Optimal. Leaf size=266 \[ \frac{2 a^3 (568 A+759 C) \sin (c+d x)}{693 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 a^3 (232 A+297 C) \sin (c+d x)}{693 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (32 A+33 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{231 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{4 a^3 (568 A+759 C) \sin (c+d x)}{693 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{10 a A \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{99 d \cos ^{\frac{9}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{11 d \cos ^{\frac{11}{2}}(c+d x)} \]
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Rubi [A] time = 0.803021, antiderivative size = 266, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.135, Rules used = {3044, 2975, 2980, 2772, 2771} \[ \frac{2 a^3 (568 A+759 C) \sin (c+d x)}{693 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 a^3 (232 A+297 C) \sin (c+d x)}{693 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (32 A+33 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{231 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{4 a^3 (568 A+759 C) \sin (c+d x)}{693 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{10 a A \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{99 d \cos ^{\frac{9}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{11 d \cos ^{\frac{11}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Rule 3044
Rule 2975
Rule 2980
Rule 2772
Rule 2771
Rubi steps
\begin{align*} \int \frac{(a+a \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac{13}{2}}(c+d x)} \, dx &=\frac{2 A (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{11 d \cos ^{\frac{11}{2}}(c+d x)}+\frac{2 \int \frac{(a+a \cos (c+d x))^{5/2} \left (\frac{5 a A}{2}+\frac{1}{2} a (4 A+11 C) \cos (c+d x)\right )}{\cos ^{\frac{11}{2}}(c+d x)} \, dx}{11 a}\\ &=\frac{10 a A (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{99 d \cos ^{\frac{9}{2}}(c+d x)}+\frac{2 A (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{11 d \cos ^{\frac{11}{2}}(c+d x)}+\frac{4 \int \frac{(a+a \cos (c+d x))^{3/2} \left (\frac{3}{4} a^2 (32 A+33 C)+\frac{1}{4} a^2 (56 A+99 C) \cos (c+d x)\right )}{\cos ^{\frac{9}{2}}(c+d x)} \, dx}{99 a}\\ &=\frac{2 a^2 (32 A+33 C) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{231 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{10 a A (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{99 d \cos ^{\frac{9}{2}}(c+d x)}+\frac{2 A (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{11 d \cos ^{\frac{11}{2}}(c+d x)}+\frac{8 \int \frac{\sqrt{a+a \cos (c+d x)} \left (\frac{5}{8} a^3 (232 A+297 C)+\frac{1}{8} a^3 (776 A+1089 C) \cos (c+d x)\right )}{\cos ^{\frac{7}{2}}(c+d x)} \, dx}{693 a}\\ &=\frac{2 a^3 (232 A+297 C) \sin (c+d x)}{693 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+a \cos (c+d x)}}+\frac{2 a^2 (32 A+33 C) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{231 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{10 a A (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{99 d \cos ^{\frac{9}{2}}(c+d x)}+\frac{2 A (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{11 d \cos ^{\frac{11}{2}}(c+d x)}+\frac{1}{231} \left (a^2 (568 A+759 C)\right ) \int \frac{\sqrt{a+a \cos (c+d x)}}{\cos ^{\frac{5}{2}}(c+d x)} \, dx\\ &=\frac{2 a^3 (232 A+297 C) \sin (c+d x)}{693 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+a \cos (c+d x)}}+\frac{2 a^3 (568 A+759 C) \sin (c+d x)}{693 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+a \cos (c+d x)}}+\frac{2 a^2 (32 A+33 C) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{231 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{10 a A (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{99 d \cos ^{\frac{9}{2}}(c+d x)}+\frac{2 A (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{11 d \cos ^{\frac{11}{2}}(c+d x)}+\frac{1}{693} \left (2 a^2 (568 A+759 C)\right ) \int \frac{\sqrt{a+a \cos (c+d x)}}{\cos ^{\frac{3}{2}}(c+d x)} \, dx\\ &=\frac{2 a^3 (232 A+297 C) \sin (c+d x)}{693 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+a \cos (c+d x)}}+\frac{2 a^3 (568 A+759 C) \sin (c+d x)}{693 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+a \cos (c+d x)}}+\frac{4 a^3 (568 A+759 C) \sin (c+d x)}{693 d \sqrt{\cos (c+d x)} \sqrt{a+a \cos (c+d x)}}+\frac{2 a^2 (32 A+33 C) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{231 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{10 a A (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{99 d \cos ^{\frac{9}{2}}(c+d x)}+\frac{2 A (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{11 d \cos ^{\frac{11}{2}}(c+d x)}\\ \end{align*}
Mathematica [A] time = 1.00817, size = 149, normalized size = 0.56 \[ \frac{a^2 \tan \left (\frac{1}{2} (c+d x)\right ) \sqrt{a (\cos (c+d x)+1)} (2 (5014 A+4983 C) \cos (c+d x)+52 (71 A+66 C) \cos (2 (c+d x))+3692 A \cos (3 (c+d x))+568 A \cos (4 (c+d x))+568 A \cos (5 (c+d x))+3628 A+4587 C \cos (3 (c+d x))+759 C \cos (4 (c+d x))+759 C \cos (5 (c+d x))+2673 C)}{2772 d \cos ^{\frac{11}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.127, size = 146, normalized size = 0.6 \begin{align*} -{\frac{2\,{a}^{2} \left ( -1+\cos \left ( dx+c \right ) \right ) \left ( 1136\,A \left ( \cos \left ( dx+c \right ) \right ) ^{5}+1518\,C \left ( \cos \left ( dx+c \right ) \right ) ^{5}+568\,A \left ( \cos \left ( dx+c \right ) \right ) ^{4}+759\,C \left ( \cos \left ( dx+c \right ) \right ) ^{4}+426\,A \left ( \cos \left ( dx+c \right ) \right ) ^{3}+396\,C \left ( \cos \left ( dx+c \right ) \right ) ^{3}+355\,A \left ( \cos \left ( dx+c \right ) \right ) ^{2}+99\,C \left ( \cos \left ( dx+c \right ) \right ) ^{2}+224\,A\cos \left ( dx+c \right ) +63\,A \right ) }{693\,d\sin \left ( dx+c \right ) }\sqrt{a \left ( 1+\cos \left ( dx+c \right ) \right ) } \left ( \cos \left ( dx+c \right ) \right ) ^{-{\frac{11}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.72205, size = 782, normalized size = 2.94 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.61442, size = 387, normalized size = 1.45 \begin{align*} \frac{2 \,{\left (2 \,{\left (568 \, A + 759 \, C\right )} a^{2} \cos \left (d x + c\right )^{5} +{\left (568 \, A + 759 \, C\right )} a^{2} \cos \left (d x + c\right )^{4} + 6 \,{\left (71 \, A + 66 \, C\right )} a^{2} \cos \left (d x + c\right )^{3} +{\left (355 \, A + 99 \, C\right )} a^{2} \cos \left (d x + c\right )^{2} + 224 \, A a^{2} \cos \left (d x + c\right ) + 63 \, A a^{2}\right )} \sqrt{a \cos \left (d x + c\right ) + a} \sqrt{\cos \left (d x + c\right )} \sin \left (d x + c\right )}{693 \,{\left (d \cos \left (d x + c\right )^{7} + d \cos \left (d x + c\right )^{6}\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 (C \cos \left (d x + c\right )^{2} + A\right )}{\left (a \cos \left (d x + c\right ) + a\right )}^{\frac{5}{2}}}{\cos \left (d x + c\right )^{\frac{13}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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