Optimal. Leaf size=182 \[ \frac {\tanh ^{-1}\left (\frac {2 a-b+(b-2 c) \cot ^2(d+e x)}{2 \sqrt {a-b+c} \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right )}{2 \sqrt {a-b+c} e}+\frac {(b+2 c) \tanh ^{-1}\left (\frac {b+2 c \cot ^2(d+e x)}{2 \sqrt {c} \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right )}{4 c^{3/2} e}-\frac {\sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}{2 c e} \]
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Rubi [A]
time = 0.27, antiderivative size = 182, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 7, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3782, 1265,
1667, 857, 635, 212, 738} \begin {gather*} \frac {(b+2 c) \tanh ^{-1}\left (\frac {b+2 c \cot ^2(d+e x)}{2 \sqrt {c} \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right )}{4 c^{3/2} e}-\frac {\sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}{2 c e}+\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 \sqrt {a-b+c}} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 635
Rule 738
Rule 857
Rule 1265
Rule 1667
Rule 3782
Rubi steps
\begin {align*} \int \frac {\cot ^5(d+e x)}{\sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}} \, dx &=-\frac {\text {Subst}\left (\int \frac {x^5}{\left (1+x^2\right ) \sqrt {a+b x^2+c x^4}} \, dx,x,\cot (d+e x)\right )}{e}\\ &=-\frac {\text {Subst}\left (\int \frac {x^2}{(1+x) \sqrt {a+b x+c x^2}} \, dx,x,\cot ^2(d+e x)\right )}{2 e}\\ &=-\frac {\sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}{2 c e}-\frac {\text {Subst}\left (\int \frac {-\frac {b}{2}-\frac {1}{2} (b+2 c) x}{(1+x) \sqrt {a+b x+c x^2}} \, dx,x,\cot ^2(d+e x)\right )}{2 c e}\\ &=-\frac {\sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}{2 c e}-\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 e}+\frac {(b+2 c) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x+c x^2}} \, dx,x,\cot ^2(d+e x)\right )}{4 c e}\\ &=-\frac {\sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}{2 c e}+\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 )}{e}+\frac {(b+2 c) \text {Subst}\left (\int \frac {1}{4 c-x^2} \, dx,x,\frac {b+2 c \cot ^2(d+e x)}{\sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right )}{2 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 \sqrt {a-b+c} e}+\frac {(b+2 c) \tanh ^{-1}\left (\frac {b+2 c \cot ^2(d+e x)}{2 \sqrt {c} \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right )}{4 c^{3/2} e}-\frac {\sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}{2 c e}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in
optimal.
time = 34.24, size = 179905, normalized size = 988.49 \begin {gather*} \text {Result too large to show} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.40, size = 232, normalized size = 1.27
method | result | size |
derivativedivides | \(\frac {-\frac {\sqrt {a +b \left (\cot ^{2}\left (e x +d \right )\right )+c \left (\cot ^{4}\left (e x +d \right )\right )}}{2 c}+\frac {b \ln \left (\frac {\frac {b}{2}+c \left (\cot ^{2}\left (e x +d \right )\right )}{\sqrt {c}}+\sqrt {a +b \left (\cot ^{2}\left (e x +d \right )\right )+c \left (\cot ^{4}\left (e x +d \right )\right )}\right )}{4 c^{\frac {3}{2}}}+\frac {\ln \left (\frac {\frac {b}{2}+c \left (\cot ^{2}\left (e x +d \right )\right )}{\sqrt {c}}+\sqrt {a +b \left (\cot ^{2}\left (e x +d \right )\right )+c \left (\cot ^{4}\left (e x +d \right )\right )}\right )}{2 \sqrt {c}}+\frac {\ln \left (\frac {2 a -2 b +2 c +\left (b -2 c \right ) \left (\cot ^{2}\left (e x +d \right )+1\right )+2 \sqrt {a -b +c}\, \sqrt {c \left (\cot ^{2}\left (e x +d \right )+1\right )^{2}+\left (b -2 c \right ) \left (\cot ^{2}\left (e x +d \right )+1\right )+a -b +c}}{\cot ^{2}\left (e x +d \right )+1}\right )}{2 \sqrt {a -b +c}}}{e}\) | \(232\) |
default | \(\frac {-\frac {\sqrt {a +b \left (\cot ^{2}\left (e x +d \right )\right )+c \left (\cot ^{4}\left (e x +d \right )\right )}}{2 c}+\frac {b \ln \left (\frac {\frac {b}{2}+c \left (\cot ^{2}\left (e x +d \right )\right )}{\sqrt {c}}+\sqrt {a +b \left (\cot ^{2}\left (e x +d \right )\right )+c \left (\cot ^{4}\left (e x +d \right )\right )}\right )}{4 c^{\frac {3}{2}}}+\frac {\ln \left (\frac {\frac {b}{2}+c \left (\cot ^{2}\left (e x +d \right )\right )}{\sqrt {c}}+\sqrt {a +b \left (\cot ^{2}\left (e x +d \right )\right )+c \left (\cot ^{4}\left (e x +d \right )\right )}\right )}{2 \sqrt {c}}+\frac {\ln \left (\frac {2 a -2 b +2 c +\left (b -2 c \right ) \left (\cot ^{2}\left (e x +d \right )+1\right )+2 \sqrt {a -b +c}\, \sqrt {c \left (\cot ^{2}\left (e x +d \right )+1\right )^{2}+\left (b -2 c \right ) \left (\cot ^{2}\left (e x +d \right )+1\right )+a -b +c}}{\cot ^{2}\left (e x +d \right )+1}\right )}{2 \sqrt {a -b +c}}}{e}\) | \(232\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [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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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 527 vs.
\(2 (163) = 326\).
time = 5.21, size = 2180, normalized size = 11.98 \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 ^{5}{\left (d + e x \right )}}{\sqrt {a + b \cot ^{2}{\left (d + e x \right )} + c \cot ^{4}{\left (d + e 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 (d+e\,x\right )}^5}{\sqrt {c\,{\mathrm {cot}\left (d+e\,x\right )}^4+b\,{\mathrm {cot}\left (d+e\,x\right )}^2+a}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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