Optimal. Leaf size=357 \[ -\frac {\left (a^2+2 a b-b^2\right ) \text {ArcTan}\left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^2 d}+\frac {\left (a^2+2 a b-b^2\right ) \text {ArcTan}\left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^2 d}+\frac {b^{5/2} \left (7 a^2+3 b^2\right ) \text {ArcTan}\left (\frac {\sqrt {a} \sqrt {\cot (c+d x)}}{\sqrt {b}}\right )}{a^{5/2} \left (a^2+b^2\right )^2 d}-\frac {\left (2 a^2+3 b^2\right ) \sqrt {\cot (c+d x)}}{a^2 \left (a^2+b^2\right ) d}+\frac {b^2 \cot ^{\frac {3}{2}}(c+d x)}{a \left (a^2+b^2\right ) d (b+a \cot (c+d x))}-\frac {\left (a^2-2 a b-b^2\right ) \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right )^2 d}+\frac {\left (a^2-2 a b-b^2\right ) \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right )^2 d} \]
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Rubi [A]
time = 0.56, 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, 3646, 3728, 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 \cot ^{\frac {3}{2}}(c+d x)}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}-\frac {\left (2 a^2+3 b^2\right ) \sqrt {\cot (c+d x)}}{a^2 d \left (a^2+b^2\right )}-\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^{5/2} \left (7 a^2+3 b^2\right ) \text {ArcTan}\left (\frac {\sqrt {a} \sqrt {\cot (c+d x)}}{\sqrt {b}}\right )}{a^{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 3646
Rule 3715
Rule 3728
Rule 3734
Rule 3754
Rubi steps
\begin {align*} \int \frac {\cot ^{\frac {3}{2}}(c+d x)}{(a+b \tan (c+d x))^2} \, dx &=\int \frac {\cot ^{\frac {7}{2}}(c+d x)}{(b+a \cot (c+d x))^2} \, dx\\ &=\frac {b^2 \cot ^{\frac {3}{2}}(c+d x)}{a \left (a^2+b^2\right ) d (b+a \cot (c+d x))}-\frac {\int \frac {\sqrt {\cot (c+d x)} \left (-\frac {3 b^2}{2}+a b \cot (c+d x)-\frac {1}{2} \left (2 a^2+3 b^2\right ) \cot ^2(c+d x)\right )}{b+a \cot (c+d x)} \, dx}{a \left (a^2+b^2\right )}\\ &=-\frac {\left (2 a^2+3 b^2\right ) \sqrt {\cot (c+d x)}}{a^2 \left (a^2+b^2\right ) d}+\frac {b^2 \cot ^{\frac {3}{2}}(c+d x)}{a \left (a^2+b^2\right ) d (b+a \cot (c+d x))}+\frac {2 \int \frac {-\frac {1}{4} b \left (2 a^2+3 b^2\right )-\frac {1}{2} a^3 \cot (c+d x)-\frac {1}{4} b \left (4 a^2+3 b^2\right ) \cot ^2(c+d x)}{\sqrt {\cot (c+d x)} (b+a \cot (c+d x))} \, dx}{a^2 \left (a^2+b^2\right )}\\ &=-\frac {\left (2 a^2+3 b^2\right ) \sqrt {\cot (c+d x)}}{a^2 \left (a^2+b^2\right ) d}+\frac {b^2 \cot ^{\frac {3}{2}}(c+d x)}{a \left (a^2+b^2\right ) d (b+a \cot (c+d x))}+\frac {2 \int \frac {-\frac {1}{2} a^2 \left (a^2-b^2\right )-a^3 b \cot (c+d x)}{\sqrt {\cot (c+d x)}} \, dx}{a^2 \left (a^2+b^2\right )^2}-\frac {\left (b^3 \left (7 a^2+3 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^2 \left (a^2+b^2\right )^2}\\ &=-\frac {\left (2 a^2+3 b^2\right ) \sqrt {\cot (c+d x)}}{a^2 \left (a^2+b^2\right ) d}+\frac {b^2 \cot ^{\frac {3}{2}}(c+d x)}{a \left (a^2+b^2\right ) d (b+a \cot (c+d x))}+\frac {4 \text {Subst}\left (\int \frac {\frac {1}{2} a^2 \left (a^2-b^2\right )+a^3 b x^2}{1+x^4} \, dx,x,\sqrt {\cot (c+d x)}\right )}{a^2 \left (a^2+b^2\right )^2 d}-\frac {\left (b^3 \left (7 a^2+3 b^2\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-x} (b-a x)} \, dx,x,-\cot (c+d x)\right )}{2 a^2 \left (a^2+b^2\right )^2 d}\\ &=-\frac {\left (2 a^2+3 b^2\right ) \sqrt {\cot (c+d x)}}{a^2 \left (a^2+b^2\right ) d}+\frac {b^2 \cot ^{\frac {3}{2}}(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^3 \left (7 a^2+3 b^2\right )\right ) \text {Subst}\left (\int \frac {1}{b+a x^2} \, dx,x,\sqrt {\cot (c+d x)}\right )}{a^2 \left (a^2+b^2\right )^2 d}\\ &=\frac {b^{5/2} \left (7 a^2+3 b^2\right ) \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {\cot (c+d x)}}{\sqrt {b}}\right )}{a^{5/2} \left (a^2+b^2\right )^2 d}-\frac {\left (2 a^2+3 b^2\right ) \sqrt {\cot (c+d x)}}{a^2 \left (a^2+b^2\right ) d}+\frac {b^2 \cot ^{\frac {3}{2}}(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 {\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 {\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 {b^{5/2} \left (7 a^2+3 b^2\right ) \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {\cot (c+d x)}}{\sqrt {b}}\right )}{a^{5/2} \left (a^2+b^2\right )^2 d}-\frac {\left (2 a^2+3 b^2\right ) \sqrt {\cot (c+d x)}}{a^2 \left (a^2+b^2\right ) d}+\frac {b^2 \cot ^{\frac {3}{2}}(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^{5/2} \left (7 a^2+3 b^2\right ) \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {\cot (c+d x)}}{\sqrt {b}}\right )}{a^{5/2} \left (a^2+b^2\right )^2 d}-\frac {\left (2 a^2+3 b^2\right ) \sqrt {\cot (c+d x)}}{a^2 \left (a^2+b^2\right ) d}+\frac {b^2 \cot ^{\frac {3}{2}}(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 = 6.12, size = 424, normalized size = 1.19 \begin {gather*} -\frac {\frac {4 a b \cot ^{\frac {7}{2}}(c+d x)}{7 \left (a^2+b^2\right )^2}-\frac {4 b^2 \left (3 \cot ^{\frac {5}{2}}(c+d x)-5 b \left (-\frac {3 b \left (-\frac {\sqrt {b} \text {ArcTan}\left (\frac {\sqrt {a} \sqrt {\cot (c+d x)}}{\sqrt {b}}\right )}{a^{3/2}}+\frac {\sqrt {\cot (c+d x)}}{a}\right )}{a}+\frac {\cot ^{\frac {3}{2}}(c+d x)}{a}\right )\right )}{15 \left (a^2+b^2\right )^2}+\frac {4 a b \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 )^2}+\frac {2 a^2 \cot ^{\frac {9}{2}}(c+d x) \, _2F_1\left (2,\frac {9}{2};\frac {11}{2};-\frac {a \cot (c+d x)}{b}\right )}{9 b^2 \left (a^2+b^2\right )}+\frac {\left (a^2-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 )^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 = 23.15, size = 28980, normalized size = 81.18
method | result | size |
default | \(\text {Expression too large to display}\) | \(28980\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.52, size = 303, normalized size = 0.85 \begin {gather*} -\frac {\frac {4 \, b^{3}}{{\left (a^{4} b + a^{2} b^{3} + \frac {a^{5} + a^{3} b^{2}}{\tan \left (d x + c\right )}\right )} \sqrt {\tan \left (d x + c\right )}} - \frac {4 \, {\left (7 \, a^{2} b^{3} + 3 \, b^{5}\right )} \arctan \left (\frac {a}{\sqrt {a b} \sqrt {\tan \left (d x + c\right )}}\right )}{{\left (a^{6} + 2 \, a^{4} b^{2} + a^{2} b^{4}\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 {8}{a^{2} \sqrt {\tan \left (d x + c\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 {\cot ^{\frac {3}{2}}{\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 {{\mathrm {cot}\left (c+d\,x\right )}^{3/2}}{{\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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