Optimal. Leaf size=181 \[ -\frac{13 F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{6 a^3 d}-\frac{49 E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{10 a^3 d}+\frac{49 \sin (c+d x)}{10 a^3 d \sqrt{\cos (c+d x)}}-\frac{13 \sin (c+d x)}{6 d \sqrt{\cos (c+d x)} \left (a^3 \cos (c+d x)+a^3\right )}-\frac{8 \sin (c+d x)}{15 a d \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^2}-\frac{\sin (c+d x)}{5 d \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^3} \]
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Rubi [A] time = 0.347249, antiderivative size = 181, 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 = {2766, 2978, 2748, 2636, 2639, 2641} \[ -\frac{13 F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{6 a^3 d}-\frac{49 E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{10 a^3 d}+\frac{49 \sin (c+d x)}{10 a^3 d \sqrt{\cos (c+d x)}}-\frac{13 \sin (c+d x)}{6 d \sqrt{\cos (c+d x)} \left (a^3 \cos (c+d x)+a^3\right )}-\frac{8 \sin (c+d x)}{15 a d \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^2}-\frac{\sin (c+d x)}{5 d \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^3} \]
Antiderivative was successfully verified.
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Rule 2766
Rule 2978
Rule 2748
Rule 2636
Rule 2639
Rule 2641
Rubi steps
\begin{align*} \int \frac{1}{\cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))^3} \, dx &=-\frac{\sin (c+d x)}{5 d \sqrt{\cos (c+d x)} (a+a \cos (c+d x))^3}+\frac{\int \frac{\frac{11 a}{2}-\frac{5}{2} a \cos (c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))^2} \, dx}{5 a^2}\\ &=-\frac{\sin (c+d x)}{5 d \sqrt{\cos (c+d x)} (a+a \cos (c+d x))^3}-\frac{8 \sin (c+d x)}{15 a d \sqrt{\cos (c+d x)} (a+a \cos (c+d x))^2}+\frac{\int \frac{\frac{41 a^2}{2}-12 a^2 \cos (c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))} \, dx}{15 a^4}\\ &=-\frac{\sin (c+d x)}{5 d \sqrt{\cos (c+d x)} (a+a \cos (c+d x))^3}-\frac{8 \sin (c+d x)}{15 a d \sqrt{\cos (c+d x)} (a+a \cos (c+d x))^2}-\frac{13 \sin (c+d x)}{6 d \sqrt{\cos (c+d x)} \left (a^3+a^3 \cos (c+d x)\right )}+\frac{\int \frac{\frac{147 a^3}{4}-\frac{65}{4} a^3 \cos (c+d x)}{\cos ^{\frac{3}{2}}(c+d x)} \, dx}{15 a^6}\\ &=-\frac{\sin (c+d x)}{5 d \sqrt{\cos (c+d x)} (a+a \cos (c+d x))^3}-\frac{8 \sin (c+d x)}{15 a d \sqrt{\cos (c+d x)} (a+a \cos (c+d x))^2}-\frac{13 \sin (c+d x)}{6 d \sqrt{\cos (c+d x)} \left (a^3+a^3 \cos (c+d x)\right )}-\frac{13 \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{12 a^3}+\frac{49 \int \frac{1}{\cos ^{\frac{3}{2}}(c+d x)} \, dx}{20 a^3}\\ &=-\frac{13 F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{6 a^3 d}+\frac{49 \sin (c+d x)}{10 a^3 d \sqrt{\cos (c+d x)}}-\frac{\sin (c+d x)}{5 d \sqrt{\cos (c+d x)} (a+a \cos (c+d x))^3}-\frac{8 \sin (c+d x)}{15 a d \sqrt{\cos (c+d x)} (a+a \cos (c+d x))^2}-\frac{13 \sin (c+d x)}{6 d \sqrt{\cos (c+d x)} \left (a^3+a^3 \cos (c+d x)\right )}-\frac{49 \int \sqrt{\cos (c+d x)} \, dx}{20 a^3}\\ &=-\frac{49 E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{10 a^3 d}-\frac{13 F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{6 a^3 d}+\frac{49 \sin (c+d x)}{10 a^3 d \sqrt{\cos (c+d x)}}-\frac{\sin (c+d x)}{5 d \sqrt{\cos (c+d x)} (a+a \cos (c+d x))^3}-\frac{8 \sin (c+d x)}{15 a d \sqrt{\cos (c+d x)} (a+a \cos (c+d x))^2}-\frac{13 \sin (c+d x)}{6 d \sqrt{\cos (c+d x)} \left (a^3+a^3 \cos (c+d x)\right )}\\ \end{align*}
Mathematica [C] time = 1.89995, size = 364, normalized size = 2.01 \[ \frac{\cos ^6\left (\frac{1}{2} (c+d x)\right ) \left (\frac{\csc \left (\frac{c}{2}\right ) \sec \left (\frac{c}{2}\right ) \left (1284 \cos \left (\frac{1}{2} (c-d x)\right )+921 \cos \left (\frac{1}{2} (3 c+d x)\right )+1243 \cos \left (\frac{1}{2} (c+3 d x)\right )+374 \cos \left (\frac{1}{2} (5 c+3 d x)\right )+670 \cos \left (\frac{1}{2} (3 c+5 d x)\right )+65 \cos \left (\frac{1}{2} (7 c+5 d x)\right )+147 \cos \left (\frac{1}{2} (5 c+7 d x)\right )\right ) \sec ^5\left (\frac{1}{2} (c+d x)\right )}{16 d \sqrt{\cos (c+d x)}}-\frac{4 i \sqrt{2} e^{-i (c+d x)} \left (147 \left (-1+e^{2 i c}\right ) \sqrt{1+e^{2 i (c+d x)}} \text{Hypergeometric2F1}\left (-\frac{1}{4},\frac{1}{2},\frac{3}{4},-e^{2 i (c+d x)}\right )-65 \left (-1+e^{2 i c}\right ) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \text{Hypergeometric2F1}\left (\frac{1}{4},\frac{1}{2},\frac{5}{4},-e^{2 i (c+d x)}\right )+147 \left (1+e^{2 i (c+d x)}\right )\right )}{\left (-1+e^{2 i c}\right ) d \sqrt{e^{-i (c+d x)} \left (1+e^{2 i (c+d x)}\right )}}\right )}{15 a^3 (\cos (c+d x)+1)^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 2.584, size = 555, 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(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \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{\sqrt{\cos \left (d x + c\right )}}{a^{3} \cos \left (d x + c\right )^{5} + 3 \, a^{3} \cos \left (d x + c\right )^{4} + 3 \, a^{3} \cos \left (d x + c\right )^{3} + a^{3} \cos \left (d x + c\right )^{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{1}{{\left (a \cos \left (d x + c\right ) + a\right )}^{3} \cos \left (d x + c\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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