Optimal. Leaf size=245 \[ \frac {(8 A+19 C) \sin ^{-1}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {a \cos (c+d x)+a}}\right )}{4 a^{3/2} d}-\frac {(5 A+13 C) \tan ^{-1}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {2} \sqrt {\cos (c+d x)} \sqrt {a \cos (c+d x)+a}}\right )}{2 \sqrt {2} a^{3/2} d}-\frac {(A+C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}+\frac {(A+2 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{2 a d \sqrt {a \cos (c+d x)+a}}-\frac {(2 A+7 C) \sin (c+d x) \sqrt {\cos (c+d x)}}{4 a d \sqrt {a \cos (c+d x)+a}} \]
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Rubi [A] time = 0.79, antiderivative size = 245, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.189, Rules used = {3042, 2983, 2982, 2782, 205, 2774, 216} \[ \frac {(8 A+19 C) \sin ^{-1}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {a \cos (c+d x)+a}}\right )}{4 a^{3/2} d}-\frac {(5 A+13 C) \tan ^{-1}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {2} \sqrt {\cos (c+d x)} \sqrt {a \cos (c+d x)+a}}\right )}{2 \sqrt {2} a^{3/2} d}-\frac {(A+C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}+\frac {(A+2 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{2 a d \sqrt {a \cos (c+d x)+a}}-\frac {(2 A+7 C) \sin (c+d x) \sqrt {\cos (c+d x)}}{4 a d \sqrt {a \cos (c+d x)+a}} \]
Antiderivative was successfully verified.
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Rule 205
Rule 216
Rule 2774
Rule 2782
Rule 2982
Rule 2983
Rule 3042
Rubi steps
\begin {align*} \int \frac {\cos ^{\frac {3}{2}}(c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a+a \cos (c+d x))^{3/2}} \, dx &=-\frac {(A+C) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{2 d (a+a \cos (c+d x))^{3/2}}+\frac {\int \frac {\cos ^{\frac {3}{2}}(c+d x) \left (-\frac {1}{2} a (A+5 C)+2 a (A+2 C) \cos (c+d x)\right )}{\sqrt {a+a \cos (c+d x)}} \, dx}{2 a^2}\\ &=-\frac {(A+C) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{2 d (a+a \cos (c+d x))^{3/2}}+\frac {(A+2 C) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{2 a d \sqrt {a+a \cos (c+d x)}}+\frac {\int \frac {\sqrt {\cos (c+d x)} \left (3 a^2 (A+2 C)-a^2 (2 A+7 C) \cos (c+d x)\right )}{\sqrt {a+a \cos (c+d x)}} \, dx}{4 a^3}\\ &=-\frac {(A+C) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{2 d (a+a \cos (c+d x))^{3/2}}-\frac {(2 A+7 C) \sqrt {\cos (c+d x)} \sin (c+d x)}{4 a d \sqrt {a+a \cos (c+d x)}}+\frac {(A+2 C) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{2 a d \sqrt {a+a \cos (c+d x)}}+\frac {\int \frac {-\frac {1}{2} a^3 (2 A+7 C)+\frac {1}{2} a^3 (8 A+19 C) \cos (c+d x)}{\sqrt {\cos (c+d x)} \sqrt {a+a \cos (c+d x)}} \, dx}{4 a^4}\\ &=-\frac {(A+C) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{2 d (a+a \cos (c+d x))^{3/2}}-\frac {(2 A+7 C) \sqrt {\cos (c+d x)} \sin (c+d x)}{4 a d \sqrt {a+a \cos (c+d x)}}+\frac {(A+2 C) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{2 a d \sqrt {a+a \cos (c+d x)}}-\frac {(5 A+13 C) \int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {a+a \cos (c+d x)}} \, dx}{4 a}+\frac {(8 A+19 C) \int \frac {\sqrt {a+a \cos (c+d x)}}{\sqrt {\cos (c+d x)}} \, dx}{8 a^2}\\ &=-\frac {(A+C) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{2 d (a+a \cos (c+d x))^{3/2}}-\frac {(2 A+7 C) \sqrt {\cos (c+d x)} \sin (c+d x)}{4 a d \sqrt {a+a \cos (c+d x)}}+\frac {(A+2 C) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{2 a d \sqrt {a+a \cos (c+d x)}}+\frac {(5 A+13 C) \operatorname {Subst}\left (\int \frac {1}{2 a^2+a x^2} \, dx,x,-\frac {a \sin (c+d x)}{\sqrt {\cos (c+d x)} \sqrt {a+a \cos (c+d x)}}\right )}{2 d}-\frac {(8 A+19 C) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{a}}} \, dx,x,-\frac {a \sin (c+d x)}{\sqrt {a+a \cos (c+d x)}}\right )}{4 a^2 d}\\ &=\frac {(8 A+19 C) \sin ^{-1}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {a+a \cos (c+d x)}}\right )}{4 a^{3/2} d}-\frac {(5 A+13 C) \tan ^{-1}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {2} \sqrt {\cos (c+d x)} \sqrt {a+a \cos (c+d x)}}\right )}{2 \sqrt {2} a^{3/2} d}-\frac {(A+C) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{2 d (a+a \cos (c+d x))^{3/2}}-\frac {(2 A+7 C) \sqrt {\cos (c+d x)} \sin (c+d x)}{4 a d \sqrt {a+a \cos (c+d x)}}+\frac {(A+2 C) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{2 a d \sqrt {a+a \cos (c+d x)}}\\ \end {align*}
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Mathematica [C] time = 2.51, size = 370, normalized size = 1.51 \[ \frac {\cos ^3\left (\frac {1}{2} (c+d x)\right ) \left (-2 \sqrt {\cos (c+d x)} \tan \left (\frac {1}{2} (c+d x)\right ) \sec \left (\frac {1}{2} (c+d x)\right ) (2 A+3 C \cos (c+d x)-C \cos (2 (c+d x))+6 C)+\frac {\sqrt {2} e^{\frac {1}{2} i (c+d x)} \sqrt {e^{-i (c+d x)} \left (1+e^{2 i (c+d x)}\right )} \left (2 i \sqrt {2} (5 A+13 C) \log \left (1+e^{i (c+d x)}\right )-i (8 A+19 C) \sinh ^{-1}\left (e^{i (c+d x)}\right )+8 i A \log \left (1+\sqrt {1+e^{2 i (c+d x)}}\right )-10 i \sqrt {2} A \log \left (\sqrt {2} \sqrt {1+e^{2 i (c+d x)}}-e^{i (c+d x)}+1\right )+8 A d x+19 i C \log \left (1+\sqrt {1+e^{2 i (c+d x)}}\right )-26 i \sqrt {2} C \log \left (\sqrt {2} \sqrt {1+e^{2 i (c+d x)}}-e^{i (c+d x)}+1\right )+19 C d x\right )}{\sqrt {1+e^{2 i (c+d x)}}}\right )}{4 d (a (\cos (c+d x)+1))^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 9.81, size = 247, normalized size = 1.01 \[ \frac {\sqrt {2} {\left ({\left (5 \, A + 13 \, C\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (5 \, A + 13 \, C\right )} \cos \left (d x + c\right ) + 5 \, A + 13 \, C\right )} \sqrt {a} \arctan \left (\frac {\sqrt {2} \sqrt {a \cos \left (d x + c\right ) + a} \sqrt {\cos \left (d x + c\right )}}{\sqrt {a} \sin \left (d x + c\right )}\right ) + {\left (2 \, C \cos \left (d x + c\right )^{2} - 3 \, C \cos \left (d x + c\right ) - 2 \, A - 7 \, C\right )} \sqrt {a \cos \left (d x + c\right ) + a} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) - {\left ({\left (8 \, A + 19 \, C\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (8 \, A + 19 \, C\right )} \cos \left (d x + c\right ) + 8 \, A + 19 \, C\right )} \sqrt {a} \arctan \left (\frac {\sqrt {a \cos \left (d x + c\right ) + a} \sqrt {\cos \left (d x + c\right )}}{\sqrt {a} \sin \left (d x + c\right )}\right )}{4 \, {\left (a^{2} d \cos \left (d x + c\right )^{2} + 2 \, a^{2} d \cos \left (d x + c\right ) + a^{2} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (C \cos \left (d x + c\right )^{2} + A\right )} \cos \left (d x + c\right )^{\frac {3}{2}}}{{\left (a \cos \left (d x + c\right ) + a\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.50, size = 477, normalized size = 1.95 \[ \frac {\left (\cos ^{\frac {3}{2}}\left (d x +c \right )\right ) \sqrt {a \left (1+\cos \left (d x +c \right )\right )}\, \left (-1+\cos \left (d x +c \right )\right )^{4} \left (2 A \left (\cos ^{3}\left (d x +c \right )\right ) \left (\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}\right )^{\frac {5}{2}}+2 A \left (\cos ^{2}\left (d x +c \right )\right ) \left (\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}\right )^{\frac {5}{2}}-2 A \cos \left (d x +c \right ) \left (\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}\right )^{\frac {5}{2}}-2 C \left (\cos ^{5}\left (d x +c \right )\right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}+5 A \arcsin \left (\frac {-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}\right ) \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right ) \sqrt {2}-2 A \left (\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}\right )^{\frac {5}{2}}+13 C \arcsin \left (\frac {-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}\right ) \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right ) \sqrt {2}+5 C \left (\cos ^{4}\left (d x +c \right )\right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}+8 A \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right ) \arctan \left (\frac {\sin \left (d x +c \right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}}{\cos \left (d x +c \right )}\right )+4 C \left (\cos ^{3}\left (d x +c \right )\right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}+19 C \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right ) \arctan \left (\frac {\sin \left (d x +c \right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}}{\cos \left (d x +c \right )}\right )-7 C \left (\cos ^{2}\left (d x +c \right )\right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\right )}{4 d \sin \left (d x +c \right )^{9} \left (\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}\right )^{\frac {7}{2}} a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (C \cos \left (d x + c\right )^{2} + A\right )} \cos \left (d x + c\right )^{\frac {3}{2}}}{{\left (a \cos \left (d x + c\right ) + a\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {{\cos \left (c+d\,x\right )}^{3/2}\,\left (C\,{\cos \left (c+d\,x\right )}^2+A\right )}{{\left (a+a\,\cos \left (c+d\,x\right )\right )}^{3/2}} \,d x \]
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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