Optimal. Leaf size=421 \[ \frac {2 a \left (7 a^2 A-21 a b B-18 A b^2\right ) \cot ^{\frac {3}{2}}(c+d x)}{21 d}-\frac {2 a^2 (7 a B+11 A b) \cot ^{\frac {5}{2}}(c+d x)}{35 d}+\frac {2 \left (a^3 B+3 a^2 A b-3 a b^2 B-A b^3\right ) \sqrt {\cot (c+d x)}}{d}-\frac {\left (a^3 (A-B)-3 a^2 b (A+B)-3 a b^2 (A-B)+b^3 (A+B)\right ) \log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2} d}+\frac {\left (a^3 (A-B)-3 a^2 b (A+B)-3 a b^2 (A-B)+b^3 (A+B)\right ) \log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2} d}+\frac {\left (a^3 (A+B)+3 a^2 b (A-B)-3 a b^2 (A+B)-b^3 (A-B)\right ) \tan ^{-1}\left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} d}-\frac {\left (a^3 (A+B)+3 a^2 b (A-B)-3 a b^2 (A+B)-b^3 (A-B)\right ) \tan ^{-1}\left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2} d}-\frac {2 a A \cot ^{\frac {3}{2}}(c+d x) (a \cot (c+d x)+b)^2}{7 d} \]
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Rubi [A] time = 0.86, antiderivative size = 421, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 12, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {3581, 3607, 3637, 3630, 3528, 3534, 1168, 1162, 617, 204, 1165, 628} \[ \frac {2 a \left (7 a^2 A-21 a b B-18 A b^2\right ) \cot ^{\frac {3}{2}}(c+d x)}{21 d}+\frac {2 \left (3 a^2 A b+a^3 B-3 a b^2 B-A b^3\right ) \sqrt {\cot (c+d x)}}{d}-\frac {\left (-3 a^2 b (A+B)+a^3 (A-B)-3 a b^2 (A-B)+b^3 (A+B)\right ) \log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2} d}+\frac {\left (-3 a^2 b (A+B)+a^3 (A-B)-3 a b^2 (A-B)+b^3 (A+B)\right ) \log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2} d}+\frac {\left (3 a^2 b (A-B)+a^3 (A+B)-3 a b^2 (A+B)-b^3 (A-B)\right ) \tan ^{-1}\left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} d}-\frac {\left (3 a^2 b (A-B)+a^3 (A+B)-3 a b^2 (A+B)-b^3 (A-B)\right ) \tan ^{-1}\left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2} d}-\frac {2 a^2 (7 a B+11 A b) \cot ^{\frac {5}{2}}(c+d x)}{35 d}-\frac {2 a A \cot ^{\frac {3}{2}}(c+d x) (a \cot (c+d x)+b)^2}{7 d} \]
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 3581
Rule 3607
Rule 3630
Rule 3637
Rubi steps
\begin {align*} \int \cot ^{\frac {9}{2}}(c+d x) (a+b \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx &=\int \sqrt {\cot (c+d x)} (b+a \cot (c+d x))^3 (B+A \cot (c+d x)) \, dx\\ &=-\frac {2 a A \cot ^{\frac {3}{2}}(c+d x) (b+a \cot (c+d x))^2}{7 d}-\frac {2}{7} \int \sqrt {\cot (c+d x)} (b+a \cot (c+d x)) \left (\frac {1}{2} b (3 a A-7 b B)+\frac {7}{2} \left (a^2 A-A b^2-2 a b B\right ) \cot (c+d x)-\frac {1}{2} a (11 A b+7 a B) \cot ^2(c+d x)\right ) \, dx\\ &=-\frac {2 a^2 (11 A b+7 a B) \cot ^{\frac {5}{2}}(c+d x)}{35 d}-\frac {2 a A \cot ^{\frac {3}{2}}(c+d x) (b+a \cot (c+d x))^2}{7 d}-\frac {4}{35} \int \sqrt {\cot (c+d x)} \left (\frac {5}{4} b^2 (3 a A-7 b B)+\frac {35}{4} \left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right ) \cot (c+d x)+\frac {5}{4} a \left (7 a^2 A-18 A b^2-21 a b B\right ) \cot ^2(c+d x)\right ) \, dx\\ &=\frac {2 a \left (7 a^2 A-18 A b^2-21 a b B\right ) \cot ^{\frac {3}{2}}(c+d x)}{21 d}-\frac {2 a^2 (11 A b+7 a B) \cot ^{\frac {5}{2}}(c+d x)}{35 d}-\frac {2 a A \cot ^{\frac {3}{2}}(c+d x) (b+a \cot (c+d x))^2}{7 d}-\frac {4}{35} \int \sqrt {\cot (c+d x)} \left (-\frac {35}{4} \left (a^3 A-3 a A b^2-3 a^2 b B+b^3 B\right )+\frac {35}{4} \left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right ) \cot (c+d x)\right ) \, dx\\ &=\frac {2 \left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right ) \sqrt {\cot (c+d x)}}{d}+\frac {2 a \left (7 a^2 A-18 A b^2-21 a b B\right ) \cot ^{\frac {3}{2}}(c+d x)}{21 d}-\frac {2 a^2 (11 A b+7 a B) \cot ^{\frac {5}{2}}(c+d x)}{35 d}-\frac {2 a A \cot ^{\frac {3}{2}}(c+d x) (b+a \cot (c+d x))^2}{7 d}-\frac {4}{35} \int \frac {-\frac {35}{4} \left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right )-\frac {35}{4} \left (a^3 A-3 a A b^2-3 a^2 b B+b^3 B\right ) \cot (c+d x)}{\sqrt {\cot (c+d x)}} \, dx\\ &=\frac {2 \left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right ) \sqrt {\cot (c+d x)}}{d}+\frac {2 a \left (7 a^2 A-18 A b^2-21 a b B\right ) \cot ^{\frac {3}{2}}(c+d x)}{21 d}-\frac {2 a^2 (11 A b+7 a B) \cot ^{\frac {5}{2}}(c+d x)}{35 d}-\frac {2 a A \cot ^{\frac {3}{2}}(c+d x) (b+a \cot (c+d x))^2}{7 d}-\frac {8 \operatorname {Subst}\left (\int \frac {\frac {35}{4} \left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right )+\frac {35}{4} \left (a^3 A-3 a A b^2-3 a^2 b B+b^3 B\right ) x^2}{1+x^4} \, dx,x,\sqrt {\cot (c+d x)}\right )}{35 d}\\ &=\frac {2 \left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right ) \sqrt {\cot (c+d x)}}{d}+\frac {2 a \left (7 a^2 A-18 A b^2-21 a b B\right ) \cot ^{\frac {3}{2}}(c+d x)}{21 d}-\frac {2 a^2 (11 A b+7 a B) \cot ^{\frac {5}{2}}(c+d x)}{35 d}-\frac {2 a A \cot ^{\frac {3}{2}}(c+d x) (b+a \cot (c+d x))^2}{7 d}-\frac {\left (3 a^2 b (A-B)-b^3 (A-B)+a^3 (A+B)-3 a b^2 (A+B)\right ) \operatorname {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\sqrt {\cot (c+d x)}\right )}{d}+\frac {\left (a^3 (A-B)-3 a b^2 (A-B)-3 a^2 b (A+B)+b^3 (A+B)\right ) \operatorname {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\sqrt {\cot (c+d x)}\right )}{d}\\ &=\frac {2 \left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right ) \sqrt {\cot (c+d x)}}{d}+\frac {2 a \left (7 a^2 A-18 A b^2-21 a b B\right ) \cot ^{\frac {3}{2}}(c+d x)}{21 d}-\frac {2 a^2 (11 A b+7 a B) \cot ^{\frac {5}{2}}(c+d x)}{35 d}-\frac {2 a A \cot ^{\frac {3}{2}}(c+d x) (b+a \cot (c+d x))^2}{7 d}-\frac {\left (3 a^2 b (A-B)-b^3 (A-B)+a^3 (A+B)-3 a b^2 (A+B)\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 (3 a^2 b (A-B)-b^3 (A-B)+a^3 (A+B)-3 a b^2 (A+B)\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^3 (A-B)-3 a b^2 (A-B)-3 a^2 b (A+B)+b^3 (A+B)\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^3 (A-B)-3 a b^2 (A-B)-3 a^2 b (A+B)+b^3 (A+B)\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 \left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right ) \sqrt {\cot (c+d x)}}{d}+\frac {2 a \left (7 a^2 A-18 A b^2-21 a b B\right ) \cot ^{\frac {3}{2}}(c+d x)}{21 d}-\frac {2 a^2 (11 A b+7 a B) \cot ^{\frac {5}{2}}(c+d x)}{35 d}-\frac {2 a A \cot ^{\frac {3}{2}}(c+d x) (b+a \cot (c+d x))^2}{7 d}-\frac {\left (a^3 (A-B)-3 a b^2 (A-B)-3 a^2 b (A+B)+b^3 (A+B)\right ) \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt {2} d}+\frac {\left (a^3 (A-B)-3 a b^2 (A-B)-3 a^2 b (A+B)+b^3 (A+B)\right ) \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt {2} d}-\frac {\left (3 a^2 b (A-B)-b^3 (A-B)+a^3 (A+B)-3 a b^2 (A+B)\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 (3 a^2 b (A-B)-b^3 (A-B)+a^3 (A+B)-3 a b^2 (A+B)\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 (3 a^2 b (A-B)-b^3 (A-B)+a^3 (A+B)-3 a b^2 (A+B)\right ) \tan ^{-1}\left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} d}-\frac {\left (3 a^2 b (A-B)-b^3 (A-B)+a^3 (A+B)-3 a b^2 (A+B)\right ) \tan ^{-1}\left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} d}+\frac {2 \left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right ) \sqrt {\cot (c+d x)}}{d}+\frac {2 a \left (7 a^2 A-18 A b^2-21 a b B\right ) \cot ^{\frac {3}{2}}(c+d x)}{21 d}-\frac {2 a^2 (11 A b+7 a B) \cot ^{\frac {5}{2}}(c+d x)}{35 d}-\frac {2 a A \cot ^{\frac {3}{2}}(c+d x) (b+a \cot (c+d x))^2}{7 d}-\frac {\left (a^3 (A-B)-3 a b^2 (A-B)-3 a^2 b (A+B)+b^3 (A+B)\right ) \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt {2} d}+\frac {\left (a^3 (A-B)-3 a b^2 (A-B)-3 a^2 b (A+B)+b^3 (A+B)\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 [A] time = 3.57, size = 326, normalized size = 0.77 \[ \frac {2 \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (-\frac {a^3 A}{7 \tan ^{\frac {7}{2}}(c+d x)}+\frac {a \left (a^2 A-3 a b B-3 A b^2\right )}{3 \tan ^{\frac {3}{2}}(c+d x)}-\frac {a^2 (a B+3 A b)}{5 \tan ^{\frac {5}{2}}(c+d x)}-\frac {\left (a^3 (A+B)+3 a^2 b (A-B)-3 a b^2 (A+B)+b^3 (B-A)\right ) \left (\tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )-\tan ^{-1}\left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )\right )}{2 \sqrt {2}}+\frac {a^3 B+3 a^2 A b-3 a b^2 B-A b^3}{\sqrt {\tan (c+d x)}}-\frac {\left (a^3 (A-B)-3 a^2 b (A+B)+3 a b^2 (B-A)+b^3 (A+B)\right ) \left (\log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )-\log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )\right )}{4 \sqrt {2}}\right )}{d} \]
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 \[ \int {\left (B \tan \left (d x + c\right ) + A\right )} {\left (b \tan \left (d x + c\right ) + a\right )}^{3} \cot \left (d x + c\right )^{\frac {9}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 2.68, size = 18631, normalized size = 44.25 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.90, size = 366, normalized size = 0.87 \[ -\frac {210 \, \sqrt {2} {\left ({\left (A + B\right )} a^{3} + 3 \, {\left (A - B\right )} a^{2} b - 3 \, {\left (A + B\right )} a b^{2} - {\left (A - B\right )} b^{3}\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) + 210 \, \sqrt {2} {\left ({\left (A + B\right )} a^{3} + 3 \, {\left (A - B\right )} a^{2} b - 3 \, {\left (A + B\right )} a b^{2} - {\left (A - B\right )} b^{3}\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) - 105 \, \sqrt {2} {\left ({\left (A - B\right )} a^{3} - 3 \, {\left (A + B\right )} a^{2} b - 3 \, {\left (A - B\right )} a b^{2} + {\left (A + B\right )} b^{3}\right )} \log \left (\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right ) + 105 \, \sqrt {2} {\left ({\left (A - B\right )} a^{3} - 3 \, {\left (A + B\right )} a^{2} b - 3 \, {\left (A - B\right )} a b^{2} + {\left (A + B\right )} b^{3}\right )} \log \left (-\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right ) + \frac {120 \, A a^{3}}{\tan \left (d x + c\right )^{\frac {7}{2}}} - \frac {840 \, {\left (B a^{3} + 3 \, A a^{2} b - 3 \, B a b^{2} - A b^{3}\right )}}{\sqrt {\tan \left (d x + c\right )}} - \frac {280 \, {\left (A a^{3} - 3 \, B a^{2} b - 3 \, A a b^{2}\right )}}{\tan \left (d x + c\right )^{\frac {3}{2}}} + \frac {168 \, {\left (B a^{3} + 3 \, A a^{2} b\right )}}{\tan \left (d x + c\right )^{\frac {5}{2}}}}{420 \, d} \]
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 \[ \int {\mathrm {cot}\left (c+d\,x\right )}^{9/2}\,\left (A+B\,\mathrm {tan}\left (c+d\,x\right )\right )\,{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^3 \,d x \]
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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