Optimal. Leaf size=230 \[ \frac {4 a^2 (5 A+7 C) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d}-\frac {16 a^2 (2 A+3 C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {4 a^2 (5 A+7 C) \sin (c+d x)}{21 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 a^2 (19 A+21 C) \sin (c+d x)}{105 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {16 a^2 (2 A+3 C) \sin (c+d x)}{15 d \sqrt {\cos (c+d x)}}+\frac {8 A \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )}{63 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^2}{9 d \cos ^{\frac {9}{2}}(c+d x)} \]
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Rubi [A] time = 0.51, antiderivative size = 230, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.229, Rules used = {3044, 2975, 2968, 3021, 2748, 2636, 2641, 2639} \[ \frac {4 a^2 (5 A+7 C) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d}-\frac {16 a^2 (2 A+3 C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {4 a^2 (5 A+7 C) \sin (c+d x)}{21 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 a^2 (19 A+21 C) \sin (c+d x)}{105 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {16 a^2 (2 A+3 C) \sin (c+d x)}{15 d \sqrt {\cos (c+d x)}}+\frac {8 A \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )}{63 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^2}{9 d \cos ^{\frac {9}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Rule 2636
Rule 2639
Rule 2641
Rule 2748
Rule 2968
Rule 2975
Rule 3021
Rule 3044
Rubi steps
\begin {align*} \int \frac {(a+a \cos (c+d x))^2 \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac {11}{2}}(c+d x)} \, dx &=\frac {2 A (a+a \cos (c+d x))^2 \sin (c+d x)}{9 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {2 \int \frac {(a+a \cos (c+d x))^2 \left (2 a A+\frac {3}{2} a (A+3 C) \cos (c+d x)\right )}{\cos ^{\frac {9}{2}}(c+d x)} \, dx}{9 a}\\ &=\frac {2 A (a+a \cos (c+d x))^2 \sin (c+d x)}{9 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {8 A \left (a^2+a^2 \cos (c+d x)\right ) \sin (c+d x)}{63 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {4 \int \frac {(a+a \cos (c+d x)) \left (\frac {3}{4} a^2 (19 A+21 C)+\frac {3}{4} a^2 (11 A+21 C) \cos (c+d x)\right )}{\cos ^{\frac {7}{2}}(c+d x)} \, dx}{63 a}\\ &=\frac {2 A (a+a \cos (c+d x))^2 \sin (c+d x)}{9 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {8 A \left (a^2+a^2 \cos (c+d x)\right ) \sin (c+d x)}{63 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {4 \int \frac {\frac {3}{4} a^3 (19 A+21 C)+\left (\frac {3}{4} a^3 (11 A+21 C)+\frac {3}{4} a^3 (19 A+21 C)\right ) \cos (c+d x)+\frac {3}{4} a^3 (11 A+21 C) \cos ^2(c+d x)}{\cos ^{\frac {7}{2}}(c+d x)} \, dx}{63 a}\\ &=\frac {2 a^2 (19 A+21 C) \sin (c+d x)}{105 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 A (a+a \cos (c+d x))^2 \sin (c+d x)}{9 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {8 A \left (a^2+a^2 \cos (c+d x)\right ) \sin (c+d x)}{63 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {8 \int \frac {\frac {45}{4} a^3 (5 A+7 C)+21 a^3 (2 A+3 C) \cos (c+d x)}{\cos ^{\frac {5}{2}}(c+d x)} \, dx}{315 a}\\ &=\frac {2 a^2 (19 A+21 C) \sin (c+d x)}{105 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 A (a+a \cos (c+d x))^2 \sin (c+d x)}{9 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {8 A \left (a^2+a^2 \cos (c+d x)\right ) \sin (c+d x)}{63 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {1}{15} \left (8 a^2 (2 A+3 C)\right ) \int \frac {1}{\cos ^{\frac {3}{2}}(c+d x)} \, dx+\frac {1}{7} \left (2 a^2 (5 A+7 C)\right ) \int \frac {1}{\cos ^{\frac {5}{2}}(c+d x)} \, dx\\ &=\frac {2 a^2 (19 A+21 C) \sin (c+d x)}{105 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {4 a^2 (5 A+7 C) \sin (c+d x)}{21 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {16 a^2 (2 A+3 C) \sin (c+d x)}{15 d \sqrt {\cos (c+d x)}}+\frac {2 A (a+a \cos (c+d x))^2 \sin (c+d x)}{9 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {8 A \left (a^2+a^2 \cos (c+d x)\right ) \sin (c+d x)}{63 d \cos ^{\frac {7}{2}}(c+d x)}-\frac {1}{15} \left (8 a^2 (2 A+3 C)\right ) \int \sqrt {\cos (c+d x)} \, dx+\frac {1}{21} \left (2 a^2 (5 A+7 C)\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx\\ &=-\frac {16 a^2 (2 A+3 C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {4 a^2 (5 A+7 C) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d}+\frac {2 a^2 (19 A+21 C) \sin (c+d x)}{105 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {4 a^2 (5 A+7 C) \sin (c+d x)}{21 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {16 a^2 (2 A+3 C) \sin (c+d x)}{15 d \sqrt {\cos (c+d x)}}+\frac {2 A (a+a \cos (c+d x))^2 \sin (c+d x)}{9 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {8 A \left (a^2+a^2 \cos (c+d x)\right ) \sin (c+d x)}{63 d \cos ^{\frac {7}{2}}(c+d x)}\\ \end {align*}
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Mathematica [C] time = 6.71, size = 955, normalized size = 4.15 \[ \sqrt {\cos (c+d x)} (\cos (c+d x) a+a)^2 \left (\frac {A \sec (c) \sin (d x) \sec ^5(c+d x)}{18 d}+\frac {\sec (c) (7 A \sin (c)+18 A \sin (d x)) \sec ^4(c+d x)}{126 d}+\frac {\sec (c) (90 A \sin (c)+112 A \sin (d x)+63 C \sin (d x)) \sec ^3(c+d x)}{630 d}+\frac {\sec (c) (112 A \sin (c)+63 C \sin (c)+150 A \sin (d x)+210 C \sin (d x)) \sec ^2(c+d x)}{630 d}+\frac {\sec (c) (25 A \sin (c)+35 C \sin (c)+56 A \sin (d x)+84 C \sin (d x)) \sec (c+d x)}{105 d}+\frac {4 (2 A+3 C) \csc (c) \sec (c)}{15 d}\right ) \sec ^4\left (\frac {c}{2}+\frac {d x}{2}\right )+\frac {4 A (\cos (c+d x) a+a)^2 \csc (c) \left (\frac {\, _2F_1\left (-\frac {1}{2},-\frac {1}{4};\frac {3}{4};\cos ^2\left (d x+\tan ^{-1}(\tan (c))\right )\right ) \sin \left (d x+\tan ^{-1}(\tan (c))\right ) \tan (c)}{\sqrt {1-\cos \left (d x+\tan ^{-1}(\tan (c))\right )} \sqrt {\cos \left (d x+\tan ^{-1}(\tan (c))\right )+1} \sqrt {\cos (c) \cos \left (d x+\tan ^{-1}(\tan (c))\right ) \sqrt {\tan ^2(c)+1}} \sqrt {\tan ^2(c)+1}}-\frac {\frac {2 \cos \left (d x+\tan ^{-1}(\tan (c))\right ) \sqrt {\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac {\sin \left (d x+\tan ^{-1}(\tan (c))\right ) \tan (c)}{\sqrt {\tan ^2(c)+1}}}{\sqrt {\cos (c) \cos \left (d x+\tan ^{-1}(\tan (c))\right ) \sqrt {\tan ^2(c)+1}}}\right ) \sec ^4\left (\frac {c}{2}+\frac {d x}{2}\right )}{15 d}+\frac {2 C (\cos (c+d x) a+a)^2 \csc (c) \left (\frac {\, _2F_1\left (-\frac {1}{2},-\frac {1}{4};\frac {3}{4};\cos ^2\left (d x+\tan ^{-1}(\tan (c))\right )\right ) \sin \left (d x+\tan ^{-1}(\tan (c))\right ) \tan (c)}{\sqrt {1-\cos \left (d x+\tan ^{-1}(\tan (c))\right )} \sqrt {\cos \left (d x+\tan ^{-1}(\tan (c))\right )+1} \sqrt {\cos (c) \cos \left (d x+\tan ^{-1}(\tan (c))\right ) \sqrt {\tan ^2(c)+1}} \sqrt {\tan ^2(c)+1}}-\frac {\frac {2 \cos \left (d x+\tan ^{-1}(\tan (c))\right ) \sqrt {\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac {\sin \left (d x+\tan ^{-1}(\tan (c))\right ) \tan (c)}{\sqrt {\tan ^2(c)+1}}}{\sqrt {\cos (c) \cos \left (d x+\tan ^{-1}(\tan (c))\right ) \sqrt {\tan ^2(c)+1}}}\right ) \sec ^4\left (\frac {c}{2}+\frac {d x}{2}\right )}{5 d}-\frac {5 A (\cos (c+d x) a+a)^2 \csc (c) \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};\sin ^2\left (d x-\tan ^{-1}(\cot (c))\right )\right ) \sec \left (d x-\tan ^{-1}(\cot (c))\right ) \sqrt {1-\sin \left (d x-\tan ^{-1}(\cot (c))\right )} \sqrt {-\sqrt {\cot ^2(c)+1} \sin (c) \sin \left (d x-\tan ^{-1}(\cot (c))\right )} \sqrt {\sin \left (d x-\tan ^{-1}(\cot (c))\right )+1} \sec ^4\left (\frac {c}{2}+\frac {d x}{2}\right )}{21 d \sqrt {\cot ^2(c)+1}}-\frac {C (\cos (c+d x) a+a)^2 \csc (c) \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};\sin ^2\left (d x-\tan ^{-1}(\cot (c))\right )\right ) \sec \left (d x-\tan ^{-1}(\cot (c))\right ) \sqrt {1-\sin \left (d x-\tan ^{-1}(\cot (c))\right )} \sqrt {-\sqrt {\cot ^2(c)+1} \sin (c) \sin \left (d x-\tan ^{-1}(\cot (c))\right )} \sqrt {\sin \left (d x-\tan ^{-1}(\cot (c))\right )+1} \sec ^4\left (\frac {c}{2}+\frac {d x}{2}\right )}{3 d \sqrt {\cot ^2(c)+1}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {C a^{2} \cos \left (d x + c\right )^{4} + 2 \, C a^{2} \cos \left (d x + c\right )^{3} + {\left (A + C\right )} a^{2} \cos \left (d x + c\right )^{2} + 2 \, A a^{2} \cos \left (d x + c\right ) + A a^{2}}{\cos \left (d x + c\right )^{\frac {11}{2}}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (C \cos \left (d x + c\right )^{2} + A\right )} {\left (a \cos \left (d x + c\right ) + a\right )}^{2}}{\cos \left (d x + c\right )^{\frac {11}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 7.07, size = 1168, normalized size = 5.08 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (C \cos \left (d x + c\right )^{2} + A\right )} {\left (a \cos \left (d x + c\right ) + a\right )}^{2}}{\cos \left (d x + c\right )^{\frac {11}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.11, size = 482, normalized size = 2.10 \[ \frac {4\,{{}}_2{\mathrm {F}}_1\left (-\frac {3}{4},\frac {1}{2};\ \frac {1}{4};\ {\cos \left (c+d\,x\right )}^2\right )\,\left (\frac {4\,A\,a^2\,\sin \left (c+d\,x\right )}{{\cos \left (c+d\,x\right )}^{3/2}\,\sqrt {1-{\cos \left (c+d\,x\right )}^2}}+\frac {3\,A\,a^2\,\sin \left (c+d\,x\right )}{{\cos \left (c+d\,x\right )}^{7/2}\,\sqrt {1-{\cos \left (c+d\,x\right )}^2}}+\frac {7\,C\,a^2\,\sin \left (c+d\,x\right )}{{\cos \left (c+d\,x\right )}^{3/2}\,\sqrt {1-{\cos \left (c+d\,x\right )}^2}}\right )}{21\,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 {16\,A\,a^2\,\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^2\,\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^2\,\sin \left (c+d\,x\right )}{\sqrt {\cos \left (c+d\,x\right )}\,\sqrt {1-{\cos \left (c+d\,x\right )}^2}}\right )}{135\,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 {64\,A\,a^2\,\sin \left (c+d\,x\right )}{\sqrt {\cos \left (c+d\,x\right )}\,\sqrt {1-{\cos \left (c+d\,x\right )}^2}}+\frac {21\,A\,a^2\,\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^2\,\sin \left (c+d\,x\right )}{{\cos \left (c+d\,x\right )}^{9/2}\,\sqrt {1-{\cos \left (c+d\,x\right )}^2}}+\frac {81\,C\,a^2\,\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^2\,\sin \left (c+d\,x\right )}{{\cos \left (c+d\,x\right )}^{5/2}\,\sqrt {1-{\cos \left (c+d\,x\right )}^2}}\right )}{45\,d}+\frac {16\,A\,a^2\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (-\frac {3}{4},\frac {1}{2};\ \frac {5}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{21\,d\,{\cos \left (c+d\,x\right )}^{3/2}\,\sqrt {1-{\cos \left (c+d\,x\right )}^2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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