Optimal. Leaf size=194 \[ \frac{2 a \left (5 a^2+21 b^2\right ) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d}-\frac{2 b \left (9 a^2+5 b^2\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d}+\frac{2 a \left (5 a^2+21 b^2\right ) \sin (c+d x)}{21 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 b \left (9 a^2+5 b^2\right ) \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}+\frac{2 a^2 \sin (c+d x) (a+b \cos (c+d x))}{7 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{32 a^2 b \sin (c+d x)}{35 d \cos ^{\frac{5}{2}}(c+d x)} \]
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Rubi [A] time = 0.233322, antiderivative size = 194, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261, Rules used = {2792, 3021, 2748, 2636, 2641, 2639} \[ \frac{2 a \left (5 a^2+21 b^2\right ) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d}-\frac{2 b \left (9 a^2+5 b^2\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d}+\frac{2 a \left (5 a^2+21 b^2\right ) \sin (c+d x)}{21 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 b \left (9 a^2+5 b^2\right ) \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}+\frac{2 a^2 \sin (c+d x) (a+b \cos (c+d x))}{7 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{32 a^2 b \sin (c+d x)}{35 d \cos ^{\frac{5}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Rule 2792
Rule 3021
Rule 2748
Rule 2636
Rule 2641
Rule 2639
Rubi steps
\begin{align*} \int \frac{(a+b \cos (c+d x))^3}{\cos ^{\frac{9}{2}}(c+d x)} \, dx &=\frac{2 a^2 (a+b \cos (c+d x)) \sin (c+d x)}{7 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2}{7} \int \frac{8 a^2 b+\frac{1}{2} a \left (5 a^2+21 b^2\right ) \cos (c+d x)+\frac{1}{2} b \left (3 a^2+7 b^2\right ) \cos ^2(c+d x)}{\cos ^{\frac{7}{2}}(c+d x)} \, dx\\ &=\frac{32 a^2 b \sin (c+d x)}{35 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 a^2 (a+b \cos (c+d x)) \sin (c+d x)}{7 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{4}{35} \int \frac{\frac{5}{4} a \left (5 a^2+21 b^2\right )+\frac{7}{4} b \left (9 a^2+5 b^2\right ) \cos (c+d x)}{\cos ^{\frac{5}{2}}(c+d x)} \, dx\\ &=\frac{32 a^2 b \sin (c+d x)}{35 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 a^2 (a+b \cos (c+d x)) \sin (c+d x)}{7 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{1}{5} \left (b \left (9 a^2+5 b^2\right )\right ) \int \frac{1}{\cos ^{\frac{3}{2}}(c+d x)} \, dx+\frac{1}{7} \left (a \left (5 a^2+21 b^2\right )\right ) \int \frac{1}{\cos ^{\frac{5}{2}}(c+d x)} \, dx\\ &=\frac{32 a^2 b \sin (c+d x)}{35 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 a \left (5 a^2+21 b^2\right ) \sin (c+d x)}{21 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 b \left (9 a^2+5 b^2\right ) \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}+\frac{2 a^2 (a+b \cos (c+d x)) \sin (c+d x)}{7 d \cos ^{\frac{7}{2}}(c+d x)}-\frac{1}{5} \left (b \left (9 a^2+5 b^2\right )\right ) \int \sqrt{\cos (c+d x)} \, dx+\frac{1}{21} \left (a \left (5 a^2+21 b^2\right )\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx\\ &=-\frac{2 b \left (9 a^2+5 b^2\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d}+\frac{2 a \left (5 a^2+21 b^2\right ) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d}+\frac{32 a^2 b \sin (c+d x)}{35 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 a \left (5 a^2+21 b^2\right ) \sin (c+d x)}{21 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 b \left (9 a^2+5 b^2\right ) \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}+\frac{2 a^2 (a+b \cos (c+d x)) \sin (c+d x)}{7 d \cos ^{\frac{7}{2}}(c+d x)}\\ \end{align*}
Mathematica [A] time = 0.725602, size = 177, normalized size = 0.91 \[ \frac{10 a \left (5 a^2+21 b^2\right ) \cos ^{\frac{5}{2}}(c+d x) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )-42 b \left (9 a^2+5 b^2\right ) \cos ^{\frac{5}{2}}(c+d x) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )+126 a^2 b \sin (c+d x)+378 a^2 b \sin (c+d x) \cos ^2(c+d x)+25 a^3 \sin (2 (c+d x))+30 a^3 \tan (c+d x)+105 a b^2 \sin (2 (c+d x))+210 b^3 \sin (c+d x) \cos ^2(c+d x)}{105 d \cos ^{\frac{5}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Maple [B] time = 9.352, size = 847, normalized size = 4.4 \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{{\left (b \cos \left (d x + c\right ) + a\right )}^{3}}{\cos \left (d x + c\right )^{\frac{9}{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{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}}{\cos \left (d x + c\right )^{\frac{9}{2}}}, 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 )}^{3}}{\cos \left (d x + c\right )^{\frac{9}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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