Optimal. Leaf size=210 \[ \frac {2 a^2 \left (A b^2-a (b B-a C)\right ) \Pi \left (\frac {2 b}{a+b};\left .\frac {1}{2} (c+d x)\right |2\right )}{b^4 d (a+b)}+\frac {2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (5 a^2 C-5 a b B+5 A b^2+3 b^2 C\right )}{5 b^3 d}+\frac {2 F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (-3 a^3 C+3 a^2 b B-a b^2 (3 A+C)+b^3 B\right )}{3 b^4 d}+\frac {2 (b B-a C) \sin (c+d x) \sqrt {\cos (c+d x)}}{3 b^2 d}+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 b d} \]
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Rubi [A] time = 0.88, antiderivative size = 210, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.140, Rules used = {3049, 3059, 2639, 3002, 2641, 2805} \[ \frac {2 F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (3 a^2 b B-3 a^3 C-a b^2 (3 A+C)+b^3 B\right )}{3 b^4 d}+\frac {2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (5 a^2 C-5 a b B+5 A b^2+3 b^2 C\right )}{5 b^3 d}+\frac {2 a^2 \left (A b^2-a (b B-a C)\right ) \Pi \left (\frac {2 b}{a+b};\left .\frac {1}{2} (c+d x)\right |2\right )}{b^4 d (a+b)}+\frac {2 (b B-a C) \sin (c+d x) \sqrt {\cos (c+d x)}}{3 b^2 d}+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 b d} \]
Antiderivative was successfully verified.
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Rule 2639
Rule 2641
Rule 2805
Rule 3002
Rule 3049
Rule 3059
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)} \, dx &=\frac {2 C \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{5 b d}+\frac {2 \int \frac {\sqrt {\cos (c+d x)} \left (\frac {3 a C}{2}+\frac {1}{2} b (5 A+3 C) \cos (c+d x)+\frac {5}{2} (b B-a C) \cos ^2(c+d x)\right )}{a+b \cos (c+d x)} \, dx}{5 b}\\ &=\frac {2 (b B-a C) \sqrt {\cos (c+d x)} \sin (c+d x)}{3 b^2 d}+\frac {2 C \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{5 b d}+\frac {4 \int \frac {\frac {5}{4} a (b B-a C)+\frac {1}{4} b (5 b B+4 a C) \cos (c+d x)+\frac {3}{4} \left (5 A b^2-5 a b B+5 a^2 C+3 b^2 C\right ) \cos ^2(c+d x)}{\sqrt {\cos (c+d x)} (a+b \cos (c+d x))} \, dx}{15 b^2}\\ &=\frac {2 (b B-a C) \sqrt {\cos (c+d x)} \sin (c+d x)}{3 b^2 d}+\frac {2 C \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{5 b d}-\frac {4 \int \frac {-\frac {5}{4} a b (b B-a C)-\frac {5}{4} \left (3 a^2 b B+b^3 B-3 a^3 C-a b^2 (3 A+C)\right ) \cos (c+d x)}{\sqrt {\cos (c+d x)} (a+b \cos (c+d x))} \, dx}{15 b^3}+\frac {\left (5 A b^2-5 a b B+5 a^2 C+3 b^2 C\right ) \int \sqrt {\cos (c+d x)} \, dx}{5 b^3}\\ &=\frac {2 \left (5 A b^2-5 a b B+5 a^2 C+3 b^2 C\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 b^3 d}+\frac {2 (b B-a C) \sqrt {\cos (c+d x)} \sin (c+d x)}{3 b^2 d}+\frac {2 C \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{5 b d}+\frac {\left (3 a^2 b B+b^3 B-3 a^3 C-a b^2 (3 A+C)\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{3 b^4}+\frac {\left (a^2 \left (A b^2-a (b B-a C)\right )\right ) \int \frac {1}{\sqrt {\cos (c+d x)} (a+b \cos (c+d x))} \, dx}{b^4}\\ &=\frac {2 \left (5 A b^2-5 a b B+5 a^2 C+3 b^2 C\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 b^3 d}+\frac {2 \left (3 a^2 b B+b^3 B-3 a^3 C-a b^2 (3 A+C)\right ) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{3 b^4 d}+\frac {2 a^2 \left (A b^2-a (b B-a C)\right ) \Pi \left (\frac {2 b}{a+b};\left .\frac {1}{2} (c+d x)\right |2\right )}{b^4 (a+b) d}+\frac {2 (b B-a C) \sqrt {\cos (c+d x)} \sin (c+d x)}{3 b^2 d}+\frac {2 C \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{5 b d}\\ \end {align*}
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Mathematica [A] time = 2.27, size = 272, normalized size = 1.30 \[ \frac {\frac {2 b^2 \left (5 a^2 C-5 a b B+15 A b^2+9 b^2 C\right ) \Pi \left (\frac {2 b}{a+b};\left .\frac {1}{2} (c+d x)\right |2\right )}{a+b}+\frac {6 \sin (c+d x) \left (5 a^2 C-5 a b B+5 A b^2+3 b^2 C\right ) \left (\left (b^2-2 a^2\right ) \Pi \left (-\frac {b}{a};\left .\sin ^{-1}\left (\sqrt {\cos (c+d x)}\right )\right |-1\right )+2 a (a+b) F\left (\left .\sin ^{-1}\left (\sqrt {\cos (c+d x)}\right )\right |-1\right )-2 a b E\left (\left .\sin ^{-1}\left (\sqrt {\cos (c+d x)}\right )\right |-1\right )\right )}{a \sqrt {\sin ^2(c+d x)}}+4 b^2 \sin (c+d x) \sqrt {\cos (c+d x)} (-5 a C+5 b B+3 b C \cos (c+d x))+2 b^2 (4 a C+5 b B) \left (2 F\left (\left .\frac {1}{2} (c+d x)\right |2\right )-\frac {2 a \Pi \left (\frac {2 b}{a+b};\left .\frac {1}{2} (c+d x)\right |2\right )}{a+b}\right )}{30 b^4 d} \]
Antiderivative was successfully verified.
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fricas [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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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} \cos \left (d x + c\right )^{\frac {3}{2}}}{b \cos \left (d x + c\right ) + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 6.20, size = 803, normalized size = 3.82 \[ \text {result too large to display} \]
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 \[ \int \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} \cos \left (d x + c\right )^{\frac {3}{2}}}{b \cos \left (d x + c\right ) + a}\,{d x} \]
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 \[ \int \frac {{\cos \left (c+d\,x\right )}^{3/2}\,\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A\right )}{a+b\,\cos \left (c+d\,x\right )} \,d x \]
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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