Optimal. Leaf size=305 \[ \frac {2 a \left (7 a^2 B+27 a b C+22 b^2 B\right ) \sin (c+d x)}{45 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 a^2 (9 a C+13 b B) \sin (c+d x)}{63 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 \left (5 a^3 C+15 a^2 b B+21 a b^2 C+7 b^3 B\right ) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d}-\frac {2 \left (7 a^3 B+27 a^2 b C+27 a b^2 B+15 b^3 C\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {2 \left (5 a^3 C+15 a^2 b B+21 a b^2 C+7 b^3 B\right ) \sin (c+d x)}{21 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (7 a^3 B+27 a^2 b C+27 a b^2 B+15 b^3 C\right ) \sin (c+d x)}{15 d \sqrt {\cos (c+d x)}}+\frac {2 a B \sin (c+d x) (a+b \cos (c+d x))^2}{9 d \cos ^{\frac {9}{2}}(c+d x)} \]
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Rubi [A] time = 0.63, antiderivative size = 305, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3029, 2989, 3031, 3021, 2748, 2636, 2641, 2639} \[ \frac {2 \left (15 a^2 b B+5 a^3 C+21 a b^2 C+7 b^3 B\right ) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d}-\frac {2 \left (27 a^2 b C+7 a^3 B+27 a b^2 B+15 b^3 C\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {2 a \left (7 a^2 B+27 a b C+22 b^2 B\right ) \sin (c+d x)}{45 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (15 a^2 b B+5 a^3 C+21 a b^2 C+7 b^3 B\right ) \sin (c+d x)}{21 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (27 a^2 b C+7 a^3 B+27 a b^2 B+15 b^3 C\right ) \sin (c+d x)}{15 d \sqrt {\cos (c+d x)}}+\frac {2 a^2 (9 a C+13 b B) \sin (c+d x)}{63 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 a B \sin (c+d x) (a+b \cos (c+d x))^2}{9 d \cos ^{\frac {9}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Rule 2636
Rule 2639
Rule 2641
Rule 2748
Rule 2989
Rule 3021
Rule 3029
Rule 3031
Rubi steps
\begin {align*} \int \frac {(a+b \cos (c+d x))^3 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {13}{2}}(c+d x)} \, dx &=\int \frac {(a+b \cos (c+d x))^3 (B+C \cos (c+d x))}{\cos ^{\frac {11}{2}}(c+d x)} \, dx\\ &=\frac {2 a B (a+b \cos (c+d x))^2 \sin (c+d x)}{9 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {2}{9} \int \frac {(a+b \cos (c+d x)) \left (\frac {1}{2} a (13 b B+9 a C)+\frac {1}{2} \left (7 a^2 B+9 b^2 B+18 a b C\right ) \cos (c+d x)+\frac {3}{2} b (a B+3 b C) \cos ^2(c+d x)\right )}{\cos ^{\frac {9}{2}}(c+d x)} \, dx\\ &=\frac {2 a^2 (13 b B+9 a C) \sin (c+d x)}{63 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 a B (a+b \cos (c+d x))^2 \sin (c+d x)}{9 d \cos ^{\frac {9}{2}}(c+d x)}-\frac {4}{63} \int \frac {-\frac {7}{4} a \left (7 a^2 B+22 b^2 B+27 a b C\right )-\frac {9}{4} \left (15 a^2 b B+7 b^3 B+5 a^3 C+21 a b^2 C\right ) \cos (c+d x)-\frac {21}{4} b^2 (a B+3 b C) \cos ^2(c+d x)}{\cos ^{\frac {7}{2}}(c+d x)} \, dx\\ &=\frac {2 a^2 (13 b B+9 a C) \sin (c+d x)}{63 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 a \left (7 a^2 B+22 b^2 B+27 a b C\right ) \sin (c+d x)}{45 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 a B (a+b \cos (c+d x))^2 \sin (c+d x)}{9 d \cos ^{\frac {9}{2}}(c+d x)}-\frac {8}{315} \int \frac {-\frac {45}{8} \left (15 a^2 b B+7 b^3 B+5 a^3 C+21 a b^2 C\right )-\frac {21}{8} \left (7 a^3 B+27 a b^2 B+27 a^2 b C+15 b^3 C\right ) \cos (c+d x)}{\cos ^{\frac {5}{2}}(c+d x)} \, dx\\ &=\frac {2 a^2 (13 b B+9 a C) \sin (c+d x)}{63 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 a \left (7 a^2 B+22 b^2 B+27 a b C\right ) \sin (c+d x)}{45 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 a B (a+b \cos (c+d x))^2 \sin (c+d x)}{9 d \cos ^{\frac {9}{2}}(c+d x)}-\frac {1}{7} \left (-15 a^2 b B-7 b^3 B-5 a^3 C-21 a b^2 C\right ) \int \frac {1}{\cos ^{\frac {5}{2}}(c+d x)} \, dx-\frac {1}{15} \left (-7 a^3 B-27 a b^2 B-27 a^2 b C-15 b^3 C\right ) \int \frac {1}{\cos ^{\frac {3}{2}}(c+d x)} \, dx\\ &=\frac {2 a^2 (13 b B+9 a C) \sin (c+d x)}{63 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 a \left (7 a^2 B+22 b^2 B+27 a b C\right ) \sin (c+d x)}{45 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (15 a^2 b B+7 b^3 B+5 a^3 C+21 a b^2 C\right ) \sin (c+d x)}{21 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (7 a^3 B+27 a b^2 B+27 a^2 b C+15 b^3 C\right ) \sin (c+d x)}{15 d \sqrt {\cos (c+d x)}}+\frac {2 a B (a+b \cos (c+d x))^2 \sin (c+d x)}{9 d \cos ^{\frac {9}{2}}(c+d x)}-\frac {1}{21} \left (-15 a^2 b B-7 b^3 B-5 a^3 C-21 a b^2 C\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx-\frac {1}{15} \left (7 a^3 B+27 a b^2 B+27 a^2 b C+15 b^3 C\right ) \int \sqrt {\cos (c+d x)} \, dx\\ &=-\frac {2 \left (7 a^3 B+27 a b^2 B+27 a^2 b C+15 b^3 C\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {2 \left (15 a^2 b B+7 b^3 B+5 a^3 C+21 a b^2 C\right ) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d}+\frac {2 a^2 (13 b B+9 a C) \sin (c+d x)}{63 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 a \left (7 a^2 B+22 b^2 B+27 a b C\right ) \sin (c+d x)}{45 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (15 a^2 b B+7 b^3 B+5 a^3 C+21 a b^2 C\right ) \sin (c+d x)}{21 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (7 a^3 B+27 a b^2 B+27 a^2 b C+15 b^3 C\right ) \sin (c+d x)}{15 d \sqrt {\cos (c+d x)}}+\frac {2 a B (a+b \cos (c+d x))^2 \sin (c+d x)}{9 d \cos ^{\frac {9}{2}}(c+d x)}\\ \end {align*}
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Mathematica [A] time = 5.01, size = 266, normalized size = 0.87 \[ \frac {2 \left (\frac {35 a^3 B \sin (c+d x)}{\cos ^{\frac {9}{2}}(c+d x)}+\frac {7 a \left (7 a^2 B+27 a b C+27 b^2 B\right ) \sin (c+d x)}{\cos ^{\frac {5}{2}}(c+d x)}+\frac {45 a^2 (a C+3 b B) \sin (c+d x)}{\cos ^{\frac {7}{2}}(c+d x)}+15 \left (5 a^3 C+15 a^2 b B+21 a b^2 C+7 b^3 B\right ) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )-21 \left (7 a^3 B+27 a^2 b C+27 a b^2 B+15 b^3 C\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+\frac {15 \left (5 a^3 C+15 a^2 b B+21 a b^2 C+7 b^3 B\right ) \sin (c+d x)}{\cos ^{\frac {3}{2}}(c+d x)}+\frac {21 \left (7 a^3 B+27 a^2 b C+27 a b^2 B+15 b^3 C\right ) \sin (c+d x)}{\sqrt {\cos (c+d x)}}\right )}{315 d} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.62, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {C b^{3} \cos \left (d x + c\right )^{4} + B a^{3} + {\left (3 \, C a b^{2} + B b^{3}\right )} \cos \left (d x + c\right )^{3} + 3 \, {\left (C a^{2} b + B a b^{2}\right )} \cos \left (d x + c\right )^{2} + {\left (C a^{3} + 3 \, B a^{2} b\right )} \cos \left (d x + c\right )}{\cos \left (d x + c\right )^{\frac {11}{2}}}, x\right ) \]
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 \[ \int \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right )\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{3}}{\cos \left (d x + c\right )^{\frac {13}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 10.96, size = 1193, normalized size = 3.91 \[ \text {result too large to display} \]
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 \[ \int \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right )\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{3}}{\cos \left (d x + c\right )^{\frac {13}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.94, size = 304, normalized size = 1.00 \[ \frac {70\,B\,a^3\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (-\frac {9}{4},\frac {1}{2};\ -\frac {5}{4};\ {\cos \left (c+d\,x\right )}^2\right )+210\,B\,b^3\,{\cos \left (c+d\,x\right )}^3\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (-\frac {3}{4},\frac {1}{2};\ \frac {1}{4};\ {\cos \left (c+d\,x\right )}^2\right )+378\,B\,a\,b^2\,{\cos \left (c+d\,x\right )}^2\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (-\frac {5}{4},\frac {1}{2};\ -\frac {1}{4};\ {\cos \left (c+d\,x\right )}^2\right )+270\,B\,a^2\,b\,\cos \left (c+d\,x\right )\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (-\frac {7}{4},\frac {1}{2};\ -\frac {3}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{315\,d\,{\cos \left (c+d\,x\right )}^{9/2}\,\sqrt {1-{\cos \left (c+d\,x\right )}^2}}+\frac {\frac {2\,C\,a^3\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (-\frac {7}{4},\frac {1}{2};\ -\frac {3}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{7}+2\,C\,b^3\,{\cos \left (c+d\,x\right )}^3\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{4},\frac {1}{2};\ \frac {3}{4};\ {\cos \left (c+d\,x\right )}^2\right )+2\,C\,a\,b^2\,{\cos \left (c+d\,x\right )}^2\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (-\frac {3}{4},\frac {1}{2};\ \frac {1}{4};\ {\cos \left (c+d\,x\right )}^2\right )+\frac {6\,C\,a^2\,b\,\cos \left (c+d\,x\right )\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (-\frac {5}{4},\frac {1}{2};\ -\frac {1}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{5}}{d\,{\cos \left (c+d\,x\right )}^{7/2}\,\sqrt {1-{\cos \left (c+d\,x\right )}^2}} \]
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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