Optimal. Leaf size=387 \[ -\frac {\sqrt {a} \left (a^2-3 b^2\right ) e^{3/2} \text {ArcTan}\left (\frac {\sqrt {b} \sqrt {e \cot (c+d x)}}{\sqrt {a} \sqrt {e}}\right )}{\sqrt {b} \left (a^2+b^2\right )^2 d}-\frac {\left (a^2-2 a b-b^2\right ) e^{3/2} \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \left (a^2+b^2\right )^2 d}+\frac {\left (a^2-2 a b-b^2\right ) e^{3/2} \text {ArcTan}\left (1+\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \left (a^2+b^2\right )^2 d}-\frac {a e \sqrt {e \cot (c+d x)}}{\left (a^2+b^2\right ) d (a+b \cot (c+d x))}-\frac {\left (a^2+2 a b-b^2\right ) e^{3/2} \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)-\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{2 \sqrt {2} \left (a^2+b^2\right )^2 d}+\frac {\left (a^2+2 a b-b^2\right ) e^{3/2} \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)+\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{2 \sqrt {2} \left (a^2+b^2\right )^2 d} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.45, antiderivative size = 387, normalized size of antiderivative = 1.00, number of steps
used = 15, number of rules used = 12, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.480, Rules used = {3648, 3734,
3615, 1182, 1176, 631, 210, 1179, 642, 3715, 65, 211} \begin {gather*} -\frac {\sqrt {a} e^{3/2} \left (a^2-3 b^2\right ) \text {ArcTan}\left (\frac {\sqrt {b} \sqrt {e \cot (c+d x)}}{\sqrt {a} \sqrt {e}}\right )}{\sqrt {b} d \left (a^2+b^2\right )^2}-\frac {e^{3/2} \left (a^2-2 a b-b^2\right ) \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} d \left (a^2+b^2\right )^2}+\frac {e^{3/2} \left (a^2-2 a b-b^2\right ) \text {ArcTan}\left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} d \left (a^2+b^2\right )^2}-\frac {e^{3/2} \left (a^2+2 a b-b^2\right ) \log \left (\sqrt {e} \cot (c+d x)-\sqrt {2} \sqrt {e \cot (c+d x)}+\sqrt {e}\right )}{2 \sqrt {2} d \left (a^2+b^2\right )^2}+\frac {e^{3/2} \left (a^2+2 a b-b^2\right ) \log \left (\sqrt {e} \cot (c+d x)+\sqrt {2} \sqrt {e \cot (c+d x)}+\sqrt {e}\right )}{2 \sqrt {2} d \left (a^2+b^2\right )^2}-\frac {a e \sqrt {e \cot (c+d x)}}{d \left (a^2+b^2\right ) (a+b \cot (c+d x))} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 65
Rule 210
Rule 211
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 1182
Rule 3615
Rule 3648
Rule 3715
Rule 3734
Rubi steps
\begin {align*} \int \frac {(e \cot (c+d x))^{3/2}}{(a+b \cot (c+d x))^2} \, dx &=-\frac {a e \sqrt {e \cot (c+d x)}}{\left (a^2+b^2\right ) d (a+b \cot (c+d x))}-\frac {\int \frac {\frac {a e^2}{2}-b e^2 \cot (c+d x)-\frac {1}{2} a e^2 \cot ^2(c+d x)}{\sqrt {e \cot (c+d x)} (a+b \cot (c+d x))} \, dx}{a^2+b^2}\\ &=-\frac {a e \sqrt {e \cot (c+d x)}}{\left (a^2+b^2\right ) d (a+b \cot (c+d x))}-\frac {\int \frac {\left (a^2-b^2\right ) e^2-2 a b e^2 \cot (c+d x)}{\sqrt {e \cot (c+d x)}} \, dx}{\left (a^2+b^2\right )^2}+\frac {\left (a \left (a^2-3 b^2\right ) e^2\right ) \int \frac {1+\cot ^2(c+d x)}{\sqrt {e \cot (c+d x)} (a+b \cot (c+d x))} \, dx}{2 \left (a^2+b^2\right )^2}\\ &=-\frac {a e \sqrt {e \cot (c+d x)}}{\left (a^2+b^2\right ) d (a+b \cot (c+d x))}-\frac {2 \text {Subst}\left (\int \frac {-\left (a^2-b^2\right ) e^3+2 a b e^2 x^2}{e^2+x^4} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{\left (a^2+b^2\right )^2 d}+\frac {\left (a \left (a^2-3 b^2\right ) e^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-e x} (a-b x)} \, dx,x,-\cot (c+d x)\right )}{2 \left (a^2+b^2\right )^2 d}\\ &=-\frac {a e \sqrt {e \cot (c+d x)}}{\left (a^2+b^2\right ) d (a+b \cot (c+d x))}-\frac {\left (a \left (a^2-3 b^2\right ) e\right ) \text {Subst}\left (\int \frac {1}{a+\frac {b x^2}{e}} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{\left (a^2+b^2\right )^2 d}+\frac {\left (\left (a^2-2 a b-b^2\right ) e^2\right ) \text {Subst}\left (\int \frac {e+x^2}{e^2+x^4} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{\left (a^2+b^2\right )^2 d}+\frac {\left (\left (a^2+2 a b-b^2\right ) e^2\right ) \text {Subst}\left (\int \frac {e-x^2}{e^2+x^4} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{\left (a^2+b^2\right )^2 d}\\ &=-\frac {\sqrt {a} \left (a^2-3 b^2\right ) e^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {e \cot (c+d x)}}{\sqrt {a} \sqrt {e}}\right )}{\sqrt {b} \left (a^2+b^2\right )^2 d}-\frac {a e \sqrt {e \cot (c+d x)}}{\left (a^2+b^2\right ) d (a+b \cot (c+d x))}-\frac {\left (\left (a^2+2 a b-b^2\right ) e^{3/2}\right ) \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 )}{2 \sqrt {2} \left (a^2+b^2\right )^2 d}-\frac {\left (\left (a^2+2 a b-b^2\right ) e^{3/2}\right ) \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 )}{2 \sqrt {2} \left (a^2+b^2\right )^2 d}+\frac {\left (\left (a^2-2 a b-b^2\right ) e^2\right ) \text {Subst}\left (\int \frac {1}{e-\sqrt {2} \sqrt {e} x+x^2} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{2 \left (a^2+b^2\right )^2 d}+\frac {\left (\left (a^2-2 a b-b^2\right ) e^2\right ) \text {Subst}\left (\int \frac {1}{e+\sqrt {2} \sqrt {e} x+x^2} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{2 \left (a^2+b^2\right )^2 d}\\ &=-\frac {\sqrt {a} \left (a^2-3 b^2\right ) e^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {e \cot (c+d x)}}{\sqrt {a} \sqrt {e}}\right )}{\sqrt {b} \left (a^2+b^2\right )^2 d}-\frac {a e \sqrt {e \cot (c+d x)}}{\left (a^2+b^2\right ) d (a+b \cot (c+d x))}-\frac {\left (a^2+2 a b-b^2\right ) e^{3/2} \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)-\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{2 \sqrt {2} \left (a^2+b^2\right )^2 d}+\frac {\left (a^2+2 a b-b^2\right ) e^{3/2} \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)+\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{2 \sqrt {2} \left (a^2+b^2\right )^2 d}+\frac {\left (\left (a^2-2 a b-b^2\right ) e^{3/2}\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \left (a^2+b^2\right )^2 d}-\frac {\left (\left (a^2-2 a b-b^2\right ) e^{3/2}\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \left (a^2+b^2\right )^2 d}\\ &=-\frac {\sqrt {a} \left (a^2-3 b^2\right ) e^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {e \cot (c+d x)}}{\sqrt {a} \sqrt {e}}\right )}{\sqrt {b} \left (a^2+b^2\right )^2 d}-\frac {\left (a^2-2 a b-b^2\right ) e^{3/2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \left (a^2+b^2\right )^2 d}+\frac {\left (a^2-2 a b-b^2\right ) e^{3/2} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \left (a^2+b^2\right )^2 d}-\frac {a e \sqrt {e \cot (c+d x)}}{\left (a^2+b^2\right ) d (a+b \cot (c+d x))}-\frac {\left (a^2+2 a b-b^2\right ) e^{3/2} \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)-\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{2 \sqrt {2} \left (a^2+b^2\right )^2 d}+\frac {\left (a^2+2 a b-b^2\right ) e^{3/2} \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)+\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{2 \sqrt {2} \left (a^2+b^2\right )^2 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 3.39, size = 322, normalized size = 0.83 \begin {gather*} -\frac {(e \cot (c+d x))^{3/2} \left (-240 a^2 \left (-\frac {\sqrt {a} \text {ArcTan}\left (\frac {\sqrt {b} \sqrt {\cot (c+d x)}}{\sqrt {a}}\right )}{\sqrt {b}}+\sqrt {\cot (c+d x)}\right )+80 a b \cot ^{\frac {3}{2}}(c+d x)+80 a b \cot ^{\frac {3}{2}}(c+d x) \left (-1+\, _2F_1\left (\frac {3}{4},1;\frac {7}{4};-\cot ^2(c+d x)\right )\right )+\frac {24 b^2 \left (a^2+b^2\right ) \cot ^{\frac {5}{2}}(c+d x) \, _2F_1\left (2,\frac {5}{2};\frac {7}{2};-\frac {b \cot (c+d x)}{a}\right )}{a^2}+15 (a-b) (a+b) \left (2 \sqrt {2} \text {ArcTan}\left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )-2 \sqrt {2} \text {ArcTan}\left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )+8 \sqrt {\cot (c+d x)}+\sqrt {2} \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )-\sqrt {2} \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )\right )\right )}{60 \left (a^2+b^2\right )^2 d \cot ^{\frac {3}{2}}(c+d x)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.66, size = 391, normalized size = 1.01
method | result | size |
derivativedivides | \(-\frac {2 e^{3} \left (\frac {\frac {\left (-a^{2} e +b^{2} e \right ) \left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 e^{2}}+\frac {a b \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 \left (e^{2}\right )^{\frac {1}{4}}}}{\left (a^{2}+b^{2}\right )^{2} e}+\frac {a \left (\frac {\left (\frac {a^{2}}{2}+\frac {b^{2}}{2}\right ) \sqrt {e \cot \left (d x +c \right )}}{e \cot \left (d x +c \right ) b +a e}+\frac {\left (a^{2}-3 b^{2}\right ) \arctan \left (\frac {b \sqrt {e \cot \left (d x +c \right )}}{\sqrt {a e b}}\right )}{2 \sqrt {a e b}}\right )}{\left (a^{2}+b^{2}\right )^{2} e}\right )}{d}\) | \(391\) |
default | \(-\frac {2 e^{3} \left (\frac {\frac {\left (-a^{2} e +b^{2} e \right ) \left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 e^{2}}+\frac {a b \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 \left (e^{2}\right )^{\frac {1}{4}}}}{\left (a^{2}+b^{2}\right )^{2} e}+\frac {a \left (\frac {\left (\frac {a^{2}}{2}+\frac {b^{2}}{2}\right ) \sqrt {e \cot \left (d x +c \right )}}{e \cot \left (d x +c \right ) b +a e}+\frac {\left (a^{2}-3 b^{2}\right ) \arctan \left (\frac {b \sqrt {e \cot \left (d x +c \right )}}{\sqrt {a e b}}\right )}{2 \sqrt {a e b}}\right )}{\left (a^{2}+b^{2}\right )^{2} e}\right )}{d}\) | \(391\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.51, size = 276, normalized size = 0.71 \begin {gather*} -\frac {{\left (\frac {4 \, {\left (a^{3} - 3 \, a b^{2}\right )} \arctan \left (\frac {b}{\sqrt {a b} \sqrt {\tan \left (d x + c\right )}}\right )}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \sqrt {a b}} - \frac {2 \, \sqrt {2} {\left (a^{2} - 2 \, a b - b^{2}\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) + 2 \, \sqrt {2} {\left (a^{2} - 2 \, a b - b^{2}\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) + \sqrt {2} {\left (a^{2} + 2 \, a b - b^{2}\right )} \log \left (\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right ) - \sqrt {2} {\left (a^{2} + 2 \, a b - b^{2}\right )} \log \left (-\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac {4 \, a}{{\left (a^{3} + a b^{2} + \frac {a^{2} b + b^{3}}{\tan \left (d x + c\right )}\right )} \sqrt {\tan \left (d x + c\right )}}\right )} e^{\frac {3}{2}}}{4 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [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.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (e \cot {\left (c + d x \right )}\right )^{\frac {3}{2}}}{\left (a + b \cot {\left (c + d x \right )}\right )^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 3.37, size = 2500, normalized size = 6.46 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________