Optimal. Leaf size=142 \[ \frac{2 a^2 (15 A+20 B+12 C) \sin (c+d x)}{15 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^{3/2} A \tanh ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right )}{d}+\frac{2 a (5 B+3 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{15 d}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d} \]
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Rubi [A] time = 0.427034, antiderivative size = 142, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 41, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.122, Rules used = {3045, 2976, 2981, 2773, 206} \[ \frac{2 a^2 (15 A+20 B+12 C) \sin (c+d x)}{15 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^{3/2} A \tanh ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right )}{d}+\frac{2 a (5 B+3 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{15 d}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d} \]
Antiderivative was successfully verified.
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Rule 3045
Rule 2976
Rule 2981
Rule 2773
Rule 206
Rubi steps
\begin{align*} \int (a+a \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx &=\frac{2 C (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{5 d}+\frac{2 \int (a+a \cos (c+d x))^{3/2} \left (\frac{5 a A}{2}+\frac{1}{2} a (5 B+3 C) \cos (c+d x)\right ) \sec (c+d x) \, dx}{5 a}\\ &=\frac{2 a (5 B+3 C) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{15 d}+\frac{2 C (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{5 d}+\frac{4 \int \sqrt{a+a \cos (c+d x)} \left (\frac{15 a^2 A}{4}+\frac{1}{4} a^2 (15 A+20 B+12 C) \cos (c+d x)\right ) \sec (c+d x) \, dx}{15 a}\\ &=\frac{2 a^2 (15 A+20 B+12 C) \sin (c+d x)}{15 d \sqrt{a+a \cos (c+d x)}}+\frac{2 a (5 B+3 C) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{15 d}+\frac{2 C (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{5 d}+(a A) \int \sqrt{a+a \cos (c+d x)} \sec (c+d x) \, dx\\ &=\frac{2 a^2 (15 A+20 B+12 C) \sin (c+d x)}{15 d \sqrt{a+a \cos (c+d x)}}+\frac{2 a (5 B+3 C) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{15 d}+\frac{2 C (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{5 d}-\frac{\left (2 a^2 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^{3/2} A \tanh ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a+a \cos (c+d x)}}\right )}{d}+\frac{2 a^2 (15 A+20 B+12 C) \sin (c+d x)}{15 d \sqrt{a+a \cos (c+d x)}}+\frac{2 a (5 B+3 C) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{15 d}+\frac{2 C (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{5 d}\\ \end{align*}
Mathematica [A] time = 0.420754, size = 105, normalized size = 0.74 \[ \frac{a \sec \left (\frac{1}{2} (c+d x)\right ) \sqrt{a (\cos (c+d x)+1)} \left (\sin \left (\frac{1}{2} (c+d x)\right ) (30 A+2 (5 B+9 C) \cos (c+d x)+50 B+3 C \cos (2 (c+d x))+39 C)+15 \sqrt{2} A \tanh ^{-1}\left (\sqrt{2} \sin \left (\frac{1}{2} (c+d x)\right )\right )\right )}{15 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.188, size = 335, normalized size = 2.4 \begin{align*}{\frac{1}{15\,d}\sqrt{a}\cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \sqrt{a \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}} \left ( 24\,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}-20\,\sqrt{a \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}\sqrt{a}\sqrt{2} \left ( B+3\,C \right ) \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}+30\,A\sqrt{2}\sqrt{a \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}\sqrt{a}+15\,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+15\,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+60\,B\sqrt{a}\sqrt{2}\sqrt{a \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}+60\,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 = 1.8286, size = 126, normalized size = 0.89 \begin{align*} \frac{10 \,{\left (\sqrt{2} a \sin \left (\frac{3}{2} \, d x + \frac{3}{2} \, c\right ) + 9 \, \sqrt{2} a \sin \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )} B \sqrt{a} + 3 \,{\left (\sqrt{2} a \sin \left (\frac{5}{2} \, d x + \frac{5}{2} \, c\right ) + 5 \, \sqrt{2} a \sin \left (\frac{3}{2} \, d x + \frac{3}{2} \, c\right ) + 20 \, \sqrt{2} a \sin \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )} C \sqrt{a}}{30 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.02315, size = 456, normalized size = 3.21 \begin{align*} \frac{15 \,{\left (A a \cos \left (d x + c\right ) + A a\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 \cos \left (d x + c\right )^{2} +{\left (5 \, B + 9 \, C\right )} a \cos \left (d x + c\right ) +{\left (15 \, A + 25 \, B + 18 \, C\right )} a\right )} \sqrt{a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{30 \,{\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 [B] time = 4.80781, size = 343, normalized size = 2.42 \begin{align*} \frac{\frac{15 \, A a^{\frac{5}{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 (15 \, \sqrt{2} A a^{4} + 30 \, \sqrt{2} B a^{4} + 30 \, \sqrt{2} C a^{4} +{\left (30 \, \sqrt{2} A a^{4} + 50 \, \sqrt{2} B a^{4} + 30 \, \sqrt{2} C a^{4} +{\left (15 \, \sqrt{2} A a^{4} + 20 \, \sqrt{2} B a^{4} + 12 \, \sqrt{2} C a^{4}\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{5}{2}}}}{15 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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