Optimal. Leaf size=660 \[ -\frac {2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (A b^2-a (b B-a C)\right )}{b d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}+\frac {\sin (c+d x) \sqrt {\cos (c+d x)} \left (5 a^2 C-4 a b B+4 A b^2-b^2 C\right ) \sqrt {a+b \cos (c+d x)}}{2 b^2 d \left (a^2-b^2\right )}-\frac {\sqrt {a+b} \cot (c+d x) \left (15 a^2 C-12 a b B+8 A b^2+4 b^2 C\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \Pi \left (\frac {a+b}{b};\sin ^{-1}\left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right )}{4 b^4 d}-\frac {\cot (c+d x) \left (15 a^2 C-a b (12 B-5 C)+8 A b^2-2 b^2 (2 B+C)\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} F\left (\sin ^{-1}\left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right )}{4 b^3 d \sqrt {a+b}}+\frac {\sin (c+d x) \left (-15 a^3 C+12 a^2 b B-a b^2 (8 A-7 C)-4 b^3 B\right ) \sqrt {a+b \cos (c+d x)}}{4 b^3 d \left (a^2-b^2\right ) \sqrt {\cos (c+d x)}}-\frac {\cot (c+d x) \left (-15 a^3 C+12 a^2 b B-a b^2 (8 A-7 C)-4 b^3 B\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} E\left (\sin ^{-1}\left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right )}{4 a b^3 d \sqrt {a+b}} \]
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Rubi [A] time = 2.05, antiderivative size = 660, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.178, Rules used = {3047, 3049, 3061, 3053, 2809, 2998, 2816, 2994} \[ -\frac {2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (A b^2-a (b B-a C)\right )}{b d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}+\frac {\sin (c+d x) \sqrt {\cos (c+d x)} \left (5 a^2 C-4 a b B+4 A b^2-b^2 C\right ) \sqrt {a+b \cos (c+d x)}}{2 b^2 d \left (a^2-b^2\right )}+\frac {\sin (c+d x) \left (12 a^2 b B-15 a^3 C-a b^2 (8 A-7 C)-4 b^3 B\right ) \sqrt {a+b \cos (c+d x)}}{4 b^3 d \left (a^2-b^2\right ) \sqrt {\cos (c+d x)}}-\frac {\cot (c+d x) \left (15 a^2 C-a b (12 B-5 C)+8 A b^2-2 b^2 (2 B+C)\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} F\left (\sin ^{-1}\left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right )}{4 b^3 d \sqrt {a+b}}-\frac {\cot (c+d x) \left (12 a^2 b B-15 a^3 C-a b^2 (8 A-7 C)-4 b^3 B\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} E\left (\sin ^{-1}\left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right )}{4 a b^3 d \sqrt {a+b}}-\frac {\sqrt {a+b} \cot (c+d x) \left (15 a^2 C-12 a b B+8 A b^2+4 b^2 C\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \Pi \left (\frac {a+b}{b};\sin ^{-1}\left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right )}{4 b^4 d} \]
Antiderivative was successfully verified.
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Rule 2809
Rule 2816
Rule 2994
Rule 2998
Rule 3047
Rule 3049
Rule 3053
Rule 3061
Rubi steps
\begin {align*} \int \frac {\cos ^{\frac {3}{2}}(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^{3/2}} \, dx &=-\frac {2 \left (A b^2-a (b B-a C)\right ) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{b \left (a^2-b^2\right ) d \sqrt {a+b \cos (c+d x)}}-\frac {2 \int \frac {\sqrt {\cos (c+d x)} \left (\frac {3}{2} \left (A b^2-a (b B-a C)\right )+\frac {1}{2} b (b B-a (A+C)) \cos (c+d x)-\frac {1}{2} \left (4 A b^2-4 a b B+5 a^2 C-b^2 C\right ) \cos ^2(c+d x)\right )}{\sqrt {a+b \cos (c+d x)}} \, dx}{b \left (a^2-b^2\right )}\\ &=-\frac {2 \left (A b^2-a (b B-a C)\right ) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{b \left (a^2-b^2\right ) d \sqrt {a+b \cos (c+d x)}}+\frac {\left (4 A b^2-4 a b B+5 a^2 C-b^2 C\right ) \sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{2 b^2 \left (a^2-b^2\right ) d}-\frac {\int \frac {-\frac {1}{4} a \left (4 A b^2-4 a b B+5 a^2 C-b^2 C\right )+\frac {1}{2} b \left (2 A b^2-2 a b B+a^2 C+b^2 C\right ) \cos (c+d x)-\frac {1}{4} \left (12 a^2 b B-4 b^3 B-a b^2 (8 A-7 C)-15 a^3 C\right ) \cos ^2(c+d x)}{\sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}} \, dx}{b^2 \left (a^2-b^2\right )}\\ &=-\frac {2 \left (A b^2-a (b B-a C)\right ) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{b \left (a^2-b^2\right ) d \sqrt {a+b \cos (c+d x)}}+\frac {\left (12 a^2 b B-4 b^3 B-a b^2 (8 A-7 C)-15 a^3 C\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{4 b^3 \left (a^2-b^2\right ) d \sqrt {\cos (c+d x)}}+\frac {\left (4 A b^2-4 a b B+5 a^2 C-b^2 C\right ) \sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{2 b^2 \left (a^2-b^2\right ) d}-\frac {\int \frac {\frac {1}{4} a \left (12 a^2 b B-4 b^3 B-a b^2 (8 A-7 C)-15 a^3 C\right )-\frac {1}{2} a b \left (4 A b^2-4 a b B+5 a^2 C-b^2 C\right ) \cos (c+d x)-\frac {1}{4} \left (a^2-b^2\right ) \left (8 A b^2-12 a b B+15 a^2 C+4 b^2 C\right ) \cos ^2(c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}} \, dx}{2 b^3 \left (a^2-b^2\right )}\\ &=-\frac {2 \left (A b^2-a (b B-a C)\right ) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{b \left (a^2-b^2\right ) d \sqrt {a+b \cos (c+d x)}}+\frac {\left (12 a^2 b B-4 b^3 B-a b^2 (8 A-7 C)-15 a^3 C\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{4 b^3 \left (a^2-b^2\right ) d \sqrt {\cos (c+d x)}}+\frac {\left (4 A b^2-4 a b B+5 a^2 C-b^2 C\right ) \sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{2 b^2 \left (a^2-b^2\right ) d}-\frac {\int \frac {\frac {1}{4} a \left (12 a^2 b B-4 b^3 B-a b^2 (8 A-7 C)-15 a^3 C\right )-\frac {1}{2} a b \left (4 A b^2-4 a b B+5 a^2 C-b^2 C\right ) \cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}} \, dx}{2 b^3 \left (a^2-b^2\right )}+\frac {\left (8 A b^2-12 a b B+15 a^2 C+4 b^2 C\right ) \int \frac {\sqrt {\cos (c+d x)}}{\sqrt {a+b \cos (c+d x)}} \, dx}{8 b^3}\\ &=-\frac {\sqrt {a+b} \left (8 A b^2-12 a b B+15 a^2 C+4 b^2 C\right ) \cot (c+d x) \Pi \left (\frac {a+b}{b};\sin ^{-1}\left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{4 b^4 d}-\frac {2 \left (A b^2-a (b B-a C)\right ) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{b \left (a^2-b^2\right ) d \sqrt {a+b \cos (c+d x)}}+\frac {\left (12 a^2 b B-4 b^3 B-a b^2 (8 A-7 C)-15 a^3 C\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{4 b^3 \left (a^2-b^2\right ) d \sqrt {\cos (c+d x)}}+\frac {\left (4 A b^2-4 a b B+5 a^2 C-b^2 C\right ) \sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{2 b^2 \left (a^2-b^2\right ) d}-\frac {\left (a \left (12 a^2 b B-4 b^3 B-a b^2 (8 A-7 C)-15 a^3 C\right )\right ) \int \frac {1+\cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}} \, dx}{8 b^3 \left (a^2-b^2\right )}-\frac {\left (a \left (8 A b^2-a b (12 B-5 C)+15 a^2 C-2 b^2 (2 B+C)\right )\right ) \int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}} \, dx}{8 b^3 (a+b)}\\ &=-\frac {\left (12 a^2 b B-4 b^3 B-a b^2 (8 A-7 C)-15 a^3 C\right ) \cot (c+d x) E\left (\sin ^{-1}\left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{4 a b^3 \sqrt {a+b} d}-\frac {\left (8 A b^2-a b (12 B-5 C)+15 a^2 C-2 b^2 (2 B+C)\right ) \cot (c+d x) F\left (\sin ^{-1}\left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{4 b^3 \sqrt {a+b} d}-\frac {\sqrt {a+b} \left (8 A b^2-12 a b B+15 a^2 C+4 b^2 C\right ) \cot (c+d x) \Pi \left (\frac {a+b}{b};\sin ^{-1}\left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{4 b^4 d}-\frac {2 \left (A b^2-a (b B-a C)\right ) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{b \left (a^2-b^2\right ) d \sqrt {a+b \cos (c+d x)}}+\frac {\left (12 a^2 b B-4 b^3 B-a b^2 (8 A-7 C)-15 a^3 C\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{4 b^3 \left (a^2-b^2\right ) d \sqrt {\cos (c+d x)}}+\frac {\left (4 A b^2-4 a b B+5 a^2 C-b^2 C\right ) \sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{2 b^2 \left (a^2-b^2\right ) d}\\ \end {align*}
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Mathematica [C] time = 6.76, size = 1322, normalized size = 2.00 \[ \text {result too large to display} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 75.91, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (C \cos \left (d x + c\right )^{3} + B \cos \left (d x + c\right )^{2} + A \cos \left (d x + c\right )\right )} \sqrt {b \cos \left (d x + c\right ) + a} \sqrt {\cos \left (d x + c\right )}}{b^{2} \cos \left (d x + c\right )^{2} + 2 \, a b \cos \left (d x + c\right ) + a^{2}}, x\right ) \]
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 \[ \int \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} \cos \left (d x + c\right )^{\frac {3}{2}}}{{\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.59, size = 5209, normalized size = 7.89 \[ \text {output too large to display} \]
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 \[ \int \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} \cos \left (d x + c\right )^{\frac {3}{2}}}{{\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {{\cos \left (c+d\,x\right )}^{3/2}\,\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A\right )}{{\left (a+b\,\cos \left (c+d\,x\right )\right )}^{3/2}} \,d x \]
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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