Optimal. Leaf size=215 \[ -\frac {2 a \left (8 a^2+7 b^2\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} F\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{15 b^3 d \sqrt {a+b \cos (c+d x)}}+\frac {2 \left (8 a^2+9 b^2\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{15 b^3 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {8 a \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{15 b^2 d}+\frac {2 \sin (c+d x) \cos (c+d x) \sqrt {a+b \cos (c+d x)}}{5 b d} \]
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Rubi [A] time = 0.29, antiderivative size = 215, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {2793, 3023, 2752, 2663, 2661, 2655, 2653} \[ -\frac {2 a \left (8 a^2+7 b^2\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} F\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{15 b^3 d \sqrt {a+b \cos (c+d x)}}+\frac {2 \left (8 a^2+9 b^2\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{15 b^3 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {8 a \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{15 b^2 d}+\frac {2 \sin (c+d x) \cos (c+d x) \sqrt {a+b \cos (c+d x)}}{5 b d} \]
Antiderivative was successfully verified.
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Rule 2653
Rule 2655
Rule 2661
Rule 2663
Rule 2752
Rule 2793
Rule 3023
Rubi steps
\begin {align*} \int \frac {\cos ^3(c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx &=\frac {2 \cos (c+d x) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{5 b d}+\frac {2 \int \frac {a+\frac {3}{2} b \cos (c+d x)-2 a \cos ^2(c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx}{5 b}\\ &=-\frac {8 a \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{15 b^2 d}+\frac {2 \cos (c+d x) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{5 b d}+\frac {4 \int \frac {\frac {a b}{2}+\frac {1}{4} \left (8 a^2+9 b^2\right ) \cos (c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx}{15 b^2}\\ &=-\frac {8 a \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{15 b^2 d}+\frac {2 \cos (c+d x) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{5 b d}-\frac {\left (a \left (8 a^2+7 b^2\right )\right ) \int \frac {1}{\sqrt {a+b \cos (c+d x)}} \, dx}{15 b^3}+\frac {\left (8 a^2+9 b^2\right ) \int \sqrt {a+b \cos (c+d x)} \, dx}{15 b^3}\\ &=-\frac {8 a \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{15 b^2 d}+\frac {2 \cos (c+d x) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{5 b d}+\frac {\left (\left (8 a^2+9 b^2\right ) \sqrt {a+b \cos (c+d x)}\right ) \int \sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}} \, dx}{15 b^3 \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {\left (a \left (8 a^2+7 b^2\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}}\right ) \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}} \, dx}{15 b^3 \sqrt {a+b \cos (c+d x)}}\\ &=\frac {2 \left (8 a^2+9 b^2\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{15 b^3 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {2 a \left (8 a^2+7 b^2\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} F\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{15 b^3 d \sqrt {a+b \cos (c+d x)}}-\frac {8 a \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{15 b^2 d}+\frac {2 \cos (c+d x) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{5 b d}\\ \end {align*}
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Mathematica [A] time = 0.91, size = 182, normalized size = 0.85 \[ \frac {b \sin (c+d x) \left (-8 a^2-2 a b \cos (c+d x)+3 b^2 \cos (2 (c+d x))+3 b^2\right )-2 a \left (8 a^2+7 b^2\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} F\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )+2 \left (8 a^3+8 a^2 b+9 a b^2+9 b^3\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{15 b^3 d \sqrt {a+b \cos (c+d x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.54, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\cos \left (d x + c\right )^{3}}{\sqrt {b \cos \left (d x + c\right ) + a}}, x\right ) \]
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 \[ \int \frac {\cos \left (d x + c\right )^{3}}{\sqrt {b \cos \left (d x + c\right ) + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.82, size = 665, normalized size = 3.09 \[ -\frac {2 \sqrt {\left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b +a -b \right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \left (24 \left (\cos ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{3}-4 \left (\cos ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a \,b^{2}-48 \left (\cos ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{3}-8 \left (\cos ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2} b +6 \left (\cos ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a \,b^{2}+30 \left (\cos ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{3}-8 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {\frac {2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b +a -b}{a -b}}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {-\frac {2 b}{a -b}}\right ) a^{3}-7 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {\frac {2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b +a -b}{a -b}}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {-\frac {2 b}{a -b}}\right ) a \,b^{2}+8 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {\frac {2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b +a -b}{a -b}}\, \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {-\frac {2 b}{a -b}}\right ) a^{3}-8 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {\frac {2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b +a -b}{a -b}}\, \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {-\frac {2 b}{a -b}}\right ) a^{2} b +9 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {\frac {2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b +a -b}{a -b}}\, \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {-\frac {2 b}{a -b}}\right ) a \,b^{2}-9 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {\frac {2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b +a -b}{a -b}}\, \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {-\frac {2 b}{a -b}}\right ) b^{3}+8 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{2} b -2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) a \,b^{2}-6 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{3}\right )}{15 b^{3} \sqrt {-2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b +\left (a +b \right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b +a +b}\, 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 \[ \int \frac {\cos \left (d x + c\right )^{3}}{\sqrt {b \cos \left (d x + c\right ) + a}}\,{d x} \]
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 \[ \int \frac {{\cos \left (c+d\,x\right )}^3}{\sqrt {a+b\,\cos \left (c+d\,x\right )}} \,d x \]
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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