cos \left (c+d\,x\right )}^2}}+\frac {9\,C\,a^3\,\sin \left (c+d\,x\right )}{\sqrt {\cos \left (c+d\,x\right )}\,\sqrt {1-{\cos \left (c+d\,x\right )}^2}}\right )}{45\,d}+\frac {2\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{4},\frac {1}{2};\ \frac {3}{4};\ {\cos \left (c+d\,x\right )}^2\right )\,\left (\frac {40\,A\,a^3\,\sin \left (c+d\,x\right )}{\sqrt {\cos \left (c+d\,x\right )}\,\sqrt {1-{\cos \left (c+d\,x\right )}^2}}+\frac {15\,A\,a^3\,\sin \left (c+d\,x\right )}{{\cos \left (c+d\,x\right )}^{5/2}\,\sqrt {1-{\cos \left (c+d\,x\right )}^2}}+\frac {5\,A\,a^3\,\sin \left (c+d\,x\right )}{{\cos \left (c+d\,x\right )}^{9/2}\,\sqrt {1-{\cos \left (c+d\,x\right )}^2}}+\frac {51\,C\,a^3\,\sin \left (c+d\,x\right )}{\sqrt {\cos \left (c+d\,x\right )}\,\sqrt {1-{\cos \left (c+d\,x\right )}^2}}+\frac {9\,C\,a^3\,\sin \left (c+d\,x\right )}{{\cos \left (c+d\,x\right )}^{5/2}\,\sqrt {1-{\cos \left (c+d\,x\right )}^2}}\right )}{15\,d}+\frac {2\,{{}}_2{\mathrm {F}}_1\left (-\frac {3}{4},\frac {1}{2};\ \frac {1}{4};\ {\cos \left (c+d\,x\right )}^2\right )\,\left (\frac {168\,A\,a^3\,\sin \left (c+d\,x\right )}{{\cos \left (c+d\,x\right )}^{3/2}\,\sqrt {1-{\cos \left (c+d\,x\right )}^2}}+\frac {119\,A\,a^3\,\sin \left (c+d\,x\right )}{{\cos \left (c+d\,x\right )}^{7/2}\,\sqrt {1-{\cos \left (c+d\,x\right )}^2}}+\frac {21\,A\,a^3\,\sin \left (c+d\,x\right )}{{\cos \left (c+d\,x\right )}^{11/2}\,\sqrt {1-{\cos \left (c+d\,x\right )}^2}}+\frac {275\,C\,a^3\,\sin \left (c+d\,x\right )}{{\cos \left (c+d\,x\right )}^{3/2}\,\sqrt {1-{\cos \left (c+d\,x\right )}^2}}+\frac {33\,C\,a^3\,\sin \left (c+d\,x\right )}{{\cos \left (c+d\,x\right )}^{7/2}\,\sqrt {1-{\cos \left (c+d\,x\right )}^2}}\right )}{231\,d} \]
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