Optimal. Leaf size=279 \[ \frac {e^{3/2} \text {ArcTan}\left (\frac {\sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{2 a^2 d}+\frac {e^{3/2} \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{2 \sqrt {2} a^2 d}-\frac {e^{3/2} \text {ArcTan}\left (1+\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{2 \sqrt {2} a^2 d}-\frac {e \sqrt {e \cot (c+d x)}}{2 d \left (a^2+a^2 \cot (c+d x)\right )}-\frac {e^{3/2} \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)-\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{4 \sqrt {2} a^2 d}+\frac {e^{3/2} \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)+\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{4 \sqrt {2} a^2 d} \]
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Rubi [A]
time = 0.39, antiderivative size = 279, normalized size of antiderivative = 1.00, number
of steps used = 18, number of rules used = 15, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules
used = {3648, 3734, 12, 16, 3557, 335, 303, 1176, 631, 210, 1179, 642, 3715, 65, 211}
\begin {gather*} \frac {e^{3/2} \text {ArcTan}\left (\frac {\sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{2 a^2 d}+\frac {e^{3/2} \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{2 \sqrt {2} a^2 d}-\frac {e^{3/2} \text {ArcTan}\left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{2 \sqrt {2} a^2 d}-\frac {e^{3/2} \log \left (\sqrt {e} \cot (c+d x)-\sqrt {2} \sqrt {e \cot (c+d x)}+\sqrt {e}\right )}{4 \sqrt {2} a^2 d}+\frac {e^{3/2} \log \left (\sqrt {e} \cot (c+d x)+\sqrt {2} \sqrt {e \cot (c+d x)}+\sqrt {e}\right )}{4 \sqrt {2} a^2 d}-\frac {e \sqrt {e \cot (c+d x)}}{2 d \left (a^2 \cot (c+d x)+a^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 16
Rule 65
Rule 210
Rule 211
Rule 303
Rule 335
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 3557
Rule 3648
Rule 3715
Rule 3734
Rubi steps
\begin {align*} \int \frac {(e \cot (c+d x))^{3/2}}{(a+a \cot (c+d x))^2} \, dx &=-\frac {e \sqrt {e \cot (c+d x)}}{2 d \left (a^2+a^2 \cot (c+d x)\right )}-\frac {\int \frac {\frac {a e^2}{2}-a e^2 \cot (c+d x)-\frac {1}{2} a e^2 \cot ^2(c+d x)}{\sqrt {e \cot (c+d x)} (a+a \cot (c+d x))} \, dx}{2 a^2}\\ &=-\frac {e \sqrt {e \cot (c+d x)}}{2 d \left (a^2+a^2 \cot (c+d x)\right )}-\frac {\int -\frac {2 a^2 e^2 \cot (c+d x)}{\sqrt {e \cot (c+d x)}} \, dx}{4 a^4}-\frac {e^2 \int \frac {1+\cot ^2(c+d x)}{\sqrt {e \cot (c+d x)} (a+a \cot (c+d x))} \, dx}{4 a}\\ &=-\frac {e \sqrt {e \cot (c+d x)}}{2 d \left (a^2+a^2 \cot (c+d x)\right )}+\frac {e^2 \int \frac {\cot (c+d x)}{\sqrt {e \cot (c+d x)}} \, dx}{2 a^2}-\frac {e^2 \text {Subst}\left (\int \frac {1}{\sqrt {-e x} (a-a x)} \, dx,x,-\cot (c+d x)\right )}{4 a d}\\ &=-\frac {e \sqrt {e \cot (c+d x)}}{2 d \left (a^2+a^2 \cot (c+d x)\right )}+\frac {e \int \sqrt {e \cot (c+d x)} \, dx}{2 a^2}+\frac {e \text {Subst}\left (\int \frac {1}{a+\frac {a x^2}{e}} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{2 a d}\\ &=\frac {e^{3/2} \tan ^{-1}\left (\frac {\sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{2 a^2 d}-\frac {e \sqrt {e \cot (c+d x)}}{2 d \left (a^2+a^2 \cot (c+d x)\right )}-\frac {e^2 \text {Subst}\left (\int \frac {\sqrt {x}}{e^2+x^2} \, dx,x,e \cot (c+d x)\right )}{2 a^2 d}\\ &=\frac {e^{3/2} \tan ^{-1}\left (\frac {\sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{2 a^2 d}-\frac {e \sqrt {e \cot (c+d x)}}{2 d \left (a^2+a^2 \cot (c+d x)\right )}-\frac {e^2 \text {Subst}\left (\int \frac {x^2}{e^2+x^4} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{a^2 d}\\ &=\frac {e^{3/2} \tan ^{-1}\left (\frac {\sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{2 a^2 d}-\frac {e \sqrt {e \cot (c+d x)}}{2 d \left (a^2+a^2 \cot (c+d x)\right )}+\frac {e^2 \text {Subst}\left (\int \frac {e-x^2}{e^2+x^4} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{2 a^2 d}-\frac {e^2 \text {Subst}\left (\int \frac {e+x^2}{e^2+x^4} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{2 a^2 d}\\ &=\frac {e^{3/2} \tan ^{-1}\left (\frac {\sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{2 a^2 d}-\frac {e \sqrt {e \cot (c+d x)}}{2 d \left (a^2+a^2 \cot (c+d x)\right )}-\frac {e^{3/2} \text {Subst}\left (\int \frac {\sqrt {2} \sqrt {e}+2 x}{-e-\sqrt {2} \sqrt {e} x-x^2} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{4 \sqrt {2} a^2 d}-\frac {e^{3/2} \text {Subst}\left (\int \frac {\sqrt {2} \sqrt {e}-2 x}{-e+\sqrt {2} \sqrt {e} x-x^2} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{4 \sqrt {2} a^2 d}-\frac {e^2 \text {Subst}\left (\int \frac {1}{e-\sqrt {2} \sqrt {e} x+x^2} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{4 a^2 d}-\frac {e^2 \text {Subst}\left (\int \frac {1}{e+\sqrt {2} \sqrt {e} x+x^2} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{4 a^2 d}\\ &=\frac {e^{3/2} \tan ^{-1}\left (\frac {\sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{2 a^2 d}-\frac {e \sqrt {e \cot (c+d x)}}{2 d \left (a^2+a^2 \cot (c+d x)\right )}-\frac {e^{3/2} \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)-\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{4 \sqrt {2} a^2 d}+\frac {e^{3/2} \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)+\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{4 \sqrt {2} a^2 d}-\frac {e^{3/2} \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{2 \sqrt {2} a^2 d}+\frac {e^{3/2} \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{2 \sqrt {2} a^2 d}\\ &=\frac {e^{3/2} \tan ^{-1}\left (\frac {\sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{2 a^2 d}+\frac {e^{3/2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{2 \sqrt {2} a^2 d}-\frac {e^{3/2} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{2 \sqrt {2} a^2 d}-\frac {e \sqrt {e \cot (c+d x)}}{2 d \left (a^2+a^2 \cot (c+d x)\right )}-\frac {e^{3/2} \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)-\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{4 \sqrt {2} a^2 d}+\frac {e^{3/2} \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)+\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{4 \sqrt {2} a^2 d}\\ \end {align*}
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Mathematica [A]
time = 3.02, size = 312, normalized size = 1.12 \begin {gather*} -\frac {(e \cot (c+d x))^{3/2} \left (-4 \sqrt {\cot (c+d x)}+4 \cot ^{\frac {3}{2}}(c+d x)-4 \cot ^{\frac {5}{2}}(c+d x)+4 \cot ^{\frac {7}{2}}(c+d x)-2 \sqrt {2} \text {ArcTan}\left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right ) \cos (2 (c+d x)) \csc ^4(c+d x)+2 \sqrt {2} \text {ArcTan}\left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right ) \cos (2 (c+d x)) \csc ^4(c+d x)-4 \text {ArcTan}\left (\sqrt {\cot (c+d x)}\right ) \cos (2 (c+d x)) \csc ^4(c+d x)+\sqrt {2} \cos (2 (c+d x)) \csc ^4(c+d x) \log \left (-1+\sqrt {2} \sqrt {\cot (c+d x)}-\cot (c+d x)\right )-\sqrt {2} \cos (2 (c+d x)) \csc ^4(c+d x) \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )\right ) \sin ^2(c+d x)}{8 a^2 d \cot ^{\frac {3}{2}}(c+d x) \left (-1+\cot ^2(c+d x)\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.52, size = 197, normalized size = 0.71
method | result | size |
derivativedivides | \(-\frac {2 e^{3} \left (-\frac {-\frac {\sqrt {e \cot \left (d x +c \right )}}{2 \left (e \cot \left (d x +c \right )+e \right )}+\frac {\arctan \left (\frac {\sqrt {e \cot \left (d x +c \right )}}{\sqrt {e}}\right )}{2 \sqrt {e}}}{2 e}+\frac {\sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{16 e \left (e^{2}\right )^{\frac {1}{4}}}\right )}{d \,a^{2}}\) | \(197\) |
default | \(-\frac {2 e^{3} \left (-\frac {-\frac {\sqrt {e \cot \left (d x +c \right )}}{2 \left (e \cot \left (d x +c \right )+e \right )}+\frac {\arctan \left (\frac {\sqrt {e \cot \left (d x +c \right )}}{\sqrt {e}}\right )}{2 \sqrt {e}}}{2 e}+\frac {\sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{16 e \left (e^{2}\right )^{\frac {1}{4}}}\right )}{d \,a^{2}}\) | \(197\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 161, normalized size = 0.58 \begin {gather*} -\frac {{\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) + 2 \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) - \sqrt {2} \log \left (\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right ) + \sqrt {2} \log \left (-\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right )}{a^{2}} - \frac {4 \, \arctan \left (\frac {1}{\sqrt {\tan \left (d x + c\right )}}\right )}{a^{2}} + \frac {4}{{\left (a^{2} + \frac {a^{2}}{\tan \left (d x + c\right )}\right )} \sqrt {\tan \left (d x + c\right )}}\right )} e^{\frac {3}{2}}}{8 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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*} \frac {\int \frac {\left (e \cot {\left (c + d x \right )}\right )^{\frac {3}{2}}}{\cot ^{2}{\left (c + d x \right )} + 2 \cot {\left (c + d x \right )} + 1}\, dx}{a^{2}} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.82, size = 376, normalized size = 1.35 \begin {gather*} -\frac {\mathrm {atan}\left (\frac {4\,e^{16}\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}\,{\left (-\frac {e^6}{a^8\,d^4}\right )}^{1/4}}{\frac {4\,e^{18}}{a^2\,d}+4\,a^2\,d\,e^{15}\,\sqrt {-\frac {e^6}{a^8\,d^4}}}+\frac {4\,e^{13}\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}\,{\left (-\frac {e^6}{a^8\,d^4}\right )}^{3/4}}{\frac {4\,e^{18}}{a^6\,d^3}+\frac {4\,e^{15}\,\sqrt {-\frac {e^6}{a^8\,d^4}}}{a^2\,d}}\right )\,{\left (-\frac {e^6}{a^8\,d^4}\right )}^{1/4}}{2}-\mathrm {atan}\left (\frac {e^{16}\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}\,{\left (-\frac {e^6}{256\,a^8\,d^4}\right )}^{1/4}\,16{}\mathrm {i}}{\frac {4\,e^{18}}{a^2\,d}-64\,a^2\,d\,e^{15}\,\sqrt {-\frac {e^6}{256\,a^8\,d^4}}}-\frac {e^{13}\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}\,{\left (-\frac {e^6}{256\,a^8\,d^4}\right )}^{3/4}\,256{}\mathrm {i}}{\frac {4\,e^{18}}{a^6\,d^3}-\frac {64\,e^{15}\,\sqrt {-\frac {e^6}{256\,a^8\,d^4}}}{a^2\,d}}\right )\,{\left (-\frac {e^6}{256\,a^8\,d^4}\right )}^{1/4}\,2{}\mathrm {i}-\frac {e^2\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{2\,\left (a^2\,d\,e+a^2\,d\,e\,\mathrm {cot}\left (c+d\,x\right )\right )}+\frac {\mathrm {atan}\left (\frac {\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}\,\sqrt {-e^3}\,1{}\mathrm {i}}{e^2}\right )\,\sqrt {-e^3}\,1{}\mathrm {i}}{2\,a^2\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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