Optimal. Leaf size=156 \[ \frac {\tanh ^{-1}\left (\frac {2 a-b+(b-2 c) \cot ^2(d+e x)}{2 \sqrt {a-b+c} \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right )}{2 (a-b+c)^{3/2} e}-\frac {b^2-2 a c-b c+(b-2 c) c \cot ^2(d+e x)}{(a-b+c) \left (b^2-4 a c\right ) e \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}} \]
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Rubi [A]
time = 0.16, antiderivative size = 156, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {3782, 1261,
754, 12, 738, 212} \begin {gather*} \frac {\tanh ^{-1}\left (\frac {2 a+(b-2 c) \cot ^2(d+e x)-b}{2 \sqrt {a-b+c} \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right )}{2 e (a-b+c)^{3/2}}-\frac {-2 a c+b^2+c (b-2 c) \cot ^2(d+e x)-b c}{e (a-b+c) \left (b^2-4 a c\right ) \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 212
Rule 738
Rule 754
Rule 1261
Rule 3782
Rubi steps
\begin {align*} \int \frac {\cot (d+e x)}{\left (a+b \cot ^2(d+e x)+c \cot ^4(d+e x)\right )^{3/2}} \, dx &=-\frac {\text {Subst}\left (\int \frac {x}{\left (1+x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}} \, dx,x,\cot (d+e x)\right )}{e}\\ &=-\frac {\text {Subst}\left (\int \frac {1}{(1+x) \left (a+b x+c x^2\right )^{3/2}} \, dx,x,\cot ^2(d+e x)\right )}{2 e}\\ &=-\frac {b^2-2 a c-b c+(b-2 c) c \cot ^2(d+e x)}{(a-b+c) \left (b^2-4 a c\right ) e \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}+\frac {\text {Subst}\left (\int \frac {-\frac {b^2}{2}+2 a c}{(1+x) \sqrt {a+b x+c x^2}} \, dx,x,\cot ^2(d+e x)\right )}{(a-b+c) \left (b^2-4 a c\right ) e}\\ &=-\frac {b^2-2 a c-b c+(b-2 c) c \cot ^2(d+e x)}{(a-b+c) \left (b^2-4 a c\right ) e \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}-\frac {\text {Subst}\left (\int \frac {1}{(1+x) \sqrt {a+b x+c x^2}} \, dx,x,\cot ^2(d+e x)\right )}{2 (a-b+c) e}\\ &=-\frac {b^2-2 a c-b c+(b-2 c) c \cot ^2(d+e x)}{(a-b+c) \left (b^2-4 a c\right ) e \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}+\frac {\text {Subst}\left (\int \frac {1}{4 a-4 b+4 c-x^2} \, dx,x,\frac {2 a-b-(-b+2 c) \cot ^2(d+e x)}{\sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right )}{(a-b+c) e}\\ &=\frac {\tanh ^{-1}\left (\frac {2 a-b+(b-2 c) \cot ^2(d+e x)}{2 \sqrt {a-b+c} \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right )}{2 (a-b+c)^{3/2} e}-\frac {b^2-2 a c-b c+(b-2 c) c \cot ^2(d+e x)}{(a-b+c) \left (b^2-4 a c\right ) e \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in
optimal.
time = 7.67, size = 25149, normalized size = 161.21 \begin {gather*} \text {Result too large to show} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(405\) vs.
\(2(145)=290\).
time = 0.27, size = 406, normalized size = 2.60
method | result | size |
derivativedivides | \(\frac {-\frac {2 c \ln \left (\frac {2 a -2 b +2 c +\left (b -2 c \right ) \left (\cot ^{2}\left (e x +d \right )+1\right )+2 \sqrt {a -b +c}\, \sqrt {c \left (\cot ^{2}\left (e x +d \right )+1\right )^{2}+\left (b -2 c \right ) \left (\cot ^{2}\left (e x +d \right )+1\right )+a -b +c}}{\cot ^{2}\left (e x +d \right )+1}\right )}{\left (\sqrt {-4 a c +b^{2}}-b +2 c \right ) \left (\sqrt {-4 a c +b^{2}}+b -2 c \right ) \sqrt {a -b +c}}-\frac {2 c \sqrt {\left (\cot ^{2}\left (e x +d \right )+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right )^{2} c -\sqrt {-4 a c +b^{2}}\, \left (\cot ^{2}\left (e x +d \right )+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}}{\left (\sqrt {-4 a c +b^{2}}+b -2 c \right ) \left (-4 a c +b^{2}\right ) \left (\cot ^{2}\left (e x +d \right )+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}+\frac {2 c \sqrt {\left (\cot ^{2}\left (e x +d \right )-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )^{2} c +\sqrt {-4 a c +b^{2}}\, \left (\cot ^{2}\left (e x +d \right )-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}}{\left (\sqrt {-4 a c +b^{2}}-b +2 c \right ) \left (-4 a c +b^{2}\right ) \left (\cot ^{2}\left (e x +d \right )-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}}{e}\) | \(406\) |
default | \(\frac {-\frac {2 c \ln \left (\frac {2 a -2 b +2 c +\left (b -2 c \right ) \left (\cot ^{2}\left (e x +d \right )+1\right )+2 \sqrt {a -b +c}\, \sqrt {c \left (\cot ^{2}\left (e x +d \right )+1\right )^{2}+\left (b -2 c \right ) \left (\cot ^{2}\left (e x +d \right )+1\right )+a -b +c}}{\cot ^{2}\left (e x +d \right )+1}\right )}{\left (\sqrt {-4 a c +b^{2}}-b +2 c \right ) \left (\sqrt {-4 a c +b^{2}}+b -2 c \right ) \sqrt {a -b +c}}-\frac {2 c \sqrt {\left (\cot ^{2}\left (e x +d \right )+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right )^{2} c -\sqrt {-4 a c +b^{2}}\, \left (\cot ^{2}\left (e x +d \right )+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}}{\left (\sqrt {-4 a c +b^{2}}+b -2 c \right ) \left (-4 a c +b^{2}\right ) \left (\cot ^{2}\left (e x +d \right )+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}+\frac {2 c \sqrt {\left (\cot ^{2}\left (e x +d \right )-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )^{2} c +\sqrt {-4 a c +b^{2}}\, \left (\cot ^{2}\left (e x +d \right )-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}}{\left (\sqrt {-4 a c +b^{2}}-b +2 c \right ) \left (-4 a c +b^{2}\right ) \left (\cot ^{2}\left (e x +d \right )-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}}{e}\) | \(406\) |
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 [B] Leaf count of result is larger than twice the leaf count of optimal. 908 vs.
\(2 (148) = 296\).
time = 4.18, size = 1813, normalized size = 11.62 \begin {gather*} \text {Too large to display} \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 {\left (d + e x \right )}}{\left (a + b \cot ^{2}{\left (d + e x \right )} + c \cot ^{4}{\left (d + e x \right )}\right )^{\frac {3}{2}}}\, 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 (d+e\,x\right )}{{\left (c\,{\mathrm {cot}\left (d+e\,x\right )}^4+b\,{\mathrm {cot}\left (d+e\,x\right )}^2+a\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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