Optimal. Leaf size=444 \[ \frac {b^2 \left (13 a^2+5 b^2\right ) \cot ^{\frac {3}{2}}(c+d x)}{4 a^2 d \left (a^2+b^2\right )^2 (a \cot (c+d x)+b)}+\frac {b^2 \cot ^{\frac {5}{2}}(c+d x)}{2 a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}-\frac {(a+b) \left (a^2-4 a b+b^2\right ) \log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )^3}+\frac {(a+b) \left (a^2-4 a b+b^2\right ) \log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )^3}-\frac {(a-b) \left (a^2+4 a b+b^2\right ) \tan ^{-1}\left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} d \left (a^2+b^2\right )^3}+\frac {(a-b) \left (a^2+4 a b+b^2\right ) \tan ^{-1}\left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2} d \left (a^2+b^2\right )^3}+\frac {b^{5/2} \left (63 a^4+46 a^2 b^2+15 b^4\right ) \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {\cot (c+d x)}}{\sqrt {b}}\right )}{4 a^{7/2} d \left (a^2+b^2\right )^3}-\frac {\left (8 a^4+31 a^2 b^2+15 b^4\right ) \sqrt {\cot (c+d x)}}{4 a^3 d \left (a^2+b^2\right )^2} \]
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Rubi [A] time = 1.21, antiderivative size = 444, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 15, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.652, Rules used = {3673, 3565, 3645, 3647, 3653, 3534, 1168, 1162, 617, 204, 1165, 628, 3634, 63, 205} \[ \frac {b^2 \left (13 a^2+5 b^2\right ) \cot ^{\frac {3}{2}}(c+d x)}{4 a^2 d \left (a^2+b^2\right )^2 (a \cot (c+d x)+b)}+\frac {b^2 \cot ^{\frac {5}{2}}(c+d x)}{2 a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}-\frac {\left (31 a^2 b^2+8 a^4+15 b^4\right ) \sqrt {\cot (c+d x)}}{4 a^3 d \left (a^2+b^2\right )^2}-\frac {(a+b) \left (a^2-4 a b+b^2\right ) \log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )^3}+\frac {(a+b) \left (a^2-4 a b+b^2\right ) \log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )^3}+\frac {b^{5/2} \left (46 a^2 b^2+63 a^4+15 b^4\right ) \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {\cot (c+d x)}}{\sqrt {b}}\right )}{4 a^{7/2} d \left (a^2+b^2\right )^3}-\frac {(a-b) \left (a^2+4 a b+b^2\right ) \tan ^{-1}\left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} d \left (a^2+b^2\right )^3}+\frac {(a-b) \left (a^2+4 a b+b^2\right ) \tan ^{-1}\left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2} d \left (a^2+b^2\right )^3} \]
Antiderivative was successfully verified.
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Rule 63
Rule 204
Rule 205
Rule 617
Rule 628
Rule 1162
Rule 1165
Rule 1168
Rule 3534
Rule 3565
Rule 3634
Rule 3645
Rule 3647
Rule 3653
Rule 3673
Rubi steps
\begin {align*} \int \frac {\cot ^{\frac {3}{2}}(c+d x)}{(a+b \tan (c+d x))^3} \, dx &=\int \frac {\cot ^{\frac {9}{2}}(c+d x)}{(b+a \cot (c+d x))^3} \, dx\\ &=\frac {b^2 \cot ^{\frac {5}{2}}(c+d x)}{2 a \left (a^2+b^2\right ) d (b+a \cot (c+d x))^2}-\frac {\int \frac {\cot ^{\frac {3}{2}}(c+d x) \left (-\frac {5 b^2}{2}+2 a b \cot (c+d x)-\frac {1}{2} \left (4 a^2+5 b^2\right ) \cot ^2(c+d x)\right )}{(b+a \cot (c+d x))^2} \, dx}{2 a \left (a^2+b^2\right )}\\ &=\frac {b^2 \cot ^{\frac {5}{2}}(c+d x)}{2 a \left (a^2+b^2\right ) d (b+a \cot (c+d x))^2}+\frac {b^2 \left (13 a^2+5 b^2\right ) \cot ^{\frac {3}{2}}(c+d x)}{4 a^2 \left (a^2+b^2\right )^2 d (b+a \cot (c+d x))}+\frac {\int \frac {\sqrt {\cot (c+d x)} \left (\frac {3}{4} b^2 \left (13 a^2+5 b^2\right )-4 a^3 b \cot (c+d x)+\frac {1}{4} \left (8 a^4+31 a^2 b^2+15 b^4\right ) \cot ^2(c+d x)\right )}{b+a \cot (c+d x)} \, dx}{2 a^2 \left (a^2+b^2\right )^2}\\ &=-\frac {\left (8 a^4+31 a^2 b^2+15 b^4\right ) \sqrt {\cot (c+d x)}}{4 a^3 \left (a^2+b^2\right )^2 d}+\frac {b^2 \cot ^{\frac {5}{2}}(c+d x)}{2 a \left (a^2+b^2\right ) d (b+a \cot (c+d x))^2}+\frac {b^2 \left (13 a^2+5 b^2\right ) \cot ^{\frac {3}{2}}(c+d x)}{4 a^2 \left (a^2+b^2\right )^2 d (b+a \cot (c+d x))}-\frac {\int \frac {\frac {1}{8} b \left (8 a^4+31 a^2 b^2+15 b^4\right )+a^3 \left (a^2-b^2\right ) \cot (c+d x)+\frac {1}{8} b \left (24 a^4+31 a^2 b^2+15 b^4\right ) \cot ^2(c+d x)}{\sqrt {\cot (c+d x)} (b+a \cot (c+d x))} \, dx}{a^3 \left (a^2+b^2\right )^2}\\ &=-\frac {\left (8 a^4+31 a^2 b^2+15 b^4\right ) \sqrt {\cot (c+d x)}}{4 a^3 \left (a^2+b^2\right )^2 d}+\frac {b^2 \cot ^{\frac {5}{2}}(c+d x)}{2 a \left (a^2+b^2\right ) d (b+a \cot (c+d x))^2}+\frac {b^2 \left (13 a^2+5 b^2\right ) \cot ^{\frac {3}{2}}(c+d x)}{4 a^2 \left (a^2+b^2\right )^2 d (b+a \cot (c+d x))}-\frac {\int \frac {a^4 \left (a^2-3 b^2\right )+a^3 b \left (3 a^2-b^2\right ) \cot (c+d x)}{\sqrt {\cot (c+d x)}} \, dx}{a^3 \left (a^2+b^2\right )^3}-\frac {\left (b^3 \left (63 a^4+46 a^2 b^2+15 b^4\right )\right ) \int \frac {1+\cot ^2(c+d x)}{\sqrt {\cot (c+d x)} (b+a \cot (c+d x))} \, dx}{8 a^3 \left (a^2+b^2\right )^3}\\ &=-\frac {\left (8 a^4+31 a^2 b^2+15 b^4\right ) \sqrt {\cot (c+d x)}}{4 a^3 \left (a^2+b^2\right )^2 d}+\frac {b^2 \cot ^{\frac {5}{2}}(c+d x)}{2 a \left (a^2+b^2\right ) d (b+a \cot (c+d x))^2}+\frac {b^2 \left (13 a^2+5 b^2\right ) \cot ^{\frac {3}{2}}(c+d x)}{4 a^2 \left (a^2+b^2\right )^2 d (b+a \cot (c+d x))}-\frac {2 \operatorname {Subst}\left (\int \frac {-a^4 \left (a^2-3 b^2\right )-a^3 b \left (3 a^2-b^2\right ) x^2}{1+x^4} \, dx,x,\sqrt {\cot (c+d x)}\right )}{a^3 \left (a^2+b^2\right )^3 d}-\frac {\left (b^3 \left (63 a^4+46 a^2 b^2+15 b^4\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-x} (b-a x)} \, dx,x,-\cot (c+d x)\right )}{8 a^3 \left (a^2+b^2\right )^3 d}\\ &=-\frac {\left (8 a^4+31 a^2 b^2+15 b^4\right ) \sqrt {\cot (c+d x)}}{4 a^3 \left (a^2+b^2\right )^2 d}+\frac {b^2 \cot ^{\frac {5}{2}}(c+d x)}{2 a \left (a^2+b^2\right ) d (b+a \cot (c+d x))^2}+\frac {b^2 \left (13 a^2+5 b^2\right ) \cot ^{\frac {3}{2}}(c+d x)}{4 a^2 \left (a^2+b^2\right )^2 d (b+a \cot (c+d x))}+\frac {\left ((a+b) \left (a^2-4 a b+b^2\right )\right ) \operatorname {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\sqrt {\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 )\right ) \operatorname {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\sqrt {\cot (c+d x)}\right )}{\left (a^2+b^2\right )^3 d}+\frac {\left (b^3 \left (63 a^4+46 a^2 b^2+15 b^4\right )\right ) \operatorname {Subst}\left (\int \frac {1}{b+a x^2} \, dx,x,\sqrt {\cot (c+d x)}\right )}{4 a^3 \left (a^2+b^2\right )^3 d}\\ &=\frac {b^{5/2} \left (63 a^4+46 a^2 b^2+15 b^4\right ) \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {\cot (c+d x)}}{\sqrt {b}}\right )}{4 a^{7/2} \left (a^2+b^2\right )^3 d}-\frac {\left (8 a^4+31 a^2 b^2+15 b^4\right ) \sqrt {\cot (c+d x)}}{4 a^3 \left (a^2+b^2\right )^2 d}+\frac {b^2 \cot ^{\frac {5}{2}}(c+d x)}{2 a \left (a^2+b^2\right ) d (b+a \cot (c+d x))^2}+\frac {b^2 \left (13 a^2+5 b^2\right ) \cot ^{\frac {3}{2}}(c+d x)}{4 a^2 \left (a^2+b^2\right )^2 d (b+a \cot (c+d x))}-\frac {\left ((a+b) \left (a^2-4 a b+b^2\right )\right ) \operatorname {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 )^3 d}-\frac {\left ((a+b) \left (a^2-4 a b+b^2\right )\right ) \operatorname {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 )^3 d}+\frac {\left ((a-b) \left (a^2+4 a b+b^2\right )\right ) \operatorname {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 )^3 d}+\frac {\left ((a-b) \left (a^2+4 a b+b^2\right )\right ) \operatorname {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 )^3 d}\\ &=\frac {b^{5/2} \left (63 a^4+46 a^2 b^2+15 b^4\right ) \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {\cot (c+d x)}}{\sqrt {b}}\right )}{4 a^{7/2} \left (a^2+b^2\right )^3 d}-\frac {\left (8 a^4+31 a^2 b^2+15 b^4\right ) \sqrt {\cot (c+d x)}}{4 a^3 \left (a^2+b^2\right )^2 d}+\frac {b^2 \cot ^{\frac {5}{2}}(c+d x)}{2 a \left (a^2+b^2\right ) d (b+a \cot (c+d x))^2}+\frac {b^2 \left (13 a^2+5 b^2\right ) \cot ^{\frac {3}{2}}(c+d x)}{4 a^2 \left (a^2+b^2\right )^2 d (b+a \cot (c+d x))}-\frac {(a+b) \left (a^2-4 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 )^3 d}+\frac {(a+b) \left (a^2-4 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 )^3 d}+\frac {\left ((a-b) \left (a^2+4 a b+b^2\right )\right ) \operatorname {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 )^3 d}-\frac {\left ((a-b) \left (a^2+4 a b+b^2\right )\right ) \operatorname {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 )^3 d}\\ &=-\frac {(a-b) \left (a^2+4 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 )^3 d}+\frac {(a-b) \left (a^2+4 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 )^3 d}+\frac {b^{5/2} \left (63 a^4+46 a^2 b^2+15 b^4\right ) \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {\cot (c+d x)}}{\sqrt {b}}\right )}{4 a^{7/2} \left (a^2+b^2\right )^3 d}-\frac {\left (8 a^4+31 a^2 b^2+15 b^4\right ) \sqrt {\cot (c+d x)}}{4 a^3 \left (a^2+b^2\right )^2 d}+\frac {b^2 \cot ^{\frac {5}{2}}(c+d x)}{2 a \left (a^2+b^2\right ) d (b+a \cot (c+d x))^2}+\frac {b^2 \left (13 a^2+5 b^2\right ) \cot ^{\frac {3}{2}}(c+d x)}{4 a^2 \left (a^2+b^2\right )^2 d (b+a \cot (c+d x))}-\frac {(a+b) \left (a^2-4 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 )^3 d}+\frac {(a+b) \left (a^2-4 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 )^3 d}\\ \end {align*}
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Mathematica [C] time = 6.23, size = 530, normalized size = 1.19 \[ -\frac {\frac {4 a^2 \cot ^{\frac {11}{2}}(c+d x) \, _2F_1\left (2,\frac {11}{2};\frac {13}{2};-\frac {a \cot (c+d x)}{b}\right )}{11 b \left (a^2+b^2\right )^2}+\frac {2 b \left (3 a^2-b^2\right ) \left (-7 \cot ^{\frac {3}{2}}(c+d x) \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};-\cot ^2(c+d x)\right )-3 \cot ^{\frac {7}{2}}(c+d x)+7 \cot ^{\frac {3}{2}}(c+d x)\right )}{21 \left (a^2+b^2\right )^3}-\frac {2 a \left (a^2-3 b^2\right ) \cot ^{\frac {9}{2}}(c+d x)}{9 \left (a^2+b^2\right )^3}+\frac {a \left (a^2-3 b^2\right ) \left (40 \cot ^{\frac {9}{2}}(c+d x)-72 \cot ^{\frac {5}{2}}(c+d x)+360 \sqrt {\cot (c+d x)}+45 \left (\sqrt {2} \log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )-\sqrt {2} \log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )+2 \left (\sqrt {2} \tan ^{-1}\left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )-\sqrt {2} \tan ^{-1}\left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )\right )\right )\right )}{180 \left (a^2+b^2\right )^3}+\frac {2 a^2 \cot ^{\frac {11}{2}}(c+d x) \, _2F_1\left (3,\frac {11}{2};\frac {13}{2};-\frac {a \cot (c+d x)}{b}\right )}{11 b^3 \left (a^2+b^2\right )}+\frac {2 b \left (a^2-3 b^2\right ) \left (15 \cot ^{\frac {7}{2}}(c+d x)-7 b \left (\frac {3 \cot ^{\frac {5}{2}}(c+d x)}{a}-\frac {5 b \left (\frac {\cot ^{\frac {3}{2}}(c+d x)}{a}-\frac {3 b \left (\frac {\sqrt {\cot (c+d x)}}{a}-\frac {\sqrt {b} \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {\cot (c+d x)}}{\sqrt {b}}\right )}{a^{3/2}}\right )}{a}\right )}{a}\right )\right )}{105 \left (a^2+b^2\right )^3}}{d} \]
Antiderivative was successfully verified.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
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 \[ \int \frac {\cot \left (d x + c\right )^{\frac {3}{2}}}{{\left (b \tan \left (d x + c\right ) + a\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 5.88, size = 78085, normalized size = 175.87 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.94, size = 439, normalized size = 0.99 \[ \frac {\frac {{\left (63 \, a^{4} b^{3} + 46 \, a^{2} b^{5} + 15 \, b^{7}\right )} \arctan \left (\frac {a}{\sqrt {a b} \sqrt {\tan \left (d x + c\right )}}\right )}{{\left (a^{9} + 3 \, a^{7} b^{2} + 3 \, a^{5} b^{4} + a^{3} b^{6}\right )} \sqrt {a b}} + \frac {2 \, \sqrt {2} {\left (a^{3} + 3 \, a^{2} b - 3 \, a b^{2} - b^{3}\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) + 2 \, \sqrt {2} {\left (a^{3} + 3 \, a^{2} b - 3 \, a b^{2} - b^{3}\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) + \sqrt {2} {\left (a^{3} - 3 \, a^{2} b - 3 \, a b^{2} + b^{3}\right )} \log \left (\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right ) - \sqrt {2} {\left (a^{3} - 3 \, a^{2} b - 3 \, a b^{2} + b^{3}\right )} \log \left (-\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac {\frac {15 \, a^{2} b^{4} + 7 \, b^{6}}{\sqrt {\tan \left (d x + c\right )}} + \frac {17 \, a^{3} b^{3} + 9 \, a b^{5}}{\tan \left (d x + c\right )^{\frac {3}{2}}}}{a^{7} b^{2} + 2 \, a^{5} b^{4} + a^{3} b^{6} + \frac {2 \, {\left (a^{8} b + 2 \, a^{6} b^{3} + a^{4} b^{5}\right )}}{\tan \left (d x + c\right )} + \frac {a^{9} + 2 \, a^{7} b^{2} + a^{5} b^{4}}{\tan \left (d x + c\right )^{2}}} - \frac {8}{a^{3} \sqrt {\tan \left (d x + c\right )}}}{4 \, d} \]
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 \[ \int \frac {{\mathrm {cot}\left (c+d\,x\right )}^{3/2}}{{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^3} \,d x \]
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 \[ \int \frac {\cot ^{\frac {3}{2}}{\left (c + d x \right )}}{\left (a + b \tan {\left (c + d x \right )}\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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