Optimal. Leaf size=126 \[ \frac {2 \left (a^2 B+6 a b C+3 b^2 B\right ) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{3 d}-\frac {2 \left (a^2 C+2 a b B-b^2 C\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}+\frac {2 a^2 B \sin (c+d x)}{3 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 a (a C+2 b B) \sin (c+d x)}{d \sqrt {\cos (c+d x)}} \]
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Rubi [A] time = 0.36, antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3029, 2988, 3021, 2748, 2641, 2639} \[ \frac {2 \left (a^2 B+6 a b C+3 b^2 B\right ) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{3 d}-\frac {2 \left (a^2 C+2 a b B-b^2 C\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}+\frac {2 a^2 B \sin (c+d x)}{3 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 a (a C+2 b B) \sin (c+d x)}{d \sqrt {\cos (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 2639
Rule 2641
Rule 2748
Rule 2988
Rule 3021
Rule 3029
Rubi steps
\begin {align*} \int \frac {(a+b \cos (c+d x))^2 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {7}{2}}(c+d x)} \, dx &=\int \frac {(a+b \cos (c+d x))^2 (B+C \cos (c+d x))}{\cos ^{\frac {5}{2}}(c+d x)} \, dx\\ &=\frac {2 a^2 B \sin (c+d x)}{3 d \cos ^{\frac {3}{2}}(c+d x)}-\frac {2}{3} \int \frac {-\frac {3}{2} a (2 b B+a C)-\frac {1}{2} \left (a^2 B+3 b^2 B+6 a b C\right ) \cos (c+d x)-\frac {3}{2} b^2 C \cos ^2(c+d x)}{\cos ^{\frac {3}{2}}(c+d x)} \, dx\\ &=\frac {2 a^2 B \sin (c+d x)}{3 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 a (2 b B+a C) \sin (c+d x)}{d \sqrt {\cos (c+d x)}}-\frac {4}{3} \int \frac {\frac {1}{4} \left (-a^2 B-3 b^2 B-6 a b C\right )+\frac {3}{4} \left (2 a b B+a^2 C-b^2 C\right ) \cos (c+d x)}{\sqrt {\cos (c+d x)}} \, dx\\ &=\frac {2 a^2 B \sin (c+d x)}{3 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 a (2 b B+a C) \sin (c+d x)}{d \sqrt {\cos (c+d x)}}-\frac {1}{3} \left (-a^2 B-3 b^2 B-6 a b C\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx-\left (2 a b B+a^2 C-b^2 C\right ) \int \sqrt {\cos (c+d x)} \, dx\\ &=-\frac {2 \left (2 a b B+a^2 C-b^2 C\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}+\frac {2 \left (a^2 B+3 b^2 B+6 a b C\right ) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{3 d}+\frac {2 a^2 B \sin (c+d x)}{3 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 a (2 b B+a C) \sin (c+d x)}{d \sqrt {\cos (c+d x)}}\\ \end {align*}
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Mathematica [A] time = 1.16, size = 105, normalized size = 0.83 \[ \frac {2 \left (\left (a^2 B+6 a b C+3 b^2 B\right ) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )-3 \left (a^2 C+2 a b B-b^2 C\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+\frac {a \sin (c+d x) (3 (a C+2 b B) \cos (c+d x)+a B)}{\cos ^{\frac {3}{2}}(c+d x)}\right )}{3 d} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.64, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {C b^{2} \cos \left (d x + c\right )^{3} + B a^{2} + {\left (2 \, C a b + B b^{2}\right )} \cos \left (d x + c\right )^{2} + {\left (C a^{2} + 2 \, B a b\right )} \cos \left (d x + c\right )}{\cos \left (d x + c\right )^{\frac {5}{2}}}, x\right ) \]
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 \[ \int \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right )\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{2}}{\cos \left (d x + c\right )^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 5.54, size = 677, normalized size = 5.37 \[ \text {result too large to display} \]
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 \[ \int \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right )\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{2}}{\cos \left (d x + c\right )^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.53, size = 194, normalized size = 1.54 \[ \frac {2\,B\,b^2\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )}{d}+\frac {2\,C\,b^2\,\mathrm {E}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )}{d}+\frac {4\,C\,a\,b\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )}{d}+\frac {2\,B\,a^2\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (-\frac {3}{4},\frac {1}{2};\ \frac {1}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{3\,d\,{\cos \left (c+d\,x\right )}^{3/2}\,\sqrt {{\sin \left (c+d\,x\right )}^2}}+\frac {2\,C\,a^2\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{4},\frac {1}{2};\ \frac {3}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{d\,\sqrt {\cos \left (c+d\,x\right )}\,\sqrt {{\sin \left (c+d\,x\right )}^2}}+\frac {4\,B\,a\,b\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{4},\frac {1}{2};\ \frac {3}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{d\,\sqrt {\cos \left (c+d\,x\right )}\,\sqrt {{\sin \left (c+d\,x\right )}^2}} \]
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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