Optimal. Leaf size=351 \[ -\frac {2 b^{7/2} \text {ArcTan}\left (\frac {\sqrt {b} \sqrt {e \cot (c+d x)}}{\sqrt {a} \sqrt {e}}\right )}{a^{5/2} \left (a^2+b^2\right ) d e^{5/2}}-\frac {(a-b) \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \left (a^2+b^2\right ) d e^{5/2}}+\frac {(a-b) \text {ArcTan}\left (1+\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \left (a^2+b^2\right ) d e^{5/2}}+\frac {2}{3 a d e (e \cot (c+d x))^{3/2}}-\frac {2 b}{a^2 d e^2 \sqrt {e \cot (c+d x)}}-\frac {(a+b) \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)-\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{2 \sqrt {2} \left (a^2+b^2\right ) d e^{5/2}}+\frac {(a+b) \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)+\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{2 \sqrt {2} \left (a^2+b^2\right ) d e^{5/2}} \]
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Rubi [A]
time = 0.59, antiderivative size = 351, normalized size of antiderivative = 1.00, number of steps
used = 16, number of rules used = 13, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.520, Rules used = {3650, 3730,
3735, 3615, 1182, 1176, 631, 210, 1179, 642, 3715, 65, 211} \begin {gather*} -\frac {(a-b) \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} d e^{5/2} \left (a^2+b^2\right )}+\frac {(a-b) \text {ArcTan}\left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} d e^{5/2} \left (a^2+b^2\right )}-\frac {(a+b) \log \left (\sqrt {e} \cot (c+d x)-\sqrt {2} \sqrt {e \cot (c+d x)}+\sqrt {e}\right )}{2 \sqrt {2} d e^{5/2} \left (a^2+b^2\right )}+\frac {(a+b) \log \left (\sqrt {e} \cot (c+d x)+\sqrt {2} \sqrt {e \cot (c+d x)}+\sqrt {e}\right )}{2 \sqrt {2} d e^{5/2} \left (a^2+b^2\right )}-\frac {2 b}{a^2 d e^2 \sqrt {e \cot (c+d x)}}-\frac {2 b^{7/2} \text {ArcTan}\left (\frac {\sqrt {b} \sqrt {e \cot (c+d x)}}{\sqrt {a} \sqrt {e}}\right )}{a^{5/2} d e^{5/2} \left (a^2+b^2\right )}+\frac {2}{3 a d e (e \cot (c+d x))^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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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 3735
Rubi steps
\begin {align*} \int \frac {1}{(e \cot (c+d x))^{5/2} (a+b \cot (c+d x))} \, dx &=\frac {2}{3 a d e (e \cot (c+d x))^{3/2}}+\frac {2 \int \frac {-\frac {3 b e^2}{2}-\frac {3}{2} a e^2 \cot (c+d x)-\frac {3}{2} b e^2 \cot ^2(c+d x)}{(e \cot (c+d x))^{3/2} (a+b \cot (c+d x))} \, dx}{3 a e^3}\\ &=\frac {2}{3 a d e (e \cot (c+d x))^{3/2}}-\frac {2 b}{a^2 d e^2 \sqrt {e \cot (c+d x)}}+\frac {4 \int \frac {-\frac {3}{4} \left (a^2-b^2\right ) e^4+\frac {3}{4} b^2 e^4 \cot ^2(c+d x)}{\sqrt {e \cot (c+d x)} (a+b \cot (c+d x))} \, dx}{3 a^2 e^6}\\ &=\frac {2}{3 a d e (e \cot (c+d x))^{3/2}}-\frac {2 b}{a^2 d e^2 \sqrt {e \cot (c+d x)}}+\frac {4 \int \frac {-\frac {3}{4} a^3 e^4+\frac {3}{4} a^2 b e^4 \cot (c+d x)}{\sqrt {e \cot (c+d x)}} \, dx}{3 a^2 \left (a^2+b^2\right ) e^6}+\frac {b^4 \int \frac {1+\cot ^2(c+d x)}{\sqrt {e \cot (c+d x)} (a+b \cot (c+d x))} \, dx}{a^2 \left (a^2+b^2\right ) e^2}\\ &=\frac {2}{3 a d e (e \cot (c+d x))^{3/2}}-\frac {2 b}{a^2 d e^2 \sqrt {e \cot (c+d x)}}+\frac {8 \text {Subst}\left (\int \frac {\frac {3 a^3 e^5}{4}-\frac {3}{4} a^2 b e^4 x^2}{e^2+x^4} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{3 a^2 \left (a^2+b^2\right ) d e^6}+\frac {b^4 \text {Subst}\left (\int \frac {1}{\sqrt {-e x} (a-b x)} \, dx,x,-\cot (c+d x)\right )}{a^2 \left (a^2+b^2\right ) d e^2}\\ &=\frac {2}{3 a d e (e \cot (c+d x))^{3/2}}-\frac {2 b}{a^2 d e^2 \sqrt {e \cot (c+d x)}}-\frac {\left (2 b^4\right ) \text {Subst}\left (\int \frac {1}{a+\frac {b x^2}{e}} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{a^2 \left (a^2+b^2\right ) d e^3}+\frac {(a-b) \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 ) d e^2}+\frac {(a+b) \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 ) d e^2}\\ &=-\frac {2 b^{7/2} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {e \cot (c+d x)}}{\sqrt {a} \sqrt {e}}\right )}{a^{5/2} \left (a^2+b^2\right ) d e^{5/2}}+\frac {2}{3 a d e (e \cot (c+d x))^{3/2}}-\frac {2 b}{a^2 d e^2 \sqrt {e \cot (c+d x)}}-\frac {(a+b) \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 ) d e^{5/2}}-\frac {(a+b) \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 ) d e^{5/2}}+\frac {(a-b) \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 ) d e^2}+\frac {(a-b) \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 ) d e^2}\\ &=-\frac {2 b^{7/2} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {e \cot (c+d x)}}{\sqrt {a} \sqrt {e}}\right )}{a^{5/2} \left (a^2+b^2\right ) d e^{5/2}}+\frac {2}{3 a d e (e \cot (c+d x))^{3/2}}-\frac {2 b}{a^2 d e^2 \sqrt {e \cot (c+d x)}}-\frac {(a+b) \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 ) d e^{5/2}}+\frac {(a+b) \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 ) d e^{5/2}}+\frac {(a-b) \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 ) d e^{5/2}}-\frac {(a-b) \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 ) d e^{5/2}}\\ &=-\frac {2 b^{7/2} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {e \cot (c+d x)}}{\sqrt {a} \sqrt {e}}\right )}{a^{5/2} \left (a^2+b^2\right ) d e^{5/2}}-\frac {(a-b) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \left (a^2+b^2\right ) d e^{5/2}}+\frac {(a-b) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \left (a^2+b^2\right ) d e^{5/2}}+\frac {2}{3 a d e (e \cot (c+d x))^{3/2}}-\frac {2 b}{a^2 d e^2 \sqrt {e \cot (c+d x)}}-\frac {(a+b) \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 ) d e^{5/2}}+\frac {(a+b) \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 ) d e^{5/2}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 0.26, size = 109, normalized size = 0.31 \begin {gather*} \frac {2 \left (b^2 \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};-\frac {b \cot (c+d x)}{a}\right )+a \left (a \, _2F_1\left (-\frac {3}{4},1;\frac {1}{4};-\cot ^2(c+d x)\right )-3 b \cot (c+d x) \, _2F_1\left (-\frac {1}{4},1;\frac {3}{4};-\cot ^2(c+d x)\right )\right )\right )}{3 a \left (a^2+b^2\right ) d e (e \cot (c+d x))^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.58, size = 371, normalized size = 1.06
method | result | size |
derivativedivides | \(-\frac {2 e^{2} \left (\frac {b^{4} \arctan \left (\frac {b \sqrt {e \cot \left (d x +c \right )}}{\sqrt {a e b}}\right )}{a^{2} e^{4} \left (a^{2}+b^{2}\right ) \sqrt {a e b}}-\frac {1}{3 a \,e^{3} \left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}}+\frac {b}{a^{2} e^{4} \sqrt {e \cot \left (d x +c \right )}}+\frac {-\frac {a \left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 e}+\frac {b \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (e^{2}\right )^{\frac {1}{4}}}}{\left (a^{2}+b^{2}\right ) e^{4}}\right )}{d}\) | \(371\) |
default | \(-\frac {2 e^{2} \left (\frac {b^{4} \arctan \left (\frac {b \sqrt {e \cot \left (d x +c \right )}}{\sqrt {a e b}}\right )}{a^{2} e^{4} \left (a^{2}+b^{2}\right ) \sqrt {a e b}}-\frac {1}{3 a \,e^{3} \left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}}+\frac {b}{a^{2} e^{4} \sqrt {e \cot \left (d x +c \right )}}+\frac {-\frac {a \left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 e}+\frac {b \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (e^{2}\right )^{\frac {1}{4}}}}{\left (a^{2}+b^{2}\right ) e^{4}}\right )}{d}\) | \(371\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.52, size = 207, normalized size = 0.59 \begin {gather*} -\frac {{\left (\frac {24 \, b^{4} \arctan \left (\frac {b}{\sqrt {a b} \sqrt {\tan \left (d x + c\right )}}\right )}{{\left (a^{4} + a^{2} b^{2}\right )} \sqrt {a b}} - \frac {8 \, {\left (a - \frac {3 \, b}{\tan \left (d x + c\right )}\right )} \tan \left (d x + c\right )^{\frac {3}{2}}}{a^{2}} - \frac {3 \, {\left (2 \, \sqrt {2} {\left (a - b\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 - b\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 + b\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 + b\right )} \log \left (-\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right )\right )}}{a^{2} + b^{2}}\right )} e^{\left (-\frac {5}{2}\right )}}{12 \, 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 {5}{2}} \left (a + b \cot {\left (c + d x \right )}\right )}\, 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 = 2.81, size = 2500, normalized size = 7.12 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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