Optimal. Leaf size=374 \[ -\frac {2 a \left (a^2 B+3 a A b+2 b^2 B\right ) \sqrt {\cot (c+d x)}}{d}-\frac {2 a^2 (a A+3 b B) \cot ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {\left (a^3 (A-B)-3 a^2 b (A+B)-3 a b^2 (A-B)+b^3 (A+B)\right ) \log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2} d}-\frac {\left (a^3 (A-B)-3 a^2 b (A+B)-3 a b^2 (A-B)+b^3 (A+B)\right ) \log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2} d}-\frac {\left (a^3 (A+B)+3 a^2 b (A-B)-3 a b^2 (A+B)-b^3 (A-B)\right ) \tan ^{-1}\left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} d}+\frac {\left (a^3 (A+B)+3 a^2 b (A-B)-3 a b^2 (A+B)-b^3 (A-B)\right ) \tan ^{-1}\left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2} d}+\frac {2 b B (a \cot (c+d x)+b)^2}{d \sqrt {\cot (c+d x)}} \]
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Rubi [A] time = 0.77, antiderivative size = 374, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 11, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3581, 3605, 3637, 3630, 3534, 1168, 1162, 617, 204, 1165, 628} \[ -\frac {2 a \left (a^2 B+3 a A b+2 b^2 B\right ) \sqrt {\cot (c+d x)}}{d}+\frac {\left (-3 a^2 b (A+B)+a^3 (A-B)-3 a b^2 (A-B)+b^3 (A+B)\right ) \log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2} d}-\frac {\left (-3 a^2 b (A+B)+a^3 (A-B)-3 a b^2 (A-B)+b^3 (A+B)\right ) \log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2} d}-\frac {\left (3 a^2 b (A-B)+a^3 (A+B)-3 a b^2 (A+B)-b^3 (A-B)\right ) \tan ^{-1}\left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} d}+\frac {\left (3 a^2 b (A-B)+a^3 (A+B)-3 a b^2 (A+B)-b^3 (A-B)\right ) \tan ^{-1}\left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2} d}-\frac {2 a^2 (a A+3 b B) \cot ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {2 b B (a \cot (c+d x)+b)^2}{d \sqrt {\cot (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 204
Rule 617
Rule 628
Rule 1162
Rule 1165
Rule 1168
Rule 3534
Rule 3581
Rule 3605
Rule 3630
Rule 3637
Rubi steps
\begin {align*} \int \cot ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx &=\int \frac {(b+a \cot (c+d x))^3 (B+A \cot (c+d x))}{\cot ^{\frac {3}{2}}(c+d x)} \, dx\\ &=\frac {2 b B (b+a \cot (c+d x))^2}{d \sqrt {\cot (c+d x)}}-2 \int \frac {(b+a \cot (c+d x)) \left (-\frac {1}{2} b (A b+5 a B)-\frac {1}{2} \left (2 a A b+a^2 B-b^2 B\right ) \cot (c+d x)-\frac {1}{2} a (a A+3 b B) \cot ^2(c+d x)\right )}{\sqrt {\cot (c+d x)}} \, dx\\ &=-\frac {2 a^2 (a A+3 b B) \cot ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {2 b B (b+a \cot (c+d x))^2}{d \sqrt {\cot (c+d x)}}-\frac {4}{3} \int \frac {-\frac {3}{4} b^2 (A b+5 a B)+\frac {3}{4} \left (a^3 A-3 a A b^2-3 a^2 b B+b^3 B\right ) \cot (c+d x)-\frac {3}{4} a \left (3 a A b+a^2 B+2 b^2 B\right ) \cot ^2(c+d x)}{\sqrt {\cot (c+d x)}} \, dx\\ &=-\frac {2 a \left (3 a A b+a^2 B+2 b^2 B\right ) \sqrt {\cot (c+d x)}}{d}-\frac {2 a^2 (a A+3 b B) \cot ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {2 b B (b+a \cot (c+d x))^2}{d \sqrt {\cot (c+d x)}}-\frac {4}{3} \int \frac {\frac {3}{4} \left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right )+\frac {3}{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 a \left (3 a A b+a^2 B+2 b^2 B\right ) \sqrt {\cot (c+d x)}}{d}-\frac {2 a^2 (a A+3 b B) \cot ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {2 b B (b+a \cot (c+d x))^2}{d \sqrt {\cot (c+d x)}}-\frac {8 \operatorname {Subst}\left (\int \frac {-\frac {3}{4} \left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right )-\frac {3}{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 )}{3 d}\\ &=-\frac {2 a \left (3 a A b+a^2 B+2 b^2 B\right ) \sqrt {\cot (c+d x)}}{d}-\frac {2 a^2 (a A+3 b B) \cot ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {2 b B (b+a \cot (c+d x))^2}{d \sqrt {\cot (c+d x)}}+\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 a \left (3 a A b+a^2 B+2 b^2 B\right ) \sqrt {\cot (c+d x)}}{d}-\frac {2 a^2 (a A+3 b B) \cot ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {2 b B (b+a \cot (c+d x))^2}{d \sqrt {\cot (c+d x)}}+\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 a \left (3 a A b+a^2 B+2 b^2 B\right ) \sqrt {\cot (c+d x)}}{d}-\frac {2 a^2 (a A+3 b B) \cot ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {2 b B (b+a \cot (c+d x))^2}{d \sqrt {\cot (c+d x)}}+\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 a \left (3 a A b+a^2 B+2 b^2 B\right ) \sqrt {\cot (c+d x)}}{d}-\frac {2 a^2 (a A+3 b B) \cot ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {2 b B (b+a \cot (c+d x))^2}{d \sqrt {\cot (c+d x)}}+\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 = 2.12, size = 270, normalized size = 0.72 \[ \frac {2 \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (-\frac {a^3 A}{3 \tan ^{\frac {3}{2}}(c+d x)}-\frac {a^2 (a B+3 A b)}{\sqrt {\tan (c+d x)}}+\frac {\left (a^3 (A+B)+3 a^2 b (A-B)-3 a b^2 (A+B)+b^3 (B-A)\right ) \left (\tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )-\tan ^{-1}\left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )\right )}{2 \sqrt {2}}+\frac {\left (a^3 (A-B)-3 a^2 b (A+B)+3 a b^2 (B-A)+b^3 (A+B)\right ) \left (\log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )-\log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )\right )}{4 \sqrt {2}}+b^3 B \sqrt {\tan (c+d x)}\right )}{d} \]
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 \[ \int {\left (B \tan \left (d x + c\right ) + A\right )} {\left (b \tan \left (d x + c\right ) + a\right )}^{3} \cot \left (d x + c\right )^{\frac {5}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 2.24, size = 9099, normalized size = 24.33 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.59, size = 314, normalized size = 0.84 \[ \frac {24 \, B b^{3} \sqrt {\tan \left (d x + c\right )} + 6 \, \sqrt {2} {\left ({\left (A + B\right )} a^{3} + 3 \, {\left (A - B\right )} a^{2} b - 3 \, {\left (A + B\right )} a b^{2} - {\left (A - B\right )} b^{3}\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) + 6 \, \sqrt {2} {\left ({\left (A + B\right )} a^{3} + 3 \, {\left (A - B\right )} a^{2} b - 3 \, {\left (A + B\right )} a b^{2} - {\left (A - B\right )} b^{3}\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) - 3 \, \sqrt {2} {\left ({\left (A - B\right )} a^{3} - 3 \, {\left (A + B\right )} a^{2} b - 3 \, {\left (A - B\right )} a b^{2} + {\left (A + B\right )} b^{3}\right )} \log \left (\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right ) + 3 \, \sqrt {2} {\left ({\left (A - B\right )} a^{3} - 3 \, {\left (A + B\right )} a^{2} b - 3 \, {\left (A - B\right )} a b^{2} + {\left (A + B\right )} b^{3}\right )} \log \left (-\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right ) - \frac {8 \, A a^{3}}{\tan \left (d x + c\right )^{\frac {3}{2}}} - \frac {24 \, {\left (B a^{3} + 3 \, A a^{2} b\right )}}{\sqrt {\tan \left (d x + c\right )}}}{12 \, d} \]
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 \[ \int {\mathrm {cot}\left (c+d\,x\right )}^{5/2}\,\left (A+B\,\mathrm {tan}\left (c+d\,x\right )\right )\,{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^3 \,d x \]
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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