Optimal. Leaf size=288 \[ \frac {2 \left (15 a^2 C+56 a b B+25 b^2 C\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{105 d}-\frac {2 \left (a^2-b^2\right ) \left (15 a^2 C+56 a b B+25 b^2 C\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} F\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{105 b d \sqrt {a+b \cos (c+d x)}}+\frac {2 \left (15 a^3 C+161 a^2 b B+145 a b^2 C+63 b^3 B\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{105 b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}+\frac {2 (5 a C+7 b B) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{35 d}+\frac {2 C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d} \]
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Rubi [A] time = 0.58, antiderivative size = 288, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.175, Rules used = {3029, 2753, 2752, 2663, 2661, 2655, 2653} \[ \frac {2 \left (15 a^2 C+56 a b B+25 b^2 C\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{105 d}-\frac {2 \left (a^2-b^2\right ) \left (15 a^2 C+56 a b B+25 b^2 C\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} F\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{105 b d \sqrt {a+b \cos (c+d x)}}+\frac {2 \left (161 a^2 b B+15 a^3 C+145 a b^2 C+63 b^3 B\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{105 b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}+\frac {2 (5 a C+7 b B) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{35 d}+\frac {2 C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d} \]
Antiderivative was successfully verified.
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Rule 2653
Rule 2655
Rule 2661
Rule 2663
Rule 2752
Rule 2753
Rule 3029
Rubi steps
\begin {align*} \int (a+b \cos (c+d x))^{5/2} \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx &=\int (a+b \cos (c+d x))^{5/2} (B+C \cos (c+d x)) \, dx\\ &=\frac {2 C (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{7 d}+\frac {2}{7} \int (a+b \cos (c+d x))^{3/2} \left (\frac {1}{2} (7 a B+5 b C)+\frac {1}{2} (7 b B+5 a C) \cos (c+d x)\right ) \, dx\\ &=\frac {2 (7 b B+5 a C) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{35 d}+\frac {2 C (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{7 d}+\frac {4}{35} \int \sqrt {a+b \cos (c+d x)} \left (\frac {1}{4} \left (35 a^2 B+21 b^2 B+40 a b C\right )+\frac {1}{4} \left (56 a b B+15 a^2 C+25 b^2 C\right ) \cos (c+d x)\right ) \, dx\\ &=\frac {2 \left (56 a b B+15 a^2 C+25 b^2 C\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{105 d}+\frac {2 (7 b B+5 a C) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{35 d}+\frac {2 C (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{7 d}+\frac {8}{105} \int \frac {\frac {1}{8} \left (105 a^3 B+119 a b^2 B+135 a^2 b C+25 b^3 C\right )+\frac {1}{8} \left (161 a^2 b B+63 b^3 B+15 a^3 C+145 a b^2 C\right ) \cos (c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx\\ &=\frac {2 \left (56 a b B+15 a^2 C+25 b^2 C\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{105 d}+\frac {2 (7 b B+5 a C) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{35 d}+\frac {2 C (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{7 d}-\frac {\left (\left (a^2-b^2\right ) \left (56 a b B+15 a^2 C+25 b^2 C\right )\right ) \int \frac {1}{\sqrt {a+b \cos (c+d x)}} \, dx}{105 b}+\frac {\left (161 a^2 b B+63 b^3 B+15 a^3 C+145 a b^2 C\right ) \int \sqrt {a+b \cos (c+d x)} \, dx}{105 b}\\ &=\frac {2 \left (56 a b B+15 a^2 C+25 b^2 C\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{105 d}+\frac {2 (7 b B+5 a C) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{35 d}+\frac {2 C (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{7 d}+\frac {\left (\left (161 a^2 b B+63 b^3 B+15 a^3 C+145 a b^2 C\right ) \sqrt {a+b \cos (c+d x)}\right ) \int \sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}} \, dx}{105 b \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {\left (\left (a^2-b^2\right ) \left (56 a b B+15 a^2 C+25 b^2 C\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}}\right ) \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}} \, dx}{105 b \sqrt {a+b \cos (c+d x)}}\\ &=\frac {2 \left (161 a^2 b B+63 b^3 B+15 a^3 C+145 a b^2 C\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{105 b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {2 \left (a^2-b^2\right ) \left (56 a b B+15 a^2 C+25 b^2 C\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} F\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{105 b d \sqrt {a+b \cos (c+d x)}}+\frac {2 \left (56 a b B+15 a^2 C+25 b^2 C\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{105 d}+\frac {2 (7 b B+5 a C) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{35 d}+\frac {2 C (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{7 d}\\ \end {align*}
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Mathematica [A] time = 1.10, size = 254, normalized size = 0.88 \[ \frac {b \sin (c+d x) (a+b \cos (c+d x)) \left (90 a^2 C+6 b (15 a C+7 b B) \cos (c+d x)+154 a b B+15 b^2 C \cos (2 (c+d x))+65 b^2 C\right )+2 b \left (105 a^3 B+135 a^2 b C+119 a b^2 B+25 b^3 C\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} F\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )+2 \left (15 a^3 C+161 a^2 b B+145 a b^2 C+63 b^3 B\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \left ((a+b) E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )-a F\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )\right )}{105 b d \sqrt {a+b \cos (c+d x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.92, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (C b^{2} \cos \left (d x + c\right )^{4} + B a^{2} \cos \left (d x + c\right ) + {\left (2 \, C a b + B b^{2}\right )} \cos \left (d x + c\right )^{3} + {\left (C a^{2} + 2 \, B a b\right )} \cos \left (d x + c\right )^{2}\right )} \sqrt {b \cos \left (d x + c\right ) + a} \sec \left (d x + c\right ), x\right ) \]
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 \[ \int {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right )\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \sec \left (d x + c\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 2.72, size = 1305, normalized size = 4.53 \[ \text {result too large to display} \]
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 \[ \int {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right )\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \sec \left (d x + c\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )\right )\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^{5/2}}{\cos \left (c+d\,x\right )} \,d x \]
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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