Optimal. Leaf size=622 \[ -\frac{2 \sin (c+d x) \cos ^3(c+d x) \left (A b^2-a (b B-a C)\right )}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}+\frac{2 \sin (c+d x) \cos ^2(c+d x) \left (-2 a^2 b^2 (A-6 C)+5 a^3 b B-8 a^4 C-9 a b^3 B+6 A b^4\right )}{3 b^2 d \left (a^2-b^2\right )^2 \sqrt{a+b \cos (c+d x)}}-\frac{2 \sin (c+d x) \cos (c+d x) \left (-a^2 b^2 (15 A-71 C)+30 a^3 b B-48 a^4 C-50 a b^3 B+b^4 (35 A-3 C)\right ) \sqrt{a+b \cos (c+d x)}}{15 b^3 d \left (a^2-b^2\right )^2}+\frac{2 \sin (c+d x) \left (-2 a^3 b^2 (10 A-49 C)-65 a^2 b^3 B+40 a^4 b B-64 a^5 C+2 a b^4 (20 A-7 C)+5 b^5 B\right ) \sqrt{a+b \cos (c+d x)}}{15 b^4 d \left (a^2-b^2\right )^2}+\frac{2 \left (-4 a^3 b^2 (10 A-29 C)-80 a^2 b^3 B+80 a^4 b B-128 a^5 C+a b^4 (45 A+17 C)-5 b^5 B\right ) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{15 b^5 d \left (a^2-b^2\right ) \sqrt{a+b \cos (c+d x)}}-\frac{2 \left (-4 a^4 b^2 (10 A-53 C)+5 a^2 b^4 (15 A-11 C)-140 a^3 b^3 B+80 a^5 b B-128 a^6 C+40 a b^5 B-3 b^6 (5 A+3 C)\right ) \sqrt{a+b \cos (c+d x)} E\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{15 b^5 d \left (a^2-b^2\right )^2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}}} \]
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Rubi [A] time = 1.67499, antiderivative size = 622, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 43, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.186, Rules used = {3047, 3049, 3023, 2752, 2663, 2661, 2655, 2653} \[ -\frac{2 \sin (c+d x) \cos ^3(c+d x) \left (A b^2-a (b B-a C)\right )}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}+\frac{2 \sin (c+d x) \cos ^2(c+d x) \left (-2 a^2 b^2 (A-6 C)+5 a^3 b B-8 a^4 C-9 a b^3 B+6 A b^4\right )}{3 b^2 d \left (a^2-b^2\right )^2 \sqrt{a+b \cos (c+d x)}}-\frac{2 \sin (c+d x) \cos (c+d x) \left (-a^2 b^2 (15 A-71 C)+30 a^3 b B-48 a^4 C-50 a b^3 B+b^4 (35 A-3 C)\right ) \sqrt{a+b \cos (c+d x)}}{15 b^3 d \left (a^2-b^2\right )^2}+\frac{2 \sin (c+d x) \left (-2 a^3 b^2 (10 A-49 C)-65 a^2 b^3 B+40 a^4 b B-64 a^5 C+2 a b^4 (20 A-7 C)+5 b^5 B\right ) \sqrt{a+b \cos (c+d x)}}{15 b^4 d \left (a^2-b^2\right )^2}+\frac{2 \left (-4 a^3 b^2 (10 A-29 C)-80 a^2 b^3 B+80 a^4 b B-128 a^5 C+a b^4 (45 A+17 C)-5 b^5 B\right ) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{15 b^5 d \left (a^2-b^2\right ) \sqrt{a+b \cos (c+d x)}}-\frac{2 \left (-4 a^4 b^2 (10 A-53 C)+5 a^2 b^4 (15 A-11 C)-140 a^3 b^3 B+80 a^5 b B-128 a^6 C+40 a b^5 B-3 b^6 (5 A+3 C)\right ) \sqrt{a+b \cos (c+d x)} E\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{15 b^5 d \left (a^2-b^2\right )^2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}}} \]
Antiderivative was successfully verified.
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Rule 3047
Rule 3049
Rule 3023
Rule 2752
Rule 2663
Rule 2661
Rule 2655
Rule 2653
Rubi steps
\begin{align*} \int \frac{\cos ^3(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 ^3(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 ^2(c+d x) \left (3 \left (A b^2-a (b B-a C)\right )+\frac{3}{2} b (b B-a (A+C)) \cos (c+d x)-\frac{1}{2} \left (5 A b^2-5 a b B+8 a^2 C-3 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 ^3(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 \left (6 A b^4+5 a^3 b B-9 a b^3 B-2 a^2 b^2 (A-6 C)-8 a^4 C\right ) \cos ^2(c+d x) \sin (c+d x)}{3 b^2 \left (a^2-b^2\right )^2 d \sqrt{a+b \cos (c+d x)}}+\frac{4 \int \frac{\cos (c+d x) \left (6 A b^4+5 a^3 b B-9 a b^3 B-2 a^2 b^2 (A-6 C)-8 a^4 C+\frac{1}{4} b \left (a^2 b B+3 b^3 B+2 a^3 C-2 a b^2 (2 A+3 C)\right ) \cos (c+d x)-\frac{1}{4} \left (30 a^3 b B-50 a b^3 B-a^2 b^2 (15 A-71 C)+b^4 (35 A-3 C)-48 a^4 C\right ) \cos ^2(c+d x)\right )}{\sqrt{a+b \cos (c+d x)}} \, dx}{3 b^2 \left (a^2-b^2\right )^2}\\ &=-\frac{2 \left (A b^2-a (b B-a C)\right ) \cos ^3(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 \left (6 A b^4+5 a^3 b B-9 a b^3 B-2 a^2 b^2 (A-6 C)-8 a^4 C\right ) \cos ^2(c+d x) \sin (c+d x)}{3 b^2 \left (a^2-b^2\right )^2 d \sqrt{a+b \cos (c+d x)}}-\frac{2 \left (30 a^3 b B-50 a b^3 B-a^2 b^2 (15 A-71 C)+b^4 (35 A-3 C)-48 a^4 C\right ) \cos (c+d x) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{15 b^3 \left (a^2-b^2\right )^2 d}+\frac{8 \int \frac{-\frac{1}{4} a \left (30 a^3 b B-50 a b^3 B-a^2 b^2 (15 A-71 C)+b^4 (35 A-3 C)-48 a^4 C\right )+\frac{1}{8} b \left (10 a^3 b B-30 a b^3 B-16 a^4 C+3 b^4 (5 A+3 C)+a^2 b^2 (5 A+27 C)\right ) \cos (c+d x)+\frac{3}{8} \left (40 a^4 b B-65 a^2 b^3 B+5 b^5 B-2 a^3 b^2 (10 A-49 C)+2 a b^4 (20 A-7 C)-64 a^5 C\right ) \cos ^2(c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx}{15 b^3 \left (a^2-b^2\right )^2}\\ &=-\frac{2 \left (A b^2-a (b B-a C)\right ) \cos ^3(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 \left (6 A b^4+5 a^3 b B-9 a b^3 B-2 a^2 b^2 (A-6 C)-8 a^4 C\right ) \cos ^2(c+d x) \sin (c+d x)}{3 b^2 \left (a^2-b^2\right )^2 d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left (40 a^4 b B-65 a^2 b^3 B+5 b^5 B-2 a^3 b^2 (10 A-49 C)+2 a b^4 (20 A-7 C)-64 a^5 C\right ) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{15 b^4 \left (a^2-b^2\right )^2 d}-\frac{2 \left (30 a^3 b B-50 a b^3 B-a^2 b^2 (15 A-71 C)+b^4 (35 A-3 C)-48 a^4 C\right ) \cos (c+d x) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{15 b^3 \left (a^2-b^2\right )^2 d}+\frac{16 \int \frac{-\frac{3}{16} b \left (20 a^4 b B-35 a^2 b^3 B-5 b^5 B-2 a^3 b^2 (5 A-22 C)-32 a^5 C+2 a b^4 (15 A+4 C)\right )-\frac{3}{16} \left (80 a^5 b B-140 a^3 b^3 B+40 a b^5 B-4 a^4 b^2 (10 A-53 C)+5 a^2 b^4 (15 A-11 C)-128 a^6 C-3 b^6 (5 A+3 C)\right ) \cos (c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx}{45 b^4 \left (a^2-b^2\right )^2}\\ &=-\frac{2 \left (A b^2-a (b B-a C)\right ) \cos ^3(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 \left (6 A b^4+5 a^3 b B-9 a b^3 B-2 a^2 b^2 (A-6 C)-8 a^4 C\right ) \cos ^2(c+d x) \sin (c+d x)}{3 b^2 \left (a^2-b^2\right )^2 d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left (40 a^4 b B-65 a^2 b^3 B+5 b^5 B-2 a^3 b^2 (10 A-49 C)+2 a b^4 (20 A-7 C)-64 a^5 C\right ) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{15 b^4 \left (a^2-b^2\right )^2 d}-\frac{2 \left (30 a^3 b B-50 a b^3 B-a^2 b^2 (15 A-71 C)+b^4 (35 A-3 C)-48 a^4 C\right ) \cos (c+d x) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{15 b^3 \left (a^2-b^2\right )^2 d}-\frac{\left (80 a^5 b B-140 a^3 b^3 B+40 a b^5 B-4 a^4 b^2 (10 A-53 C)+5 a^2 b^4 (15 A-11 C)-128 a^6 C-3 b^6 (5 A+3 C)\right ) \int \sqrt{a+b \cos (c+d x)} \, dx}{15 b^5 \left (a^2-b^2\right )^2}+\frac{\left (80 a^4 b B-80 a^2 b^3 B-5 b^5 B-4 a^3 b^2 (10 A-29 C)-128 a^5 C+a b^4 (45 A+17 C)\right ) \int \frac{1}{\sqrt{a+b \cos (c+d x)}} \, dx}{15 b^5 \left (a^2-b^2\right )}\\ &=-\frac{2 \left (A b^2-a (b B-a C)\right ) \cos ^3(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 \left (6 A b^4+5 a^3 b B-9 a b^3 B-2 a^2 b^2 (A-6 C)-8 a^4 C\right ) \cos ^2(c+d x) \sin (c+d x)}{3 b^2 \left (a^2-b^2\right )^2 d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left (40 a^4 b B-65 a^2 b^3 B+5 b^5 B-2 a^3 b^2 (10 A-49 C)+2 a b^4 (20 A-7 C)-64 a^5 C\right ) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{15 b^4 \left (a^2-b^2\right )^2 d}-\frac{2 \left (30 a^3 b B-50 a b^3 B-a^2 b^2 (15 A-71 C)+b^4 (35 A-3 C)-48 a^4 C\right ) \cos (c+d x) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{15 b^3 \left (a^2-b^2\right )^2 d}-\frac{\left (\left (80 a^5 b B-140 a^3 b^3 B+40 a b^5 B-4 a^4 b^2 (10 A-53 C)+5 a^2 b^4 (15 A-11 C)-128 a^6 C-3 b^6 (5 A+3 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}{15 b^5 \left (a^2-b^2\right )^2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{\left (\left (80 a^4 b B-80 a^2 b^3 B-5 b^5 B-4 a^3 b^2 (10 A-29 C)-128 a^5 C+a b^4 (45 A+17 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}{15 b^5 \left (a^2-b^2\right ) \sqrt{a+b \cos (c+d x)}}\\ &=-\frac{2 \left (80 a^5 b B-140 a^3 b^3 B+40 a b^5 B-4 a^4 b^2 (10 A-53 C)+5 a^2 b^4 (15 A-11 C)-128 a^6 C-3 b^6 (5 A+3 C)\right ) \sqrt{a+b \cos (c+d x)} E\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{15 b^5 \left (a^2-b^2\right )^2 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 \left (80 a^4 b B-80 a^2 b^3 B-5 b^5 B-4 a^3 b^2 (10 A-29 C)-128 a^5 C+a b^4 (45 A+17 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 )}{15 b^5 \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 ^3(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 \left (6 A b^4+5 a^3 b B-9 a b^3 B-2 a^2 b^2 (A-6 C)-8 a^4 C\right ) \cos ^2(c+d x) \sin (c+d x)}{3 b^2 \left (a^2-b^2\right )^2 d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left (40 a^4 b B-65 a^2 b^3 B+5 b^5 B-2 a^3 b^2 (10 A-49 C)+2 a b^4 (20 A-7 C)-64 a^5 C\right ) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{15 b^4 \left (a^2-b^2\right )^2 d}-\frac{2 \left (30 a^3 b B-50 a b^3 B-a^2 b^2 (15 A-71 C)+b^4 (35 A-3 C)-48 a^4 C\right ) \cos (c+d x) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{15 b^3 \left (a^2-b^2\right )^2 d}\\ \end{align*}
Mathematica [A] time = 5.3527, size = 422, normalized size = 0.68 \[ \frac{b \left (\frac{10 a^3 \sin (c+d x) \left (a (a C-b B)+A b^2\right )}{a^2-b^2}-\frac{10 a^2 \sin (c+d x) \left (5 a^2 b^2 (A-3 C)-8 a^3 b B+11 a^4 C+12 a b^3 B-9 A b^4\right ) (a+b \cos (c+d x))}{\left (a^2-b^2\right )^2}+2 (5 b B-14 a C) \sin (c+d x) (a+b \cos (c+d x))^2+3 b C \sin (2 (c+d x)) (a+b \cos (c+d x))^2\right )+\frac{2 \left (\frac{a+b \cos (c+d x)}{a+b}\right )^{3/2} \left (b^2 \left (2 a^3 b^2 (5 A-22 C)+35 a^2 b^3 B-20 a^4 b B+32 a^5 C-2 a b^4 (15 A+4 C)+5 b^5 B\right ) F\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )+\left (4 a^4 b^2 (10 A-53 C)+5 a^2 b^4 (11 C-15 A)+140 a^3 b^3 B-80 a^5 b B+128 a^6 C-40 a b^5 B+3 b^6 (5 A+3 C)\right ) \left ((a+b) E\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )-a F\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )\right )\right )}{(a-b)^2 (a+b)}}{15 b^5 d (a+b \cos (c+d x))^{3/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 5.57, size = 1780, normalized size = 2.9 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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 )^{3}}{{\left (b \cos \left (d x + c\right ) + a\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (C \cos \left (d x + c\right )^{5} + B \cos \left (d x + c\right )^{4} + A \cos \left (d x + c\right )^{3}\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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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 )^{3}}{{\left (b \cos \left (d x + c\right ) + a\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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