Optimal. Leaf size=170 \[ \frac{2 a^3 (49 A+32 C) \sin (c+d x)}{21 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (7 A+8 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{21 d}+\frac{2 a^{5/2} A \tanh ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right )}{d}+\frac{2 a C \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{7 d}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{7 d} \]
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Rubi [A] time = 0.566954, antiderivative size = 170, 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 = {3046, 2976, 2981, 2773, 206} \[ \frac{2 a^3 (49 A+32 C) \sin (c+d x)}{21 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (7 A+8 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{21 d}+\frac{2 a^{5/2} A \tanh ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right )}{d}+\frac{2 a C \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{7 d}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{7 d} \]
Antiderivative was successfully verified.
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Rule 3046
Rule 2976
Rule 2981
Rule 2773
Rule 206
Rubi steps
\begin{align*} \int (a+a \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx &=\frac{2 C (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{7 d}+\frac{2 \int (a+a \cos (c+d x))^{5/2} \left (\frac{7 a A}{2}+\frac{5}{2} a C \cos (c+d x)\right ) \sec (c+d x) \, dx}{7 a}\\ &=\frac{2 a C (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{7 d}+\frac{2 C (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{7 d}+\frac{4 \int (a+a \cos (c+d x))^{3/2} \left (\frac{35 a^2 A}{4}+\frac{5}{4} a^2 (7 A+8 C) \cos (c+d x)\right ) \sec (c+d x) \, dx}{35 a}\\ &=\frac{2 a^2 (7 A+8 C) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{21 d}+\frac{2 a C (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{7 d}+\frac{2 C (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{7 d}+\frac{8 \int \sqrt{a+a \cos (c+d x)} \left (\frac{105 a^3 A}{8}+\frac{5}{8} a^3 (49 A+32 C) \cos (c+d x)\right ) \sec (c+d x) \, dx}{105 a}\\ &=\frac{2 a^3 (49 A+32 C) \sin (c+d x)}{21 d \sqrt{a+a \cos (c+d x)}}+\frac{2 a^2 (7 A+8 C) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{21 d}+\frac{2 a C (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{7 d}+\frac{2 C (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{7 d}+\left (a^2 A\right ) \int \sqrt{a+a \cos (c+d x)} \sec (c+d x) \, dx\\ &=\frac{2 a^3 (49 A+32 C) \sin (c+d x)}{21 d \sqrt{a+a \cos (c+d x)}}+\frac{2 a^2 (7 A+8 C) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{21 d}+\frac{2 a C (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{7 d}+\frac{2 C (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{7 d}-\frac{\left (2 a^3 A\right ) \operatorname{Subst}\left (\int \frac{1}{a-x^2} \, dx,x,-\frac{a \sin (c+d x)}{\sqrt{a+a \cos (c+d x)}}\right )}{d}\\ &=\frac{2 a^{5/2} A \tanh ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a+a \cos (c+d x)}}\right )}{d}+\frac{2 a^3 (49 A+32 C) \sin (c+d x)}{21 d \sqrt{a+a \cos (c+d x)}}+\frac{2 a^2 (7 A+8 C) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{21 d}+\frac{2 a C (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{7 d}+\frac{2 C (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{7 d}\\ \end{align*}
Mathematica [A] time = 0.525522, size = 115, normalized size = 0.68 \[ \frac{a^2 \sec \left (\frac{1}{2} (c+d x)\right ) \sqrt{a (\cos (c+d x)+1)} \left (2 \sin \left (\frac{1}{2} (c+d x)\right ) ((28 A+101 C) \cos (c+d x)+224 A+24 C \cos (2 (c+d x))+3 C \cos (3 (c+d x))+208 C)+84 \sqrt{2} A \tanh ^{-1}\left (\sqrt{2} \sin \left (\frac{1}{2} (c+d x)\right )\right )\right )}{84 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.084, size = 346, normalized size = 2. \begin{align*}{\frac{1}{21\,d}{a}^{{\frac{3}{2}}}\cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \sqrt{a \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}} \left ( -48\,C\sqrt{2}\sqrt{a \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}\sqrt{a} \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{6}+168\,C\sqrt{2}\sqrt{a \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}\sqrt{a} \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}-28\,\sqrt{a}\sqrt{a \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}\sqrt{2} \left ( A+8\,C \right ) \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}+126\,A\sqrt{2}\sqrt{a \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}\sqrt{a}+21\,A\ln \left ( -4\,{\frac{\sqrt{a}\sqrt{2}\sqrt{a \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}-a\sqrt{2}\cos \left ( 1/2\,dx+c/2 \right ) +2\,a}{-2\,\cos \left ( 1/2\,dx+c/2 \right ) +\sqrt{2}}} \right ) a+21\,A\ln \left ( 4\,{\frac{a\sqrt{2}\cos \left ( 1/2\,dx+c/2 \right ) +\sqrt{a}\sqrt{2}\sqrt{a \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}+2\,a}{2\,\cos \left ( 1/2\,dx+c/2 \right ) +\sqrt{2}}} \right ) a+168\,C\sqrt{2}\sqrt{a \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}\sqrt{a} \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-1}{\frac{1}{\sqrt{a \left ( \cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.93268, size = 105, normalized size = 0.62 \begin{align*} \frac{{\left (3 \, \sqrt{2} a^{2} \sin \left (\frac{7}{2} \, d x + \frac{7}{2} \, c\right ) + 21 \, \sqrt{2} a^{2} \sin \left (\frac{5}{2} \, d x + \frac{5}{2} \, c\right ) + 77 \, \sqrt{2} a^{2} \sin \left (\frac{3}{2} \, d x + \frac{3}{2} \, c\right ) + 315 \, \sqrt{2} a^{2} \sin \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )} C \sqrt{a}}{84 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.72862, size = 500, normalized size = 2.94 \begin{align*} \frac{21 \,{\left (A a^{2} \cos \left (d x + c\right ) + A a^{2}\right )} \sqrt{a} \log \left (\frac{a \cos \left (d x + c\right )^{3} - 7 \, a \cos \left (d x + c\right )^{2} - 4 \, \sqrt{a \cos \left (d x + c\right ) + a} \sqrt{a}{\left (\cos \left (d x + c\right ) - 2\right )} \sin \left (d x + c\right ) + 8 \, a}{\cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2}}\right ) + 4 \,{\left (3 \, C a^{2} \cos \left (d x + c\right )^{3} + 12 \, C a^{2} \cos \left (d x + c\right )^{2} +{\left (7 \, A + 23 \, C\right )} a^{2} \cos \left (d x + c\right ) + 2 \,{\left (28 \, A + 23 \, C\right )} a^{2}\right )} \sqrt{a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{42 \,{\left (d \cos \left (d x + c\right ) + d\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 [A] time = 3.82816, size = 348, normalized size = 2.05 \begin{align*} \frac{\frac{21 \, A a^{\frac{7}{2}} \log \left (\frac{{\left | 2 \,{\left (\sqrt{a} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - \sqrt{a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + a}\right )}^{2} - 4 \, \sqrt{2}{\left | a \right |} - 6 \, a \right |}}{{\left | 2 \,{\left (\sqrt{a} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - \sqrt{a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + a}\right )}^{2} + 4 \, \sqrt{2}{\left | a \right |} - 6 \, a \right |}}\right )}{{\left | a \right |}} + \frac{2 \,{\left (63 \, \sqrt{2} A a^{6} + 84 \, \sqrt{2} C a^{6} +{\left (175 \, \sqrt{2} A a^{6} + 140 \, \sqrt{2} C a^{6} +{\left (161 \, \sqrt{2} A a^{6} + 112 \, \sqrt{2} C a^{6} +{\left (49 \, \sqrt{2} A a^{6} + 32 \, \sqrt{2} C a^{6}\right )} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2}\right )} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2}\right )} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2}\right )} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{{\left (a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + a\right )}^{\frac{7}{2}}}}{21 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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