Optimal. Leaf size=182 \[ \frac{2 a^3 (245 A+224 B+160 C) \sin (c+d x)}{105 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (35 A+56 B+40 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{105 d}+\frac{2 a^{5/2} A \tanh ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right )}{d}+\frac{2 a (7 B+5 C) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{35 d}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{7 d} \]
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Rubi [A] time = 0.661587, antiderivative size = 182, normalized size of antiderivative = 1., number of steps used = 6, 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} \[ \frac{2 a^3 (245 A+224 B+160 C) \sin (c+d x)}{105 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (35 A+56 B+40 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{105 d}+\frac{2 a^{5/2} A \tanh ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right )}{d}+\frac{2 a (7 B+5 C) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{35 d}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{7 d} \]
Antiderivative was successfully verified.
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Rule 3045
Rule 2976
Rule 2981
Rule 2773
Rule 206
Rubi steps
\begin{align*} \int (a+a \cos (c+d x))^{5/2} \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))^{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{1}{2} a (7 B+5 C) \cos (c+d x)\right ) \sec (c+d x) \, dx}{7 a}\\ &=\frac{2 a (7 B+5 C) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{35 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{1}{4} a^2 (35 A+56 B+40 C) \cos (c+d x)\right ) \sec (c+d x) \, dx}{35 a}\\ &=\frac{2 a^2 (35 A+56 B+40 C) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{105 d}+\frac{2 a (7 B+5 C) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{35 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{1}{8} a^3 (245 A+224 B+160 C) \cos (c+d x)\right ) \sec (c+d x) \, dx}{105 a}\\ &=\frac{2 a^3 (245 A+224 B+160 C) \sin (c+d x)}{105 d \sqrt{a+a \cos (c+d x)}}+\frac{2 a^2 (35 A+56 B+40 C) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{105 d}+\frac{2 a (7 B+5 C) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{35 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 (245 A+224 B+160 C) \sin (c+d x)}{105 d \sqrt{a+a \cos (c+d x)}}+\frac{2 a^2 (35 A+56 B+40 C) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{105 d}+\frac{2 a (7 B+5 C) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{35 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 (245 A+224 B+160 C) \sin (c+d x)}{105 d \sqrt{a+a \cos (c+d x)}}+\frac{2 a^2 (35 A+56 B+40 C) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{105 d}+\frac{2 a (7 B+5 C) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{35 d}+\frac{2 C (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{7 d}\\ \end{align*}
Mathematica [A] time = 0.734871, size = 127, normalized size = 0.7 \[ \frac{a^2 \sec \left (\frac{1}{2} (c+d x)\right ) \sqrt{a (\cos (c+d x)+1)} \left (2 \sin \left (\frac{1}{2} (c+d x)\right ) ((140 A+392 B+505 C) \cos (c+d x)+1120 A+6 (7 B+20 C) \cos (2 (c+d x))+1246 B+15 C \cos (3 (c+d x))+1040 C)+420 \sqrt{2} A \tanh ^{-1}\left (\sqrt{2} \sin \left (\frac{1}{2} (c+d x)\right )\right )\right )}{420 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.201, size = 377, normalized size = 2.1 \begin{align*}{\frac{1}{105\,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 ( -240\,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\,\sqrt{a \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}\sqrt{a}\sqrt{2} \left ( B+5\,C \right ) \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}-140\,\sqrt{a \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}\sqrt{a}\sqrt{2} \left ( A+4\,B+8\,C \right ) \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}+630\,A\sqrt{2}\sqrt{a \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}\sqrt{a}+105\,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+105\,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+840\,B\sqrt{a}\sqrt{2}\sqrt{a \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}+840\,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 = 1.87061, size = 188, normalized size = 1.03 \begin{align*} \frac{14 \,{\left (3 \, \sqrt{2} a^{2} \sin \left (\frac{5}{2} \, d x + \frac{5}{2} \, c\right ) + 25 \, \sqrt{2} a^{2} \sin \left (\frac{3}{2} \, d x + \frac{3}{2} \, c\right ) + 150 \, \sqrt{2} a^{2} \sin \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )} B \sqrt{a} + 5 \,{\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}}{420 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.16476, size = 540, normalized size = 2.97 \begin{align*} \frac{105 \,{\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 (15 \, C a^{2} \cos \left (d x + c\right )^{3} + 3 \,{\left (7 \, B + 20 \, C\right )} a^{2} \cos \left (d x + c\right )^{2} +{\left (35 \, A + 98 \, B + 115 \, C\right )} a^{2} \cos \left (d x + c\right ) +{\left (280 \, A + 301 \, B + 230 \, C\right )} a^{2}\right )} \sqrt{a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{210 \,{\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 = 3.93448, size = 397, normalized size = 2.18 \begin{align*} \frac{\frac{105 \, 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 (315 \, \sqrt{2} A a^{6} + 420 \, \sqrt{2} B a^{6} + 420 \, \sqrt{2} C a^{6} +{\left (875 \, \sqrt{2} A a^{6} + 980 \, \sqrt{2} B a^{6} + 700 \, \sqrt{2} C a^{6} +{\left (805 \, \sqrt{2} A a^{6} + 784 \, \sqrt{2} B a^{6} + 560 \, \sqrt{2} C a^{6} +{\left (245 \, \sqrt{2} A a^{6} + 224 \, \sqrt{2} B a^{6} + 160 \, \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}}}}{105 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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