Optimal. Leaf size=264 \[ \frac {2 a \left (8 a^2+19 b^2\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{105 b^3 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {2 \left (8 a^4+17 a^2 b^2-25 b^4\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} F\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{105 b^3 d \sqrt {a+b \cos (c+d x)}}+\frac {2 \left (8 a^2+25 b^2\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{105 b^2 d}-\frac {8 a (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{35 b^2 d}+\frac {2 \cos (c+d x) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{7 b d} \]
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Rubi [A]
time = 0.27, 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 = {2872, 3102,
2832, 2831, 2742, 2740, 2734, 2732} \begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 2732
Rule 2734
Rule 2740
Rule 2742
Rule 2831
Rule 2832
Rule 2872
Rule 3102
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 = 1.17, size = 214, normalized size = 0.81 \begin {gather*} \frac {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 )-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 \left (-16 a^3+136 a b^2+\left (-4 a^2 b+145 b^3\right ) \cos (c+d x)+36 a b^2 \cos (2 (c+d x))+15 b^3 \cos (3 (c+d x))\right ) \sin (c+d x)}{210 b^3 d \sqrt {a+b \cos (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(826\) vs.
\(2(298)=596\).
time = 0.31, size = 827, normalized size = 3.13
method | result | size |
default | \(-\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 (240 \left (\cos ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{4}+144 \left (\cos ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a \,b^{3}-600 \left (\cos ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{4}-4 \left (\cos ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2} b^{2}-288 \left (\cos ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a \,b^{3}+640 \left (\cos ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{4}-8 \left (\cos ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{3} b +6 \left (\cos ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2} b^{2}+230 \left (\cos ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a \,b^{3}-360 \left (\cos ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{4}-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^{4}-17 \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^{2} b^{2}+25 \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 ) b^{4}+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^{4}-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} b +19 \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^{2}-19 \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^{3}+8 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{3} b -2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{2} b^{2}-86 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) a \,b^{3}+80 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{4}\right )}{105 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}\) | \(827\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.16, size = 474, normalized size = 1.80 \begin {gather*} \frac {\sqrt {2} {\left (16 i \, a^{4} + 32 i \, a^{2} b^{2} - 75 i \, b^{4}\right )} \sqrt {b} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) + 3 i \, b \sin \left (d x + c\right ) + 2 \, a}{3 \, b}\right ) + \sqrt {2} {\left (-16 i \, a^{4} - 32 i \, a^{2} b^{2} + 75 i \, b^{4}\right )} \sqrt {b} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) - 3 i \, b \sin \left (d x + c\right ) + 2 \, a}{3 \, b}\right ) - 3 \, \sqrt {2} {\left (-8 i \, a^{3} b - 19 i \, a b^{3}\right )} \sqrt {b} {\rm weierstrassZeta}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) + 3 i \, b \sin \left (d x + c\right ) + 2 \, a}{3 \, b}\right )\right ) - 3 \, \sqrt {2} {\left (8 i \, a^{3} b + 19 i \, a b^{3}\right )} \sqrt {b} {\rm weierstrassZeta}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) - 3 i \, b \sin \left (d x + c\right ) + 2 \, a}{3 \, b}\right )\right ) + 6 \, {\left (15 \, b^{4} \cos \left (d x + c\right )^{2} + 3 \, a b^{3} \cos \left (d x + c\right ) - 4 \, a^{2} b^{2} + 25 \, b^{4}\right )} \sqrt {b \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{315 \, b^{4} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int {\cos \left (c+d\,x\right )}^3\,\sqrt {a+b\,\cos \left (c+d\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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