Optimal. Leaf size=704 \[ \frac {\sin (c+d x) \left (-9 a^3 C+24 a^2 b B+12 a b^2 (20 A+13 C)+128 b^3 B\right ) \sqrt {a+b \cos (c+d x)}}{192 b^2 d \sqrt {\cos (c+d x)}}-\frac {\sqrt {a+b} \cot (c+d x) \left (9 a^3 C-6 a^2 b (4 B+C)-4 a b^2 (60 A+28 B+39 C)-8 b^3 (12 A+16 B+9 C)\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} F\left (\sin ^{-1}\left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right )}{192 b^2 d}-\frac {(a-b) \sqrt {a+b} \cot (c+d x) \left (-9 a^3 C+24 a^2 b B+12 a b^2 (20 A+13 C)+128 b^3 B\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} E\left (\sin ^{-1}\left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right )}{192 a b^2 d}+\frac {\sqrt {a+b} \cot (c+d x) \left (-3 a^4 C+8 a^3 b B-24 a^2 b^2 (2 A+C)-96 a b^3 B-16 b^4 (4 A+3 C)\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \Pi \left (\frac {a+b}{b};\sin ^{-1}\left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right )}{64 b^3 d}+\frac {\sin (c+d x) \sqrt {\cos (c+d x)} \left (a (8 b B-3 a C)+4 b^2 (4 A+3 C)\right ) \sqrt {a+b \cos (c+d x)}}{32 b d}+\frac {(8 b B-3 a C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{24 b d}+\frac {C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{4 b d} \]
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Rubi [A] time = 2.49, antiderivative size = 704, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.156, Rules used = {3049, 3061, 3053, 2809, 2998, 2816, 2994} \[ \frac {\sin (c+d x) \left (24 a^2 b B-9 a^3 C+12 a b^2 (20 A+13 C)+128 b^3 B\right ) \sqrt {a+b \cos (c+d x)}}{192 b^2 d \sqrt {\cos (c+d x)}}-\frac {\sqrt {a+b} \cot (c+d x) \left (-6 a^2 b (4 B+C)+9 a^3 C-4 a b^2 (60 A+28 B+39 C)-8 b^3 (12 A+16 B+9 C)\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} F\left (\sin ^{-1}\left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right )}{192 b^2 d}-\frac {(a-b) \sqrt {a+b} \cot (c+d x) \left (24 a^2 b B-9 a^3 C+12 a b^2 (20 A+13 C)+128 b^3 B\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} E\left (\sin ^{-1}\left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right )}{192 a b^2 d}+\frac {\sqrt {a+b} \cot (c+d x) \left (-24 a^2 b^2 (2 A+C)+8 a^3 b B-3 a^4 C-96 a b^3 B-16 b^4 (4 A+3 C)\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \Pi \left (\frac {a+b}{b};\sin ^{-1}\left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right )}{64 b^3 d}+\frac {\sin (c+d x) \sqrt {\cos (c+d x)} \left (a (8 b B-3 a C)+4 b^2 (4 A+3 C)\right ) \sqrt {a+b \cos (c+d x)}}{32 b d}+\frac {(8 b B-3 a C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{24 b d}+\frac {C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{4 b d} \]
Antiderivative was successfully verified.
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Rule 2809
Rule 2816
Rule 2994
Rule 2998
Rule 3049
Rule 3053
Rule 3061
Rubi steps
\begin {align*} \int \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx &=\frac {C \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{4 b d}+\frac {\int \frac {(a+b \cos (c+d x))^{3/2} \left (\frac {a C}{2}+b (4 A+3 C) \cos (c+d x)+\frac {1}{2} (8 b B-3 a C) \cos ^2(c+d x)\right )}{\sqrt {\cos (c+d x)}} \, dx}{4 b}\\ &=\frac {(8 b B-3 a C) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{24 b d}+\frac {C \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{4 b d}+\frac {\int \frac {\sqrt {a+b \cos (c+d x)} \left (\frac {1}{4} a (8 b B+3 a C)+\frac {1}{2} b (24 a A+16 b B+15 a C) \cos (c+d x)+\frac {3}{4} \left (4 b^2 (4 A+3 C)+a (8 b B-3 a C)\right ) \cos ^2(c+d x)\right )}{\sqrt {\cos (c+d x)}} \, dx}{12 b}\\ &=\frac {\left (4 b^2 (4 A+3 C)+a (8 b B-3 a C)\right ) \sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{32 b d}+\frac {(8 b B-3 a C) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{24 b d}+\frac {C \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{4 b d}+\frac {\int \frac {\frac {1}{8} a \left (48 A b^2+56 a b B+3 a^2 C+36 b^2 C\right )+\frac {1}{4} b \left (104 a b B+12 b^2 (4 A+3 C)+a^2 (96 A+57 C)\right ) \cos (c+d x)+\frac {1}{8} \left (24 a^2 b B+128 b^3 B-9 a^3 C+12 a b^2 (20 A+13 C)\right ) \cos ^2(c+d x)}{\sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}} \, dx}{24 b}\\ &=\frac {\left (24 a^2 b B+128 b^3 B-9 a^3 C+12 a b^2 (20 A+13 C)\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{192 b^2 d \sqrt {\cos (c+d x)}}+\frac {\left (4 b^2 (4 A+3 C)+a (8 b B-3 a C)\right ) \sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{32 b d}+\frac {(8 b B-3 a C) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{24 b d}+\frac {C \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{4 b d}+\frac {\int \frac {-\frac {1}{8} a \left (24 a^2 b B+128 b^3 B-9 a^3 C+12 a b^2 (20 A+13 C)\right )+\frac {1}{4} a b \left (48 A b^2+56 a b B+3 a^2 C+36 b^2 C\right ) \cos (c+d x)-\frac {3}{8} \left (8 a^3 b B-96 a b^3 B-3 a^4 C-24 a^2 b^2 (2 A+C)-16 b^4 (4 A+3 C)\right ) \cos ^2(c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}} \, dx}{48 b^2}\\ &=\frac {\left (24 a^2 b B+128 b^3 B-9 a^3 C+12 a b^2 (20 A+13 C)\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{192 b^2 d \sqrt {\cos (c+d x)}}+\frac {\left (4 b^2 (4 A+3 C)+a (8 b B-3 a C)\right ) \sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{32 b d}+\frac {(8 b B-3 a C) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{24 b d}+\frac {C \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{4 b d}+\frac {\int \frac {-\frac {1}{8} a \left (24 a^2 b B+128 b^3 B-9 a^3 C+12 a b^2 (20 A+13 C)\right )+\frac {1}{4} a b \left (48 A b^2+56 a b B+3 a^2 C+36 b^2 C\right ) \cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}} \, dx}{48 b^2}-\frac {\left (8 a^3 b B-96 a b^3 B-3 a^4 C-24 a^2 b^2 (2 A+C)-16 b^4 (4 A+3 C)\right ) \int \frac {\sqrt {\cos (c+d x)}}{\sqrt {a+b \cos (c+d x)}} \, dx}{128 b^2}\\ &=\frac {\sqrt {a+b} \left (8 a^3 b B-96 a b^3 B-3 a^4 C-24 a^2 b^2 (2 A+C)-16 b^4 (4 A+3 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}}}{64 b^3 d}+\frac {\left (24 a^2 b B+128 b^3 B-9 a^3 C+12 a b^2 (20 A+13 C)\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{192 b^2 d \sqrt {\cos (c+d x)}}+\frac {\left (4 b^2 (4 A+3 C)+a (8 b B-3 a C)\right ) \sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{32 b d}+\frac {(8 b B-3 a C) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{24 b d}+\frac {C \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{4 b d}-\frac {\left (a \left (24 a^2 b B+128 b^3 B-9 a^3 C+12 a b^2 (20 A+13 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}{384 b^2}-\frac {\left (a \left (9 a^3 C-6 a^2 b (4 B+C)-8 b^3 (12 A+16 B+9 C)-4 a b^2 (60 A+28 B+39 C)\right )\right ) \int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}} \, dx}{384 b^2}\\ &=-\frac {(a-b) \sqrt {a+b} \left (24 a^2 b B+128 b^3 B-9 a^3 C+12 a b^2 (20 A+13 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}}}{192 a b^2 d}-\frac {\sqrt {a+b} \left (9 a^3 C-6 a^2 b (4 B+C)-8 b^3 (12 A+16 B+9 C)-4 a b^2 (60 A+28 B+39 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}}}{192 b^2 d}+\frac {\sqrt {a+b} \left (8 a^3 b B-96 a b^3 B-3 a^4 C-24 a^2 b^2 (2 A+C)-16 b^4 (4 A+3 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}}}{64 b^3 d}+\frac {\left (24 a^2 b B+128 b^3 B-9 a^3 C+12 a b^2 (20 A+13 C)\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{192 b^2 d \sqrt {\cos (c+d x)}}+\frac {\left (4 b^2 (4 A+3 C)+a (8 b B-3 a C)\right ) \sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{32 b d}+\frac {(8 b B-3 a C) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{24 b d}+\frac {C \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{4 b d}\\ \end {align*}
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Mathematica [C] time = 6.56, size = 1317, normalized size = 1.87 \[ \text {result too large to display} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 93.34, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (C b \cos \left (d x + c\right )^{3} + {\left (C a + B b\right )} \cos \left (d x + c\right )^{2} + A a + {\left (B a + A b\right )} \cos \left (d x + c\right )\right )} \sqrt {b \cos \left (d x + c\right ) + a} \sqrt {\cos \left (d x + c\right )}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [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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maple [B] time = 0.83, size = 5493, normalized size = 7.80 \[ \text {output too large to display} \]
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 \[ \int {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \sqrt {\cos \left (d x + c\right )}\,{d x} \]
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 \[ \int \sqrt {\cos \left (c+d\,x\right )}\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^{3/2}\,\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A\right ) \,d x \]
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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