Optimal. Leaf size=270 \[ \frac {2 a \left (a^2-3 b^2\right ) \sqrt {\cot (c+d x)}}{d}+\frac {(a-b) \left (a^2+4 a b+b^2\right ) \log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2} d}-\frac {(a-b) \left (a^2+4 a b+b^2\right ) \log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2} d}+\frac {(a+b) \left (a^2-4 a b+b^2\right ) \tan ^{-1}\left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} d}-\frac {(a+b) \left (a^2-4 a b+b^2\right ) \tan ^{-1}\left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2} d}-\frac {8 a^2 b \cot ^{\frac {3}{2}}(c+d x)}{5 d}-\frac {2 a^2 \cot ^{\frac {3}{2}}(c+d x) (a \cot (c+d x)+b)}{5 d} \]
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Rubi [A] time = 0.38, antiderivative size = 270, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 11, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.478, Rules used = {3673, 3566, 3630, 3528, 3534, 1168, 1162, 617, 204, 1165, 628} \[ \frac {2 a \left (a^2-3 b^2\right ) \sqrt {\cot (c+d x)}}{d}+\frac {(a-b) \left (a^2+4 a b+b^2\right ) \log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2} d}-\frac {(a-b) \left (a^2+4 a b+b^2\right ) \log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2} d}+\frac {(a+b) \left (a^2-4 a b+b^2\right ) \tan ^{-1}\left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} d}-\frac {(a+b) \left (a^2-4 a b+b^2\right ) \tan ^{-1}\left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2} d}-\frac {8 a^2 b \cot ^{\frac {3}{2}}(c+d x)}{5 d}-\frac {2 a^2 \cot ^{\frac {3}{2}}(c+d x) (a \cot (c+d x)+b)}{5 d} \]
Antiderivative was successfully verified.
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Rule 204
Rule 617
Rule 628
Rule 1162
Rule 1165
Rule 1168
Rule 3528
Rule 3534
Rule 3566
Rule 3630
Rule 3673
Rubi steps
\begin {align*} \int \cot ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))^3 \, dx &=\int \sqrt {\cot (c+d x)} (b+a \cot (c+d x))^3 \, dx\\ &=-\frac {2 a^2 \cot ^{\frac {3}{2}}(c+d x) (b+a \cot (c+d x))}{5 d}-\frac {2}{5} \int \sqrt {\cot (c+d x)} \left (\frac {1}{2} b \left (3 a^2-5 b^2\right )+\frac {5}{2} a \left (a^2-3 b^2\right ) \cot (c+d x)-6 a^2 b \cot ^2(c+d x)\right ) \, dx\\ &=-\frac {8 a^2 b \cot ^{\frac {3}{2}}(c+d x)}{5 d}-\frac {2 a^2 \cot ^{\frac {3}{2}}(c+d x) (b+a \cot (c+d x))}{5 d}-\frac {2}{5} \int \sqrt {\cot (c+d x)} \left (\frac {5}{2} b \left (3 a^2-b^2\right )+\frac {5}{2} a \left (a^2-3 b^2\right ) \cot (c+d x)\right ) \, dx\\ &=\frac {2 a \left (a^2-3 b^2\right ) \sqrt {\cot (c+d x)}}{d}-\frac {8 a^2 b \cot ^{\frac {3}{2}}(c+d x)}{5 d}-\frac {2 a^2 \cot ^{\frac {3}{2}}(c+d x) (b+a \cot (c+d x))}{5 d}-\frac {2}{5} \int \frac {-\frac {5}{2} a \left (a^2-3 b^2\right )+\frac {5}{2} b \left (3 a^2-b^2\right ) \cot (c+d x)}{\sqrt {\cot (c+d x)}} \, dx\\ &=\frac {2 a \left (a^2-3 b^2\right ) \sqrt {\cot (c+d x)}}{d}-\frac {8 a^2 b \cot ^{\frac {3}{2}}(c+d x)}{5 d}-\frac {2 a^2 \cot ^{\frac {3}{2}}(c+d x) (b+a \cot (c+d x))}{5 d}-\frac {4 \operatorname {Subst}\left (\int \frac {\frac {5}{2} a \left (a^2-3 b^2\right )-\frac {5}{2} b \left (3 a^2-b^2\right ) x^2}{1+x^4} \, dx,x,\sqrt {\cot (c+d x)}\right )}{5 d}\\ &=\frac {2 a \left (a^2-3 b^2\right ) \sqrt {\cot (c+d x)}}{d}-\frac {8 a^2 b \cot ^{\frac {3}{2}}(c+d x)}{5 d}-\frac {2 a^2 \cot ^{\frac {3}{2}}(c+d x) (b+a \cot (c+d x))}{5 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 )}{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 )}{d}\\ &=\frac {2 a \left (a^2-3 b^2\right ) \sqrt {\cot (c+d x)}}{d}-\frac {8 a^2 b \cot ^{\frac {3}{2}}(c+d x)}{5 d}-\frac {2 a^2 \cot ^{\frac {3}{2}}(c+d x) (b+a \cot (c+d x))}{5 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 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 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} 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} d}\\ &=\frac {2 a \left (a^2-3 b^2\right ) \sqrt {\cot (c+d x)}}{d}-\frac {8 a^2 b \cot ^{\frac {3}{2}}(c+d x)}{5 d}-\frac {2 a^2 \cot ^{\frac {3}{2}}(c+d x) (b+a \cot (c+d x))}{5 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} 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} 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} 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} 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} 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} d}+\frac {2 a \left (a^2-3 b^2\right ) \sqrt {\cot (c+d x)}}{d}-\frac {8 a^2 b \cot ^{\frac {3}{2}}(c+d x)}{5 d}-\frac {2 a^2 \cot ^{\frac {3}{2}}(c+d x) (b+a \cot (c+d x))}{5 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} 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} d}\\ \end {align*}
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Mathematica [C] time = 2.35, size = 225, normalized size = 0.83 \[ -\frac {\frac {2}{5} a^3 \cot ^{\frac {5}{2}}(c+d x)+\frac {2}{3} b \left (b^2-3 a^2\right ) \cot ^{\frac {3}{2}}(c+d x) \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};-\cot ^2(c+d x)\right )-\frac {1}{4} a \left (a^2-3 b^2\right ) \left (8 \sqrt {\cot (c+d x)}+\sqrt {2} \log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )-\sqrt {2} \log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )+2 \sqrt {2} \tan ^{-1}\left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )-2 \sqrt {2} \tan ^{-1}\left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )\right )+2 a^2 b \cot ^{\frac {3}{2}}(c+d x)}{d} \]
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 \[ \int {\left (b \tan \left (d x + c\right ) + a\right )}^{3} \cot \left (d x + c\right )^{\frac {7}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 1.53, size = 8630, normalized size = 31.96 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.87, size = 240, normalized size = 0.89 \[ -\frac {10 \, \sqrt {2} {\left (a^{3} - 3 \, a^{2} b - 3 \, a b^{2} + b^{3}\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) + 10 \, \sqrt {2} {\left (a^{3} - 3 \, a^{2} b - 3 \, a b^{2} + b^{3}\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) + 5 \, \sqrt {2} {\left (a^{3} + 3 \, a^{2} b - 3 \, a b^{2} - b^{3}\right )} \log \left (\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right ) - 5 \, \sqrt {2} {\left (a^{3} + 3 \, a^{2} b - 3 \, a b^{2} - b^{3}\right )} \log \left (-\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right ) + \frac {40 \, a^{2} b}{\tan \left (d x + c\right )^{\frac {3}{2}}} + \frac {8 \, a^{3}}{\tan \left (d x + c\right )^{\frac {5}{2}}} - \frac {40 \, {\left (a^{3} - 3 \, a b^{2}\right )}}{\sqrt {\tan \left (d x + c\right )}}}{20 \, d} \]
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 \[ \int {\mathrm {cot}\left (c+d\,x\right )}^{7/2}\,{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^3 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-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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