Optimal. Leaf size=138 \[ -\frac {8 a^3 (e \cot (c+d x))^{3/2}}{5 d e}-\frac {4 a^3 \sqrt {e \cot (c+d x)}}{d}-\frac {2 \left (a^3 \cot (c+d x)+a^3\right ) (e \cot (c+d x))^{3/2}}{5 d e}-\frac {2 \sqrt {2} a^3 \sqrt {e} \tan ^{-1}\left (\frac {\sqrt {e}-\sqrt {e} \cot (c+d x)}{\sqrt {2} \sqrt {e \cot (c+d x)}}\right )}{d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.20, antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3566, 3630, 3528, 3532, 205} \[ -\frac {8 a^3 (e \cot (c+d x))^{3/2}}{5 d e}-\frac {4 a^3 \sqrt {e \cot (c+d x)}}{d}-\frac {2 \left (a^3 \cot (c+d x)+a^3\right ) (e \cot (c+d x))^{3/2}}{5 d e}-\frac {2 \sqrt {2} a^3 \sqrt {e} \tan ^{-1}\left (\frac {\sqrt {e}-\sqrt {e} \cot (c+d x)}{\sqrt {2} \sqrt {e \cot (c+d x)}}\right )}{d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 205
Rule 3528
Rule 3532
Rule 3566
Rule 3630
Rubi steps
\begin {align*} \int \sqrt {e \cot (c+d x)} (a+a \cot (c+d x))^3 \, dx &=-\frac {2 (e \cot (c+d x))^{3/2} \left (a^3+a^3 \cot (c+d x)\right )}{5 d e}-\frac {2 \int \sqrt {e \cot (c+d x)} \left (-a^3 e-5 a^3 e \cot (c+d x)-6 a^3 e \cot ^2(c+d x)\right ) \, dx}{5 e}\\ &=-\frac {8 a^3 (e \cot (c+d x))^{3/2}}{5 d e}-\frac {2 (e \cot (c+d x))^{3/2} \left (a^3+a^3 \cot (c+d x)\right )}{5 d e}-\frac {2 \int \sqrt {e \cot (c+d x)} \left (5 a^3 e-5 a^3 e \cot (c+d x)\right ) \, dx}{5 e}\\ &=-\frac {4 a^3 \sqrt {e \cot (c+d x)}}{d}-\frac {8 a^3 (e \cot (c+d x))^{3/2}}{5 d e}-\frac {2 (e \cot (c+d x))^{3/2} \left (a^3+a^3 \cot (c+d x)\right )}{5 d e}-\frac {2 \int \frac {5 a^3 e^2+5 a^3 e^2 \cot (c+d x)}{\sqrt {e \cot (c+d x)}} \, dx}{5 e}\\ &=-\frac {4 a^3 \sqrt {e \cot (c+d x)}}{d}-\frac {8 a^3 (e \cot (c+d x))^{3/2}}{5 d e}-\frac {2 (e \cot (c+d x))^{3/2} \left (a^3+a^3 \cot (c+d x)\right )}{5 d e}+\frac {\left (20 a^6 e^3\right ) \operatorname {Subst}\left (\int \frac {1}{-50 a^6 e^4-e x^2} \, dx,x,\frac {5 a^3 e^2-5 a^3 e^2 \cot (c+d x)}{\sqrt {e \cot (c+d x)}}\right )}{d}\\ &=-\frac {2 \sqrt {2} a^3 \sqrt {e} \tan ^{-1}\left (\frac {\sqrt {e}-\sqrt {e} \cot (c+d x)}{\sqrt {2} \sqrt {e \cot (c+d x)}}\right )}{d}-\frac {4 a^3 \sqrt {e \cot (c+d x)}}{d}-\frac {8 a^3 (e \cot (c+d x))^{3/2}}{5 d e}-\frac {2 (e \cot (c+d x))^{3/2} \left (a^3+a^3 \cot (c+d x)\right )}{5 d e}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 1.55, size = 315, normalized size = 2.28 \[ -\frac {a^3 \sin (c+d x) (\cot (c+d x)+1)^3 \sqrt {e \cot (c+d x)} \left (3 \left (4 \cos ^2(c+d x) \sqrt {\cot (c+d x)}+40 \sin ^2(c+d x) \sqrt {\cot (c+d x)}+10 \sin (2 (c+d x)) \sqrt {\cot (c+d x)}+5 \sqrt {2} \sin ^2(c+d x) \log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )-5 \sqrt {2} \sin ^2(c+d x) \log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )+10 \sqrt {2} \sin ^2(c+d x) \tan ^{-1}\left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )-10 \sqrt {2} \sin ^2(c+d x) \tan ^{-1}\left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )\right )-20 \sin (2 (c+d x)) \sqrt {\cot (c+d x)} \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};-\cot ^2(c+d x)\right )\right )}{30 d \sqrt {\cot (c+d x)} (\sin (c+d x)+\cos (c+d x))^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.72, size = 366, normalized size = 2.65 \[ \left [\frac {5 \, \sqrt {2} {\left (a^{3} \cos \left (2 \, d x + 2 \, c\right ) - a^{3}\right )} \sqrt {-e} \log \left (\sqrt {2} \sqrt {-e} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}} {\left (\cos \left (2 \, d x + 2 \, c\right ) + \sin \left (2 \, d x + 2 \, c\right ) - 1\right )} - 2 \, e \sin \left (2 \, d x + 2 \, c\right ) + e\right ) - 2 \, {\left (9 \, a^{3} \cos \left (2 \, d x + 2 \, c\right ) - 5 \, a^{3} \sin \left (2 \, d x + 2 \, c\right ) - 11 \, a^{3}\right )} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}}}{5 \, {\left (d \cos \left (2 \, d x + 2 \, c\right ) - d\right )}}, -\frac {2 \, {\left (5 \, \sqrt {2} {\left (a^{3} \cos \left (2 \, d x + 2 \, c\right ) - a^{3}\right )} \sqrt {e} \arctan \left (-\frac {\sqrt {2} \sqrt {e} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}} {\left (\cos \left (2 \, d x + 2 \, c\right ) - \sin \left (2 \, d x + 2 \, c\right ) + 1\right )}}{2 \, {\left (e \cos \left (2 \, d x + 2 \, c\right ) + e\right )}}\right ) + {\left (9 \, a^{3} \cos \left (2 \, d x + 2 \, c\right ) - 5 \, a^{3} \sin \left (2 \, d x + 2 \, c\right ) - 11 \, a^{3}\right )} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}}\right )}}{5 \, {\left (d \cos \left (2 \, d x + 2 \, c\right ) - d\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (a \cot \left (d x + c\right ) + a\right )}^{3} \sqrt {e \cot \left (d x + c\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.96, size = 391, normalized size = 2.83 \[ -\frac {2 a^{3} \left (e \cot \left (d x +c \right )\right )^{\frac {5}{2}}}{5 d \,e^{2}}-\frac {2 a^{3} \left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}}{d e}-\frac {4 a^{3} \sqrt {e \cot \left (d x +c \right )}}{d}+\frac {a^{3} \left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )}{2 d}+\frac {a^{3} \left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )}{d}-\frac {a^{3} \left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )}{d}+\frac {a^{3} e \sqrt {2}\, \ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )}{2 d \left (e^{2}\right )^{\frac {1}{4}}}+\frac {a^{3} e \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )}{d \left (e^{2}\right )^{\frac {1}{4}}}-\frac {a^{3} e \sqrt {2}\, \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )}{d \left (e^{2}\right )^{\frac {1}{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.76, size = 149, normalized size = 1.08 \[ \frac {2 \, {\left (5 \, a^{3} {\left (\frac {\sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {e} + 2 \, \sqrt {\frac {e}{\tan \left (d x + c\right )}}\right )}}{2 \, \sqrt {e}}\right )}{\sqrt {e}} + \frac {\sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {e} - 2 \, \sqrt {\frac {e}{\tan \left (d x + c\right )}}\right )}}{2 \, \sqrt {e}}\right )}{\sqrt {e}}\right )} - \frac {10 \, a^{3} e^{2} \sqrt {\frac {e}{\tan \left (d x + c\right )}} + 5 \, a^{3} e \left (\frac {e}{\tan \left (d x + c\right )}\right )^{\frac {3}{2}} + a^{3} \left (\frac {e}{\tan \left (d x + c\right )}\right )^{\frac {5}{2}}}{e^{3}}\right )} e}{5 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.99, size = 136, normalized size = 0.99 \[ \frac {\sqrt {2}\,a^3\,\sqrt {e}\,\left (2\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{2\,\sqrt {e}}\right )+2\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{2\,\sqrt {e}}+\frac {\sqrt {2}\,{\left (e\,\mathrm {cot}\left (c+d\,x\right )\right )}^{3/2}}{2\,e^{3/2}}\right )\right )}{d}-\frac {2\,a^3\,{\left (e\,\mathrm {cot}\left (c+d\,x\right )\right )}^{3/2}}{d\,e}-\frac {2\,a^3\,{\left (e\,\mathrm {cot}\left (c+d\,x\right )\right )}^{5/2}}{5\,d\,e^2}-\frac {4\,a^3\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ a^{3} \left (\int \sqrt {e \cot {\left (c + d x \right )}}\, dx + \int 3 \sqrt {e \cot {\left (c + d x \right )}} \cot {\left (c + d x \right )}\, dx + \int 3 \sqrt {e \cot {\left (c + d x \right )}} \cot ^{2}{\left (c + d x \right )}\, dx + \int \sqrt {e \cot {\left (c + d x \right )}} \cot ^{3}{\left (c + d x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________