Optimal. Leaf size=214 \[ \frac {\log \left (\sqrt {d} \cot (e+f x)-\sqrt {2} \sqrt {d \cot (e+f x)}+\sqrt {d}\right )}{2 \sqrt {2} d^{3/2} f}-\frac {\log \left (\sqrt {d} \cot (e+f x)+\sqrt {2} \sqrt {d \cot (e+f x)}+\sqrt {d}\right )}{2 \sqrt {2} d^{3/2} f}-\frac {\tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {d \cot (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} d^{3/2} f}+\frac {\tan ^{-1}\left (\frac {\sqrt {2} \sqrt {d \cot (e+f x)}}{\sqrt {d}}+1\right )}{\sqrt {2} d^{3/2} f}-\frac {2 (d \cot (e+f x))^{3/2}}{3 d^3 f} \]
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Rubi [A] time = 0.16, antiderivative size = 214, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 10, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, Rules used = {16, 3473, 3476, 329, 297, 1162, 617, 204, 1165, 628} \[ -\frac {2 (d \cot (e+f x))^{3/2}}{3 d^3 f}+\frac {\log \left (\sqrt {d} \cot (e+f x)-\sqrt {2} \sqrt {d \cot (e+f x)}+\sqrt {d}\right )}{2 \sqrt {2} d^{3/2} f}-\frac {\log \left (\sqrt {d} \cot (e+f x)+\sqrt {2} \sqrt {d \cot (e+f x)}+\sqrt {d}\right )}{2 \sqrt {2} d^{3/2} f}-\frac {\tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {d \cot (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} d^{3/2} f}+\frac {\tan ^{-1}\left (\frac {\sqrt {2} \sqrt {d \cot (e+f x)}}{\sqrt {d}}+1\right )}{\sqrt {2} d^{3/2} f} \]
Antiderivative was successfully verified.
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Rule 16
Rule 204
Rule 297
Rule 329
Rule 617
Rule 628
Rule 1162
Rule 1165
Rule 3473
Rule 3476
Rubi steps
\begin {align*} \int \frac {\cot ^4(e+f x)}{(d \cot (e+f x))^{3/2}} \, dx &=\frac {\int (d \cot (e+f x))^{5/2} \, dx}{d^4}\\ &=-\frac {2 (d \cot (e+f x))^{3/2}}{3 d^3 f}-\frac {\int \sqrt {d \cot (e+f x)} \, dx}{d^2}\\ &=-\frac {2 (d \cot (e+f x))^{3/2}}{3 d^3 f}+\frac {\operatorname {Subst}\left (\int \frac {\sqrt {x}}{d^2+x^2} \, dx,x,d \cot (e+f x)\right )}{d f}\\ &=-\frac {2 (d \cot (e+f x))^{3/2}}{3 d^3 f}+\frac {2 \operatorname {Subst}\left (\int \frac {x^2}{d^2+x^4} \, dx,x,\sqrt {d \cot (e+f x)}\right )}{d f}\\ &=-\frac {2 (d \cot (e+f x))^{3/2}}{3 d^3 f}-\frac {\operatorname {Subst}\left (\int \frac {d-x^2}{d^2+x^4} \, dx,x,\sqrt {d \cot (e+f x)}\right )}{d f}+\frac {\operatorname {Subst}\left (\int \frac {d+x^2}{d^2+x^4} \, dx,x,\sqrt {d \cot (e+f x)}\right )}{d f}\\ &=-\frac {2 (d \cot (e+f x))^{3/2}}{3 d^3 f}+\frac {\operatorname {Subst}\left (\int \frac {\sqrt {2} \sqrt {d}+2 x}{-d-\sqrt {2} \sqrt {d} x-x^2} \, dx,x,\sqrt {d \cot (e+f x)}\right )}{2 \sqrt {2} d^{3/2} f}+\frac {\operatorname {Subst}\left (\int \frac {\sqrt {2} \sqrt {d}-2 x}{-d+\sqrt {2} \sqrt {d} x-x^2} \, dx,x,\sqrt {d \cot (e+f x)}\right )}{2 \sqrt {2} d^{3/2} f}+\frac {\operatorname {Subst}\left (\int \frac {1}{d-\sqrt {2} \sqrt {d} x+x^2} \, dx,x,\sqrt {d \cot (e+f x)}\right )}{2 d f}+\frac {\operatorname {Subst}\left (\int \frac {1}{d+\sqrt {2} \sqrt {d} x+x^2} \, dx,x,\sqrt {d \cot (e+f x)}\right )}{2 d f}\\ &=-\frac {2 (d \cot (e+f x))^{3/2}}{3 d^3 f}+\frac {\log \left (\sqrt {d}+\sqrt {d} \cot (e+f x)-\sqrt {2} \sqrt {d \cot (e+f x)}\right )}{2 \sqrt {2} d^{3/2} f}-\frac {\log \left (\sqrt {d}+\sqrt {d} \cot (e+f x)+\sqrt {2} \sqrt {d \cot (e+f x)}\right )}{2 \sqrt {2} d^{3/2} f}+\frac {\operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt {d \cot (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} d^{3/2} f}-\frac {\operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt {d \cot (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} d^{3/2} f}\\ &=-\frac {\tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {d \cot (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} d^{3/2} f}+\frac {\tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt {d \cot (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} d^{3/2} f}-\frac {2 (d \cot (e+f x))^{3/2}}{3 d^3 f}+\frac {\log \left (\sqrt {d}+\sqrt {d} \cot (e+f x)-\sqrt {2} \sqrt {d \cot (e+f x)}\right )}{2 \sqrt {2} d^{3/2} f}-\frac {\log \left (\sqrt {d}+\sqrt {d} \cot (e+f x)+\sqrt {2} \sqrt {d \cot (e+f x)}\right )}{2 \sqrt {2} d^{3/2} f}\\ \end {align*}
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Mathematica [C] time = 0.09, size = 47, normalized size = 0.22 \[ \frac {2 \cot ^3(e+f x) \left (\, _2F_1\left (\frac {3}{4},1;\frac {7}{4};-\cot ^2(e+f x)\right )-1\right )}{3 f (d \cot (e+f x))^{3/2}} \]
Antiderivative was successfully verified.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cot \left (f x + e\right )^{4}}{\left (d \cot \left (f x + e\right )\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.14, size = 184, normalized size = 0.86 \[ -\frac {2 \left (d \cot \left (f x +e \right )\right )^{\frac {3}{2}}}{3 d^{3} f}+\frac {\sqrt {2}\, \ln \left (\frac {d \cot \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \cot \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \cot \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \cot \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )}{4 f d \left (d^{2}\right )^{\frac {1}{4}}}+\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {d \cot \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )}{2 f d \left (d^{2}\right )^{\frac {1}{4}}}-\frac {\sqrt {2}\, \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \cot \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )}{2 f d \left (d^{2}\right )^{\frac {1}{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.63, size = 187, normalized size = 0.87 \[ \frac {3 \, d^{2} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {d} + 2 \, \sqrt {\frac {d}{\tan \left (f x + e\right )}}\right )}}{2 \, \sqrt {d}}\right )}{\sqrt {d}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {d} - 2 \, \sqrt {\frac {d}{\tan \left (f x + e\right )}}\right )}}{2 \, \sqrt {d}}\right )}{\sqrt {d}} - \frac {\sqrt {2} \log \left (\sqrt {2} \sqrt {d} \sqrt {\frac {d}{\tan \left (f x + e\right )}} + d + \frac {d}{\tan \left (f x + e\right )}\right )}{\sqrt {d}} + \frac {\sqrt {2} \log \left (-\sqrt {2} \sqrt {d} \sqrt {\frac {d}{\tan \left (f x + e\right )}} + d + \frac {d}{\tan \left (f x + e\right )}\right )}{\sqrt {d}}\right )} - 8 \, \left (\frac {d}{\tan \left (f x + e\right )}\right )^{\frac {3}{2}}}{12 \, d^{3} f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.65, size = 76, normalized size = 0.36 \[ \frac {{\left (-1\right )}^{1/4}\,\mathrm {atan}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {d\,\mathrm {cot}\left (e+f\,x\right )}}{\sqrt {d}}\right )}{d^{3/2}\,f}-\frac {2\,{\left (d\,\mathrm {cot}\left (e+f\,x\right )\right )}^{3/2}}{3\,d^3\,f}-\frac {{\left (-1\right )}^{1/4}\,\mathrm {atanh}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {d\,\mathrm {cot}\left (e+f\,x\right )}}{\sqrt {d}}\right )}{d^{3/2}\,f} \]
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 \[ \int \frac {\cot ^{4}{\left (e + f x \right )}}{\left (d \cot {\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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