Optimal. Leaf size=377 \[ -\frac {8 a b \left (a^2 (7 A+9 C)+3 b^2 (3 A+5 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {2 \left (3 a^2 (9 A+11 C)+16 A b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^2}{231 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {4 a b \left (a^2 (673 A+891 C)+96 A b^2\right ) \sin (c+d x)}{3465 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {8 a b \left (a^2 (7 A+9 C)+3 b^2 (3 A+5 C)\right ) \sin (c+d x)}{15 d \sqrt {\cos (c+d x)}}+\frac {2 \left (5 a^4 (9 A+11 C)+66 a^2 b^2 (5 A+7 C)+77 b^4 (A+3 C)\right ) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{231 d}+\frac {2 \left (15 a^4 (9 A+11 C)+9 a^2 b^2 (101 A+143 C)+64 A b^4\right ) \sin (c+d x)}{693 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{11 d \cos ^{\frac {11}{2}}(c+d x)}+\frac {16 A b \sin (c+d x) (a+b \cos (c+d x))^3}{99 d \cos ^{\frac {9}{2}}(c+d x)} \]
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Rubi [A] time = 1.23, antiderivative size = 377, 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 = {3048, 3047, 3031, 3021, 2748, 2636, 2639, 2641} \[ \frac {2 \left (66 a^2 b^2 (5 A+7 C)+5 a^4 (9 A+11 C)+77 b^4 (A+3 C)\right ) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{231 d}-\frac {8 a b \left (a^2 (7 A+9 C)+3 b^2 (3 A+5 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {2 \left (3 a^2 (9 A+11 C)+16 A b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^2}{231 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 \left (9 a^2 b^2 (101 A+143 C)+15 a^4 (9 A+11 C)+64 A b^4\right ) \sin (c+d x)}{693 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {4 a b \left (a^2 (673 A+891 C)+96 A b^2\right ) \sin (c+d x)}{3465 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {8 a b \left (a^2 (7 A+9 C)+3 b^2 (3 A+5 C)\right ) \sin (c+d x)}{15 d \sqrt {\cos (c+d x)}}+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{11 d \cos ^{\frac {11}{2}}(c+d x)}+\frac {16 A b \sin (c+d x) (a+b \cos (c+d x))^3}{99 d \cos ^{\frac {9}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Rule 2636
Rule 2639
Rule 2641
Rule 2748
Rule 3021
Rule 3031
Rule 3047
Rule 3048
Rubi steps
\begin {align*} \int \frac {(a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac {13}{2}}(c+d x)} \, dx &=\frac {2 A (a+b \cos (c+d x))^4 \sin (c+d x)}{11 d \cos ^{\frac {11}{2}}(c+d x)}+\frac {2}{11} \int \frac {(a+b \cos (c+d x))^3 \left (4 A b+\frac {1}{2} a (9 A+11 C) \cos (c+d x)+\frac {1}{2} b (A+11 C) \cos ^2(c+d x)\right )}{\cos ^{\frac {11}{2}}(c+d x)} \, dx\\ &=\frac {16 A b (a+b \cos (c+d x))^3 \sin (c+d x)}{99 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {2 A (a+b \cos (c+d x))^4 \sin (c+d x)}{11 d \cos ^{\frac {11}{2}}(c+d x)}+\frac {4}{99} \int \frac {(a+b \cos (c+d x))^2 \left (\frac {3}{4} \left (16 A b^2+3 a^2 (9 A+11 C)\right )+\frac {1}{2} a b (73 A+99 C) \cos (c+d x)+\frac {1}{4} b^2 (17 A+99 C) \cos ^2(c+d x)\right )}{\cos ^{\frac {9}{2}}(c+d x)} \, dx\\ &=\frac {2 \left (16 A b^2+3 a^2 (9 A+11 C)\right ) (a+b \cos (c+d x))^2 \sin (c+d x)}{231 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {16 A b (a+b \cos (c+d x))^3 \sin (c+d x)}{99 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {2 A (a+b \cos (c+d x))^4 \sin (c+d x)}{11 d \cos ^{\frac {11}{2}}(c+d x)}+\frac {8}{693} \int \frac {(a+b \cos (c+d x)) \left (\frac {1}{4} b \left (96 A b^2+a^2 (673 A+891 C)\right )+\frac {1}{8} a \left (45 a^2 (9 A+11 C)+b^2 (1381 A+2079 C)\right ) \cos (c+d x)+\frac {1}{8} b \left (9 a^2 (9 A+11 C)+b^2 (167 A+693 C)\right ) \cos ^2(c+d x)\right )}{\cos ^{\frac {7}{2}}(c+d x)} \, dx\\ &=\frac {4 a b \left (96 A b^2+a^2 (673 A+891 C)\right ) \sin (c+d x)}{3465 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (16 A b^2+3 a^2 (9 A+11 C)\right ) (a+b \cos (c+d x))^2 \sin (c+d x)}{231 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {16 A b (a+b \cos (c+d x))^3 \sin (c+d x)}{99 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {2 A (a+b \cos (c+d x))^4 \sin (c+d x)}{11 d \cos ^{\frac {11}{2}}(c+d x)}-\frac {16 \int \frac {-\frac {15}{16} \left (64 A b^4+15 a^4 (9 A+11 C)+9 a^2 b^2 (101 A+143 C)\right )-\frac {231}{4} a b \left (3 b^2 (3 A+5 C)+a^2 (7 A+9 C)\right ) \cos (c+d x)-\frac {5}{16} b^2 \left (9 a^2 (9 A+11 C)+b^2 (167 A+693 C)\right ) \cos ^2(c+d x)}{\cos ^{\frac {5}{2}}(c+d x)} \, dx}{3465}\\ &=\frac {4 a b \left (96 A b^2+a^2 (673 A+891 C)\right ) \sin (c+d x)}{3465 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (64 A b^4+15 a^4 (9 A+11 C)+9 a^2 b^2 (101 A+143 C)\right ) \sin (c+d x)}{693 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (16 A b^2+3 a^2 (9 A+11 C)\right ) (a+b \cos (c+d x))^2 \sin (c+d x)}{231 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {16 A b (a+b \cos (c+d x))^3 \sin (c+d x)}{99 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {2 A (a+b \cos (c+d x))^4 \sin (c+d x)}{11 d \cos ^{\frac {11}{2}}(c+d x)}-\frac {32 \int \frac {-\frac {693}{8} a b \left (3 b^2 (3 A+5 C)+a^2 (7 A+9 C)\right )-\frac {45}{32} \left (77 b^4 (A+3 C)+66 a^2 b^2 (5 A+7 C)+5 a^4 (9 A+11 C)\right ) \cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x)} \, dx}{10395}\\ &=\frac {4 a b \left (96 A b^2+a^2 (673 A+891 C)\right ) \sin (c+d x)}{3465 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (64 A b^4+15 a^4 (9 A+11 C)+9 a^2 b^2 (101 A+143 C)\right ) \sin (c+d x)}{693 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (16 A b^2+3 a^2 (9 A+11 C)\right ) (a+b \cos (c+d x))^2 \sin (c+d x)}{231 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {16 A b (a+b \cos (c+d x))^3 \sin (c+d x)}{99 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {2 A (a+b \cos (c+d x))^4 \sin (c+d x)}{11 d \cos ^{\frac {11}{2}}(c+d x)}+\frac {1}{15} \left (4 a b \left (3 b^2 (3 A+5 C)+a^2 (7 A+9 C)\right )\right ) \int \frac {1}{\cos ^{\frac {3}{2}}(c+d x)} \, dx-\frac {1}{231} \left (-77 b^4 (A+3 C)-66 a^2 b^2 (5 A+7 C)-5 a^4 (9 A+11 C)\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx\\ &=\frac {2 \left (77 b^4 (A+3 C)+66 a^2 b^2 (5 A+7 C)+5 a^4 (9 A+11 C)\right ) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{231 d}+\frac {4 a b \left (96 A b^2+a^2 (673 A+891 C)\right ) \sin (c+d x)}{3465 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (64 A b^4+15 a^4 (9 A+11 C)+9 a^2 b^2 (101 A+143 C)\right ) \sin (c+d x)}{693 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {8 a b \left (3 b^2 (3 A+5 C)+a^2 (7 A+9 C)\right ) \sin (c+d x)}{15 d \sqrt {\cos (c+d x)}}+\frac {2 \left (16 A b^2+3 a^2 (9 A+11 C)\right ) (a+b \cos (c+d x))^2 \sin (c+d x)}{231 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {16 A b (a+b \cos (c+d x))^3 \sin (c+d x)}{99 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {2 A (a+b \cos (c+d x))^4 \sin (c+d x)}{11 d \cos ^{\frac {11}{2}}(c+d x)}-\frac {1}{15} \left (4 a b \left (3 b^2 (3 A+5 C)+a^2 (7 A+9 C)\right )\right ) \int \sqrt {\cos (c+d x)} \, dx\\ &=-\frac {8 a b \left (3 b^2 (3 A+5 C)+a^2 (7 A+9 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {2 \left (77 b^4 (A+3 C)+66 a^2 b^2 (5 A+7 C)+5 a^4 (9 A+11 C)\right ) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{231 d}+\frac {4 a b \left (96 A b^2+a^2 (673 A+891 C)\right ) \sin (c+d x)}{3465 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (64 A b^4+15 a^4 (9 A+11 C)+9 a^2 b^2 (101 A+143 C)\right ) \sin (c+d x)}{693 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {8 a b \left (3 b^2 (3 A+5 C)+a^2 (7 A+9 C)\right ) \sin (c+d x)}{15 d \sqrt {\cos (c+d x)}}+\frac {2 \left (16 A b^2+3 a^2 (9 A+11 C)\right ) (a+b \cos (c+d x))^2 \sin (c+d x)}{231 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {16 A b (a+b \cos (c+d x))^3 \sin (c+d x)}{99 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {2 A (a+b \cos (c+d x))^4 \sin (c+d x)}{11 d \cos ^{\frac {11}{2}}(c+d x)}\\ \end {align*}
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Mathematica [A] time = 4.60, size = 284, normalized size = 0.75 \[ \frac {-616 \left (a^3 b (7 A+9 C)+3 a b^3 (3 A+5 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+10 \left (5 a^4 (9 A+11 C)+66 a^2 b^2 (5 A+7 C)+77 b^4 (A+3 C)\right ) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )+\frac {45 \left (14 a^4 A \tan (c+d x)+\left (a^4 (9 A+11 C)+66 a^2 A b^2\right ) \sin (2 (c+d x))\right )+2 \sin (c+d x) \left (1540 a^3 A b+924 a b \left (a^2 (7 A+9 C)+3 b^2 (3 A+5 C)\right ) \cos ^4(c+d x)+308 a b \left (a^2 (7 A+9 C)+9 A b^2\right ) \cos ^2(c+d x)+15 \left (5 a^4 (9 A+11 C)+66 a^2 b^2 (5 A+7 C)+77 A b^4\right ) \cos ^3(c+d x)\right )}{3 \cos ^{\frac {9}{2}}(c+d x)}}{1155 d} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.71, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {C b^{4} \cos \left (d x + c\right )^{6} + 4 \, C a b^{3} \cos \left (d x + c\right )^{5} + 4 \, A a^{3} b \cos \left (d x + c\right ) + A a^{4} + {\left (6 \, C a^{2} b^{2} + A b^{4}\right )} \cos \left (d x + c\right )^{4} + 4 \, {\left (C a^{3} b + A a b^{3}\right )} \cos \left (d x + c\right )^{3} + {\left (C a^{4} + 6 \, A a^{2} b^{2}\right )} \cos \left (d x + c\right )^{2}}{\cos \left (d x + c\right )^{\frac {13}{2}}}, x\right ) \]
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 \[ \int \frac {{\left (C \cos \left (d x + c\right )^{2} + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{4}}{\cos \left (d x + c\right )^{\frac {13}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 12.60, size = 1521, normalized size = 4.03 \[ \text {result too large to display} \]
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 \[ \int \frac {{\left (C \cos \left (d x + c\right )^{2} + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{4}}{\cos \left (d x + c\right )^{\frac {13}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 7.73, size = 685, normalized size = 1.82 \[ \frac {8\,{{}}_2{\mathrm {F}}_1\left (-\frac {5}{4},\frac {1}{2};\ -\frac {1}{4};\ {\cos \left (c+d\,x\right )}^2\right )\,\left (\frac {9\,A\,a\,b^3\,\sin \left (c+d\,x\right )}{{\cos \left (c+d\,x\right )}^{5/2}\,\sqrt {{\sin \left (c+d\,x\right )}^2}}+\frac {4\,A\,a^3\,b\,\sin \left (c+d\,x\right )}{{\cos \left (c+d\,x\right )}^{5/2}\,\sqrt {{\sin \left (c+d\,x\right )}^2}}+\frac {5\,A\,a^3\,b\,\sin \left (c+d\,x\right )}{{\cos \left (c+d\,x\right )}^{9/2}\,\sqrt {{\sin \left (c+d\,x\right )}^2}}\right )}{45\,d}+\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 {9\,A\,a^4\,\sin \left (c+d\,x\right )}{{\cos \left (c+d\,x\right )}^{3/2}\,\sqrt {{\sin \left (c+d\,x\right )}^2}}+\frac {7\,A\,a^4\,\sin \left (c+d\,x\right )}{{\cos \left (c+d\,x\right )}^{7/2}\,\sqrt {{\sin \left (c+d\,x\right )}^2}}+\frac {66\,A\,a^2\,b^2\,\sin \left (c+d\,x\right )}{{\cos \left (c+d\,x\right )}^{3/2}\,\sqrt {{\sin \left (c+d\,x\right )}^2}}\right )}{231\,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 {36\,A\,a^4\,\sin \left (c+d\,x\right )}{{\cos \left (c+d\,x\right )}^{3/2}\,\sqrt {{\sin \left (c+d\,x\right )}^2}}+\frac {20\,A\,a^4\,\sin \left (c+d\,x\right )}{{\cos \left (c+d\,x\right )}^{7/2}\,\sqrt {{\sin \left (c+d\,x\right )}^2}}+\frac {21\,A\,a^4\,\sin \left (c+d\,x\right )}{{\cos \left (c+d\,x\right )}^{11/2}\,\sqrt {{\sin \left (c+d\,x\right )}^2}}+\frac {77\,A\,b^4\,\sin \left (c+d\,x\right )}{{\cos \left (c+d\,x\right )}^{3/2}\,\sqrt {{\sin \left (c+d\,x\right )}^2}}+\frac {264\,A\,a^2\,b^2\,\sin \left (c+d\,x\right )}{{\cos \left (c+d\,x\right )}^{3/2}\,\sqrt {{\sin \left (c+d\,x\right )}^2}}+\frac {198\,A\,a^2\,b^2\,\sin \left (c+d\,x\right )}{{\cos \left (c+d\,x\right )}^{7/2}\,\sqrt {{\sin \left (c+d\,x\right )}^2}}\right )}{231\,d}+\frac {2\,C\,b^4\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )}{d}+\frac {2\,C\,a^4\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (-\frac {7}{4},\frac {1}{2};\ -\frac {3}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{7\,d\,{\cos \left (c+d\,x\right )}^{7/2}\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {32\,A\,a^3\,b\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (-\frac {5}{4},\frac {1}{2};\ \frac {3}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{15\,d\,{\cos \left (c+d\,x\right )}^{5/2}\,\sqrt {{\sin \left (c+d\,x\right )}^2}}+\frac {8\,C\,a\,b^3\,\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 {8\,C\,a^3\,b\,\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 {4\,C\,a^2\,b^2\,\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}} \]
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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