Optimal. Leaf size=372 \[ -\frac{\left (3 a^2 b (A-B)+a^3 (A+B)-3 a b^2 (A+B)-b^3 (A-B)\right ) \log \left (\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right )}{2 \sqrt{2} d}+\frac{\left (3 a^2 b (A-B)+a^3 (A+B)-3 a b^2 (A+B)-b^3 (A-B)\right ) \log \left (\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right )}{2 \sqrt{2} d}-\frac{\left (-3 a^2 b (A+B)+a^3 (A-B)-3 a b^2 (A-B)+b^3 (A+B)\right ) \tan ^{-1}\left (1-\sqrt{2} \sqrt{\cot (c+d x)}\right )}{\sqrt{2} d}+\frac{\left (-3 a^2 b (A+B)+a^3 (A-B)-3 a b^2 (A-B)+b^3 (A+B)\right ) \tan ^{-1}\left (\sqrt{2} \sqrt{\cot (c+d x)}+1\right )}{\sqrt{2} d}-\frac{2 a^2 (3 a A+b B) \sqrt{\cot (c+d x)}}{3 d}+\frac{2 b^2 (7 a B+3 A b)}{3 d \sqrt{\cot (c+d x)}}+\frac{2 b B (a \cot (c+d x)+b)^2}{3 d \cot ^{\frac{3}{2}}(c+d x)} \]
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Rubi [A] time = 0.694769, antiderivative size = 372, normalized size of antiderivative = 1., 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, 3635, 3630, 3534, 1168, 1162, 617, 204, 1165, 628} \[ -\frac{\left (3 a^2 b (A-B)+a^3 (A+B)-3 a b^2 (A+B)-b^3 (A-B)\right ) \log \left (\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right )}{2 \sqrt{2} d}+\frac{\left (3 a^2 b (A-B)+a^3 (A+B)-3 a b^2 (A+B)-b^3 (A-B)\right ) \log \left (\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right )}{2 \sqrt{2} d}-\frac{\left (-3 a^2 b (A+B)+a^3 (A-B)-3 a b^2 (A-B)+b^3 (A+B)\right ) \tan ^{-1}\left (1-\sqrt{2} \sqrt{\cot (c+d x)}\right )}{\sqrt{2} d}+\frac{\left (-3 a^2 b (A+B)+a^3 (A-B)-3 a b^2 (A-B)+b^3 (A+B)\right ) \tan ^{-1}\left (\sqrt{2} \sqrt{\cot (c+d x)}+1\right )}{\sqrt{2} d}-\frac{2 a^2 (3 a A+b B) \sqrt{\cot (c+d x)}}{3 d}+\frac{2 b^2 (7 a B+3 A b)}{3 d \sqrt{\cot (c+d x)}}+\frac{2 b B (a \cot (c+d x)+b)^2}{3 d \cot ^{\frac{3}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Rule 3581
Rule 3605
Rule 3635
Rule 3630
Rule 3534
Rule 1168
Rule 1162
Rule 617
Rule 204
Rule 1165
Rule 628
Rubi steps
\begin{align*} \int \cot ^{\frac{3}{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{5}{2}}(c+d x)} \, dx\\ &=\frac{2 b B (b+a \cot (c+d x))^2}{3 d \cot ^{\frac{3}{2}}(c+d x)}-\frac{2}{3} \int \frac{(b+a \cot (c+d x)) \left (-\frac{1}{2} b (3 A b+7 a B)-\frac{3}{2} \left (2 a A b+a^2 B-b^2 B\right ) \cot (c+d x)-\frac{1}{2} a (3 a A+b B) \cot ^2(c+d x)\right )}{\cot ^{\frac{3}{2}}(c+d x)} \, dx\\ &=\frac{2 b^2 (3 A b+7 a B)}{3 d \sqrt{\cot (c+d x)}}+\frac{2 b B (b+a \cot (c+d x))^2}{3 d \cot ^{\frac{3}{2}}(c+d x)}+\frac{2}{3} \int \frac{\frac{1}{2} b \left (9 a A b+10 a^2 B-3 b^2 B\right )+\frac{3}{2} \left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right ) \cot (c+d x)+\frac{1}{2} a^2 (3 a A+b B) \cot ^2(c+d x)}{\sqrt{\cot (c+d x)}} \, dx\\ &=\frac{2 b^2 (3 A b+7 a B)}{3 d \sqrt{\cot (c+d x)}}-\frac{2 a^2 (3 a A+b B) \sqrt{\cot (c+d x)}}{3 d}+\frac{2 b B (b+a \cot (c+d x))^2}{3 d \cot ^{\frac{3}{2}}(c+d x)}+\frac{2}{3} \int \frac{-\frac{3}{2} \left (a^3 A-3 a A b^2-3 a^2 b B+b^3 B\right )+\frac{3}{2} \left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right ) \cot (c+d x)}{\sqrt{\cot (c+d x)}} \, dx\\ &=\frac{2 b^2 (3 A b+7 a B)}{3 d \sqrt{\cot (c+d x)}}-\frac{2 a^2 (3 a A+b B) \sqrt{\cot (c+d x)}}{3 d}+\frac{2 b B (b+a \cot (c+d x))^2}{3 d \cot ^{\frac{3}{2}}(c+d x)}+\frac{4 \operatorname{Subst}\left (\int \frac{\frac{3}{2} \left (a^3 A-3 a A b^2-3 a^2 b B+b^3 B\right )-\frac{3}{2} \left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right ) x^2}{1+x^4} \, dx,x,\sqrt{\cot (c+d x)}\right )}{3 d}\\ &=\frac{2 b^2 (3 A b+7 a B)}{3 d \sqrt{\cot (c+d x)}}-\frac{2 a^2 (3 a A+b B) \sqrt{\cot (c+d x)}}{3 d}+\frac{2 b B (b+a \cot (c+d x))^2}{3 d \cot ^{\frac{3}{2}}(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 b^2 (3 A b+7 a B)}{3 d \sqrt{\cot (c+d x)}}-\frac{2 a^2 (3 a A+b B) \sqrt{\cot (c+d x)}}{3 d}+\frac{2 b B (b+a \cot (c+d x))^2}{3 d \cot ^{\frac{3}{2}}(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{\sqrt{2}+2 x}{-1-\sqrt{2} x-x^2} \, dx,x,\sqrt{\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{\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{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{1}{1+\sqrt{2} x+x^2} \, dx,x,\sqrt{\cot (c+d x)}\right )}{2 d}\\ &=\frac{2 b^2 (3 A b+7 a B)}{3 d \sqrt{\cot (c+d x)}}-\frac{2 a^2 (3 a A+b B) \sqrt{\cot (c+d x)}}{3 d}+\frac{2 b B (b+a \cot (c+d x))^2}{3 d \cot ^{\frac{3}{2}}(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 ) \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 ) \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 ) \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^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}{-1-x^2} \, dx,x,1+\sqrt{2} \sqrt{\cot (c+d x)}\right )}{\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 ) \tan ^{-1}\left (1-\sqrt{2} \sqrt{\cot (c+d x)}\right )}{\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 ) \tan ^{-1}\left (1+\sqrt{2} \sqrt{\cot (c+d x)}\right )}{\sqrt{2} d}+\frac{2 b^2 (3 A b+7 a B)}{3 d \sqrt{\cot (c+d x)}}-\frac{2 a^2 (3 a A+b B) \sqrt{\cot (c+d x)}}{3 d}+\frac{2 b B (b+a \cot (c+d x))^2}{3 d \cot ^{\frac{3}{2}}(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 ) \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 ) \log \left (1+\sqrt{2} \sqrt{\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt{2} d}\\ \end{align*}
Mathematica [A] time = 2.07442, size = 270, normalized size = 0.73 \[ \frac{2 \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \left (\frac{\left (-3 a^2 b (A+B)+a^3 (A-B)+3 a b^2 (B-A)+b^3 (A+B)\right ) \left (\tan ^{-1}\left (1-\sqrt{2} \sqrt{\tan (c+d x)}\right )-\tan ^{-1}\left (\sqrt{2} \sqrt{\tan (c+d x)}+1\right )\right )}{2 \sqrt{2}}-\frac{\left (3 a^2 b (A-B)+a^3 (A+B)-3 a b^2 (A+B)+b^3 (B-A)\right ) \left (\log \left (\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right )-\log \left (\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right )\right )}{4 \sqrt{2}}-\frac{a^3 A}{\sqrt{\tan (c+d x)}}+b^2 (3 a B+A b) \sqrt{\tan (c+d x)}+\frac{1}{3} b^3 B \tan ^{\frac{3}{2}}(c+d x)\right )}{d} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.63, size = 8955, normalized size = 24.1 \begin{align*} \text{output too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.5331, size = 427, normalized size = 1.15 \begin{align*} -\frac{\frac{24 \, A a^{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 ) - 8 \,{\left (B b^{3} + \frac{3 \,{\left (3 \, B a b^{2} + A b^{3}\right )}}{\tan \left (d x + c\right )}\right )} \tan \left (d x + c\right )^{\frac{3}{2}}}{12 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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