Optimal. Leaf size=318 \[ \frac {\left (a^2-2 a b-b^2\right ) \text {ArcTan}\left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^2 d}-\frac {\left (a^2-2 a b-b^2\right ) \text {ArcTan}\left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^2 d}-\frac {b^{3/2} \left (5 a^2+b^2\right ) \text {ArcTan}\left (\frac {\sqrt {a} \sqrt {\cot (c+d x)}}{\sqrt {b}}\right )}{a^{3/2} \left (a^2+b^2\right )^2 d}+\frac {b^2 \sqrt {\cot (c+d x)}}{a \left (a^2+b^2\right ) d (b+a \cot (c+d x))}-\frac {\left (a^2+2 a b-b^2\right ) \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right )^2 d}+\frac {\left (a^2+2 a b-b^2\right ) \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right )^2 d} \]
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Rubi [A]
time = 0.38, antiderivative size = 318, normalized size of antiderivative = 1.00, number of steps
used = 16, number of rules used = 13, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.565, Rules used = {3754, 3646,
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 {b^2 \sqrt {\cot (c+d x)}}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}-\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 {b^{3/2} \left (5 a^2+b^2\right ) \text {ArcTan}\left (\frac {\sqrt {a} \sqrt {\cot (c+d x)}}{\sqrt {b}}\right )}{a^{3/2} 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 3646
Rule 3715
Rule 3734
Rule 3754
Rubi steps
\begin {align*} \int \frac {\sqrt {\cot (c+d x)}}{(a+b \tan (c+d x))^2} \, dx &=\int \frac {\cot ^{\frac {5}{2}}(c+d x)}{(b+a \cot (c+d x))^2} \, dx\\ &=\frac {b^2 \sqrt {\cot (c+d x)}}{a \left (a^2+b^2\right ) d (b+a \cot (c+d x))}-\frac {\int \frac {-\frac {b^2}{2}+a b \cot (c+d x)-\frac {1}{2} \left (2 a^2+b^2\right ) \cot ^2(c+d x)}{\sqrt {\cot (c+d x)} (b+a \cot (c+d x))} \, dx}{a \left (a^2+b^2\right )}\\ &=\frac {b^2 \sqrt {\cot (c+d x)}}{a \left (a^2+b^2\right ) d (b+a \cot (c+d x))}-\frac {\int \frac {2 a^2 b-a \left (a^2-b^2\right ) \cot (c+d x)}{\sqrt {\cot (c+d x)}} \, dx}{a \left (a^2+b^2\right )^2}+\frac {\left (b^2 \left (5 a^2+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 a \left (a^2+b^2\right )^2}\\ &=\frac {b^2 \sqrt {\cot (c+d x)}}{a \left (a^2+b^2\right ) d (b+a \cot (c+d x))}-\frac {2 \text {Subst}\left (\int \frac {-2 a^2 b+a \left (a^2-b^2\right ) x^2}{1+x^4} \, dx,x,\sqrt {\cot (c+d x)}\right )}{a \left (a^2+b^2\right )^2 d}+\frac {\left (b^2 \left (5 a^2+b^2\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-x} (b-a x)} \, dx,x,-\cot (c+d x)\right )}{2 a \left (a^2+b^2\right )^2 d}\\ &=\frac {b^2 \sqrt {\cot (c+d x)}}{a \left (a^2+b^2\right ) d (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 (b^2 \left (5 a^2+b^2\right )\right ) \text {Subst}\left (\int \frac {1}{b+a x^2} \, dx,x,\sqrt {\cot (c+d x)}\right )}{a \left (a^2+b^2\right )^2 d}\\ &=-\frac {b^{3/2} \left (5 a^2+b^2\right ) \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {\cot (c+d x)}}{\sqrt {b}}\right )}{a^{3/2} \left (a^2+b^2\right )^2 d}+\frac {b^2 \sqrt {\cot (c+d x)}}{a \left (a^2+b^2\right ) d (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 {b^{3/2} \left (5 a^2+b^2\right ) \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {\cot (c+d x)}}{\sqrt {b}}\right )}{a^{3/2} \left (a^2+b^2\right )^2 d}+\frac {b^2 \sqrt {\cot (c+d x)}}{a \left (a^2+b^2\right ) d (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 {b^{3/2} \left (5 a^2+b^2\right ) \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {\cot (c+d x)}}{\sqrt {b}}\right )}{a^{3/2} \left (a^2+b^2\right )^2 d}+\frac {b^2 \sqrt {\cot (c+d x)}}{a \left (a^2+b^2\right ) d (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 = 1.72, size = 368, normalized size = 1.16 \begin {gather*} \frac {-28 a^{3/2} b^2 \left (a^2-b^2\right ) \cot ^{\frac {3}{2}}(c+d x) \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};-\cot ^2(c+d x)\right )-12 a^{7/2} \left (a^2+b^2\right ) \cot ^{\frac {7}{2}}(c+d x) \, _2F_1\left (2,\frac {7}{2};\frac {9}{2};-\frac {a \cot (c+d x)}{b}\right )+7 b^2 \left (-6 \sqrt {2} a^{5/2} b \text {ArcTan}\left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )+6 \sqrt {2} a^{5/2} b \text {ArcTan}\left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )+24 b^{7/2} \text {ArcTan}\left (\frac {\sqrt {a} \sqrt {\cot (c+d x)}}{\sqrt {b}}\right )-24 a^{5/2} b \sqrt {\cot (c+d x)}-24 \sqrt {a} b^3 \sqrt {\cot (c+d x)}+4 a^{7/2} \cot ^{\frac {3}{2}}(c+d x)+4 a^{3/2} b^2 \cot ^{\frac {3}{2}}(c+d x)-3 \sqrt {2} a^{5/2} b \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )+3 \sqrt {2} a^{5/2} b \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )\right )}{42 a^{3/2} b^2 \left (a^2+b^2\right )^2 d} \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 = 20.42, size = 18333, normalized size = 57.65
method | result | size |
default | \(\text {Expression too large to display}\) | \(18333\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 283, normalized size = 0.89 \begin {gather*} -\frac {\frac {4 \, {\left (5 \, a^{2} b^{2} + b^{4}\right )} \arctan \left (\frac {a}{\sqrt {a b} \sqrt {\tan \left (d x + c\right )}}\right )}{{\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} \sqrt {a b}} - \frac {4 \, b^{2}}{{\left (a^{3} b + a b^{3} + \frac {a^{4} + a^{2} b^{2}}{\tan \left (d x + c\right )}\right )} \sqrt {\tan \left (d x + c\right )}} + \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}}}{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 {\sqrt {\cot {\left (c + d x \right )}}}{\left (a + b \tan {\left (c + d x \right )}\right )^{2}}\, 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 [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\sqrt {\mathrm {cot}\left (c+d\,x\right )}}{{\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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