Optimal. Leaf size=264 \[ \frac {2 \left (8 a^2+25 b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{105 b^2 d}+\frac {2 a \left (8 a^2+19 b^2\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{105 b^3 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {2 \left (8 a^4+17 a^2 b^2-25 b^4\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} F\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{105 b^3 d \sqrt {a+b \cos (c+d x)}}-\frac {8 a \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{35 b^2 d}+\frac {2 \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{3/2}}{7 b d} \]
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Rubi [A] time = 0.41, antiderivative size = 264, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {2793, 3023, 2753, 2752, 2663, 2661, 2655, 2653} \[ \frac {2 \left (8 a^2+25 b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{105 b^2 d}-\frac {2 \left (17 a^2 b^2+8 a^4-25 b^4\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} F\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{105 b^3 d \sqrt {a+b \cos (c+d x)}}+\frac {2 a \left (8 a^2+19 b^2\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{105 b^3 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {8 a \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{35 b^2 d}+\frac {2 \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{3/2}}{7 b d} \]
Antiderivative was successfully verified.
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Rule 2653
Rule 2655
Rule 2661
Rule 2663
Rule 2752
Rule 2753
Rule 2793
Rule 3023
Rubi steps
\begin {align*} \int \cos ^3(c+d x) \sqrt {a+b \cos (c+d x)} \, dx &=\frac {2 \cos (c+d x) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{7 b d}+\frac {2 \int \sqrt {a+b \cos (c+d x)} \left (a+\frac {5}{2} b \cos (c+d x)-2 a \cos ^2(c+d x)\right ) \, dx}{7 b}\\ &=-\frac {8 a (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{35 b^2 d}+\frac {2 \cos (c+d x) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{7 b d}+\frac {4 \int \sqrt {a+b \cos (c+d x)} \left (-\frac {a b}{2}+\frac {1}{4} \left (8 a^2+25 b^2\right ) \cos (c+d x)\right ) \, dx}{35 b^2}\\ &=\frac {2 \left (8 a^2+25 b^2\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{105 b^2 d}-\frac {8 a (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{35 b^2 d}+\frac {2 \cos (c+d x) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{7 b d}+\frac {8 \int \frac {\frac {1}{8} b \left (2 a^2+25 b^2\right )+\frac {1}{8} a \left (8 a^2+19 b^2\right ) \cos (c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx}{105 b^2}\\ &=\frac {2 \left (8 a^2+25 b^2\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{105 b^2 d}-\frac {8 a (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{35 b^2 d}+\frac {2 \cos (c+d x) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{7 b d}+\frac {\left (a \left (8 a^2+19 b^2\right )\right ) \int \sqrt {a+b \cos (c+d x)} \, dx}{105 b^3}-\frac {\left (8 a^4+17 a^2 b^2-25 b^4\right ) \int \frac {1}{\sqrt {a+b \cos (c+d x)}} \, dx}{105 b^3}\\ &=\frac {2 \left (8 a^2+25 b^2\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{105 b^2 d}-\frac {8 a (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{35 b^2 d}+\frac {2 \cos (c+d x) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{7 b d}+\frac {\left (a \left (8 a^2+19 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}{105 b^3 \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {\left (\left (8 a^4+17 a^2 b^2-25 b^4\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}{105 b^3 \sqrt {a+b \cos (c+d x)}}\\ &=\frac {2 a \left (8 a^2+19 b^2\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{105 b^3 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {2 \left (8 a^4+17 a^2 b^2-25 b^4\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 )}{105 b^3 d \sqrt {a+b \cos (c+d x)}}+\frac {2 \left (8 a^2+25 b^2\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{105 b^2 d}-\frac {8 a (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{35 b^2 d}+\frac {2 \cos (c+d x) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{7 b d}\\ \end {align*}
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Mathematica [A] time = 1.14, size = 214, normalized size = 0.81 \[ \frac {-4 \left (8 a^4+17 a^2 b^2-25 b^4\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} F\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )+b \sin (c+d x) \left (-16 a^3+\left (145 b^3-4 a^2 b\right ) \cos (c+d x)+36 a b^2 \cos (2 (c+d x))+136 a b^2+15 b^3 \cos (3 (c+d x))\right )+4 a \left (8 a^3+8 a^2 b+19 a b^2+19 b^3\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{210 b^3 d \sqrt {a+b \cos (c+d x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {b \cos \left (d x + c\right ) + a} \cos \left (d x + c\right )^{3}, x\right ) \]
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 \[ \int \sqrt {b \cos \left (d x + c\right ) + a} \cos \left (d x + c\right )^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 1.09, size = 827, normalized size = 3.13 \[ \text {result too large to display} \]
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 \[ \int \sqrt {b \cos \left (d x + c\right ) + a} \cos \left (d x + c\right )^{3}\,{d x} \]
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 \[ \int {\cos \left (c+d\,x\right )}^3\,\sqrt {a+b\,\cos \left (c+d\,x\right )} \,d x \]
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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