Optimal. Leaf size=227 \[ \frac {(19 A-75 B+163 C) \tanh ^{-1}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {2} \sqrt {a \cos (c+d x)+a}}\right )}{16 \sqrt {2} a^{5/2} d}+\frac {(15 A-39 B+95 C) \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{48 a^3 d}-\frac {(21 A-93 B+197 C) \sin (c+d x)}{24 a^2 d \sqrt {a \cos (c+d x)+a}}-\frac {(A-B+C) \sin (c+d x) \cos ^3(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}-\frac {(A-9 B+17 C) \sin (c+d x) \cos ^2(c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}} \]
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Rubi [A] time = 0.69, antiderivative size = 227, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.163, Rules used = {3041, 2977, 2968, 3023, 2751, 2649, 206} \[ \frac {(15 A-39 B+95 C) \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{48 a^3 d}-\frac {(21 A-93 B+197 C) \sin (c+d x)}{24 a^2 d \sqrt {a \cos (c+d x)+a}}+\frac {(19 A-75 B+163 C) \tanh ^{-1}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {2} \sqrt {a \cos (c+d x)+a}}\right )}{16 \sqrt {2} a^{5/2} d}-\frac {(A-B+C) \sin (c+d x) \cos ^3(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}-\frac {(A-9 B+17 C) \sin (c+d x) \cos ^2(c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 2649
Rule 2751
Rule 2968
Rule 2977
Rule 3023
Rule 3041
Rubi steps
\begin {align*} \int \frac {\cos ^2(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+a \cos (c+d x))^{5/2}} \, dx &=-\frac {(A-B+C) \cos ^3(c+d x) \sin (c+d x)}{4 d (a+a \cos (c+d x))^{5/2}}+\frac {\int \frac {\cos ^2(c+d x) \left (a (A+3 B-3 C)+\frac {1}{2} a (3 A-3 B+11 C) \cos (c+d x)\right )}{(a+a \cos (c+d x))^{3/2}} \, dx}{4 a^2}\\ &=-\frac {(A-B+C) \cos ^3(c+d x) \sin (c+d x)}{4 d (a+a \cos (c+d x))^{5/2}}-\frac {(A-9 B+17 C) \cos ^2(c+d x) \sin (c+d x)}{16 a d (a+a \cos (c+d x))^{3/2}}+\frac {\int \frac {\cos (c+d x) \left (-a^2 (A-9 B+17 C)+\frac {1}{4} a^2 (15 A-39 B+95 C) \cos (c+d x)\right )}{\sqrt {a+a \cos (c+d x)}} \, dx}{8 a^4}\\ &=-\frac {(A-B+C) \cos ^3(c+d x) \sin (c+d x)}{4 d (a+a \cos (c+d x))^{5/2}}-\frac {(A-9 B+17 C) \cos ^2(c+d x) \sin (c+d x)}{16 a d (a+a \cos (c+d x))^{3/2}}+\frac {\int \frac {-a^2 (A-9 B+17 C) \cos (c+d x)+\frac {1}{4} a^2 (15 A-39 B+95 C) \cos ^2(c+d x)}{\sqrt {a+a \cos (c+d x)}} \, dx}{8 a^4}\\ &=-\frac {(A-B+C) \cos ^3(c+d x) \sin (c+d x)}{4 d (a+a \cos (c+d x))^{5/2}}-\frac {(A-9 B+17 C) \cos ^2(c+d x) \sin (c+d x)}{16 a d (a+a \cos (c+d x))^{3/2}}+\frac {(15 A-39 B+95 C) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{48 a^3 d}+\frac {\int \frac {\frac {1}{8} a^3 (15 A-39 B+95 C)-\frac {1}{4} a^3 (21 A-93 B+197 C) \cos (c+d x)}{\sqrt {a+a \cos (c+d x)}} \, dx}{12 a^5}\\ &=-\frac {(A-B+C) \cos ^3(c+d x) \sin (c+d x)}{4 d (a+a \cos (c+d x))^{5/2}}-\frac {(A-9 B+17 C) \cos ^2(c+d x) \sin (c+d x)}{16 a d (a+a \cos (c+d x))^{3/2}}-\frac {(21 A-93 B+197 C) \sin (c+d x)}{24 a^2 d \sqrt {a+a \cos (c+d x)}}+\frac {(15 A-39 B+95 C) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{48 a^3 d}+\frac {(19 A-75 B+163 C) \int \frac {1}{\sqrt {a+a \cos (c+d x)}} \, dx}{32 a^2}\\ &=-\frac {(A-B+C) \cos ^3(c+d x) \sin (c+d x)}{4 d (a+a \cos (c+d x))^{5/2}}-\frac {(A-9 B+17 C) \cos ^2(c+d x) \sin (c+d x)}{16 a d (a+a \cos (c+d x))^{3/2}}-\frac {(21 A-93 B+197 C) \sin (c+d x)}{24 a^2 d \sqrt {a+a \cos (c+d x)}}+\frac {(15 A-39 B+95 C) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{48 a^3 d}-\frac {(19 A-75 B+163 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 )}{16 a^2 d}\\ &=\frac {(19 A-75 B+163 C) \tanh ^{-1}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {2} \sqrt {a+a \cos (c+d x)}}\right )}{16 \sqrt {2} a^{5/2} d}-\frac {(A-B+C) \cos ^3(c+d x) \sin (c+d x)}{4 d (a+a \cos (c+d x))^{5/2}}-\frac {(A-9 B+17 C) \cos ^2(c+d x) \sin (c+d x)}{16 a d (a+a \cos (c+d x))^{3/2}}-\frac {(21 A-93 B+197 C) \sin (c+d x)}{24 a^2 d \sqrt {a+a \cos (c+d x)}}+\frac {(15 A-39 B+95 C) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{48 a^3 d}\\ \end {align*}
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Mathematica [A] time = 1.20, size = 126, normalized size = 0.56 \[ \frac {\tan \left (\frac {1}{2} (c+d x)\right ) ((-39 A+255 B-479 C) \cos (c+d x)-27 A+16 (3 B-5 C) \cos (2 (c+d x))+195 B+8 C \cos (3 (c+d x))-379 C)+6 (19 A-75 B+163 C) \cos ^3\left (\frac {1}{2} (c+d x)\right ) \tanh ^{-1}\left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{48 a d (a (\cos (c+d x)+1))^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 273, normalized size = 1.20 \[ \frac {3 \, \sqrt {2} {\left ({\left (19 \, A - 75 \, B + 163 \, C\right )} \cos \left (d x + c\right )^{3} + 3 \, {\left (19 \, A - 75 \, B + 163 \, C\right )} \cos \left (d x + c\right )^{2} + 3 \, {\left (19 \, A - 75 \, B + 163 \, C\right )} \cos \left (d x + c\right ) + 19 \, A - 75 \, B + 163 \, C\right )} \sqrt {a} \log \left (-\frac {a \cos \left (d x + c\right )^{2} - 2 \, \sqrt {2} \sqrt {a \cos \left (d x + c\right ) + a} \sqrt {a} \sin \left (d x + c\right ) - 2 \, a \cos \left (d x + c\right ) - 3 \, a}{\cos \left (d x + c\right )^{2} + 2 \, \cos \left (d x + c\right ) + 1}\right ) + 4 \, {\left (32 \, C \cos \left (d x + c\right )^{3} + 32 \, {\left (3 \, B - 5 \, C\right )} \cos \left (d x + c\right )^{2} - {\left (39 \, A - 255 \, B + 503 \, C\right )} \cos \left (d x + c\right ) - 27 \, A + 147 \, B - 299 \, C\right )} \sqrt {a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{192 \, {\left (a^{3} d \cos \left (d x + c\right )^{3} + 3 \, a^{3} d \cos \left (d x + c\right )^{2} + 3 \, a^{3} d \cos \left (d x + c\right ) + a^{3} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 2.37, size = 230, normalized size = 1.01 \[ \frac {\frac {{\left ({\left (3 \, {\left (\frac {2 \, \sqrt {2} {\left (A a^{5} - B a^{5} + C a^{5}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}}{a^{6}} - \frac {\sqrt {2} {\left (7 \, A a^{5} - 15 \, B a^{5} + 23 \, C a^{5}\right )}}{a^{6}}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - \frac {4 \, \sqrt {2} {\left (15 \, A a^{5} - 75 \, B a^{5} + 167 \, C a^{5}\right )}}{a^{6}}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - \frac {3 \, \sqrt {2} {\left (11 \, A a^{5} - 83 \, B a^{5} + 155 \, C a^{5}\right )}}{a^{6}}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{{\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a\right )}^{\frac {3}{2}}} - \frac {3 \, \sqrt {2} {\left (19 \, A - 75 \, B + 163 \, C\right )} \log \left ({\left | -\sqrt {a} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \sqrt {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a} \right |}\right )}{a^{\frac {5}{2}}}}{96 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 1.39, size = 512, normalized size = 2.26 \[ \frac {\sqrt {a \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \left (128 C \sqrt {2}\, \sqrt {a \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \sqrt {a}\, \left (\cos ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+57 A \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 ) \sqrt {2}\, \left (\cos ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a -225 B \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 ) \sqrt {2}\, \left (\cos ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +489 C \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 ) \left (\cos ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +192 B \sqrt {2}\, \sqrt {a \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \sqrt {a}\, \left (\cos ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-512 C \sqrt {2}\, \sqrt {a \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \sqrt {a}\, \left (\cos ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-39 A \sqrt {a}\, \sqrt {2}\, \sqrt {a \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+63 B \sqrt {a}\, \sqrt {2}\, \sqrt {a \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-87 C \sqrt {2}\, \sqrt {a \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \sqrt {a}\, \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+6 A \sqrt {2}\, \sqrt {a \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \sqrt {a}-6 B \sqrt {2}\, \sqrt {a \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \sqrt {a}+6 C \sqrt {2}\, \sqrt {a \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \sqrt {a}\right )}{96 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} a^{\frac {7}{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 = -0.00 \[ \int \frac {{\cos \left (c+d\,x\right )}^2\,\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A\right )}{{\left (a+a\,\cos \left (c+d\,x\right )\right )}^{5/2}} \,d x \]
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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