Optimal. Leaf size=357 \[ \frac {\left (a^2-2 a b-b^2\right ) \text {ArcTan}\left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^2 d}-\frac {\left (a^2-2 a b-b^2\right ) \text {ArcTan}\left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^2 d}+\frac {a^{5/2} \left (3 a^2+7 b^2\right ) \text {ArcTan}\left (\frac {\sqrt {a} \sqrt {\cot (c+d x)}}{\sqrt {b}}\right )}{b^{5/2} \left (a^2+b^2\right )^2 d}+\frac {3 a^2+2 b^2}{b^2 \left (a^2+b^2\right ) d \sqrt {\cot (c+d x)}}-\frac {a^2}{b \left (a^2+b^2\right ) d \sqrt {\cot (c+d x)} (b+a \cot (c+d x))}-\frac {\left (a^2+2 a b-b^2\right ) \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right )^2 d}+\frac {\left (a^2+2 a b-b^2\right ) \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right )^2 d} \]
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Rubi [A]
time = 0.58, antiderivative size = 357, normalized size of antiderivative = 1.00, number
of steps used = 17, number of rules used = 14, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.609, Rules
used = {3754, 3650, 3730, 3734, 3615, 1182, 1176, 631, 210, 1179, 642, 3715, 65, 211}
\begin {gather*} \frac {\left (a^2-2 a b-b^2\right ) \text {ArcTan}\left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} d \left (a^2+b^2\right )^2}-\frac {\left (a^2-2 a b-b^2\right ) \text {ArcTan}\left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2} d \left (a^2+b^2\right )^2}-\frac {a^2}{b d \left (a^2+b^2\right ) \sqrt {\cot (c+d x)} (a \cot (c+d x)+b)}+\frac {3 a^2+2 b^2}{b^2 d \left (a^2+b^2\right ) \sqrt {\cot (c+d x)}}-\frac {\left (a^2+2 a b-b^2\right ) \log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )^2}+\frac {\left (a^2+2 a b-b^2\right ) \log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )^2}+\frac {a^{5/2} \left (3 a^2+7 b^2\right ) \text {ArcTan}\left (\frac {\sqrt {a} \sqrt {\cot (c+d x)}}{\sqrt {b}}\right )}{b^{5/2} d \left (a^2+b^2\right )^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 3734
Rule 3754
Rubi steps
\begin {align*} \int \frac {1}{\cot ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))^2} \, dx &=\int \frac {1}{\cot ^{\frac {3}{2}}(c+d x) (b+a \cot (c+d x))^2} \, dx\\ &=-\frac {a^2}{b \left (a^2+b^2\right ) d \sqrt {\cot (c+d x)} (b+a \cot (c+d x))}-\frac {\int \frac {\frac {1}{2} \left (-3 a^2-2 b^2\right )+a b \cot (c+d x)-\frac {3}{2} a^2 \cot ^2(c+d x)}{\cot ^{\frac {3}{2}}(c+d x) (b+a \cot (c+d x))} \, dx}{b \left (a^2+b^2\right )}\\ &=\frac {3 a^2+2 b^2}{b^2 \left (a^2+b^2\right ) d \sqrt {\cot (c+d x)}}-\frac {a^2}{b \left (a^2+b^2\right ) d \sqrt {\cot (c+d x)} (b+a \cot (c+d x))}-\frac {2 \int \frac {\frac {1}{4} a \left (3 a^2+4 b^2\right )+\frac {1}{2} b^3 \cot (c+d x)+\frac {1}{4} a \left (3 a^2+2 b^2\right ) \cot ^2(c+d x)}{\sqrt {\cot (c+d x)} (b+a \cot (c+d x))} \, dx}{b^2 \left (a^2+b^2\right )}\\ &=\frac {3 a^2+2 b^2}{b^2 \left (a^2+b^2\right ) d \sqrt {\cot (c+d x)}}-\frac {a^2}{b \left (a^2+b^2\right ) d \sqrt {\cot (c+d x)} (b+a \cot (c+d x))}-\frac {2 \int \frac {a b^3-\frac {1}{2} b^2 \left (a^2-b^2\right ) \cot (c+d x)}{\sqrt {\cot (c+d x)}} \, dx}{b^2 \left (a^2+b^2\right )^2}-\frac {\left (a^3 \left (3 a^2+7 b^2\right )\right ) \int \frac {1+\cot ^2(c+d x)}{\sqrt {\cot (c+d x)} (b+a \cot (c+d x))} \, dx}{2 b^2 \left (a^2+b^2\right )^2}\\ &=\frac {3 a^2+2 b^2}{b^2 \left (a^2+b^2\right ) d \sqrt {\cot (c+d x)}}-\frac {a^2}{b \left (a^2+b^2\right ) d \sqrt {\cot (c+d x)} (b+a \cot (c+d x))}-\frac {4 \text {Subst}\left (\int \frac {-a b^3+\frac {1}{2} b^2 \left (a^2-b^2\right ) x^2}{1+x^4} \, dx,x,\sqrt {\cot (c+d x)}\right )}{b^2 \left (a^2+b^2\right )^2 d}-\frac {\left (a^3 \left (3 a^2+7 b^2\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-x} (b-a x)} \, dx,x,-\cot (c+d x)\right )}{2 b^2 \left (a^2+b^2\right )^2 d}\\ &=\frac {3 a^2+2 b^2}{b^2 \left (a^2+b^2\right ) d \sqrt {\cot (c+d x)}}-\frac {a^2}{b \left (a^2+b^2\right ) d \sqrt {\cot (c+d x)} (b+a \cot (c+d x))}-\frac {\left (a^2-2 a b-b^2\right ) \text {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\sqrt {\cot (c+d x)}\right )}{\left (a^2+b^2\right )^2 d}+\frac {\left (a^2+2 a b-b^2\right ) \text {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\sqrt {\cot (c+d x)}\right )}{\left (a^2+b^2\right )^2 d}+\frac {\left (a^3 \left (3 a^2+7 b^2\right )\right ) \text {Subst}\left (\int \frac {1}{b+a x^2} \, dx,x,\sqrt {\cot (c+d x)}\right )}{b^2 \left (a^2+b^2\right )^2 d}\\ &=\frac {a^{5/2} \left (3 a^2+7 b^2\right ) \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {\cot (c+d x)}}{\sqrt {b}}\right )}{b^{5/2} \left (a^2+b^2\right )^2 d}+\frac {3 a^2+2 b^2}{b^2 \left (a^2+b^2\right ) d \sqrt {\cot (c+d x)}}-\frac {a^2}{b \left (a^2+b^2\right ) d \sqrt {\cot (c+d x)} (b+a \cot (c+d x))}-\frac {\left (a^2-2 a b-b^2\right ) \text {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\sqrt {\cot (c+d x)}\right )}{2 \left (a^2+b^2\right )^2 d}-\frac {\left (a^2-2 a b-b^2\right ) \text {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\sqrt {\cot (c+d x)}\right )}{2 \left (a^2+b^2\right )^2 d}-\frac {\left (a^2+2 a b-b^2\right ) \text {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\sqrt {\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 ) \text {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\sqrt {\cot (c+d x)}\right )}{2 \sqrt {2} \left (a^2+b^2\right )^2 d}\\ &=\frac {a^{5/2} \left (3 a^2+7 b^2\right ) \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {\cot (c+d x)}}{\sqrt {b}}\right )}{b^{5/2} \left (a^2+b^2\right )^2 d}+\frac {3 a^2+2 b^2}{b^2 \left (a^2+b^2\right ) d \sqrt {\cot (c+d x)}}-\frac {a^2}{b \left (a^2+b^2\right ) d \sqrt {\cot (c+d x)} (b+a \cot (c+d x))}-\frac {\left (a^2+2 a b-b^2\right ) \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\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 ) \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\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 ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^2 d}+\frac {\left (a^2-2 a b-b^2\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^2 d}\\ &=\frac {\left (a^2-2 a b-b^2\right ) \tan ^{-1}\left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^2 d}-\frac {\left (a^2-2 a b-b^2\right ) \tan ^{-1}\left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^2 d}+\frac {a^{5/2} \left (3 a^2+7 b^2\right ) \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {\cot (c+d x)}}{\sqrt {b}}\right )}{b^{5/2} \left (a^2+b^2\right )^2 d}+\frac {3 a^2+2 b^2}{b^2 \left (a^2+b^2\right ) d \sqrt {\cot (c+d x)}}-\frac {a^2}{b \left (a^2+b^2\right ) d \sqrt {\cot (c+d x)} (b+a \cot (c+d x))}-\frac {\left (a^2+2 a b-b^2\right ) \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\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 ) \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right )^2 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 = 0.61, size = 239, normalized size = 0.67 \begin {gather*} \frac {8 a^2 b^2 \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};-\frac {a \cot (c+d x)}{b}\right )+4 a^2 \left (a^2+b^2\right ) \, _2F_1\left (-\frac {1}{2},2;\frac {1}{2};-\frac {a \cot (c+d x)}{b}\right )+b^2 \left (-4 \left (a^2-b^2\right ) \, _2F_1\left (-\frac {1}{4},1;\frac {3}{4};-\cot ^2(c+d x)\right )+\sqrt {2} a b \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 )\right )}{2 b^2 \left (a^2+b^2\right )^2 d \sqrt {\cot (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 4 vs. order
3.
time = 59.64, size = 21870, normalized size = 61.26
method | result | size |
default | \(\text {Expression too large to display}\) | \(21870\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.52, size = 322, normalized size = 0.90 \begin {gather*} \frac {\frac {4 \, {\left (3 \, a^{5} + 7 \, a^{3} b^{2}\right )} \arctan \left (\frac {a}{\sqrt {a b} \sqrt {\tan \left (d x + c\right )}}\right )}{{\left (a^{4} b^{2} + 2 \, a^{2} b^{4} + b^{6}\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 \, {\left (2 \, a^{2} b + 2 \, b^{3} + \frac {3 \, a^{3} + 2 \, a b^{2}}{\tan \left (d x + c\right )}\right )}}{\frac {a^{2} b^{3} + b^{5}}{\sqrt {\tan \left (d x + c\right )}} + \frac {a^{3} b^{2} + a b^{4}}{\tan \left (d x + c\right )^{\frac {3}{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(-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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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {1}{{\mathrm {cot}\left (c+d\,x\right )}^{7/2}\,{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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