Optimal. Leaf size=529 \[ \frac {b^{5/2} \left (63 a^4+46 a^2 b^2+15 b^4\right ) \text {ArcTan}\left (\frac {\sqrt {b} \sqrt {e \cot (c+d x)}}{\sqrt {a} \sqrt {e}}\right )}{4 a^{7/2} \left (a^2+b^2\right )^3 d e^{3/2}}-\frac {(a-b) \left (a^2+4 a b+b^2\right ) \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \left (a^2+b^2\right )^3 d e^{3/2}}+\frac {(a-b) \left (a^2+4 a b+b^2\right ) \text {ArcTan}\left (1+\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \left (a^2+b^2\right )^3 d e^{3/2}}+\frac {8 a^4+31 a^2 b^2+15 b^4}{4 a^3 \left (a^2+b^2\right )^2 d e \sqrt {e \cot (c+d x)}}-\frac {b^2}{2 a \left (a^2+b^2\right ) d e \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))^2}-\frac {b^2 \left (13 a^2+5 b^2\right )}{4 a^2 \left (a^2+b^2\right )^2 d e \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))}+\frac {(a+b) \left (a^2-4 a b+b^2\right ) \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)-\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{2 \sqrt {2} \left (a^2+b^2\right )^3 d e^{3/2}}-\frac {(a+b) \left (a^2-4 a b+b^2\right ) \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)+\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{2 \sqrt {2} \left (a^2+b^2\right )^3 d e^{3/2}} \]
[Out]
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Rubi [A]
time = 1.13, antiderivative size = 529, normalized size of antiderivative = 1.00, number of steps
used = 17, number of rules used = 13, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.520, Rules used = {3650, 3730,
3734, 3615, 1182, 1176, 631, 210, 1179, 642, 3715, 65, 211} \begin {gather*} -\frac {(a-b) \left (a^2+4 a b+b^2\right ) \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} d e^{3/2} \left (a^2+b^2\right )^3}+\frac {(a-b) \left (a^2+4 a b+b^2\right ) \text {ArcTan}\left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} d e^{3/2} \left (a^2+b^2\right )^3}+\frac {(a+b) \left (a^2-4 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 e^{3/2} \left (a^2+b^2\right )^3}-\frac {(a+b) \left (a^2-4 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 e^{3/2} \left (a^2+b^2\right )^3}-\frac {b^2 \left (13 a^2+5 b^2\right )}{4 a^2 d e \left (a^2+b^2\right )^2 \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))}-\frac {b^2}{2 a d e \left (a^2+b^2\right ) \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))^2}+\frac {b^{5/2} \left (63 a^4+46 a^2 b^2+15 b^4\right ) \text {ArcTan}\left (\frac {\sqrt {b} \sqrt {e \cot (c+d x)}}{\sqrt {a} \sqrt {e}}\right )}{4 a^{7/2} d e^{3/2} \left (a^2+b^2\right )^3}+\frac {8 a^4+31 a^2 b^2+15 b^4}{4 a^3 d e \left (a^2+b^2\right )^2 \sqrt {e \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 3650
Rule 3715
Rule 3730
Rule 3734
Rubi steps
\begin {align*} \int \frac {1}{(e \cot (c+d x))^{3/2} (a+b \cot (c+d x))^3} \, dx &=-\frac {b^2}{2 a \left (a^2+b^2\right ) d e \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))^2}-\frac {\int \frac {-\frac {1}{2} \left (4 a^2+5 b^2\right ) e+2 a b e \cot (c+d x)-\frac {5}{2} b^2 e \cot ^2(c+d x)}{(e \cot (c+d x))^{3/2} (a+b \cot (c+d x))^2} \, dx}{2 a \left (a^2+b^2\right ) e}\\ &=-\frac {b^2}{2 a \left (a^2+b^2\right ) d e \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))^2}-\frac {b^2 \left (13 a^2+5 b^2\right )}{4 a^2 \left (a^2+b^2\right )^2 d e \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))}+\frac {\int \frac {\frac {1}{4} \left (8 a^4+31 a^2 b^2+15 b^4\right ) e^2-4 a^3 b e^2 \cot (c+d x)+\frac {3}{4} b^2 \left (13 a^2+5 b^2\right ) e^2 \cot ^2(c+d x)}{(e \cot (c+d x))^{3/2} (a+b \cot (c+d x))} \, dx}{2 a^2 \left (a^2+b^2\right )^2 e^2}\\ &=\frac {8 a^4+31 a^2 b^2+15 b^4}{4 a^3 \left (a^2+b^2\right )^2 d e \sqrt {e \cot (c+d x)}}-\frac {b^2}{2 a \left (a^2+b^2\right ) d e \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))^2}-\frac {b^2 \left (13 a^2+5 b^2\right )}{4 a^2 \left (a^2+b^2\right )^2 d e \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))}+\frac {\int \frac {-\frac {1}{8} b \left (24 a^4+31 a^2 b^2+15 b^4\right ) e^4-a^3 \left (a^2-b^2\right ) e^4 \cot (c+d x)-\frac {1}{8} b \left (8 a^4+31 a^2 b^2+15 b^4\right ) e^4 \cot ^2(c+d x)}{\sqrt {e \cot (c+d x)} (a+b \cot (c+d x))} \, dx}{a^3 \left (a^2+b^2\right )^2 e^5}\\ &=\frac {8 a^4+31 a^2 b^2+15 b^4}{4 a^3 \left (a^2+b^2\right )^2 d e \sqrt {e \cot (c+d x)}}-\frac {b^2}{2 a \left (a^2+b^2\right ) d e \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))^2}-\frac {b^2 \left (13 a^2+5 b^2\right )}{4 a^2 \left (a^2+b^2\right )^2 d e \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))}+\frac {\int \frac {-a^3 b \left (3 a^2-b^2\right ) e^4-a^4 \left (a^2-3 b^2\right ) e^4 \cot (c+d x)}{\sqrt {e \cot (c+d x)}} \, dx}{a^3 \left (a^2+b^2\right )^3 e^5}-\frac {\left (b^3 \left (63 a^4+46 a^2 b^2+15 b^4\right )\right ) \int \frac {1+\cot ^2(c+d x)}{\sqrt {e \cot (c+d x)} (a+b \cot (c+d x))} \, dx}{8 a^3 \left (a^2+b^2\right )^3 e}\\ &=\frac {8 a^4+31 a^2 b^2+15 b^4}{4 a^3 \left (a^2+b^2\right )^2 d e \sqrt {e \cot (c+d x)}}-\frac {b^2}{2 a \left (a^2+b^2\right ) d e \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))^2}-\frac {b^2 \left (13 a^2+5 b^2\right )}{4 a^2 \left (a^2+b^2\right )^2 d e \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))}+\frac {2 \text {Subst}\left (\int \frac {a^3 b \left (3 a^2-b^2\right ) e^5+a^4 \left (a^2-3 b^2\right ) e^4 x^2}{e^2+x^4} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{a^3 \left (a^2+b^2\right )^3 d e^5}-\frac {\left (b^3 \left (63 a^4+46 a^2 b^2+15 b^4\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-e x} (a-b x)} \, dx,x,-\cot (c+d x)\right )}{8 a^3 \left (a^2+b^2\right )^3 d e}\\ &=\frac {8 a^4+31 a^2 b^2+15 b^4}{4 a^3 \left (a^2+b^2\right )^2 d e \sqrt {e \cot (c+d x)}}-\frac {b^2}{2 a \left (a^2+b^2\right ) d e \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))^2}-\frac {b^2 \left (13 a^2+5 b^2\right )}{4 a^2 \left (a^2+b^2\right )^2 d e \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))}+\frac {\left (b^3 \left (63 a^4+46 a^2 b^2+15 b^4\right )\right ) \text {Subst}\left (\int \frac {1}{a+\frac {b x^2}{e}} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{4 a^3 \left (a^2+b^2\right )^3 d e^2}-\frac {\left ((a+b) \left (a^2-4 a b+b^2\right )\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 )^3 d e}+\frac {\left ((a-b) \left (a^2+4 a b+b^2\right )\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 )^3 d e}\\ &=\frac {b^{5/2} \left (63 a^4+46 a^2 b^2+15 b^4\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {e \cot (c+d x)}}{\sqrt {a} \sqrt {e}}\right )}{4 a^{7/2} \left (a^2+b^2\right )^3 d e^{3/2}}+\frac {8 a^4+31 a^2 b^2+15 b^4}{4 a^3 \left (a^2+b^2\right )^2 d e \sqrt {e \cot (c+d x)}}-\frac {b^2}{2 a \left (a^2+b^2\right ) d e \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))^2}-\frac {b^2 \left (13 a^2+5 b^2\right )}{4 a^2 \left (a^2+b^2\right )^2 d e \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))}+\frac {\left ((a+b) \left (a^2-4 a b+b^2\right )\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 )^3 d e^{3/2}}+\frac {\left ((a+b) \left (a^2-4 a b+b^2\right )\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 )^3 d e^{3/2}}+\frac {\left ((a-b) \left (a^2+4 a b+b^2\right )\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 )^3 d e}+\frac {\left ((a-b) \left (a^2+4 a b+b^2\right )\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 )^3 d e}\\ &=\frac {b^{5/2} \left (63 a^4+46 a^2 b^2+15 b^4\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {e \cot (c+d x)}}{\sqrt {a} \sqrt {e}}\right )}{4 a^{7/2} \left (a^2+b^2\right )^3 d e^{3/2}}+\frac {8 a^4+31 a^2 b^2+15 b^4}{4 a^3 \left (a^2+b^2\right )^2 d e \sqrt {e \cot (c+d x)}}-\frac {b^2}{2 a \left (a^2+b^2\right ) d e \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))^2}-\frac {b^2 \left (13 a^2+5 b^2\right )}{4 a^2 \left (a^2+b^2\right )^2 d e \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))}+\frac {(a+b) \left (a^2-4 a b+b^2\right ) \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 )^3 d e^{3/2}}-\frac {(a+b) \left (a^2-4 a b+b^2\right ) \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 )^3 d e^{3/2}}+\frac {\left ((a-b) \left (a^2+4 a b+b^2\right )\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 )^3 d e^{3/2}}-\frac {\left ((a-b) \left (a^2+4 a b+b^2\right )\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 )^3 d e^{3/2}}\\ &=\frac {b^{5/2} \left (63 a^4+46 a^2 b^2+15 b^4\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {e \cot (c+d x)}}{\sqrt {a} \sqrt {e}}\right )}{4 a^{7/2} \left (a^2+b^2\right )^3 d e^{3/2}}-\frac {(a-b) \left (a^2+4 a b+b^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \left (a^2+b^2\right )^3 d e^{3/2}}+\frac {(a-b) \left (a^2+4 a b+b^2\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \left (a^2+b^2\right )^3 d e^{3/2}}+\frac {8 a^4+31 a^2 b^2+15 b^4}{4 a^3 \left (a^2+b^2\right )^2 d e \sqrt {e \cot (c+d x)}}-\frac {b^2}{2 a \left (a^2+b^2\right ) d e \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))^2}-\frac {b^2 \left (13 a^2+5 b^2\right )}{4 a^2 \left (a^2+b^2\right )^2 d e \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))}+\frac {(a+b) \left (a^2-4 a b+b^2\right ) \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 )^3 d e^{3/2}}-\frac {(a+b) \left (a^2-4 a b+b^2\right ) \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 )^3 d e^{3/2}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 1.90, size = 303, normalized size = 0.57 \begin {gather*} -\frac {-8 a^2 b^2 \left (3 a^2-b^2\right ) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};-\frac {b \cot (c+d x)}{a}\right )-16 a^2 b^2 \left (a^2+b^2\right ) \, _2F_1\left (-\frac {1}{2},2;\frac {1}{2};-\frac {b \cot (c+d x)}{a}\right )-8 b^2 \left (a^2+b^2\right )^2 \, _2F_1\left (-\frac {1}{2},3;\frac {1}{2};-\frac {b \cot (c+d x)}{a}\right )-8 a^4 \left (a^2-3 b^2\right ) \, _2F_1\left (-\frac {1}{4},1;\frac {3}{4};-\cot ^2(c+d x)\right )+\sqrt {2} a^3 b \left (3 a^2-b^2\right ) \sqrt {\cot (c+d x)} \left (2 \text {ArcTan}\left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )-2 \text {ArcTan}\left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )+\log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )-\log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )\right )}{4 a^3 \left (a^2+b^2\right )^3 d e \sqrt {e \cot (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
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Maple [A]
time = 0.65, size = 480, normalized size = 0.91
method | result | size |
derivativedivides | \(-\frac {2 e^{4} \left (-\frac {b^{3} \left (\frac {\left (\frac {15}{8} a^{4} b +\frac {11}{4} a^{2} b^{3}+\frac {7}{8} b^{5}\right ) \left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}+\frac {a e \left (17 a^{4}+26 a^{2} b^{2}+9 b^{4}\right ) \sqrt {e \cot \left (d x +c \right )}}{8}}{\left (e \cot \left (d x +c \right ) b +a e \right )^{2}}+\frac {\left (63 a^{4}+46 a^{2} b^{2}+15 b^{4}\right ) \arctan \left (\frac {b \sqrt {e \cot \left (d x +c \right )}}{\sqrt {a e b}}\right )}{8 \sqrt {a e b}}\right )}{a^{3} e^{5} \left (a^{2}+b^{2}\right )^{3}}+\frac {\frac {\left (-3 a^{2} b e +b^{3} e \right ) \left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 e^{2}}+\frac {\left (-a^{3}+3 a \,b^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (e^{2}\right )^{\frac {1}{4}}}}{\left (a^{2}+b^{2}\right )^{3} e^{5}}-\frac {1}{a^{3} e^{5} \sqrt {e \cot \left (d x +c \right )}}\right )}{d}\) | \(480\) |
default | \(-\frac {2 e^{4} \left (-\frac {b^{3} \left (\frac {\left (\frac {15}{8} a^{4} b +\frac {11}{4} a^{2} b^{3}+\frac {7}{8} b^{5}\right ) \left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}+\frac {a e \left (17 a^{4}+26 a^{2} b^{2}+9 b^{4}\right ) \sqrt {e \cot \left (d x +c \right )}}{8}}{\left (e \cot \left (d x +c \right ) b +a e \right )^{2}}+\frac {\left (63 a^{4}+46 a^{2} b^{2}+15 b^{4}\right ) \arctan \left (\frac {b \sqrt {e \cot \left (d x +c \right )}}{\sqrt {a e b}}\right )}{8 \sqrt {a e b}}\right )}{a^{3} e^{5} \left (a^{2}+b^{2}\right )^{3}}+\frac {\frac {\left (-3 a^{2} b e +b^{3} e \right ) \left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 e^{2}}+\frac {\left (-a^{3}+3 a \,b^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (e^{2}\right )^{\frac {1}{4}}}}{\left (a^{2}+b^{2}\right )^{3} e^{5}}-\frac {1}{a^{3} e^{5} \sqrt {e \cot \left (d x +c \right )}}\right )}{d}\) | \(480\) |
Verification of antiderivative is not currently implemented for this CAS.
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[Out]
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Maxima [A]
time = 0.53, size = 472, normalized size = 0.89 \begin {gather*} \frac {{\left (\frac {{\left (63 \, a^{4} b^{3} + 46 \, a^{2} b^{5} + 15 \, b^{7}\right )} \arctan \left (\frac {b}{\sqrt {a b} \sqrt {\tan \left (d x + c\right )}}\right )}{{\left (a^{9} + 3 \, a^{7} b^{2} + 3 \, a^{5} b^{4} + a^{3} b^{6}\right )} \sqrt {a b}} + \frac {2 \, \sqrt {2} {\left (a^{3} + 3 \, a^{2} b - 3 \, a b^{2} - b^{3}\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^{3} + 3 \, a^{2} b - 3 \, a b^{2} - b^{3}\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^{3} - 3 \, a^{2} b - 3 \, a b^{2} + b^{3}\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^{3} - 3 \, a^{2} b - 3 \, a b^{2} + b^{3}\right )} \log \left (-\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac {8 \, a^{6} + 16 \, a^{4} b^{2} + 8 \, a^{2} b^{4} + \frac {16 \, a^{5} b + 49 \, a^{3} b^{3} + 25 \, a b^{5}}{\tan \left (d x + c\right )} + \frac {8 \, a^{4} b^{2} + 31 \, a^{2} b^{4} + 15 \, b^{6}}{\tan \left (d x + c\right )^{2}}}{\frac {a^{9} + 2 \, a^{7} b^{2} + a^{5} b^{4}}{\sqrt {\tan \left (d x + c\right )}} + \frac {2 \, {\left (a^{8} b + 2 \, a^{6} b^{3} + a^{4} b^{5}\right )}}{\tan \left (d x + c\right )^{\frac {3}{2}}} + \frac {a^{7} b^{2} + 2 \, a^{5} b^{4} + a^{3} b^{6}}{\tan \left (d x + c\right )^{\frac {5}{2}}}}\right )} e^{\left (-\frac {3}{2}\right )}}{4 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (e \cot {\left (c + d x \right )}\right )^{\frac {3}{2}} \left (a + b \cot {\left (c + d x \right )}\right )^{3}}\, dx \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 = 10.00, size = 2500, normalized size = 4.73 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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