Optimal. Leaf size=674 \[ \frac {2 a (b B-a C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}+\frac {2 a \left (-5 a^3 C+2 a^2 b B+9 a b^2 C-6 b^3 B\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{3 b^2 d \left (a^2-b^2\right )^2 \sqrt {a+b \cos (c+d x)}}-\frac {\left (-15 a^3 C+a^2 b (6 B-5 C)+a b^2 (2 B+21 C)-3 b^3 (4 B-C)\right ) \cot (c+d x) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} F\left (\sin ^{-1}\left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right )}{3 b^3 d \sqrt {a+b} \left (a^2-b^2\right )}-\frac {\left (-15 a^4 C+6 a^3 b B+26 a^2 b^2 C-14 a b^3 B-3 b^4 C\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 b^3 d \left (a^2-b^2\right )^2 \sqrt {\cos (c+d x)}}+\frac {\left (-15 a^4 C+6 a^3 b B+26 a^2 b^2 C-14 a b^3 B-3 b^4 C\right ) \cot (c+d x) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} E\left (\sin ^{-1}\left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right )}{3 a b^3 d (a-b) (a+b)^{3/2}}-\frac {\sqrt {a+b} (2 b B-5 a C) \cot (c+d x) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \Pi \left (\frac {a+b}{b};\sin ^{-1}\left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right )}{b^4 d} \]
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Rubi [A] time = 2.12, antiderivative size = 674, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, integrand size = 44, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.204, Rules used = {3029, 2989, 3047, 3061, 3053, 2809, 2998, 2816, 2994} \[ \frac {2 a (b B-a C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}+\frac {2 a \left (2 a^2 b B-5 a^3 C+9 a b^2 C-6 b^3 B\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{3 b^2 d \left (a^2-b^2\right )^2 \sqrt {a+b \cos (c+d x)}}-\frac {\left (26 a^2 b^2 C+6 a^3 b B-15 a^4 C-14 a b^3 B-3 b^4 C\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 b^3 d \left (a^2-b^2\right )^2 \sqrt {\cos (c+d x)}}-\frac {\left (a^2 b (6 B-5 C)-15 a^3 C+a b^2 (2 B+21 C)-3 b^3 (4 B-C)\right ) \cot (c+d x) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} F\left (\sin ^{-1}\left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right )}{3 b^3 d \sqrt {a+b} \left (a^2-b^2\right )}+\frac {\left (26 a^2 b^2 C+6 a^3 b B-15 a^4 C-14 a b^3 B-3 b^4 C\right ) \cot (c+d x) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} E\left (\sin ^{-1}\left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right )}{3 a b^3 d (a-b) (a+b)^{3/2}}-\frac {\sqrt {a+b} (2 b B-5 a C) \cot (c+d x) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \Pi \left (\frac {a+b}{b};\sin ^{-1}\left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right )}{b^4 d} \]
Antiderivative was successfully verified.
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Rule 2809
Rule 2816
Rule 2989
Rule 2994
Rule 2998
Rule 3029
Rule 3047
Rule 3053
Rule 3061
Rubi steps
\begin {align*} \int \frac {\cos ^{\frac {3}{2}}(c+d x) \left (B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^{5/2}} \, dx &=\int \frac {\cos ^{\frac {5}{2}}(c+d x) (B+C \cos (c+d x))}{(a+b \cos (c+d x))^{5/2}} \, dx\\ &=\frac {2 a (b B-a C) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^{3/2}}-\frac {2 \int \frac {\sqrt {\cos (c+d x)} \left (-\frac {3}{2} a (b B-a C)+\frac {3}{2} b (b B-a C) \cos (c+d x)+\frac {1}{2} \left (2 a b B-5 a^2 C+3 b^2 C\right ) \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^{3/2}} \, dx}{3 b \left (a^2-b^2\right )}\\ &=\frac {2 a (b B-a C) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^{3/2}}+\frac {2 a \left (2 a^2 b B-6 b^3 B-5 a^3 C+9 a b^2 C\right ) \sqrt {\cos (c+d x)} \sin (c+d x)}{3 b^2 \left (a^2-b^2\right )^2 d \sqrt {a+b \cos (c+d x)}}+\frac {4 \int \frac {\frac {1}{4} a \left (2 a^2 b B-6 b^3 B-5 a^3 C+9 a b^2 C\right )+\frac {1}{4} b \left (a^2 b B+3 b^3 B+2 a^3 C-6 a b^2 C\right ) \cos (c+d x)-\frac {1}{4} \left (6 a^3 b B-14 a b^3 B-15 a^4 C+26 a^2 b^2 C-3 b^4 C\right ) \cos ^2(c+d x)}{\sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}} \, dx}{3 b^2 \left (a^2-b^2\right )^2}\\ &=\frac {2 a (b B-a C) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^{3/2}}+\frac {2 a \left (2 a^2 b B-6 b^3 B-5 a^3 C+9 a b^2 C\right ) \sqrt {\cos (c+d x)} \sin (c+d x)}{3 b^2 \left (a^2-b^2\right )^2 d \sqrt {a+b \cos (c+d x)}}-\frac {\left (6 a^3 b B-14 a b^3 B-15 a^4 C+26 a^2 b^2 C-3 b^4 C\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{3 b^3 \left (a^2-b^2\right )^2 d \sqrt {\cos (c+d x)}}+\frac {2 \int \frac {\frac {1}{4} a \left (6 a^3 b B-14 a b^3 B-15 a^4 C+26 a^2 b^2 C-3 b^4 C\right )+\frac {1}{2} a b \left (2 a^2 b B-6 b^3 B-5 a^3 C+9 a b^2 C\right ) \cos (c+d x)+\frac {3}{4} \left (a^2-b^2\right )^2 (2 b B-5 a C) \cos ^2(c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}} \, dx}{3 b^3 \left (a^2-b^2\right )^2}\\ &=\frac {2 a (b B-a C) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^{3/2}}+\frac {2 a \left (2 a^2 b B-6 b^3 B-5 a^3 C+9 a b^2 C\right ) \sqrt {\cos (c+d x)} \sin (c+d x)}{3 b^2 \left (a^2-b^2\right )^2 d \sqrt {a+b \cos (c+d x)}}-\frac {\left (6 a^3 b B-14 a b^3 B-15 a^4 C+26 a^2 b^2 C-3 b^4 C\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{3 b^3 \left (a^2-b^2\right )^2 d \sqrt {\cos (c+d x)}}+\frac {2 \int \frac {\frac {1}{4} a \left (6 a^3 b B-14 a b^3 B-15 a^4 C+26 a^2 b^2 C-3 b^4 C\right )+\frac {1}{2} a b \left (2 a^2 b B-6 b^3 B-5 a^3 C+9 a b^2 C\right ) \cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}} \, dx}{3 b^3 \left (a^2-b^2\right )^2}+\frac {(2 b B-5 a C) \int \frac {\sqrt {\cos (c+d x)}}{\sqrt {a+b \cos (c+d x)}} \, dx}{2 b^3}\\ &=-\frac {\sqrt {a+b} (2 b B-5 a C) \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}}}{b^4 d}+\frac {2 a (b B-a C) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^{3/2}}+\frac {2 a \left (2 a^2 b B-6 b^3 B-5 a^3 C+9 a b^2 C\right ) \sqrt {\cos (c+d x)} \sin (c+d x)}{3 b^2 \left (a^2-b^2\right )^2 d \sqrt {a+b \cos (c+d x)}}-\frac {\left (6 a^3 b B-14 a b^3 B-15 a^4 C+26 a^2 b^2 C-3 b^4 C\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{3 b^3 \left (a^2-b^2\right )^2 d \sqrt {\cos (c+d x)}}+\frac {\left (a \left (6 a^3 b B-14 a b^3 B-15 a^4 C+26 a^2 b^2 C-3 b^4 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}{6 b^3 \left (a^2-b^2\right )^2}-\frac {\left (a \left (a^2 b (6 B-5 C)-3 b^3 (4 B-C)-15 a^3 C+a b^2 (2 B+21 C)\right )\right ) \int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}} \, dx}{6 (a-b) b^3 (a+b)^2}\\ &=\frac {\left (6 a^3 b B-14 a b^3 B-15 a^4 C+26 a^2 b^2 C-3 b^4 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}}}{3 a (a-b) b^3 (a+b)^{3/2} d}-\frac {\left (a^2 b (6 B-5 C)-3 b^3 (4 B-C)-15 a^3 C+a b^2 (2 B+21 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}}}{3 (a-b) b^3 (a+b)^{3/2} d}-\frac {\sqrt {a+b} (2 b B-5 a C) \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}}}{b^4 d}+\frac {2 a (b B-a C) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^{3/2}}+\frac {2 a \left (2 a^2 b B-6 b^3 B-5 a^3 C+9 a b^2 C\right ) \sqrt {\cos (c+d x)} \sin (c+d x)}{3 b^2 \left (a^2-b^2\right )^2 d \sqrt {a+b \cos (c+d x)}}-\frac {\left (6 a^3 b B-14 a b^3 B-15 a^4 C+26 a^2 b^2 C-3 b^4 C\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{3 b^3 \left (a^2-b^2\right )^2 d \sqrt {\cos (c+d x)}}\\ \end {align*}
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Mathematica [C] time = 6.71, size = 1396, normalized size = 2.07 \[ \text {result too large to display} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 1.38, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (C \cos \left (d x + c\right )^{3} + B \cos \left (d x + c\right )^{2}\right )} \sqrt {b \cos \left (d x + c\right ) + a} \sqrt {\cos \left (d x + c\right )}}{b^{3} \cos \left (d x + c\right )^{3} + 3 \, a b^{2} \cos \left (d x + c\right )^{2} + 3 \, a^{2} b \cos \left (d x + c\right ) + a^{3}}, x\right ) \]
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 \[ \int \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right )\right )} \cos \left (d x + c\right )^{\frac {3}{2}}}{{\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.63, size = 8611, normalized size = 12.78 \[ \text {output too large to display} \]
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 \[ \int \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right )\right )} \cos \left (d x + c\right )^{\frac {3}{2}}}{{\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}}}\,{d x} \]
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 \[ \int \frac {{\cos \left (c+d\,x\right )}^{3/2}\,\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )\right )}{{\left (a+b\,\cos \left (c+d\,x\right )\right )}^{5/2}} \,d x \]
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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