Optimal. Leaf size=193 \[ \frac {2 (35 A+31 C) \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{105 a d}-\frac {4 (35 A+37 C) \sin (c+d x)}{105 d \sqrt {a \cos (c+d x)+a}}+\frac {\sqrt {2} (A+C) \tanh ^{-1}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {2} \sqrt {a \cos (c+d x)+a}}\right )}{\sqrt {a} d}+\frac {2 C \sin (c+d x) \cos ^3(c+d x)}{7 d \sqrt {a \cos (c+d x)+a}}-\frac {2 C \sin (c+d x) \cos ^2(c+d x)}{35 d \sqrt {a \cos (c+d x)+a}} \]
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Rubi [A] time = 0.56, antiderivative size = 193, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3046, 2983, 2968, 3023, 2751, 2649, 206} \[ \frac {2 (35 A+31 C) \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{105 a d}-\frac {4 (35 A+37 C) \sin (c+d x)}{105 d \sqrt {a \cos (c+d x)+a}}+\frac {\sqrt {2} (A+C) \tanh ^{-1}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {2} \sqrt {a \cos (c+d x)+a}}\right )}{\sqrt {a} d}+\frac {2 C \sin (c+d x) \cos ^3(c+d x)}{7 d \sqrt {a \cos (c+d x)+a}}-\frac {2 C \sin (c+d x) \cos ^2(c+d x)}{35 d \sqrt {a \cos (c+d x)+a}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 2649
Rule 2751
Rule 2968
Rule 2983
Rule 3023
Rule 3046
Rubi steps
\begin {align*} \int \frac {\cos ^2(c+d x) \left (A+C \cos ^2(c+d x)\right )}{\sqrt {a+a \cos (c+d x)}} \, dx &=\frac {2 C \cos ^3(c+d x) \sin (c+d x)}{7 d \sqrt {a+a \cos (c+d x)}}+\frac {2 \int \frac {\cos ^2(c+d x) \left (\frac {1}{2} a (7 A+6 C)-\frac {1}{2} a C \cos (c+d x)\right )}{\sqrt {a+a \cos (c+d x)}} \, dx}{7 a}\\ &=-\frac {2 C \cos ^2(c+d x) \sin (c+d x)}{35 d \sqrt {a+a \cos (c+d x)}}+\frac {2 C \cos ^3(c+d x) \sin (c+d x)}{7 d \sqrt {a+a \cos (c+d x)}}+\frac {4 \int \frac {\cos (c+d x) \left (-a^2 C+\frac {1}{4} a^2 (35 A+31 C) \cos (c+d x)\right )}{\sqrt {a+a \cos (c+d x)}} \, dx}{35 a^2}\\ &=-\frac {2 C \cos ^2(c+d x) \sin (c+d x)}{35 d \sqrt {a+a \cos (c+d x)}}+\frac {2 C \cos ^3(c+d x) \sin (c+d x)}{7 d \sqrt {a+a \cos (c+d x)}}+\frac {4 \int \frac {-a^2 C \cos (c+d x)+\frac {1}{4} a^2 (35 A+31 C) \cos ^2(c+d x)}{\sqrt {a+a \cos (c+d x)}} \, dx}{35 a^2}\\ &=-\frac {2 C \cos ^2(c+d x) \sin (c+d x)}{35 d \sqrt {a+a \cos (c+d x)}}+\frac {2 C \cos ^3(c+d x) \sin (c+d x)}{7 d \sqrt {a+a \cos (c+d x)}}+\frac {2 (35 A+31 C) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{105 a d}+\frac {8 \int \frac {\frac {1}{8} a^3 (35 A+31 C)-\frac {1}{4} a^3 (35 A+37 C) \cos (c+d x)}{\sqrt {a+a \cos (c+d x)}} \, dx}{105 a^3}\\ &=-\frac {4 (35 A+37 C) \sin (c+d x)}{105 d \sqrt {a+a \cos (c+d x)}}-\frac {2 C \cos ^2(c+d x) \sin (c+d x)}{35 d \sqrt {a+a \cos (c+d x)}}+\frac {2 C \cos ^3(c+d x) \sin (c+d x)}{7 d \sqrt {a+a \cos (c+d x)}}+\frac {2 (35 A+31 C) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{105 a d}+(A+C) \int \frac {1}{\sqrt {a+a \cos (c+d x)}} \, dx\\ &=-\frac {4 (35 A+37 C) \sin (c+d x)}{105 d \sqrt {a+a \cos (c+d x)}}-\frac {2 C \cos ^2(c+d x) \sin (c+d x)}{35 d \sqrt {a+a \cos (c+d x)}}+\frac {2 C \cos ^3(c+d x) \sin (c+d x)}{7 d \sqrt {a+a \cos (c+d x)}}+\frac {2 (35 A+31 C) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{105 a d}-\frac {(2 (A+C)) \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 )}{d}\\ &=\frac {\sqrt {2} (A+C) \tanh ^{-1}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {2} \sqrt {a+a \cos (c+d x)}}\right )}{\sqrt {a} d}-\frac {4 (35 A+37 C) \sin (c+d x)}{105 d \sqrt {a+a \cos (c+d x)}}-\frac {2 C \cos ^2(c+d x) \sin (c+d x)}{35 d \sqrt {a+a \cos (c+d x)}}+\frac {2 C \cos ^3(c+d x) \sin (c+d x)}{7 d \sqrt {a+a \cos (c+d x)}}+\frac {2 (35 A+31 C) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{105 a d}\\ \end {align*}
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Mathematica [A] time = 0.33, size = 89, normalized size = 0.46 \[ \frac {2 \cos \left (\frac {1}{2} (c+d x)\right ) \left (105 (A+C) \tanh ^{-1}\left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-2 \sin ^3\left (\frac {1}{2} (c+d x)\right ) (70 A+24 C \cos (c+d x)+15 C \cos (2 (c+d x))+101 C)\right )}{105 d \sqrt {a (\cos (c+d x)+1)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 173, normalized size = 0.90 \[ \frac {4 \, {\left (15 \, C \cos \left (d x + c\right )^{3} - 3 \, C \cos \left (d x + c\right )^{2} + {\left (35 \, A + 31 \, C\right )} \cos \left (d x + c\right ) - 35 \, A - 43 \, C\right )} \sqrt {a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right ) + \frac {105 \, \sqrt {2} {\left ({\left (A + C\right )} a \cos \left (d x + c\right ) + {\left (A + C\right )} a\right )} \log \left (-\frac {\cos \left (d x + c\right )^{2} - \frac {2 \, \sqrt {2} \sqrt {a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{\sqrt {a}} - 2 \, \cos \left (d x + c\right ) - 3}{\cos \left (d x + c\right )^{2} + 2 \, \cos \left (d x + c\right ) + 1}\right )}{\sqrt {a}}}{210 \, {\left (a d \cos \left (d x + c\right ) + a d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.88, size = 158, normalized size = 0.82 \[ -\frac {\frac {105 \, \sqrt {2} {\left (A + C\right )} \log \left ({\left | -\sqrt {a} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \sqrt {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a} \right |}\right )}{\sqrt {a}} + \frac {4 \, {\left ({\left (\sqrt {2} {\left (35 \, A a^{3} + 46 \, C a^{3}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 14 \, \sqrt {2} {\left (5 \, A a^{3} + 4 \, C a^{3}\right )}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 35 \, \sqrt {2} {\left (A a^{3} + 2 \, C a^{3}\right )}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3}}{{\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a\right )}^{\frac {7}{2}}}}{105 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.18, size = 253, normalized size = 1.31 \[ \frac {\cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {a \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \left (-240 C \sqrt {a}\, \sqrt {2}\, \sqrt {a \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+336 C \sqrt {a}\, \sqrt {2}\, \sqrt {a \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-140 \sqrt {2}\, \sqrt {a}\, \sqrt {a \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \left (A +2 C \right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+105 \sqrt {2}\, \ln \left (\frac {4 \sqrt {a}\, \sqrt {a \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}+4 a}{\cos \left (\frac {d x}{2}+\frac {c}{2}\right )}\right ) a A +105 \sqrt {2}\, \ln \left (\frac {4 \sqrt {a}\, \sqrt {a \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}+4 a}{\cos \left (\frac {d x}{2}+\frac {c}{2}\right )}\right ) a C \right )}{105 a^{\frac {3}{2}} \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {a \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\cos \left (c+d\,x\right )}^2\,\left (C\,{\cos \left (c+d\,x\right )}^2+A\right )}{\sqrt {a+a\,\cos \left (c+d\,x\right )}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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