Optimal. Leaf size=59 \[ \frac {\sqrt {a+b \sin ^4(c+d x)}}{2 d}-\frac {\sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a+b \sin ^4(c+d x)}}{\sqrt {a}}\right )}{2 d} \]
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Rubi [A] time = 0.09, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {3229, 266, 50, 63, 208} \[ \frac {\sqrt {a+b \sin ^4(c+d x)}}{2 d}-\frac {\sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a+b \sin ^4(c+d x)}}{\sqrt {a}}\right )}{2 d} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 208
Rule 266
Rule 3229
Rubi steps
\begin {align*} \int \cot (c+d x) \sqrt {a+b \sin ^4(c+d x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\sqrt {a+b x^2}}{x} \, dx,x,\sin ^2(c+d x)\right )}{2 d}\\ &=\frac {\operatorname {Subst}\left (\int \frac {\sqrt {a+b x}}{x} \, dx,x,\sin ^4(c+d x)\right )}{4 d}\\ &=\frac {\sqrt {a+b \sin ^4(c+d x)}}{2 d}+\frac {a \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\sin ^4(c+d x)\right )}{4 d}\\ &=\frac {\sqrt {a+b \sin ^4(c+d x)}}{2 d}+\frac {a \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \sin ^4(c+d x)}\right )}{2 b d}\\ &=-\frac {\sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a+b \sin ^4(c+d x)}}{\sqrt {a}}\right )}{2 d}+\frac {\sqrt {a+b \sin ^4(c+d x)}}{2 d}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 55, normalized size = 0.93 \[ -\frac {\sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a+b \sin ^4(c+d x)}}{\sqrt {a}}\right )-\sqrt {a+b \sin ^4(c+d x)}}{2 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.65, size = 195, normalized size = 3.31 \[ \left [\frac {\sqrt {a} \log \left (\frac {8 \, {\left (b \cos \left (d x + c\right )^{4} - 2 \, b \cos \left (d x + c\right )^{2} - 2 \, \sqrt {b \cos \left (d x + c\right )^{4} - 2 \, b \cos \left (d x + c\right )^{2} + a + b} \sqrt {a} + 2 \, a + b\right )}}{\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1}\right ) + 2 \, \sqrt {b \cos \left (d x + c\right )^{4} - 2 \, b \cos \left (d x + c\right )^{2} + a + b}}{4 \, d}, \frac {\sqrt {-a} \arctan \left (\frac {\sqrt {b \cos \left (d x + c\right )^{4} - 2 \, b \cos \left (d x + c\right )^{2} + a + b} \sqrt {-a}}{a}\right ) + \sqrt {b \cos \left (d x + c\right )^{4} - 2 \, b \cos \left (d x + c\right )^{2} + a + b}}{2 \, d}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [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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maple [F] time = 1.73, size = 0, normalized size = 0.00 \[ \int \cot \left (d x +c \right ) \sqrt {a +b \left (\sin ^{4}\left (d x +c \right )\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.61, size = 68, normalized size = 1.15 \[ \frac {\sqrt {a} \log \left (\frac {\sqrt {b \sin \left (d x + c\right )^{4} + a} - \sqrt {a}}{\sqrt {b \sin \left (d x + c\right )^{4} + a} + \sqrt {a}}\right ) + 2 \, \sqrt {b \sin \left (d x + c\right )^{4} + a}}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \mathrm {cot}\left (c+d\,x\right )\,\sqrt {b\,{\sin \left (c+d\,x\right )}^4+a} \,d x \]
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 \[ \int \sqrt {a + b \sin ^{4}{\left (c + d x \right )}} \cot {\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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