Optimal. Leaf size=247 \[ \frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)}}{7 b d}-\frac {32 a \left (a^2-2 b^2\right ) E\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {a+b \sin (c+d x)}}{35 b^4 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {8 \left (4 a^4-9 a^2 b^2+5 b^4\right ) F\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{35 b^4 d \sqrt {a+b \sin (c+d x)}}-\frac {4 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (4 a^2-5 b^2-3 a b \sin (c+d x)\right )}{35 b^3 d} \]
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Rubi [A]
time = 0.24, antiderivative size = 247, 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 = {2774, 2944,
2831, 2742, 2740, 2734, 2732} \begin {gather*} -\frac {32 a \left (a^2-2 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{35 b^4 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {4 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (4 a^2-3 a b \sin (c+d x)-5 b^2\right )}{35 b^3 d}+\frac {8 \left (4 a^4-9 a^2 b^2+5 b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} F\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{35 b^4 d \sqrt {a+b \sin (c+d x)}}+\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)}}{7 b d} \end {gather*}
Antiderivative was successfully verified.
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Rule 2732
Rule 2734
Rule 2740
Rule 2742
Rule 2774
Rule 2831
Rule 2944
Rubi steps
\begin {align*} \int \frac {\cos ^4(c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx &=\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)}}{7 b d}+\frac {6 \int \frac {\cos ^2(c+d x) (b+a \sin (c+d x))}{\sqrt {a+b \sin (c+d x)}} \, dx}{7 b}\\ &=\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)}}{7 b d}-\frac {4 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (4 a^2-5 b^2-3 a b \sin (c+d x)\right )}{35 b^3 d}+\frac {8 \int \frac {-\frac {1}{2} b \left (a^2-5 b^2\right )-2 a \left (a^2-2 b^2\right ) \sin (c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx}{35 b^3}\\ &=\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)}}{7 b d}-\frac {4 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (4 a^2-5 b^2-3 a b \sin (c+d x)\right )}{35 b^3 d}-\frac {\left (16 a \left (a^2-2 b^2\right )\right ) \int \sqrt {a+b \sin (c+d x)} \, dx}{35 b^4}+\frac {\left (4 \left (4 a^4-9 a^2 b^2+5 b^4\right )\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}} \, dx}{35 b^4}\\ &=\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)}}{7 b d}-\frac {4 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (4 a^2-5 b^2-3 a b \sin (c+d x)\right )}{35 b^3 d}-\frac {\left (16 a \left (a^2-2 b^2\right ) \sqrt {a+b \sin (c+d x)}\right ) \int \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}} \, dx}{35 b^4 \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {\left (4 \left (4 a^4-9 a^2 b^2+5 b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}\right ) \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}} \, dx}{35 b^4 \sqrt {a+b \sin (c+d x)}}\\ &=\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)}}{7 b d}-\frac {32 a \left (a^2-2 b^2\right ) E\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {a+b \sin (c+d x)}}{35 b^4 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {8 \left (4 a^4-9 a^2 b^2+5 b^4\right ) F\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{35 b^4 d \sqrt {a+b \sin (c+d x)}}-\frac {4 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (4 a^2-5 b^2-3 a b \sin (c+d x)\right )}{35 b^3 d}\\ \end {align*}
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Mathematica [A]
time = 1.07, size = 219, normalized size = 0.89 \begin {gather*} \frac {64 a \left (a^3+a^2 b-2 a b^2-2 b^3\right ) E\left (\frac {1}{4} (-2 c+\pi -2 d x)|\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}-16 \left (4 a^4-9 a^2 b^2+5 b^4\right ) F\left (\frac {1}{4} (-2 c+\pi -2 d x)|\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}+b \cos (c+d x) \left (-32 a^3+62 a b^2-2 a b^2 \cos (2 (c+d x))+\left (-8 a^2 b+45 b^3\right ) \sin (c+d x)+5 b^3 \sin (3 (c+d x))\right )}{70 b^4 d \sqrt {a+b \sin (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. \(941\) vs.
\(2(293)=586\).
time = 2.11, size = 942, normalized size = 3.81
method | result | size |
default | \(-\frac {2 \left (-5 b^{5} \left (\sin ^{5}\left (d x +c \right )\right )+16 \sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}\, \sqrt {-\frac {\left (\sin \left (d x +c \right )-1\right ) b}{a +b}}\, \sqrt {-\frac {\left (1+\sin \left (d x +c \right )\right ) b}{a -b}}\, \EllipticF \left (\sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right ) a^{4} b -12 \sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}\, \sqrt {-\frac {\left (\sin \left (d x +c \right )-1\right ) b}{a +b}}\, \sqrt {-\frac {\left (1+\sin \left (d x +c \right )\right ) b}{a -b}}\, \EllipticF \left (\sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right ) a^{3} b^{2}-36 \sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}\, \sqrt {-\frac {\left (\sin \left (d x +c \right )-1\right ) b}{a +b}}\, \sqrt {-\frac {\left (1+\sin \left (d x +c \right )\right ) b}{a -b}}\, \EllipticF \left (\sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right ) a^{2} b^{3}+12 \sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}\, \sqrt {-\frac {\left (\sin \left (d x +c \right )-1\right ) b}{a +b}}\, \sqrt {-\frac {\left (1+\sin \left (d x +c \right )\right ) b}{a -b}}\, \EllipticF \left (\sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right ) a \,b^{4}+20 \sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}\, \sqrt {-\frac {\left (\sin \left (d x +c \right )-1\right ) b}{a +b}}\, \sqrt {-\frac {\left (1+\sin \left (d x +c \right )\right ) b}{a -b}}\, \EllipticF \left (\sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right ) b^{5}-16 \sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}\, \sqrt {-\frac {\left (\sin \left (d x +c \right )-1\right ) b}{a +b}}\, \sqrt {-\frac {\left (1+\sin \left (d x +c \right )\right ) b}{a -b}}\, \EllipticE \left (\sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right ) a^{5}+48 \sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}\, \sqrt {-\frac {\left (\sin \left (d x +c \right )-1\right ) b}{a +b}}\, \sqrt {-\frac {\left (1+\sin \left (d x +c \right )\right ) b}{a -b}}\, \EllipticE \left (\sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right ) a^{3} b^{2}-32 \sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}\, \sqrt {-\frac {\left (\sin \left (d x +c \right )-1\right ) b}{a +b}}\, \sqrt {-\frac {\left (1+\sin \left (d x +c \right )\right ) b}{a -b}}\, \EllipticE \left (\sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right ) a \,b^{4}+a \,b^{4} \left (\sin ^{4}\left (d x +c \right )\right )-2 a^{2} b^{3} \left (\sin ^{3}\left (d x +c \right )\right )+20 b^{5} \left (\sin ^{3}\left (d x +c \right )\right )-8 a^{3} b^{2} \left (\sin ^{2}\left (d x +c \right )\right )+14 a \,b^{4} \left (\sin ^{2}\left (d x +c \right )\right )+2 a^{2} b^{3} \sin \left (d x +c \right )-15 b^{5} \sin \left (d x +c \right )+8 a^{3} b^{2}-15 a \,b^{4}\right )}{35 b^{5} \cos \left (d x +c \right ) \sqrt {a +b \sin \left (d x +c \right )}\, d}\) | \(942\) |
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 = 0.14, size = 493, normalized size = 2.00 \begin {gather*} \frac {2 \, {\left (2 \, \sqrt {2} {\left (8 \, a^{4} - 19 \, a^{2} b^{2} + 15 \, b^{4}\right )} \sqrt {i \, b} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 i \, a^{3} - 9 i \, 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 i \, a}{3 \, b}\right ) + 2 \, \sqrt {2} {\left (8 \, a^{4} - 19 \, a^{2} b^{2} + 15 \, b^{4}\right )} \sqrt {-i \, b} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (-8 i \, a^{3} + 9 i \, 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 i \, a}{3 \, b}\right ) - 24 \, \sqrt {2} {\left (-i \, a^{3} b + 2 i \, a b^{3}\right )} \sqrt {i \, b} {\rm weierstrassZeta}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 i \, a^{3} - 9 i \, 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 i \, a^{3} - 9 i \, 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 i \, a}{3 \, b}\right )\right ) - 24 \, \sqrt {2} {\left (i \, a^{3} b - 2 i \, a b^{3}\right )} \sqrt {-i \, b} {\rm weierstrassZeta}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (-8 i \, a^{3} + 9 i \, 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 i \, a^{3} + 9 i \, 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 i \, a}{3 \, b}\right )\right ) + 3 \, {\left (5 \, b^{4} \cos \left (d x + c\right )^{3} + 6 \, a b^{3} \cos \left (d x + c\right ) \sin \left (d x + c\right ) - 2 \, {\left (4 \, a^{2} b^{2} - 5 \, b^{4}\right )} \cos \left (d x + c\right )\right )} \sqrt {b \sin \left (d x + c\right ) + a}\right )}}{105 \, b^{5} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\cos ^{4}{\left (c + d x \right )}}{\sqrt {a + b \sin {\left (c + d x \right )}}}\, dx \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 \frac {{\cos \left (c+d\,x\right )}^4}{\sqrt {a+b\,\sin \left (c+d\,x\right )}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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