Optimal. Leaf size=261 \[ -\frac{(273 A-397 B) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{210 a^2 d}-\frac{(15 A-19 B) \tanh ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right )}{2 \sqrt{2} a^{3/2} d}+\frac{(A-B) \sin (c+d x) \cos ^4(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}-\frac{(7 A-11 B) \sin (c+d x) \cos ^3(c+d x)}{14 a d \sqrt{a \cos (c+d x)+a}}+\frac{(63 A-67 B) \sin (c+d x) \cos ^2(c+d x)}{70 a d \sqrt{a \cos (c+d x)+a}}+\frac{(651 A-799 B) \sin (c+d x)}{105 a d \sqrt{a \cos (c+d x)+a}} \]
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Rubi [A] time = 0.785725, antiderivative size = 261, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.212, Rules used = {2977, 2983, 2968, 3023, 2751, 2649, 206} \[ -\frac{(273 A-397 B) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{210 a^2 d}-\frac{(15 A-19 B) \tanh ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right )}{2 \sqrt{2} a^{3/2} d}+\frac{(A-B) \sin (c+d x) \cos ^4(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}-\frac{(7 A-11 B) \sin (c+d x) \cos ^3(c+d x)}{14 a d \sqrt{a \cos (c+d x)+a}}+\frac{(63 A-67 B) \sin (c+d x) \cos ^2(c+d x)}{70 a d \sqrt{a \cos (c+d x)+a}}+\frac{(651 A-799 B) \sin (c+d x)}{105 a d \sqrt{a \cos (c+d x)+a}} \]
Antiderivative was successfully verified.
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Rule 2977
Rule 2983
Rule 2968
Rule 3023
Rule 2751
Rule 2649
Rule 206
Rubi steps
\begin{align*} \int \frac{\cos ^4(c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^{3/2}} \, dx &=\frac{(A-B) \cos ^4(c+d x) \sin (c+d x)}{2 d (a+a \cos (c+d x))^{3/2}}+\frac{\int \frac{\cos ^3(c+d x) \left (4 a (A-B)-\frac{1}{2} a (7 A-11 B) \cos (c+d x)\right )}{\sqrt{a+a \cos (c+d x)}} \, dx}{2 a^2}\\ &=\frac{(A-B) \cos ^4(c+d x) \sin (c+d x)}{2 d (a+a \cos (c+d x))^{3/2}}-\frac{(7 A-11 B) \cos ^3(c+d x) \sin (c+d x)}{14 a d \sqrt{a+a \cos (c+d x)}}+\frac{\int \frac{\cos ^2(c+d x) \left (-\frac{3}{2} a^2 (7 A-11 B)+\frac{1}{4} a^2 (63 A-67 B) \cos (c+d x)\right )}{\sqrt{a+a \cos (c+d x)}} \, dx}{7 a^3}\\ &=\frac{(A-B) \cos ^4(c+d x) \sin (c+d x)}{2 d (a+a \cos (c+d x))^{3/2}}+\frac{(63 A-67 B) \cos ^2(c+d x) \sin (c+d x)}{70 a d \sqrt{a+a \cos (c+d x)}}-\frac{(7 A-11 B) \cos ^3(c+d x) \sin (c+d x)}{14 a d \sqrt{a+a \cos (c+d x)}}+\frac{2 \int \frac{\cos (c+d x) \left (\frac{1}{2} a^3 (63 A-67 B)-\frac{1}{8} a^3 (273 A-397 B) \cos (c+d x)\right )}{\sqrt{a+a \cos (c+d x)}} \, dx}{35 a^4}\\ &=\frac{(A-B) \cos ^4(c+d x) \sin (c+d x)}{2 d (a+a \cos (c+d x))^{3/2}}+\frac{(63 A-67 B) \cos ^2(c+d x) \sin (c+d x)}{70 a d \sqrt{a+a \cos (c+d x)}}-\frac{(7 A-11 B) \cos ^3(c+d x) \sin (c+d x)}{14 a d \sqrt{a+a \cos (c+d x)}}+\frac{2 \int \frac{\frac{1}{2} a^3 (63 A-67 B) \cos (c+d x)-\frac{1}{8} a^3 (273 A-397 B) \cos ^2(c+d x)}{\sqrt{a+a \cos (c+d x)}} \, dx}{35 a^4}\\ &=\frac{(A-B) \cos ^4(c+d x) \sin (c+d x)}{2 d (a+a \cos (c+d x))^{3/2}}+\frac{(63 A-67 B) \cos ^2(c+d x) \sin (c+d x)}{70 a d \sqrt{a+a \cos (c+d x)}}-\frac{(7 A-11 B) \cos ^3(c+d x) \sin (c+d x)}{14 a d \sqrt{a+a \cos (c+d x)}}-\frac{(273 A-397 B) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{210 a^2 d}+\frac{4 \int \frac{-\frac{1}{16} a^4 (273 A-397 B)+\frac{1}{8} a^4 (651 A-799 B) \cos (c+d x)}{\sqrt{a+a \cos (c+d x)}} \, dx}{105 a^5}\\ &=\frac{(A-B) \cos ^4(c+d x) \sin (c+d x)}{2 d (a+a \cos (c+d x))^{3/2}}+\frac{(651 A-799 B) \sin (c+d x)}{105 a d \sqrt{a+a \cos (c+d x)}}+\frac{(63 A-67 B) \cos ^2(c+d x) \sin (c+d x)}{70 a d \sqrt{a+a \cos (c+d x)}}-\frac{(7 A-11 B) \cos ^3(c+d x) \sin (c+d x)}{14 a d \sqrt{a+a \cos (c+d x)}}-\frac{(273 A-397 B) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{210 a^2 d}-\frac{(15 A-19 B) \int \frac{1}{\sqrt{a+a \cos (c+d x)}} \, dx}{4 a}\\ &=\frac{(A-B) \cos ^4(c+d x) \sin (c+d x)}{2 d (a+a \cos (c+d x))^{3/2}}+\frac{(651 A-799 B) \sin (c+d x)}{105 a d \sqrt{a+a \cos (c+d x)}}+\frac{(63 A-67 B) \cos ^2(c+d x) \sin (c+d x)}{70 a d \sqrt{a+a \cos (c+d x)}}-\frac{(7 A-11 B) \cos ^3(c+d x) \sin (c+d x)}{14 a d \sqrt{a+a \cos (c+d x)}}-\frac{(273 A-397 B) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{210 a^2 d}+\frac{(15 A-19 B) \operatorname{Subst}\left (\int \frac{1}{2 a-x^2} \, dx,x,-\frac{a \sin (c+d x)}{\sqrt{a+a \cos (c+d x)}}\right )}{2 a d}\\ &=-\frac{(15 A-19 B) \tanh ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a+a \cos (c+d x)}}\right )}{2 \sqrt{2} a^{3/2} d}+\frac{(A-B) \cos ^4(c+d x) \sin (c+d x)}{2 d (a+a \cos (c+d x))^{3/2}}+\frac{(651 A-799 B) \sin (c+d x)}{105 a d \sqrt{a+a \cos (c+d x)}}+\frac{(63 A-67 B) \cos ^2(c+d x) \sin (c+d x)}{70 a d \sqrt{a+a \cos (c+d x)}}-\frac{(7 A-11 B) \cos ^3(c+d x) \sin (c+d x)}{14 a d \sqrt{a+a \cos (c+d x)}}-\frac{(273 A-397 B) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{210 a^2 d}\\ \end{align*}
Mathematica [A] time = 1.06607, size = 167, normalized size = 0.64 \[ \frac{105 (15 A-19 B) \cos ^5\left (\frac{1}{2} (c+d x)\right ) \tanh ^{-1}\left (\sin \left (\frac{1}{2} (c+d x)\right )\right )-\frac{1}{2} \sin \left (\frac{1}{2} (c+d x)\right ) \cos ^3\left (\frac{1}{2} (c+d x)\right ) (6 (273 A-277 B) \cos (c+d x)+(256 B-84 A) \cos (2 (c+d x))+42 A \cos (3 (c+d x))+1974 A-18 B \cos (3 (c+d x))+15 B \cos (4 (c+d x))-2161 B)}{105 d \left (\sin ^2\left (\frac{1}{2} (c+d x)\right )-1\right ) (a (\cos (c+d x)+1))^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 2.718, size = 448, normalized size = 1.7 \begin{align*}{\frac{1}{420\,d}\sqrt{a \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}} \left ( 960\,B\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 ) ^{8}-96\,\sqrt{2}\sqrt{a}\sqrt{a \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}} \left ( 7\,A+17\,B \right ) \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{6}+224\,\sqrt{2}\sqrt{a}\sqrt{a \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}} \left ( 3\,A+8\,B \right ) \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}-35\,\sqrt{2} \left ( 48\,A\sqrt{a}\sqrt{a \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}-45\,A\ln \left ( 4\,{\frac{\sqrt{a}\sqrt{a \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}+a}{\cos \left ( 1/2\,dx+c/2 \right ) }} \right ) a-16\,B\sqrt{a}\sqrt{a \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}+57\,B\ln \left ( 4\,{\frac{\sqrt{a}\sqrt{a \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}+a}{\cos \left ( 1/2\,dx+c/2 \right ) }} \right ) a \right ) \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1575\,\sqrt{2}\ln \left ( 4\,{\frac{\sqrt{a}\sqrt{a \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}+a}{\cos \left ( 1/2\,dx+c/2 \right ) }} \right ) aA+1995\,\sqrt{2}\ln \left ( 4\,{\frac{\sqrt{a}\sqrt{a \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}+a}{\cos \left ( 1/2\,dx+c/2 \right ) }} \right ) aB+1785\,A\sqrt{a}\sqrt{2}\sqrt{a \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}-1785\,B\sqrt{2}\sqrt{a \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}\sqrt{a} \right ) \left ( \cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-1}{a}^{-{\frac{5}{2}}} \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-1}{\frac{1}{\sqrt{ \left ( \cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}a}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [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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Fricas [A] time = 1.79064, size = 653, normalized size = 2.5 \begin{align*} -\frac{105 \, \sqrt{2}{\left ({\left (15 \, A - 19 \, B\right )} \cos \left (d x + c\right )^{2} + 2 \,{\left (15 \, A - 19 \, B\right )} \cos \left (d x + c\right ) + 15 \, A - 19 \, B\right )} \sqrt{a} \log \left (-\frac{a \cos \left (d x + c\right )^{2} - 2 \, \sqrt{2} \sqrt{a \cos \left (d x + c\right ) + a} \sqrt{a} \sin \left (d x + c\right ) - 2 \, a \cos \left (d x + c\right ) - 3 \, a}{\cos \left (d x + c\right )^{2} + 2 \, \cos \left (d x + c\right ) + 1}\right ) - 4 \,{\left (60 \, B \cos \left (d x + c\right )^{4} + 12 \,{\left (7 \, A - 3 \, B\right )} \cos \left (d x + c\right )^{3} - 28 \,{\left (3 \, A - 7 \, B\right )} \cos \left (d x + c\right )^{2} + 12 \,{\left (63 \, A - 67 \, B\right )} \cos \left (d x + c\right ) + 1029 \, A - 1201 \, B\right )} \sqrt{a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{840 \,{\left (a^{2} d \cos \left (d x + c\right )^{2} + 2 \, a^{2} d \cos \left (d x + c\right ) + a^{2} 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 = 1.99236, size = 343, normalized size = 1.31 \begin{align*} \frac{\frac{105 \,{\left (15 \, \sqrt{2} A - 19 \, \sqrt{2} B\right )} \log \left ({\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 |}\right )}{a^{\frac{3}{2}}} + \frac{{\left ({\left ({\left ({\left (\frac{105 \,{\left (\sqrt{2} A a^{5} - \sqrt{2} B a^{5}\right )} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2}}{a^{3}} + \frac{4 \,{\left (693 \, \sqrt{2} A a^{5} - 877 \, \sqrt{2} B a^{5}\right )}}{a^{3}}\right )} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + \frac{14 \,{\left (453 \, \sqrt{2} A a^{5} - 517 \, \sqrt{2} B a^{5}\right )}}{a^{3}}\right )} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + \frac{140 \,{\left (39 \, \sqrt{2} A a^{5} - 47 \, \sqrt{2} B a^{5}\right )}}{a^{3}}\right )} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + \frac{1785 \,{\left (\sqrt{2} A a^{5} - \sqrt{2} B a^{5}\right )}}{a^{3}}\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}}}}{420 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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