Optimal. Leaf size=550 \[ \frac{2 a (A b-a B) \sin (c+d x) \cos ^3(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}+\frac{2 a \left (5 a^2 A b-8 a^3 B+12 a b^2 B-9 A b^3\right ) \sin (c+d x) \cos ^2(c+d x)}{3 b^2 d \left (a^2-b^2\right )^2 \sqrt{a+b \cos (c+d x)}}-\frac{2 \left (30 a^3 A b+71 a^2 b^2 B-48 a^4 B-50 a A b^3-3 b^4 B\right ) \sin (c+d x) \cos (c+d x) \sqrt{a+b \cos (c+d x)}}{15 b^3 d \left (a^2-b^2\right )^2}+\frac{2 \left (-65 a^2 A b^3+40 a^4 A b+98 a^3 b^2 B-64 a^5 B-14 a b^4 B+5 A b^5\right ) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{15 b^4 d \left (a^2-b^2\right )^2}+\frac{2 \left (-80 a^2 A b^3+80 a^4 A b+116 a^3 b^2 B-128 a^5 B+17 a b^4 B-5 A b^5\right ) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{15 b^5 d \left (a^2-b^2\right ) \sqrt{a+b \cos (c+d x)}}-\frac{2 \left (-140 a^3 A b^3+80 a^5 A b+212 a^4 b^2 B-55 a^2 b^4 B-128 a^6 B+40 a A b^5-9 b^6 B\right ) \sqrt{a+b \cos (c+d x)} E\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{15 b^5 d \left (a^2-b^2\right )^2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}}} \]
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Rubi [A] time = 1.18714, antiderivative size = 550, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {2989, 3047, 3049, 3023, 2752, 2663, 2661, 2655, 2653} \[ \frac{2 a (A b-a B) \sin (c+d x) \cos ^3(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}+\frac{2 a \left (5 a^2 A b-8 a^3 B+12 a b^2 B-9 A b^3\right ) \sin (c+d x) \cos ^2(c+d x)}{3 b^2 d \left (a^2-b^2\right )^2 \sqrt{a+b \cos (c+d x)}}-\frac{2 \left (30 a^3 A b+71 a^2 b^2 B-48 a^4 B-50 a A b^3-3 b^4 B\right ) \sin (c+d x) \cos (c+d x) \sqrt{a+b \cos (c+d x)}}{15 b^3 d \left (a^2-b^2\right )^2}+\frac{2 \left (-65 a^2 A b^3+40 a^4 A b+98 a^3 b^2 B-64 a^5 B-14 a b^4 B+5 A b^5\right ) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{15 b^4 d \left (a^2-b^2\right )^2}+\frac{2 \left (-80 a^2 A b^3+80 a^4 A b+116 a^3 b^2 B-128 a^5 B+17 a b^4 B-5 A b^5\right ) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{15 b^5 d \left (a^2-b^2\right ) \sqrt{a+b \cos (c+d x)}}-\frac{2 \left (-140 a^3 A b^3+80 a^5 A b+212 a^4 b^2 B-55 a^2 b^4 B-128 a^6 B+40 a A b^5-9 b^6 B\right ) \sqrt{a+b \cos (c+d x)} E\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{15 b^5 d \left (a^2-b^2\right )^2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}}} \]
Antiderivative was successfully verified.
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Rule 2989
Rule 3047
Rule 3049
Rule 3023
Rule 2752
Rule 2663
Rule 2661
Rule 2655
Rule 2653
Rubi steps
\begin{align*} \int \frac{\cos ^4(c+d x) (A+B \cos (c+d x))}{(a+b \cos (c+d x))^{5/2}} \, dx &=\frac{2 a (A b-a B) \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 a (A b-a B)+\frac{3}{2} b (A b-a B) \cos (c+d x)+\frac{1}{2} \left (5 a A b-8 a^2 B+3 b^2 B\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 a (A b-a B) \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 a \left (5 a^2 A b-9 A b^3-8 a^3 B+12 a b^2 B\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 (a \left (5 a^2 A b-9 A b^3-8 a^3 B+12 a b^2 B\right )+\frac{1}{4} b \left (a^2 A b+3 A b^3+2 a^3 B-6 a b^2 B\right ) \cos (c+d x)-\frac{1}{4} \left (30 a^3 A b-50 a A b^3-48 a^4 B+71 a^2 b^2 B-3 b^4 B\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 a (A b-a B) \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 a \left (5 a^2 A b-9 A b^3-8 a^3 B+12 a b^2 B\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 A b-50 a A b^3-48 a^4 B+71 a^2 b^2 B-3 b^4 B\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 A b-50 a A b^3-48 a^4 B+71 a^2 b^2 B-3 b^4 B\right )+\frac{1}{8} b \left (10 a^3 A b-30 a A b^3-16 a^4 B+27 a^2 b^2 B+9 b^4 B\right ) \cos (c+d x)+\frac{3}{8} \left (40 a^4 A b-65 a^2 A b^3+5 A b^5-64 a^5 B+98 a^3 b^2 B-14 a b^4 B\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 a (A b-a B) \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 a \left (5 a^2 A b-9 A b^3-8 a^3 B+12 a b^2 B\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 A b-65 a^2 A b^3+5 A b^5-64 a^5 B+98 a^3 b^2 B-14 a b^4 B\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 A b-50 a A b^3-48 a^4 B+71 a^2 b^2 B-3 b^4 B\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 A b-35 a^2 A b^3-5 A b^5-32 a^5 B+44 a^3 b^2 B+8 a b^4 B\right )-\frac{3}{16} \left (80 a^5 A b-140 a^3 A b^3+40 a A b^5-128 a^6 B+212 a^4 b^2 B-55 a^2 b^4 B-9 b^6 B\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 a (A b-a B) \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 a \left (5 a^2 A b-9 A b^3-8 a^3 B+12 a b^2 B\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 A b-65 a^2 A b^3+5 A b^5-64 a^5 B+98 a^3 b^2 B-14 a b^4 B\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 A b-50 a A b^3-48 a^4 B+71 a^2 b^2 B-3 b^4 B\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^4 A b-80 a^2 A b^3-5 A b^5-128 a^5 B+116 a^3 b^2 B+17 a b^4 B\right ) \int \frac{1}{\sqrt{a+b \cos (c+d x)}} \, dx}{15 b^5 \left (a^2-b^2\right )}-\frac{\left (80 a^5 A b-140 a^3 A b^3+40 a A b^5-128 a^6 B+212 a^4 b^2 B-55 a^2 b^4 B-9 b^6 B\right ) \int \sqrt{a+b \cos (c+d x)} \, dx}{15 b^5 \left (a^2-b^2\right )^2}\\ &=\frac{2 a (A b-a B) \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 a \left (5 a^2 A b-9 A b^3-8 a^3 B+12 a b^2 B\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 A b-65 a^2 A b^3+5 A b^5-64 a^5 B+98 a^3 b^2 B-14 a b^4 B\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 A b-50 a A b^3-48 a^4 B+71 a^2 b^2 B-3 b^4 B\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 A b-140 a^3 A b^3+40 a A b^5-128 a^6 B+212 a^4 b^2 B-55 a^2 b^4 B-9 b^6 B\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 A b-80 a^2 A b^3-5 A b^5-128 a^5 B+116 a^3 b^2 B+17 a b^4 B\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 A b-140 a^3 A b^3+40 a A b^5-128 a^6 B+212 a^4 b^2 B-55 a^2 b^4 B-9 b^6 B\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 A b-80 a^2 A b^3-5 A b^5-128 a^5 B+116 a^3 b^2 B+17 a b^4 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 \left (a^2-b^2\right ) d \sqrt{a+b \cos (c+d x)}}+\frac{2 a (A b-a B) \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 a \left (5 a^2 A b-9 A b^3-8 a^3 B+12 a b^2 B\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 A b-65 a^2 A b^3+5 A b^5-64 a^5 B+98 a^3 b^2 B-14 a b^4 B\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 A b-50 a A b^3-48 a^4 B+71 a^2 b^2 B-3 b^4 B\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 = 3.89872, size = 372, normalized size = 0.68 \[ \frac{b \left (\frac{10 a^4 (a B-A b) \sin (c+d x)}{a^2-b^2}-\frac{10 a^3 \left (-8 a^2 A b+11 a^3 B-15 a b^2 B+12 A b^3\right ) \sin (c+d x) (a+b \cos (c+d x))}{\left (a^2-b^2\right )^2}+2 (5 A b-14 a B) \sin (c+d x) (a+b \cos (c+d x))^2+3 b B \sin (2 (c+d x)) (a+b \cos (c+d x))^2\right )-\frac{2 \left (\frac{a+b \cos (c+d x)}{a+b}\right )^{3/2} \left (b^2 \left (-35 a^2 A b^3+20 a^4 A b+44 a^3 b^2 B-32 a^5 B+8 a b^4 B-5 A b^5\right ) F\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )-\left (140 a^3 A b^3-80 a^5 A b-212 a^4 b^2 B+55 a^2 b^4 B+128 a^6 B-40 a A b^5+9 b^6 B\right ) \left ((a+b) E\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )-a F\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )\right )\right )}{(a-b)^2 (a+b)}}{15 b^5 d (a+b \cos (c+d x))^{3/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 21.393, size = 1746, normalized size = 3.2 \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 (B \cos \left (d x + c\right ) + A\right )} \cos \left (d x + c\right )^{4}}{{\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 (B \cos \left (d x + c\right )^{5} + A \cos \left (d x + c\right )^{4}\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 (B \cos \left (d x + c\right ) + A\right )} \cos \left (d x + c\right )^{4}}{{\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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