Optimal. Leaf size=165 \[ -\frac {11 \text {ArcTan}\left (\frac {\sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{8 a^3 d \sqrt {e}}-\frac {\text {ArcTan}\left (\frac {\sqrt {e}-\sqrt {e} \cot (c+d x)}{\sqrt {2} \sqrt {e \cot (c+d x)}}\right )}{2 \sqrt {2} a^3 d \sqrt {e}}-\frac {7 \sqrt {e \cot (c+d x)}}{8 a^3 d e (1+\cot (c+d x))}-\frac {\sqrt {e \cot (c+d x)}}{4 a d e (a+a \cot (c+d x))^2} \]
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Rubi [A]
time = 0.43, antiderivative size = 165, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 7, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {3650, 3730,
3734, 3613, 211, 3715, 65} \begin {gather*} -\frac {11 \text {ArcTan}\left (\frac {\sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{8 a^3 d \sqrt {e}}-\frac {\text {ArcTan}\left (\frac {\sqrt {e}-\sqrt {e} \cot (c+d x)}{\sqrt {2} \sqrt {e \cot (c+d x)}}\right )}{2 \sqrt {2} a^3 d \sqrt {e}}-\frac {7 \sqrt {e \cot (c+d x)}}{8 a^3 d e (\cot (c+d x)+1)}-\frac {\sqrt {e \cot (c+d x)}}{4 a d e (a \cot (c+d x)+a)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 211
Rule 3613
Rule 3650
Rule 3715
Rule 3730
Rule 3734
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {e \cot (c+d x)} (a+a \cot (c+d x))^3} \, dx &=-\frac {\sqrt {e \cot (c+d x)}}{4 a d e (a+a \cot (c+d x))^2}-\frac {\int \frac {-\frac {7 a^2 e}{2}+2 a^2 e \cot (c+d x)-\frac {3}{2} a^2 e \cot ^2(c+d x)}{\sqrt {e \cot (c+d x)} (a+a \cot (c+d x))^2} \, dx}{4 a^3 e}\\ &=-\frac {7 \sqrt {e \cot (c+d x)}}{8 a^3 d e (1+\cot (c+d x))}-\frac {\sqrt {e \cot (c+d x)}}{4 a d e (a+a \cot (c+d x))^2}+\frac {\int \frac {\frac {7 a^4 e^2}{2}-4 a^4 e^2 \cot (c+d x)+\frac {7}{2} a^4 e^2 \cot ^2(c+d x)}{\sqrt {e \cot (c+d x)} (a+a \cot (c+d x))} \, dx}{8 a^6 e^2}\\ &=-\frac {7 \sqrt {e \cot (c+d x)}}{8 a^3 d e (1+\cot (c+d x))}-\frac {\sqrt {e \cot (c+d x)}}{4 a d e (a+a \cot (c+d x))^2}+\frac {11 \int \frac {1+\cot ^2(c+d x)}{\sqrt {e \cot (c+d x)} (a+a \cot (c+d x))} \, dx}{16 a^2}+\frac {\int \frac {-4 a^5 e^2-4 a^5 e^2 \cot (c+d x)}{\sqrt {e \cot (c+d x)}} \, dx}{16 a^8 e^2}\\ &=-\frac {7 \sqrt {e \cot (c+d x)}}{8 a^3 d e (1+\cot (c+d x))}-\frac {\sqrt {e \cot (c+d x)}}{4 a d e (a+a \cot (c+d x))^2}+\frac {11 \text {Subst}\left (\int \frac {1}{\sqrt {-e x} (a-a x)} \, dx,x,-\cot (c+d x)\right )}{16 a^2 d}-\frac {\left (2 a^2 e^2\right ) \text {Subst}\left (\int \frac {1}{-32 a^{10} e^4-e x^2} \, dx,x,\frac {-4 a^5 e^2+4 a^5 e^2 \cot (c+d x)}{\sqrt {e \cot (c+d x)}}\right )}{d}\\ &=-\frac {\tan ^{-1}\left (\frac {\sqrt {e}-\sqrt {e} \cot (c+d x)}{\sqrt {2} \sqrt {e \cot (c+d x)}}\right )}{2 \sqrt {2} a^3 d \sqrt {e}}-\frac {7 \sqrt {e \cot (c+d x)}}{8 a^3 d e (1+\cot (c+d x))}-\frac {\sqrt {e \cot (c+d x)}}{4 a d e (a+a \cot (c+d x))^2}-\frac {11 \text {Subst}\left (\int \frac {1}{a+\frac {a x^2}{e}} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{8 a^2 d e}\\ &=-\frac {11 \tan ^{-1}\left (\frac {\sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{8 a^3 d \sqrt {e}}-\frac {\tan ^{-1}\left (\frac {\sqrt {e}-\sqrt {e} \cot (c+d x)}{\sqrt {2} \sqrt {e \cot (c+d x)}}\right )}{2 \sqrt {2} a^3 d \sqrt {e}}-\frac {7 \sqrt {e \cot (c+d x)}}{8 a^3 d e (1+\cot (c+d x))}-\frac {\sqrt {e \cot (c+d x)}}{4 a d e (a+a \cot (c+d x))^2}\\ \end {align*}
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Mathematica [A]
time = 1.32, size = 217, normalized size = 1.32 \begin {gather*} \frac {\sqrt {\cot (c+d x)} \left (-22 \text {ArcTan}\left (\sqrt {\cot (c+d x)}\right )-9 \sqrt {\cot (c+d x)}+9 \cos (2 (c+d x)) \sqrt {\cot (c+d x)}-4 \sqrt {2} \text {ArcTan}\left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right ) (\cos (c+d x)+\sin (c+d x))^2+4 \sqrt {2} \text {ArcTan}\left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right ) (\cos (c+d x)+\sin (c+d x))^2-22 \text {ArcTan}\left (\sqrt {\cot (c+d x)}\right ) \sin (2 (c+d x))-7 \sqrt {\cot (c+d x)} \sin (2 (c+d x))\right )}{16 a^3 d \sqrt {e \cot (c+d x)} (\cos (c+d x)+\sin (c+d x))^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(348\) vs.
\(2(136)=272\).
time = 0.60, size = 349, normalized size = 2.12
method | result | size |
derivativedivides | \(-\frac {2 e^{4} \left (\frac {-\frac {\left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 e}-\frac {\sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (e^{2}\right )^{\frac {1}{4}}}}{4 e^{4}}+\frac {\frac {\frac {7 \left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}}{4}+\frac {9 e \sqrt {e \cot \left (d x +c \right )}}{4}}{\left (e \cot \left (d x +c \right )+e \right )^{2}}+\frac {11 \arctan \left (\frac {\sqrt {e \cot \left (d x +c \right )}}{\sqrt {e}}\right )}{4 \sqrt {e}}}{4 e^{4}}\right )}{d \,a^{3}}\) | \(349\) |
default | \(-\frac {2 e^{4} \left (\frac {-\frac {\left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 e}-\frac {\sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (e^{2}\right )^{\frac {1}{4}}}}{4 e^{4}}+\frac {\frac {\frac {7 \left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}}{4}+\frac {9 e \sqrt {e \cot \left (d x +c \right )}}{4}}{\left (e \cot \left (d x +c \right )+e \right )^{2}}+\frac {11 \arctan \left (\frac {\sqrt {e \cot \left (d x +c \right )}}{\sqrt {e}}\right )}{4 \sqrt {e}}}{4 e^{4}}\right )}{d \,a^{3}}\) | \(349\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.54, size = 129, normalized size = 0.78 \begin {gather*} -\frac {{\left (\frac {\frac {9}{\sqrt {\tan \left (d x + c\right )}} + \frac {7}{\tan \left (d x + c\right )^{\frac {3}{2}}}}{a^{3} + \frac {2 \, a^{3}}{\tan \left (d x + c\right )} + \frac {a^{3}}{\tan \left (d x + c\right )^{2}}} - \frac {2 \, {\left (\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} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right )\right )}}{a^{3}} + \frac {11 \, \arctan \left (\frac {1}{\sqrt {\tan \left (d x + c\right )}}\right )}{a^{3}}\right )} e^{\left (-\frac {1}{2}\right )}}{8 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.50, size = 233, normalized size = 1.41 \begin {gather*} -\frac {4 \, {\left (\sqrt {2} \sin \left (2 \, d x + 2 \, c\right ) + \sqrt {2}\right )} \arctan \left (-\frac {{\left (\sqrt {2} \cos \left (2 \, d x + 2 \, c\right ) - \sqrt {2} \sin \left (2 \, d x + 2 \, c\right ) + \sqrt {2}\right )} \sqrt {\frac {\cos \left (2 \, d x + 2 \, c\right ) + 1}{\sin \left (2 \, d x + 2 \, c\right )}}}{2 \, {\left (\cos \left (2 \, d x + 2 \, c\right ) + 1\right )}}\right ) - 22 \, {\left (\sin \left (2 \, d x + 2 \, c\right ) + 1\right )} \arctan \left (\frac {\sqrt {\frac {\cos \left (2 \, d x + 2 \, c\right ) + 1}{\sin \left (2 \, d x + 2 \, c\right )}} \sin \left (2 \, d x + 2 \, c\right )}{\cos \left (2 \, d x + 2 \, c\right ) + 1}\right ) - \sqrt {\frac {\cos \left (2 \, d x + 2 \, c\right ) + 1}{\sin \left (2 \, d x + 2 \, c\right )}} {\left (9 \, \cos \left (2 \, d x + 2 \, c\right ) - 7 \, \sin \left (2 \, d x + 2 \, c\right ) - 9\right )}}{16 \, {\left (a^{3} d e^{\frac {1}{2}} \sin \left (2 \, d x + 2 \, c\right ) + a^{3} d e^{\frac {1}{2}}\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*} \frac {\int \frac {1}{\sqrt {e \cot {\left (c + d x \right )}} \cot ^{3}{\left (c + d x \right )} + 3 \sqrt {e \cot {\left (c + d x \right )}} \cot ^{2}{\left (c + d x \right )} + 3 \sqrt {e \cot {\left (c + d x \right )}} \cot {\left (c + d x \right )} + \sqrt {e \cot {\left (c + d x \right )}}}\, dx}{a^{3}} \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 = 0.94, size = 173, normalized size = 1.05 \begin {gather*} \frac {\sqrt {2}\,\left (2\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{2\,\sqrt {e}}\right )+2\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{2\,\sqrt {e}}+\frac {\sqrt {2}\,{\left (e\,\mathrm {cot}\left (c+d\,x\right )\right )}^{3/2}}{2\,e^{3/2}}\right )\right )}{8\,a^3\,d\,\sqrt {e}}-\frac {11\,\mathrm {atan}\left (\frac {\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{\sqrt {e}}\right )}{8\,a^3\,d\,\sqrt {e}}-\frac {\frac {9\,e\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{8}+\frac {7\,{\left (e\,\mathrm {cot}\left (c+d\,x\right )\right )}^{3/2}}{8}}{d\,a^3\,e^2\,{\mathrm {cot}\left (c+d\,x\right )}^2+2\,d\,a^3\,e^2\,\mathrm {cot}\left (c+d\,x\right )+d\,a^3\,e^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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