Optimal. Leaf size=70 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \sin ^4(c+d x)}}{\sqrt {a}}\right )}{2 \sqrt {a} d}-\frac {\csc ^2(c+d x) \sqrt {a+b \sin ^4(c+d x)}}{2 a d} \]
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Rubi [A]
time = 0.07, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3308, 821, 272,
65, 214} \begin {gather*} \frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \sin ^4(c+d x)}}{\sqrt {a}}\right )}{2 \sqrt {a} d}-\frac {\csc ^2(c+d x) \sqrt {a+b \sin ^4(c+d x)}}{2 a d} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 214
Rule 272
Rule 821
Rule 3308
Rubi steps
\begin {align*} \int \frac {\cot ^3(c+d x)}{\sqrt {a+b \sin ^4(c+d x)}} \, dx &=\frac {\text {Subst}\left (\int \frac {1-x}{x^2 \sqrt {a+b x^2}} \, dx,x,\sin ^2(c+d x)\right )}{2 d}\\ &=-\frac {\csc ^2(c+d x) \sqrt {a+b \sin ^4(c+d x)}}{2 a d}-\frac {\text {Subst}\left (\int \frac {1}{x \sqrt {a+b x^2}} \, dx,x,\sin ^2(c+d x)\right )}{2 d}\\ &=-\frac {\csc ^2(c+d x) \sqrt {a+b \sin ^4(c+d x)}}{2 a d}-\frac {\text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\sin ^4(c+d x)\right )}{4 d}\\ &=-\frac {\csc ^2(c+d x) \sqrt {a+b \sin ^4(c+d x)}}{2 a d}-\frac {\text {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 {\tanh ^{-1}\left (\frac {\sqrt {a+b \sin ^4(c+d x)}}{\sqrt {a}}\right )}{2 \sqrt {a} d}-\frac {\csc ^2(c+d x) \sqrt {a+b \sin ^4(c+d x)}}{2 a d}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 66, normalized size = 0.94 \begin {gather*} \frac {\sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a+b \sin ^4(c+d x)}}{\sqrt {a}}\right )-\csc ^2(c+d x) \sqrt {a+b \sin ^4(c+d x)}}{2 a d} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 2.70, size = 0, normalized size = 0.00 \[\int \frac {\cot ^{3}\left (d x +c \right )}{\sqrt {a +b \left (\sin ^{4}\left (d x +c \right )\right )}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.56, size = 79, normalized size = 1.13 \begin {gather*} -\frac {\frac {\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 )}{\sqrt {a}} + \frac {2 \, \sqrt {b \sin \left (d x + c\right )^{4} + a}}{a \sin \left (d x + c\right )^{2}}}{4 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.55, size = 247, normalized size = 3.53 \begin {gather*} \left [\frac {{\left (\cos \left (d x + c\right )^{2} - 1\right )} \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 \, {\left (a d \cos \left (d x + c\right )^{2} - a d\right )}}, -\frac {{\left (\cos \left (d x + c\right )^{2} - 1\right )} \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 \, {\left (a d \cos \left (d x + c\right )^{2} - a d\right )}}\right ] \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 {\cot ^{3}{\left (c + d x \right )}}{\sqrt {a + b \sin ^{4}{\left (c + d x \right )}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [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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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\mathrm {cot}\left (c+d\,x\right )}^3}{\sqrt {b\,{\sin \left (c+d\,x\right )}^4+a}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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