Optimal. Leaf size=193 \[ -\frac {2 b \left (5 a^2-4 b^2\right ) \sqrt {e \sin (c+d x)}}{21 d e^5}+\frac {2 a \left (5 a^2-6 b^2\right ) \sqrt {\sin (c+d x)} F\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right )}{21 d e^4 \sqrt {e \sin (c+d x)}}-\frac {2 \left (\left (5 a^2-4 b^2\right ) \cos (c+d x)+a b\right ) (a+b \cos (c+d x))}{21 d e^3 (e \sin (c+d x))^{3/2}}-\frac {2 (a \cos (c+d x)+b) (a+b \cos (c+d x))^2}{7 d e (e \sin (c+d x))^{7/2}} \]
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Rubi [A] time = 0.27, antiderivative size = 193, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2691, 2861, 2669, 2642, 2641} \[ -\frac {2 b \left (5 a^2-4 b^2\right ) \sqrt {e \sin (c+d x)}}{21 d e^5}-\frac {2 \left (\left (5 a^2-4 b^2\right ) \cos (c+d x)+a b\right ) (a+b \cos (c+d x))}{21 d e^3 (e \sin (c+d x))^{3/2}}+\frac {2 a \left (5 a^2-6 b^2\right ) \sqrt {\sin (c+d x)} F\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right )}{21 d e^4 \sqrt {e \sin (c+d x)}}-\frac {2 (a \cos (c+d x)+b) (a+b \cos (c+d x))^2}{7 d e (e \sin (c+d x))^{7/2}} \]
Antiderivative was successfully verified.
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Rule 2641
Rule 2642
Rule 2669
Rule 2691
Rule 2861
Rubi steps
\begin {align*} \int \frac {(a+b \cos (c+d x))^3}{(e \sin (c+d x))^{9/2}} \, dx &=-\frac {2 (b+a \cos (c+d x)) (a+b \cos (c+d x))^2}{7 d e (e \sin (c+d x))^{7/2}}-\frac {2 \int \frac {(a+b \cos (c+d x)) \left (-\frac {5 a^2}{2}+2 b^2-\frac {1}{2} a b \cos (c+d x)\right )}{(e \sin (c+d x))^{5/2}} \, dx}{7 e^2}\\ &=-\frac {2 (b+a \cos (c+d x)) (a+b \cos (c+d x))^2}{7 d e (e \sin (c+d x))^{7/2}}-\frac {2 (a+b \cos (c+d x)) \left (a b+\left (5 a^2-4 b^2\right ) \cos (c+d x)\right )}{21 d e^3 (e \sin (c+d x))^{3/2}}+\frac {4 \int \frac {\frac {1}{4} a \left (5 a^2-6 b^2\right )-\frac {1}{4} b \left (5 a^2-4 b^2\right ) \cos (c+d x)}{\sqrt {e \sin (c+d x)}} \, dx}{21 e^4}\\ &=-\frac {2 (b+a \cos (c+d x)) (a+b \cos (c+d x))^2}{7 d e (e \sin (c+d x))^{7/2}}-\frac {2 (a+b \cos (c+d x)) \left (a b+\left (5 a^2-4 b^2\right ) \cos (c+d x)\right )}{21 d e^3 (e \sin (c+d x))^{3/2}}-\frac {2 b \left (5 a^2-4 b^2\right ) \sqrt {e \sin (c+d x)}}{21 d e^5}+\frac {\left (a \left (5 a^2-6 b^2\right )\right ) \int \frac {1}{\sqrt {e \sin (c+d x)}} \, dx}{21 e^4}\\ &=-\frac {2 (b+a \cos (c+d x)) (a+b \cos (c+d x))^2}{7 d e (e \sin (c+d x))^{7/2}}-\frac {2 (a+b \cos (c+d x)) \left (a b+\left (5 a^2-4 b^2\right ) \cos (c+d x)\right )}{21 d e^3 (e \sin (c+d x))^{3/2}}-\frac {2 b \left (5 a^2-4 b^2\right ) \sqrt {e \sin (c+d x)}}{21 d e^5}+\frac {\left (a \left (5 a^2-6 b^2\right ) \sqrt {\sin (c+d x)}\right ) \int \frac {1}{\sqrt {\sin (c+d x)}} \, dx}{21 e^4 \sqrt {e \sin (c+d x)}}\\ &=-\frac {2 (b+a \cos (c+d x)) (a+b \cos (c+d x))^2}{7 d e (e \sin (c+d x))^{7/2}}-\frac {2 (a+b \cos (c+d x)) \left (a b+\left (5 a^2-4 b^2\right ) \cos (c+d x)\right )}{21 d e^3 (e \sin (c+d x))^{3/2}}+\frac {2 a \left (5 a^2-6 b^2\right ) F\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {\sin (c+d x)}}{21 d e^4 \sqrt {e \sin (c+d x)}}-\frac {2 b \left (5 a^2-4 b^2\right ) \sqrt {e \sin (c+d x)}}{21 d e^5}\\ \end {align*}
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Mathematica [A] time = 0.62, size = 144, normalized size = 0.75 \[ -\frac {2 \csc ^4(c+d x) \sqrt {e \sin (c+d x)} \left (a \left (5 a^2-6 b^2\right ) \sin ^{\frac {7}{2}}(c+d x) F\left (\left .\frac {1}{4} (-2 c-2 d x+\pi )\right |2\right )+\frac {1}{4} \left (-5 a^3 \cos (3 (c+d x))+a \left (17 a^2+30 b^2\right ) \cos (c+d x)+36 a^2 b+6 a b^2 \cos (3 (c+d x))+14 b^3 \cos (2 (c+d x))-2 b^3\right )\right )}{21 d e^5} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.10, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b^{3} \cos \left (d x + c\right )^{3} + 3 \, a b^{2} \cos \left (d x + c\right )^{2} + 3 \, a^{2} b \cos \left (d x + c\right ) + a^{3}\right )} \sqrt {e \sin \left (d x + c\right )}}{{\left (e^{5} \cos \left (d x + c\right )^{4} - 2 \, e^{5} \cos \left (d x + c\right )^{2} + e^{5}\right )} \sin \left (d x + c\right )}, x\right ) \]
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 \[ \int \frac {{\left (b \cos \left (d x + c\right ) + a\right )}^{3}}{\left (e \sin \left (d x + c\right )\right )^{\frac {9}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.33, size = 241, normalized size = 1.25 \[ \frac {-\frac {2 b \left (7 \left (\cos ^{2}\left (d x +c \right )\right ) b^{2}+9 a^{2}-4 b^{2}\right )}{21 e \left (e \sin \left (d x +c \right )\right )^{\frac {7}{2}}}-\frac {a \left (\left (-10 a^{2}+12 b^{2}\right ) \sin \left (d x +c \right ) \left (\cos ^{4}\left (d x +c \right )\right )+\left (16 a^{2}+6 b^{2}\right ) \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )+5 \sqrt {-\sin \left (d x +c \right )+1}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \left (\sin ^{\frac {9}{2}}\left (d x +c \right )\right ) \EllipticF \left (\sqrt {-\sin \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right ) a^{2}-6 \sqrt {-\sin \left (d x +c \right )+1}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \left (\sin ^{\frac {9}{2}}\left (d x +c \right )\right ) \EllipticF \left (\sqrt {-\sin \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right ) b^{2}\right )}{21 e^{4} \sin \left (d x +c \right )^{4} \cos \left (d x +c \right ) \sqrt {e \sin \left (d x +c \right )}}}{d} \]
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 \[ \int \frac {{\left (b \cos \left (d x + c\right ) + a\right )}^{3}}{\left (e \sin \left (d x + c\right )\right )^{\frac {9}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (a+b\,\cos \left (c+d\,x\right )\right )}^3}{{\left (e\,\sin \left (c+d\,x\right )\right )}^{9/2}} \,d x \]
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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