Optimal. Leaf size=476 \[ -\frac {a^{3/2} \left (3 a^4+6 a^2 b^2+35 b^4\right ) e^{7/2} \text {ArcTan}\left (\frac {\sqrt {b} \sqrt {e \cot (c+d x)}}{\sqrt {a} \sqrt {e}}\right )}{4 b^{5/2} \left (a^2+b^2\right )^3 d}+\frac {(a+b) \left (a^2-4 a b+b^2\right ) e^{7/2} \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \left (a^2+b^2\right )^3 d}-\frac {(a+b) \left (a^2-4 a b+b^2\right ) e^{7/2} \text {ArcTan}\left (1+\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \left (a^2+b^2\right )^3 d}+\frac {a^2 e^2 (e \cot (c+d x))^{3/2}}{2 b \left (a^2+b^2\right ) d (a+b \cot (c+d x))^2}+\frac {a^2 \left (3 a^2+11 b^2\right ) e^3 \sqrt {e \cot (c+d x)}}{4 b^2 \left (a^2+b^2\right )^2 d (a+b \cot (c+d x))}+\frac {(a-b) \left (a^2+4 a b+b^2\right ) e^{7/2} \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)-\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{2 \sqrt {2} \left (a^2+b^2\right )^3 d}-\frac {(a-b) \left (a^2+4 a b+b^2\right ) e^{7/2} \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)+\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{2 \sqrt {2} \left (a^2+b^2\right )^3 d} \]
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Rubi [A]
time = 0.81, antiderivative size = 476, 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 = {3646, 3726,
3734, 3615, 1182, 1176, 631, 210, 1179, 642, 3715, 65, 211} \begin {gather*} \frac {e^{7/2} (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 \left (a^2+b^2\right )^3}-\frac {e^{7/2} (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 \left (a^2+b^2\right )^3}+\frac {e^{7/2} (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 \left (a^2+b^2\right )^3}-\frac {e^{7/2} (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 \left (a^2+b^2\right )^3}+\frac {a^2 e^3 \left (3 a^2+11 b^2\right ) \sqrt {e \cot (c+d x)}}{4 b^2 d \left (a^2+b^2\right )^2 (a+b \cot (c+d x))}+\frac {a^2 e^2 (e \cot (c+d x))^{3/2}}{2 b d \left (a^2+b^2\right ) (a+b \cot (c+d x))^2}-\frac {a^{3/2} e^{7/2} \left (3 a^4+6 a^2 b^2+35 b^4\right ) \text {ArcTan}\left (\frac {\sqrt {b} \sqrt {e \cot (c+d x)}}{\sqrt {a} \sqrt {e}}\right )}{4 b^{5/2} d \left (a^2+b^2\right )^3} \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 3646
Rule 3715
Rule 3726
Rule 3734
Rubi steps
\begin {align*} \int \frac {(e \cot (c+d x))^{7/2}}{(a+b \cot (c+d x))^3} \, dx &=\frac {a^2 e^2 (e \cot (c+d x))^{3/2}}{2 b \left (a^2+b^2\right ) d (a+b \cot (c+d x))^2}-\frac {\int \frac {\sqrt {e \cot (c+d x)} \left (-\frac {3}{2} a^2 e^3+2 a b e^3 \cot (c+d x)-\frac {1}{2} \left (3 a^2+4 b^2\right ) e^3 \cot ^2(c+d x)\right )}{(a+b \cot (c+d x))^2} \, dx}{2 b \left (a^2+b^2\right )}\\ &=\frac {a^2 e^2 (e \cot (c+d x))^{3/2}}{2 b \left (a^2+b^2\right ) d (a+b \cot (c+d x))^2}+\frac {a^2 \left (3 a^2+11 b^2\right ) e^3 \sqrt {e \cot (c+d x)}}{4 b^2 \left (a^2+b^2\right )^2 d (a+b \cot (c+d x))}+\frac {\int \frac {\frac {1}{4} a^2 \left (3 a^2+11 b^2\right ) e^4-4 a b^3 e^4 \cot (c+d x)+\frac {1}{4} \left (3 a^4+3 a^2 b^2+8 b^4\right ) e^4 \cot ^2(c+d x)}{\sqrt {e \cot (c+d x)} (a+b \cot (c+d x))} \, dx}{2 b^2 \left (a^2+b^2\right )^2}\\ &=\frac {a^2 e^2 (e \cot (c+d x))^{3/2}}{2 b \left (a^2+b^2\right ) d (a+b \cot (c+d x))^2}+\frac {a^2 \left (3 a^2+11 b^2\right ) e^3 \sqrt {e \cot (c+d x)}}{4 b^2 \left (a^2+b^2\right )^2 d (a+b \cot (c+d x))}+\frac {\int \frac {2 a b^2 \left (a^2-3 b^2\right ) e^4-2 b^3 \left (3 a^2-b^2\right ) e^4 \cot (c+d x)}{\sqrt {e \cot (c+d x)}} \, dx}{2 b^2 \left (a^2+b^2\right )^3}+\frac {\left (a^2 \left (3 a^4+6 a^2 b^2+35 b^4\right ) e^4\right ) \int \frac {1+\cot ^2(c+d x)}{\sqrt {e \cot (c+d x)} (a+b \cot (c+d x))} \, dx}{8 b^2 \left (a^2+b^2\right )^3}\\ &=\frac {a^2 e^2 (e \cot (c+d x))^{3/2}}{2 b \left (a^2+b^2\right ) d (a+b \cot (c+d x))^2}+\frac {a^2 \left (3 a^2+11 b^2\right ) e^3 \sqrt {e \cot (c+d x)}}{4 b^2 \left (a^2+b^2\right )^2 d (a+b \cot (c+d x))}+\frac {\text {Subst}\left (\int \frac {-2 a b^2 \left (a^2-3 b^2\right ) e^5+2 b^3 \left (3 a^2-b^2\right ) e^4 x^2}{e^2+x^4} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{b^2 \left (a^2+b^2\right )^3 d}+\frac {\left (a^2 \left (3 a^4+6 a^2 b^2+35 b^4\right ) e^4\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-e x} (a-b x)} \, dx,x,-\cot (c+d x)\right )}{8 b^2 \left (a^2+b^2\right )^3 d}\\ &=\frac {a^2 e^2 (e \cot (c+d x))^{3/2}}{2 b \left (a^2+b^2\right ) d (a+b \cot (c+d x))^2}+\frac {a^2 \left (3 a^2+11 b^2\right ) e^3 \sqrt {e \cot (c+d x)}}{4 b^2 \left (a^2+b^2\right )^2 d (a+b \cot (c+d x))}-\frac {\left (a^2 \left (3 a^4+6 a^2 b^2+35 b^4\right ) e^3\right ) \text {Subst}\left (\int \frac {1}{a+\frac {b x^2}{e}} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{4 b^2 \left (a^2+b^2\right )^3 d}-\frac {\left ((a+b) \left (a^2-4 a b+b^2\right ) e^4\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}-\frac {\left ((a-b) \left (a^2+4 a b+b^2\right ) e^4\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}\\ &=-\frac {a^{3/2} \left (3 a^4+6 a^2 b^2+35 b^4\right ) e^{7/2} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {e \cot (c+d x)}}{\sqrt {a} \sqrt {e}}\right )}{4 b^{5/2} \left (a^2+b^2\right )^3 d}+\frac {a^2 e^2 (e \cot (c+d x))^{3/2}}{2 b \left (a^2+b^2\right ) d (a+b \cot (c+d x))^2}+\frac {a^2 \left (3 a^2+11 b^2\right ) e^3 \sqrt {e \cot (c+d x)}}{4 b^2 \left (a^2+b^2\right )^2 d (a+b \cot (c+d x))}+\frac {\left ((a-b) \left (a^2+4 a b+b^2\right ) e^{7/2}\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 )^3 d}+\frac {\left ((a-b) \left (a^2+4 a b+b^2\right ) e^{7/2}\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 )^3 d}-\frac {\left ((a+b) \left (a^2-4 a b+b^2\right ) e^4\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}-\frac {\left ((a+b) \left (a^2-4 a b+b^2\right ) e^4\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}\\ &=-\frac {a^{3/2} \left (3 a^4+6 a^2 b^2+35 b^4\right ) e^{7/2} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {e \cot (c+d x)}}{\sqrt {a} \sqrt {e}}\right )}{4 b^{5/2} \left (a^2+b^2\right )^3 d}+\frac {a^2 e^2 (e \cot (c+d x))^{3/2}}{2 b \left (a^2+b^2\right ) d (a+b \cot (c+d x))^2}+\frac {a^2 \left (3 a^2+11 b^2\right ) e^3 \sqrt {e \cot (c+d x)}}{4 b^2 \left (a^2+b^2\right )^2 d (a+b \cot (c+d x))}+\frac {(a-b) \left (a^2+4 a b+b^2\right ) e^{7/2} \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 )^3 d}-\frac {(a-b) \left (a^2+4 a b+b^2\right ) e^{7/2} \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 )^3 d}-\frac {\left ((a+b) \left (a^2-4 a b+b^2\right ) e^{7/2}\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 )^3 d}+\frac {\left ((a+b) \left (a^2-4 a b+b^2\right ) e^{7/2}\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 )^3 d}\\ &=-\frac {a^{3/2} \left (3 a^4+6 a^2 b^2+35 b^4\right ) e^{7/2} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {e \cot (c+d x)}}{\sqrt {a} \sqrt {e}}\right )}{4 b^{5/2} \left (a^2+b^2\right )^3 d}+\frac {(a+b) \left (a^2-4 a b+b^2\right ) e^{7/2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \left (a^2+b^2\right )^3 d}-\frac {(a+b) \left (a^2-4 a b+b^2\right ) e^{7/2} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \left (a^2+b^2\right )^3 d}+\frac {a^2 e^2 (e \cot (c+d x))^{3/2}}{2 b \left (a^2+b^2\right ) d (a+b \cot (c+d x))^2}+\frac {a^2 \left (3 a^2+11 b^2\right ) e^3 \sqrt {e \cot (c+d x)}}{4 b^2 \left (a^2+b^2\right )^2 d (a+b \cot (c+d x))}+\frac {(a-b) \left (a^2+4 a b+b^2\right ) e^{7/2} \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 )^3 d}-\frac {(a-b) \left (a^2+4 a b+b^2\right ) e^{7/2} \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 )^3 d}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 6.19, size = 525, normalized size = 1.10 \begin {gather*} -\frac {(e \cot (c+d x))^{7/2} \left (\frac {2 b \left (3 a^2-b^2\right ) \cot ^{\frac {7}{2}}(c+d x)}{7 \left (a^2+b^2\right )^3}-\frac {2 a \left (3 a^2-b^2\right ) \left (3 \cot ^{\frac {5}{2}}(c+d x)-5 a \left (-\frac {3 a \left (-\frac {\sqrt {a} \text {ArcTan}\left (\frac {\sqrt {b} \sqrt {\cot (c+d x)}}{\sqrt {a}}\right )}{b^{3/2}}+\frac {\sqrt {\cot (c+d x)}}{b}\right )}{b}+\frac {\cot ^{\frac {3}{2}}(c+d x)}{b}\right )\right )}{15 \left (a^2+b^2\right )^3}+\frac {2 b \left (3 a^2-b^2\right ) \left (7 \cot ^{\frac {3}{2}}(c+d x)-3 \cot ^{\frac {7}{2}}(c+d x)-7 \cot ^{\frac {3}{2}}(c+d x) \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};-\cot ^2(c+d x)\right )\right )}{21 \left (a^2+b^2\right )^3}+\frac {4 b^2 \cot ^{\frac {9}{2}}(c+d x) \, _2F_1\left (2,\frac {9}{2};\frac {11}{2};-\frac {b \cot (c+d x)}{a}\right )}{9 a \left (a^2+b^2\right )^2}+\frac {2 b^2 \cot ^{\frac {9}{2}}(c+d x) \, _2F_1\left (3,\frac {9}{2};\frac {11}{2};-\frac {b \cot (c+d x)}{a}\right )}{9 a^3 \left (a^2+b^2\right )}-\frac {a \left (a^2-3 b^2\right ) \left (40 \sqrt {\cot (c+d x)}-8 \cot ^{\frac {5}{2}}(c+d x)+\frac {5}{2} \left (4 \left (\sqrt {2} \text {ArcTan}\left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )-\sqrt {2} \text {ArcTan}\left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )\right )+2 \sqrt {2} \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )-2 \sqrt {2} \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )\right )\right )}{20 \left (a^2+b^2\right )^3}\right )}{d \cot ^{\frac {7}{2}}(c+d x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.66, size = 460, normalized size = 0.97
method | result | size |
derivativedivides | \(-\frac {2 e^{4} \left (\frac {\frac {\left (a^{3} e -3 a \,b^{2} e \right ) \left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 e^{2}}+\frac {\left (-3 a^{2} b +b^{3}\right ) \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (e^{2}\right )^{\frac {1}{4}}}}{\left (a^{2}+b^{2}\right )^{3}}-\frac {a^{2} \left (\frac {\frac {\left (5 a^{4}+18 a^{2} b^{2}+13 b^{4}\right ) \left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}}{8 b}+\frac {a e \left (3 a^{4}+14 a^{2} b^{2}+11 b^{4}\right ) \sqrt {e \cot \left (d x +c \right )}}{8 b^{2}}}{\left (e \cot \left (d x +c \right ) b +a e \right )^{2}}-\frac {\left (3 a^{4}+6 a^{2} b^{2}+35 b^{4}\right ) \arctan \left (\frac {b \sqrt {e \cot \left (d x +c \right )}}{\sqrt {a e b}}\right )}{8 b^{2} \sqrt {a e b}}\right )}{\left (a^{2}+b^{2}\right )^{3}}\right )}{d}\) | \(460\) |
default | \(-\frac {2 e^{4} \left (\frac {\frac {\left (a^{3} e -3 a \,b^{2} e \right ) \left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 e^{2}}+\frac {\left (-3 a^{2} b +b^{3}\right ) \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (e^{2}\right )^{\frac {1}{4}}}}{\left (a^{2}+b^{2}\right )^{3}}-\frac {a^{2} \left (\frac {\frac {\left (5 a^{4}+18 a^{2} b^{2}+13 b^{4}\right ) \left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}}{8 b}+\frac {a e \left (3 a^{4}+14 a^{2} b^{2}+11 b^{4}\right ) \sqrt {e \cot \left (d x +c \right )}}{8 b^{2}}}{\left (e \cot \left (d x +c \right ) b +a e \right )^{2}}-\frac {\left (3 a^{4}+6 a^{2} b^{2}+35 b^{4}\right ) \arctan \left (\frac {b \sqrt {e \cot \left (d x +c \right )}}{\sqrt {a e b}}\right )}{8 b^{2} \sqrt {a e b}}\right )}{\left (a^{2}+b^{2}\right )^{3}}\right )}{d}\) | \(460\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.52, size = 428, normalized size = 0.90 \begin {gather*} -\frac {{\left (\frac {{\left (3 \, a^{6} + 6 \, a^{4} b^{2} + 35 \, a^{2} b^{4}\right )} \arctan \left (\frac {b}{\sqrt {a b} \sqrt {\tan \left (d x + c\right )}}\right )}{{\left (a^{6} b^{2} + 3 \, a^{4} b^{4} + 3 \, a^{2} b^{6} + b^{8}\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 {\frac {3 \, a^{5} + 11 \, a^{3} b^{2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {5 \, a^{4} b + 13 \, a^{2} b^{3}}{\tan \left (d x + c\right )^{\frac {3}{2}}}}{a^{6} b^{2} + 2 \, a^{4} b^{4} + a^{2} b^{6} + \frac {2 \, {\left (a^{5} b^{3} + 2 \, a^{3} b^{5} + a b^{7}\right )}}{\tan \left (d x + c\right )} + \frac {a^{4} b^{4} + 2 \, a^{2} b^{6} + b^{8}}{\tan \left (d x + c\right )^{2}}}\right )} e^{\frac {7}{2}}}{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(-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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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 = 7.26, size = 2500, normalized size = 5.25 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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