Optimal. Leaf size=157 \[ \frac{2 \left (a^2-b^2\right ) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{3 d \sqrt{a+b \cos (c+d x)}}-\frac{2 b \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 d}+\frac{4 a \sqrt{a+b \cos (c+d x)} E\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{3 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}} \]
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Rubi [A] time = 0.233376, antiderivative size = 157, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.219, Rules used = {3016, 2753, 2752, 2663, 2661, 2655, 2653} \[ \frac{2 \left (a^2-b^2\right ) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{3 d \sqrt{a+b \cos (c+d x)}}-\frac{2 b \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 d}+\frac{4 a \sqrt{a+b \cos (c+d x)} E\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{3 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}} \]
Antiderivative was successfully verified.
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Rule 3016
Rule 2753
Rule 2752
Rule 2663
Rule 2661
Rule 2655
Rule 2653
Rubi steps
\begin{align*} \int \frac{a^2-b^2 \cos ^2(c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx &=-\int (-a+b \cos (c+d x)) \sqrt{a+b \cos (c+d x)} \, dx\\ &=-\frac{2 b \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{3 d}-\frac{2}{3} \int \frac{\frac{1}{2} \left (-3 a^2+b^2\right )-a b \cos (c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx\\ &=-\frac{2 b \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{3 d}+\frac{1}{3} (2 a) \int \sqrt{a+b \cos (c+d x)} \, dx-\frac{1}{3} \left (-a^2+b^2\right ) \int \frac{1}{\sqrt{a+b \cos (c+d x)}} \, dx\\ &=-\frac{2 b \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{3 d}+\frac{\left (2 a \sqrt{a+b \cos (c+d x)}\right ) \int \sqrt{\frac{a}{a+b}+\frac{b \cos (c+d x)}{a+b}} \, dx}{3 \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{\left (\left (-a^2+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}{3 \sqrt{a+b \cos (c+d x)}}\\ &=\frac{4 a \sqrt{a+b \cos (c+d x)} E\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{3 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 \left (a^2-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 )}{3 d \sqrt{a+b \cos (c+d x)}}-\frac{2 b \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{3 d}\\ \end{align*}
Mathematica [A] time = 0.497941, size = 134, normalized size = 0.85 \[ \frac{2 \left (a^2-b^2\right ) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )-2 b \sin (c+d x) (a+b \cos (c+d x))+4 a (a+b) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} E\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{3 d \sqrt{a+b \cos (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.718, size = 450, normalized size = 2.9 \begin{align*}{\frac{2}{3\,d}\sqrt{ \left ( 2\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}b+a-b \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}} \left ( 4\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{5}{b}^{2}+2\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{3}ab-6\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{3}{b}^{2}-\sqrt{ \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}}\sqrt{{\frac{1}{a-b} \left ( 2\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}b+a-b \right ) }}{\it EllipticF} \left ( \cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) ,\sqrt{-2\,{\frac{b}{a-b}}} \right ){a}^{2}+\sqrt{ \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}}\sqrt{{\frac{1}{a-b} \left ( 2\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}b+a-b \right ) }}{\it EllipticF} \left ( \cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) ,\sqrt{-2\,{\frac{b}{a-b}}} \right ){b}^{2}-2\,\sqrt{ \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}\sqrt{{\frac{2\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}b+a-b}{a-b}}}{\it EllipticE} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{-2\,{\frac{b}{a-b}}} \right ){a}^{2}+2\,\sqrt{ \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}\sqrt{{\frac{2\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}b+a-b}{a-b}}}{\it EllipticE} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{-2\,{\frac{b}{a-b}}} \right ) ab-2\,\cos \left ( 1/2\,dx+c/2 \right ) ab+2\,\cos \left ( 1/2\,dx+c/2 \right ){b}^{2} \right ){\frac{1}{\sqrt{-2\,b \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}+ \left ( a+b \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}}}} \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-1}{\frac{1}{\sqrt{-2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}b+a+b}}}} \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{b^{2} \cos \left (d x + c\right )^{2} - a^{2}}{\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 (-\sqrt{b \cos \left (d x + c\right ) + a}{\left (b \cos \left (d x + c\right ) - a\right )}, 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{b^{2} \cos \left (d x + c\right )^{2} - a^{2}}{\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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