Optimal. Leaf size=202 \[ -\frac {2 b \left (6 a^2 B-3 a b C-b^2 B\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}+\frac {2 \left (3 a^3 C+9 a^2 b B+3 a b^2 C+b^3 B\right ) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{3 d}-\frac {2 \left (5 a^3 B-15 a^2 b C-15 a b^2 B-3 b^3 C\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}-\frac {2 b^2 (5 a B-b C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}+\frac {2 a B \sin (c+d x) (a+b \cos (c+d x))^2}{d \sqrt {\cos (c+d x)}} \]
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Rubi [A] time = 0.56, antiderivative size = 202, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3029, 2989, 3033, 3023, 2748, 2641, 2639} \[ \frac {2 \left (9 a^2 b B+3 a^3 C+3 a b^2 C+b^3 B\right ) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{3 d}-\frac {2 \left (-15 a^2 b C+5 a^3 B-15 a b^2 B-3 b^3 C\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}-\frac {2 b \left (6 a^2 B-3 a b C-b^2 B\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}-\frac {2 b^2 (5 a B-b C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}+\frac {2 a B \sin (c+d x) (a+b \cos (c+d x))^2}{d \sqrt {\cos (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 2639
Rule 2641
Rule 2748
Rule 2989
Rule 3023
Rule 3029
Rule 3033
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 {5}{2}}(c+d x)} \, dx &=\int \frac {(a+b \cos (c+d x))^3 (B+C \cos (c+d x))}{\cos ^{\frac {3}{2}}(c+d x)} \, dx\\ &=\frac {2 a B (a+b \cos (c+d x))^2 \sin (c+d x)}{d \sqrt {\cos (c+d x)}}+2 \int \frac {(a+b \cos (c+d x)) \left (\frac {1}{2} a (5 b B+a C)-\frac {1}{2} \left (a^2 B-b^2 B-2 a b C\right ) \cos (c+d x)-\frac {1}{2} b (5 a B-b C) \cos ^2(c+d x)\right )}{\sqrt {\cos (c+d x)}} \, dx\\ &=-\frac {2 b^2 (5 a B-b C) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{5 d}+\frac {2 a B (a+b \cos (c+d x))^2 \sin (c+d x)}{d \sqrt {\cos (c+d x)}}+\frac {4}{5} \int \frac {\frac {5}{4} a^2 (5 b B+a C)-\frac {1}{4} \left (5 a^3 B-15 a b^2 B-15 a^2 b C-3 b^3 C\right ) \cos (c+d x)-\frac {5}{4} b \left (6 a^2 B-b^2 B-3 a b C\right ) \cos ^2(c+d x)}{\sqrt {\cos (c+d x)}} \, dx\\ &=-\frac {2 b \left (6 a^2 B-b^2 B-3 a b C\right ) \sqrt {\cos (c+d x)} \sin (c+d x)}{3 d}-\frac {2 b^2 (5 a B-b C) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{5 d}+\frac {2 a B (a+b \cos (c+d x))^2 \sin (c+d x)}{d \sqrt {\cos (c+d x)}}+\frac {8}{15} \int \frac {\frac {5}{8} \left (9 a^2 b B+b^3 B+3 a^3 C+3 a b^2 C\right )-\frac {3}{8} \left (5 a^3 B-15 a b^2 B-15 a^2 b C-3 b^3 C\right ) \cos (c+d x)}{\sqrt {\cos (c+d x)}} \, dx\\ &=-\frac {2 b \left (6 a^2 B-b^2 B-3 a b C\right ) \sqrt {\cos (c+d x)} \sin (c+d x)}{3 d}-\frac {2 b^2 (5 a B-b C) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{5 d}+\frac {2 a B (a+b \cos (c+d x))^2 \sin (c+d x)}{d \sqrt {\cos (c+d x)}}+\frac {1}{3} \left (9 a^2 b B+b^3 B+3 a^3 C+3 a b^2 C\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx+\frac {1}{5} \left (-5 a^3 B+15 a b^2 B+15 a^2 b C+3 b^3 C\right ) \int \sqrt {\cos (c+d x)} \, dx\\ &=-\frac {2 \left (5 a^3 B-15 a b^2 B-15 a^2 b C-3 b^3 C\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {2 \left (9 a^2 b B+b^3 B+3 a^3 C+3 a b^2 C\right ) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{3 d}-\frac {2 b \left (6 a^2 B-b^2 B-3 a b C\right ) \sqrt {\cos (c+d x)} \sin (c+d x)}{3 d}-\frac {2 b^2 (5 a B-b C) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{5 d}+\frac {2 a B (a+b \cos (c+d x))^2 \sin (c+d x)}{d \sqrt {\cos (c+d x)}}\\ \end {align*}
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Mathematica [A] time = 1.18, size = 150, normalized size = 0.74 \[ \frac {\frac {\sin (c+d x) \left (3 \left (10 a^3 B+b^3 C \cos (2 (c+d x))+b^3 C\right )+10 b^2 (3 a C+b B) \cos (c+d x)\right )}{\sqrt {\cos (c+d x)}}+10 \left (3 a^3 C+9 a^2 b B+3 a b^2 C+b^3 B\right ) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )+\left (-30 a^3 B+90 a^2 b C+90 a b^2 B+18 b^3 C\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.52, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {C b^{3} \cos \left (d x + c\right )^{4} + B a^{3} + {\left (3 \, C a b^{2} + B b^{3}\right )} \cos \left (d x + c\right )^{3} + 3 \, {\left (C a^{2} b + B a b^{2}\right )} \cos \left (d x + c\right )^{2} + {\left (C a^{3} + 3 \, B a^{2} b\right )} \cos \left (d x + c\right )}{\cos \left (d x + c\right )^{\frac {3}{2}}}, x\right ) \]
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 \[ \int \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right )\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{3}}{\cos \left (d x + c\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 2.78, size = 867, normalized size = 4.29 \[ \text {result too large to display} \]
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 \[ \int \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right )\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{3}}{\cos \left (d x + c\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.75, size = 248, normalized size = 1.23 \[ \frac {B\,b^3\,\left (\frac {2\,\sqrt {\cos \left (c+d\,x\right )}\,\sin \left (c+d\,x\right )}{3}+\frac {2\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )}{3}\right )}{d}+\frac {2\,C\,a^3\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )}{d}+\frac {6\,B\,a\,b^2\,\mathrm {E}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )}{d}+\frac {6\,B\,a^2\,b\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )}{d}+\frac {6\,C\,a^2\,b\,\mathrm {E}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )}{d}+\frac {3\,C\,a\,b^2\,\left (\frac {2\,\sqrt {\cos \left (c+d\,x\right )}\,\sin \left (c+d\,x\right )}{3}+\frac {2\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )}{3}\right )}{d}+\frac {2\,B\,a^3\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{4},\frac {1}{2};\ \frac {3}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{d\,\sqrt {\cos \left (c+d\,x\right )}\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {2\,C\,b^3\,{\cos \left (c+d\,x\right )}^{7/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {7}{4};\ \frac {11}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{7\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}} \]
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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