Optimal. Leaf size=436 \[ -\frac{2 a^2 \sin (c+d x) \cos ^3(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}-\frac{8 a^2 \left (2 a^2-3 b^2\right ) \sin (c+d x) \cos ^2(c+d x)}{3 b^2 d \left (a^2-b^2\right )^2 \sqrt{a+b \cos (c+d x)}}+\frac{2 \left (-71 a^2 b^2+48 a^4+3 b^4\right ) \sin (c+d x) \cos (c+d x) \sqrt{a+b \cos (c+d x)}}{15 b^3 d \left (a^2-b^2\right )^2}-\frac{4 a \left (-49 a^2 b^2+32 a^4+7 b^4\right ) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{15 b^4 d \left (a^2-b^2\right )^2}-\frac{2 a \left (-116 a^2 b^2+128 a^4-17 b^4\right ) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{15 b^5 d \left (a^2-b^2\right ) \sqrt{a+b \cos (c+d x)}}+\frac{2 \left (-212 a^4 b^2+55 a^2 b^4+128 a^6+9 b^6\right ) \sqrt{a+b \cos (c+d x)} E\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{15 b^5 d \left (a^2-b^2\right )^2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}}} \]
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Rubi [A] time = 0.862177, antiderivative size = 436, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.391, Rules used = {2792, 3047, 3049, 3023, 2752, 2663, 2661, 2655, 2653} \[ -\frac{2 a^2 \sin (c+d x) \cos ^3(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}-\frac{8 a^2 \left (2 a^2-3 b^2\right ) \sin (c+d x) \cos ^2(c+d x)}{3 b^2 d \left (a^2-b^2\right )^2 \sqrt{a+b \cos (c+d x)}}+\frac{2 \left (-71 a^2 b^2+48 a^4+3 b^4\right ) \sin (c+d x) \cos (c+d x) \sqrt{a+b \cos (c+d x)}}{15 b^3 d \left (a^2-b^2\right )^2}-\frac{4 a \left (-49 a^2 b^2+32 a^4+7 b^4\right ) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{15 b^4 d \left (a^2-b^2\right )^2}-\frac{2 a \left (-116 a^2 b^2+128 a^4-17 b^4\right ) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{15 b^5 d \left (a^2-b^2\right ) \sqrt{a+b \cos (c+d x)}}+\frac{2 \left (-212 a^4 b^2+55 a^2 b^4+128 a^6+9 b^6\right ) \sqrt{a+b \cos (c+d x)} E\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{15 b^5 d \left (a^2-b^2\right )^2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}}} \]
Antiderivative was successfully verified.
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Rule 2792
Rule 3047
Rule 3049
Rule 3023
Rule 2752
Rule 2663
Rule 2661
Rule 2655
Rule 2653
Rubi steps
\begin{align*} \int \frac{\cos ^5(c+d x)}{(a+b \cos (c+d x))^{5/2}} \, dx &=-\frac{2 a^2 \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^2-\frac{3}{2} a b \cos (c+d x)-\frac{1}{2} \left (8 a^2-3 b^2\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^2 \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{8 a^2 \left (2 a^2-3 b^2\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 (-4 a^2 \left (2 a^2-3 b^2\right )+\frac{1}{2} a b \left (a^2-3 b^2\right ) \cos (c+d x)+\frac{1}{4} \left (48 a^4-71 a^2 b^2+3 b^4\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^2 \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{8 a^2 \left (2 a^2-3 b^2\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 (48 a^4-71 a^2 b^2+3 b^4\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 (48 a^4-71 a^2 b^2+3 b^4\right )-\frac{1}{8} b \left (16 a^4-27 a^2 b^2-9 b^4\right ) \cos (c+d x)-\frac{3}{4} a \left (32 a^4-49 a^2 b^2+7 b^4\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^2 \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{8 a^2 \left (2 a^2-3 b^2\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 a \left (32 a^4-49 a^2 b^2+7 b^4\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 (48 a^4-71 a^2 b^2+3 b^4\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}{4} a b \left (8 a^4-11 a^2 b^2-2 b^4\right )+\frac{3}{16} \left (128 a^6-212 a^4 b^2+55 a^2 b^4+9 b^6\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^2 \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{8 a^2 \left (2 a^2-3 b^2\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 a \left (32 a^4-49 a^2 b^2+7 b^4\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 (48 a^4-71 a^2 b^2+3 b^4\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 (a \left (128 a^4-116 a^2 b^2-17 b^4\right )\right ) \int \frac{1}{\sqrt{a+b \cos (c+d x)}} \, dx}{15 b^5 \left (a^2-b^2\right )}+\frac{\left (128 a^6-212 a^4 b^2+55 a^2 b^4+9 b^6\right ) \int \sqrt{a+b \cos (c+d x)} \, dx}{15 b^5 \left (a^2-b^2\right )^2}\\ &=-\frac{2 a^2 \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{8 a^2 \left (2 a^2-3 b^2\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 a \left (32 a^4-49 a^2 b^2+7 b^4\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 (48 a^4-71 a^2 b^2+3 b^4\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 (128 a^6-212 a^4 b^2+55 a^2 b^4+9 b^6\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 (a \left (128 a^4-116 a^2 b^2-17 b^4\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 (128 a^6-212 a^4 b^2+55 a^2 b^4+9 b^6\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 a \left (128 a^4-116 a^2 b^2-17 b^4\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^2 \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{8 a^2 \left (2 a^2-3 b^2\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 a \left (32 a^4-49 a^2 b^2+7 b^4\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 (48 a^4-71 a^2 b^2+3 b^4\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 = 1.90669, size = 272, normalized size = 0.62 \[ \frac{b \left (\frac{10 a^5 \sin (c+d x)}{a^2-b^2}-\frac{10 a^4 \left (11 a^2-15 b^2\right ) \sin (c+d x) (a+b \cos (c+d x))}{\left (a^2-b^2\right )^2}-28 a \sin (c+d x) (a+b \cos (c+d x))^2+3 b \sin (2 (c+d x)) (a+b \cos (c+d x))^2\right )+\frac{2 \left (\frac{a+b \cos (c+d x)}{a+b}\right )^{3/2} \left (a \left (116 a^3 b^2-116 a^2 b^3+128 a^4 b-128 a^5+17 a b^4-17 b^5\right ) F\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )+\left (-212 a^4 b^2+55 a^2 b^4+128 a^6+9 b^6\right ) E\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )\right )}{(a-b)^2}}{15 b^5 d (a+b \cos (c+d x))^{3/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 17.893, size = 1684, normalized size = 3.9 \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{\cos \left (d x + c\right )^{5}}{{\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{\sqrt{b \cos \left (d x + c\right ) + a} \cos \left (d x + c\right )^{5}}{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{\cos \left (d x + c\right )^{5}}{{\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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