Optimal. Leaf size=156 \[ \frac{5 (A-B) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 a d}-\frac{3 (5 A-7 B) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 a d}+\frac{(A-B) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{d (a \cos (c+d x)+a)}-\frac{(5 A-7 B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 a d}+\frac{5 (A-B) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 a d} \]
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Rubi [A] time = 0.198745, antiderivative size = 156, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.152, Rules used = {2977, 2748, 2635, 2641, 2639} \[ \frac{5 (A-B) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 a d}-\frac{3 (5 A-7 B) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 a d}+\frac{(A-B) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{d (a \cos (c+d x)+a)}-\frac{(5 A-7 B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 a d}+\frac{5 (A-B) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 a d} \]
Antiderivative was successfully verified.
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Rule 2977
Rule 2748
Rule 2635
Rule 2641
Rule 2639
Rubi steps
\begin{align*} \int \frac{\cos ^{\frac{5}{2}}(c+d x) (A+B \cos (c+d x))}{a+a \cos (c+d x)} \, dx &=\frac{(A-B) \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{d (a+a \cos (c+d x))}+\frac{\int \cos ^{\frac{3}{2}}(c+d x) \left (\frac{5}{2} a (A-B)-\frac{1}{2} a (5 A-7 B) \cos (c+d x)\right ) \, dx}{a^2}\\ &=\frac{(A-B) \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{d (a+a \cos (c+d x))}-\frac{(5 A-7 B) \int \cos ^{\frac{5}{2}}(c+d x) \, dx}{2 a}+\frac{(5 (A-B)) \int \cos ^{\frac{3}{2}}(c+d x) \, dx}{2 a}\\ &=\frac{5 (A-B) \sqrt{\cos (c+d x)} \sin (c+d x)}{3 a d}-\frac{(5 A-7 B) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{5 a d}+\frac{(A-B) \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{d (a+a \cos (c+d x))}-\frac{(3 (5 A-7 B)) \int \sqrt{\cos (c+d x)} \, dx}{10 a}+\frac{(5 (A-B)) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{6 a}\\ &=-\frac{3 (5 A-7 B) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 a d}+\frac{5 (A-B) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 a d}+\frac{5 (A-B) \sqrt{\cos (c+d x)} \sin (c+d x)}{3 a d}-\frac{(5 A-7 B) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{5 a d}+\frac{(A-B) \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{d (a+a \cos (c+d x))}\\ \end{align*}
Mathematica [C] time = 6.5567, size = 1182, normalized size = 7.58 \[ \text{result too large to display} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 3.599, size = 281, normalized size = 1.8 \begin{align*} -{\frac{1}{15\,da}\sqrt{ \left ( 2\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1 \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}} \left ( \cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \sqrt{ \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}}\sqrt{2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1} \left ( 25\,A{\it EllipticF} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) +45\,A{\it EllipticE} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) -25\,B{\it EllipticF} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) -63\,B{\it EllipticE} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) \right ) +48\,B \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{8}+ \left ( -40\,A-56\,B \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{6}+ \left ( 90\,A-30\,B \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{4}+ \left ( -35\,A+23\,B \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \right ) \left ( \cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-1}{\frac{1}{\sqrt{-2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}+ \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 ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1}}}} \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 )^{\frac{5}{2}}}{a \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{{\left (B \cos \left (d x + c\right )^{3} + A \cos \left (d x + c\right )^{2}\right )} \sqrt{\cos \left (d x + c\right )}}{a \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{{\left (B \cos \left (d x + c\right ) + A\right )} \cos \left (d x + c\right )^{\frac{5}{2}}}{a \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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