Optimal. Leaf size=215 \[ -\frac{2 a \left (8 a^2+7 b^2\right ) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{15 b^3 d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left (8 a^2+9 b^2\right ) \sqrt{a+b \cos (c+d x)} E\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{15 b^3 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{8 a \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{15 b^2 d}+\frac{2 \sin (c+d x) \cos (c+d x) \sqrt{a+b \cos (c+d x)}}{5 b d} \]
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Rubi [A] time = 0.287047, antiderivative size = 215, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.304, Rules used = {2793, 3023, 2752, 2663, 2661, 2655, 2653} \[ -\frac{2 a \left (8 a^2+7 b^2\right ) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{15 b^3 d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left (8 a^2+9 b^2\right ) \sqrt{a+b \cos (c+d x)} E\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{15 b^3 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{8 a \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{15 b^2 d}+\frac{2 \sin (c+d x) \cos (c+d x) \sqrt{a+b \cos (c+d x)}}{5 b d} \]
Antiderivative was successfully verified.
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Rule 2793
Rule 3023
Rule 2752
Rule 2663
Rule 2661
Rule 2655
Rule 2653
Rubi steps
\begin{align*} \int \frac{\cos ^3(c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx &=\frac{2 \cos (c+d x) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{5 b d}+\frac{2 \int \frac{a+\frac{3}{2} b \cos (c+d x)-2 a \cos ^2(c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx}{5 b}\\ &=-\frac{8 a \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{15 b^2 d}+\frac{2 \cos (c+d x) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{5 b d}+\frac{4 \int \frac{\frac{a b}{2}+\frac{1}{4} \left (8 a^2+9 b^2\right ) \cos (c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx}{15 b^2}\\ &=-\frac{8 a \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{15 b^2 d}+\frac{2 \cos (c+d x) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{5 b d}-\frac{\left (a \left (8 a^2+7 b^2\right )\right ) \int \frac{1}{\sqrt{a+b \cos (c+d x)}} \, dx}{15 b^3}+\frac{\left (8 a^2+9 b^2\right ) \int \sqrt{a+b \cos (c+d x)} \, dx}{15 b^3}\\ &=-\frac{8 a \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{15 b^2 d}+\frac{2 \cos (c+d x) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{5 b d}+\frac{\left (\left (8 a^2+9 b^2\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^3 \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{\left (a \left (8 a^2+7 b^2\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^3 \sqrt{a+b \cos (c+d x)}}\\ &=\frac{2 \left (8 a^2+9 b^2\right ) \sqrt{a+b \cos (c+d x)} E\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{15 b^3 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{2 a \left (8 a^2+7 b^2\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^3 d \sqrt{a+b \cos (c+d x)}}-\frac{8 a \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{15 b^2 d}+\frac{2 \cos (c+d x) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{5 b d}\\ \end{align*}
Mathematica [A] time = 0.932382, size = 182, normalized size = 0.85 \[ \frac{b \sin (c+d x) \left (-8 a^2-2 a b \cos (c+d x)+3 b^2 \cos (2 (c+d x))+3 b^2\right )-2 a \left (8 a^2+7 b^2\right ) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )+2 \left (8 a^2 b+8 a^3+9 a b^2+9 b^3\right ) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} E\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{15 b^3 d \sqrt{a+b \cos (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 3.274, size = 665, normalized size = 3.1 \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 )^{3}}{\sqrt{b \cos \left (d x + c\right ) + a}}\,{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{\cos \left (d x + c\right )^{3}}{\sqrt{b \cos \left (d x + c\right ) + a}}, 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 )^{3}}{\sqrt{b \cos \left (d x + c\right ) + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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