Optimal. Leaf size=453 \[ -\frac {2 \sin (c+d x) \cos ^2(c+d x) \left (A b^2-a (b B-a C)\right )}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}+\frac {2 \sin (c+d x) \left (2 a^2 C-a b B+A b^2-b^2 C\right ) \sqrt {a+b \cos (c+d x)}}{3 b^3 d \left (a^2-b^2\right )}-\frac {2 a \sin (c+d x) \left (a \left (-6 a^3 C+3 a^2 b B+10 a b^2 C-7 b^3 B\right )+4 A b^4\right )}{3 b^3 d \left (a^2-b^2\right )^2 \sqrt {a+b \cos (c+d x)}}-\frac {2 \left (-16 a^4 C+8 a^3 b B-2 a^2 b^2 (A-8 C)-9 a b^3 B+b^4 (3 A+C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} F\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{3 b^4 d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}+\frac {2 \left (-16 a^5 C+8 a^4 b B-2 a^3 b^2 (A-14 C)-15 a^2 b^3 B+2 a b^4 (3 A-4 C)+3 b^5 B\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{3 b^4 d \left (a^2-b^2\right )^2 \sqrt {\frac {a+b \cos (c+d x)}{a+b}}} \]
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Rubi [A] time = 1.02, antiderivative size = 453, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.186, Rules used = {3047, 3031, 3023, 2752, 2663, 2661, 2655, 2653} \[ -\frac {2 \sin (c+d x) \cos ^2(c+d x) \left (A b^2-a (b B-a C)\right )}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}+\frac {2 \sin (c+d x) \left (2 a^2 C-a b B+A b^2-b^2 C\right ) \sqrt {a+b \cos (c+d x)}}{3 b^3 d \left (a^2-b^2\right )}-\frac {2 a \sin (c+d x) \left (a \left (3 a^2 b B-6 a^3 C+10 a b^2 C-7 b^3 B\right )+4 A b^4\right )}{3 b^3 d \left (a^2-b^2\right )^2 \sqrt {a+b \cos (c+d x)}}-\frac {2 \left (-2 a^2 b^2 (A-8 C)+8 a^3 b B-16 a^4 C-9 a b^3 B+b^4 (3 A+C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} F\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{3 b^4 d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}+\frac {2 \left (-2 a^3 b^2 (A-14 C)-15 a^2 b^3 B+8 a^4 b B-16 a^5 C+2 a b^4 (3 A-4 C)+3 b^5 B\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{3 b^4 d \left (a^2-b^2\right )^2 \sqrt {\frac {a+b \cos (c+d x)}{a+b}}} \]
Antiderivative was successfully verified.
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Rule 2653
Rule 2655
Rule 2661
Rule 2663
Rule 2752
Rule 3023
Rule 3031
Rule 3047
Rubi steps
\begin {align*} \int \frac {\cos ^2(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^{5/2}} \, dx &=-\frac {2 \left (A b^2-a (b B-a C)\right ) \cos ^2(c+d x) \sin (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^{3/2}}-\frac {2 \int \frac {\cos (c+d x) \left (2 \left (A b^2-a (b B-a C)\right )+\frac {3}{2} b (b B-a (A+C)) \cos (c+d x)-\frac {3}{2} \left (A b^2-a b B+2 a^2 C-b^2 C\right ) \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^{3/2}} \, dx}{3 b \left (a^2-b^2\right )}\\ &=-\frac {2 \left (A b^2-a (b B-a C)\right ) \cos ^2(c+d x) \sin (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^{3/2}}-\frac {2 a \left (4 A b^4+a \left (3 a^2 b B-7 b^3 B-6 a^3 C+10 a b^2 C\right )\right ) \sin (c+d x)}{3 b^3 \left (a^2-b^2\right )^2 d \sqrt {a+b \cos (c+d x)}}-\frac {4 \int \frac {-\frac {1}{4} b \left (4 A b^4+a \left (3 a^2 b B-7 b^3 B-6 a^3 C+10 a b^2 C\right )\right )-\frac {1}{4} \left (6 a^4 b B-13 a^2 b^3 B+3 b^5 B+2 a b^4 (2 A-3 C)-12 a^5 C+22 a^3 b^2 C\right ) \cos (c+d x)-\frac {3}{4} b \left (a^2-b^2\right ) \left (A b^2-a b B+2 a^2 C-b^2 C\right ) \cos ^2(c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx}{3 b^3 \left (a^2-b^2\right )^2}\\ &=-\frac {2 \left (A b^2-a (b B-a C)\right ) \cos ^2(c+d x) \sin (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^{3/2}}-\frac {2 a \left (4 A b^4+a \left (3 a^2 b B-7 b^3 B-6 a^3 C+10 a b^2 C\right )\right ) \sin (c+d x)}{3 b^3 \left (a^2-b^2\right )^2 d \sqrt {a+b \cos (c+d x)}}+\frac {2 \left (A b^2-a b B+2 a^2 C-b^2 C\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{3 b^3 \left (a^2-b^2\right ) d}-\frac {8 \int \frac {-\frac {3}{8} b^2 \left (2 a^3 b B-6 a b^3 B-4 a^4 C+b^4 (3 A+C)+a^2 b^2 (A+7 C)\right )-\frac {3}{8} b \left (8 a^4 b B-15 a^2 b^3 B+3 b^5 B-2 a^3 b^2 (A-14 C)+2 a b^4 (3 A-4 C)-16 a^5 C\right ) \cos (c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx}{9 b^4 \left (a^2-b^2\right )^2}\\ &=-\frac {2 \left (A b^2-a (b B-a C)\right ) \cos ^2(c+d x) \sin (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^{3/2}}-\frac {2 a \left (4 A b^4+a \left (3 a^2 b B-7 b^3 B-6 a^3 C+10 a b^2 C\right )\right ) \sin (c+d x)}{3 b^3 \left (a^2-b^2\right )^2 d \sqrt {a+b \cos (c+d x)}}+\frac {2 \left (A b^2-a b B+2 a^2 C-b^2 C\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{3 b^3 \left (a^2-b^2\right ) d}+\frac {\left (8 a^4 b B-15 a^2 b^3 B+3 b^5 B-2 a^3 b^2 (A-14 C)+2 a b^4 (3 A-4 C)-16 a^5 C\right ) \int \sqrt {a+b \cos (c+d x)} \, dx}{3 b^4 \left (a^2-b^2\right )^2}-\frac {\left (8 a^3 b B-9 a b^3 B-2 a^2 b^2 (A-8 C)-16 a^4 C+b^4 (3 A+C)\right ) \int \frac {1}{\sqrt {a+b \cos (c+d x)}} \, dx}{3 b^4 \left (a^2-b^2\right )}\\ &=-\frac {2 \left (A b^2-a (b B-a C)\right ) \cos ^2(c+d x) \sin (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^{3/2}}-\frac {2 a \left (4 A b^4+a \left (3 a^2 b B-7 b^3 B-6 a^3 C+10 a b^2 C\right )\right ) \sin (c+d x)}{3 b^3 \left (a^2-b^2\right )^2 d \sqrt {a+b \cos (c+d x)}}+\frac {2 \left (A b^2-a b B+2 a^2 C-b^2 C\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{3 b^3 \left (a^2-b^2\right ) d}+\frac {\left (\left (8 a^4 b B-15 a^2 b^3 B+3 b^5 B-2 a^3 b^2 (A-14 C)+2 a b^4 (3 A-4 C)-16 a^5 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}{3 b^4 \left (a^2-b^2\right )^2 \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {\left (\left (8 a^3 b B-9 a b^3 B-2 a^2 b^2 (A-8 C)-16 a^4 C+b^4 (3 A+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}{3 b^4 \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}\\ &=\frac {2 \left (8 a^4 b B-15 a^2 b^3 B+3 b^5 B-2 a^3 b^2 (A-14 C)+2 a b^4 (3 A-4 C)-16 a^5 C\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{3 b^4 \left (a^2-b^2\right )^2 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {2 \left (8 a^3 b B-9 a b^3 B-2 a^2 b^2 (A-8 C)-16 a^4 C+b^4 (3 A+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 )}{3 b^4 \left (a^2-b^2\right ) d \sqrt {a+b \cos (c+d x)}}-\frac {2 \left (A b^2-a (b B-a C)\right ) \cos ^2(c+d x) \sin (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^{3/2}}-\frac {2 a \left (4 A b^4+a \left (3 a^2 b B-7 b^3 B-6 a^3 C+10 a b^2 C\right )\right ) \sin (c+d x)}{3 b^3 \left (a^2-b^2\right )^2 d \sqrt {a+b \cos (c+d x)}}+\frac {2 \left (A b^2-a b B+2 a^2 C-b^2 C\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{3 b^3 \left (a^2-b^2\right ) d}\\ \end {align*}
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Mathematica [A] time = 3.93, size = 377, normalized size = 0.83 \[ \frac {2 \left (\frac {\left (\frac {a+b \cos (c+d x)}{a+b}\right )^{3/2} \left (b^2 \left (-4 a^4 C+2 a^3 b B+a^2 b^2 (A+7 C)-6 a b^3 B+b^4 (3 A+C)\right ) F\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )-\left (16 a^5 C-8 a^4 b B+2 a^3 b^2 (A-14 C)+15 a^2 b^3 B+2 a b^4 (4 C-3 A)-3 b^5 B\right ) \left ((a+b) E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )-a F\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )\right )\right )}{(a-b)^2 (a+b)}+\frac {b \sin (c+d x) \left (16 a^6 C-8 a^5 b B+2 a^4 A b^2-25 a^4 b^2 C+16 a^3 b^3 B-10 a^2 A b^4+C \left (b^3-a^2 b\right )^2 \cos (2 (c+d x))+2 a b \cos (c+d x) \left (10 a^4 C-5 a^3 b B+2 a^2 b^2 (A-8 C)+9 a b^3 B+2 b^4 (C-3 A)\right )+b^6 C\right )}{2 \left (a^2-b^2\right )^2}\right )}{3 b^4 d (a+b \cos (c+d x))^{3/2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.05, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (C \cos \left (d x + c\right )^{4} + B \cos \left (d x + c\right )^{3} + A \cos \left (d x + c\right )^{2}\right )} \sqrt {b \cos \left (d x + c\right ) + a}}{b^{3} \cos \left (d x + c\right )^{3} + 3 \, a b^{2} \cos \left (d x + c\right )^{2} + 3 \, a^{2} b \cos \left (d x + c\right ) + a^{3}}, x\right ) \]
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 \[ \int \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} \cos \left (d x + c\right )^{2}}{{\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 14.10, size = 1480, normalized size = 3.27 \[ \text {result too large to display} \]
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 \[ \int \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} \cos \left (d x + c\right )^{2}}{{\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}}}\,{d x} \]
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 \[ \int \frac {{\cos \left (c+d\,x\right )}^2\,\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A\right )}{{\left (a+b\,\cos \left (c+d\,x\right )\right )}^{5/2}} \,d x \]
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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