Optimal. Leaf size=111 \[ \frac{6 A E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d}+\frac{2 A \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d}+\frac{10 B F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d}+\frac{2 B \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 d}+\frac{10 B \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d} \]
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Rubi [A] time = 0.074807, antiderivative size = 111, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {2748, 2635, 2639, 2641} \[ \frac{6 A E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d}+\frac{2 A \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d}+\frac{10 B F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d}+\frac{2 B \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 d}+\frac{10 B \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d} \]
Antiderivative was successfully verified.
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Rule 2748
Rule 2635
Rule 2639
Rule 2641
Rubi steps
\begin{align*} \int \cos ^{\frac{5}{2}}(c+d x) (A+B \cos (c+d x)) \, dx &=A \int \cos ^{\frac{5}{2}}(c+d x) \, dx+B \int \cos ^{\frac{7}{2}}(c+d x) \, dx\\ &=\frac{2 A \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{5 d}+\frac{2 B \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{7 d}+\frac{1}{5} (3 A) \int \sqrt{\cos (c+d x)} \, dx+\frac{1}{7} (5 B) \int \cos ^{\frac{3}{2}}(c+d x) \, dx\\ &=\frac{6 A E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d}+\frac{10 B \sqrt{\cos (c+d x)} \sin (c+d x)}{21 d}+\frac{2 A \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{5 d}+\frac{2 B \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{7 d}+\frac{1}{21} (5 B) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{6 A E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d}+\frac{10 B F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d}+\frac{10 B \sqrt{\cos (c+d x)} \sin (c+d x)}{21 d}+\frac{2 A \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{5 d}+\frac{2 B \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{7 d}\\ \end{align*}
Mathematica [A] time = 0.494371, size = 77, normalized size = 0.69 \[ \frac{\sin (c+d x) \sqrt{\cos (c+d x)} (42 A \cos (c+d x)+15 B \cos (2 (c+d x))+65 B)+126 A E\left (\left .\frac{1}{2} (c+d x)\right |2\right )+50 B F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{105 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 2.775, size = 290, normalized size = 2.6 \begin{align*} -{\frac{2}{105\,d}\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 ( 240\,B\cos \left ( 1/2\,dx+c/2 \right ) \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{8}+ \left ( -168\,A-360\,B \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{6}\cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) + \left ( 168\,A+280\,B \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{4}\cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) + \left ( -42\,A-80\,B \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}\cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -63\,A\sqrt{ \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}\sqrt{2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1}{\it EllipticE} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) +25\,B\sqrt{ \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}\sqrt{2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1}{\it EllipticF} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) \right ){\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{\left (B \cos \left (d x + c\right ) + A\right )} \cos \left (d x + c\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 ({\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 )}, 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(-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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