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 = 0.0889988, antiderivative size = 59, normalized size of antiderivative = 1., 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 3229
Rule 266
Rule 50
Rule 63
Rule 208
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*}
Mathematica [A] time = 0.053884, 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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Maple [F] time = 0.754, size = 0, normalized size = 0. \begin{align*} \int \cot \left ( dx+c \right ) \sqrt{a+b \left ( \sin \left ( dx+c \right ) \right ) ^{4}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.92246, size = 514, normalized size = 8.71 \begin{align*} \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 ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{a + b \sin ^{4}{\left (c + d x \right )}} \cot{\left (c + d x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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