Optimal. Leaf size=326 \[ \frac{2 \left (a^2 A-2 a b B-A b^2\right ) \sqrt{\cot (c+d x)}}{d}+\frac{\left (a^2 (A+B)+2 a b (A-B)-b^2 (A+B)\right ) \log \left (\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right )}{2 \sqrt{2} d}-\frac{\left (a^2 (A+B)+2 a b (A-B)-b^2 (A+B)\right ) \log \left (\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right )}{2 \sqrt{2} d}+\frac{\left (a^2 (A-B)-2 a b (A+B)-b^2 (A-B)\right ) \tan ^{-1}\left (1-\sqrt{2} \sqrt{\cot (c+d x)}\right )}{\sqrt{2} d}-\frac{\left (a^2 (A-B)-2 a b (A+B)-b^2 (A-B)\right ) \tan ^{-1}\left (\sqrt{2} \sqrt{\cot (c+d x)}+1\right )}{\sqrt{2} d}-\frac{2 a (5 a B+7 A b) \cot ^{\frac{3}{2}}(c+d x)}{15 d}-\frac{2 a A \cot ^{\frac{3}{2}}(c+d x) (a \cot (c+d x)+b)}{5 d} \]
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Rubi [A] time = 0.613995, antiderivative size = 326, 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, 3607, 3630, 3528, 3534, 1168, 1162, 617, 204, 1165, 628} \[ \frac{2 \left (a^2 A-2 a b B-A b^2\right ) \sqrt{\cot (c+d x)}}{d}+\frac{\left (a^2 (A+B)+2 a b (A-B)-b^2 (A+B)\right ) \log \left (\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right )}{2 \sqrt{2} d}-\frac{\left (a^2 (A+B)+2 a b (A-B)-b^2 (A+B)\right ) \log \left (\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right )}{2 \sqrt{2} d}+\frac{\left (a^2 (A-B)-2 a b (A+B)-b^2 (A-B)\right ) \tan ^{-1}\left (1-\sqrt{2} \sqrt{\cot (c+d x)}\right )}{\sqrt{2} d}-\frac{\left (a^2 (A-B)-2 a b (A+B)-b^2 (A-B)\right ) \tan ^{-1}\left (\sqrt{2} \sqrt{\cot (c+d x)}+1\right )}{\sqrt{2} d}-\frac{2 a (5 a B+7 A b) \cot ^{\frac{3}{2}}(c+d x)}{15 d}-\frac{2 a A \cot ^{\frac{3}{2}}(c+d x) (a \cot (c+d x)+b)}{5 d} \]
Antiderivative was successfully verified.
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Rule 3581
Rule 3607
Rule 3630
Rule 3528
Rule 3534
Rule 1168
Rule 1162
Rule 617
Rule 204
Rule 1165
Rule 628
Rubi steps
\begin{align*} \int \cot ^{\frac{7}{2}}(c+d x) (a+b \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx &=\int \sqrt{\cot (c+d x)} (b+a \cot (c+d x))^2 (B+A \cot (c+d x)) \, dx\\ &=-\frac{2 a A \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 (3 a A-5 b B)+\frac{5}{2} \left (a^2 A-A b^2-2 a b B\right ) \cot (c+d x)-\frac{1}{2} a (7 A b+5 a B) \cot ^2(c+d x)\right ) \, dx\\ &=-\frac{2 a (7 A b+5 a B) \cot ^{\frac{3}{2}}(c+d x)}{15 d}-\frac{2 a A \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} \left (2 a A b+a^2 B-b^2 B\right )+\frac{5}{2} \left (a^2 A-A b^2-2 a b B\right ) \cot (c+d x)\right ) \, dx\\ &=\frac{2 \left (a^2 A-A b^2-2 a b B\right ) \sqrt{\cot (c+d x)}}{d}-\frac{2 a (7 A b+5 a B) \cot ^{\frac{3}{2}}(c+d x)}{15 d}-\frac{2 a A \cot ^{\frac{3}{2}}(c+d x) (b+a \cot (c+d x))}{5 d}-\frac{2}{5} \int \frac{-\frac{5}{2} \left (a^2 A-A b^2-2 a b B\right )+\frac{5}{2} \left (2 a A b+a^2 B-b^2 B\right ) \cot (c+d x)}{\sqrt{\cot (c+d x)}} \, dx\\ &=\frac{2 \left (a^2 A-A b^2-2 a b B\right ) \sqrt{\cot (c+d x)}}{d}-\frac{2 a (7 A b+5 a B) \cot ^{\frac{3}{2}}(c+d x)}{15 d}-\frac{2 a A \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} \left (a^2 A-A b^2-2 a b B\right )-\frac{5}{2} \left (2 a A b+a^2 B-b^2 B\right ) x^2}{1+x^4} \, dx,x,\sqrt{\cot (c+d x)}\right )}{5 d}\\ &=\frac{2 \left (a^2 A-A b^2-2 a b B\right ) \sqrt{\cot (c+d x)}}{d}-\frac{2 a (7 A b+5 a B) \cot ^{\frac{3}{2}}(c+d x)}{15 d}-\frac{2 a A \cot ^{\frac{3}{2}}(c+d x) (b+a \cot (c+d x))}{5 d}-\frac{\left (a^2 (A-B)-b^2 (A-B)-2 a b (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 (2 a b (A-B)+a^2 (A+B)-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{2 \left (a^2 A-A b^2-2 a b B\right ) \sqrt{\cot (c+d x)}}{d}-\frac{2 a (7 A b+5 a B) \cot ^{\frac{3}{2}}(c+d x)}{15 d}-\frac{2 a A \cot ^{\frac{3}{2}}(c+d x) (b+a \cot (c+d x))}{5 d}-\frac{\left (a^2 (A-B)-b^2 (A-B)-2 a b (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^2 (A-B)-b^2 (A-B)-2 a b (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 (2 a b (A-B)+a^2 (A+B)-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 (2 a b (A-B)+a^2 (A+B)-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{2 \left (a^2 A-A b^2-2 a b B\right ) \sqrt{\cot (c+d x)}}{d}-\frac{2 a (7 A b+5 a B) \cot ^{\frac{3}{2}}(c+d x)}{15 d}-\frac{2 a A \cot ^{\frac{3}{2}}(c+d x) (b+a \cot (c+d x))}{5 d}+\frac{\left (2 a b (A-B)+a^2 (A+B)-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 (2 a b (A-B)+a^2 (A+B)-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^2 (A-B)-b^2 (A-B)-2 a b (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^2 (A-B)-b^2 (A-B)-2 a b (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^2 (A-B)-b^2 (A-B)-2 a b (A+B)\right ) \tan ^{-1}\left (1-\sqrt{2} \sqrt{\cot (c+d x)}\right )}{\sqrt{2} d}-\frac{\left (a^2 (A-B)-b^2 (A-B)-2 a b (A+B)\right ) \tan ^{-1}\left (1+\sqrt{2} \sqrt{\cot (c+d x)}\right )}{\sqrt{2} d}+\frac{2 \left (a^2 A-A b^2-2 a b B\right ) \sqrt{\cot (c+d x)}}{d}-\frac{2 a (7 A b+5 a B) \cot ^{\frac{3}{2}}(c+d x)}{15 d}-\frac{2 a A \cot ^{\frac{3}{2}}(c+d x) (b+a \cot (c+d x))}{5 d}+\frac{\left (2 a b (A-B)+a^2 (A+B)-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 (2 a b (A-B)+a^2 (A+B)-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 = 1.85171, size = 255, normalized size = 0.78 \[ -\frac{\sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \left (30 \sqrt{2} \left (a^2 (A-B)-2 a b (A+B)+b^2 (B-A)\right ) \left (\tan ^{-1}\left (1-\sqrt{2} \sqrt{\tan (c+d x)}\right )-\tan ^{-1}\left (\sqrt{2} \sqrt{\tan (c+d x)}+1\right )\right )-\frac{120 \left (a^2 A-2 a b B-A b^2\right )}{\sqrt{\tan (c+d x)}}-15 \sqrt{2} \left (a^2 (A+B)+2 a b (A-B)-b^2 (A+B)\right ) \left (\log \left (\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right )-\log \left (\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right )\right )+\frac{24 a^2 A}{\tan ^{\frac{5}{2}}(c+d x)}+\frac{40 a (a B+2 A b)}{\tan ^{\frac{3}{2}}(c+d x)}\right )}{60 d} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.641, size = 13170, normalized size = 40.4 \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 = 1.71818, size = 377, normalized size = 1.16 \begin{align*} -\frac{30 \, \sqrt{2}{\left ({\left (A - B\right )} a^{2} - 2 \,{\left (A + B\right )} a b -{\left (A - B\right )} b^{2}\right )} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} + \frac{2}{\sqrt{\tan \left (d x + c\right )}}\right )}\right ) + 30 \, \sqrt{2}{\left ({\left (A - B\right )} a^{2} - 2 \,{\left (A + B\right )} a b -{\left (A - B\right )} b^{2}\right )} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} - \frac{2}{\sqrt{\tan \left (d x + c\right )}}\right )}\right ) + 15 \, \sqrt{2}{\left ({\left (A + B\right )} a^{2} + 2 \,{\left (A - B\right )} a b -{\left (A + B\right )} b^{2}\right )} \log \left (\frac{\sqrt{2}}{\sqrt{\tan \left (d x + c\right )}} + \frac{1}{\tan \left (d x + c\right )} + 1\right ) - 15 \, \sqrt{2}{\left ({\left (A + B\right )} a^{2} + 2 \,{\left (A - B\right )} a b -{\left (A + B\right )} b^{2}\right )} \log \left (-\frac{\sqrt{2}}{\sqrt{\tan \left (d x + c\right )}} + \frac{1}{\tan \left (d x + c\right )} + 1\right ) + \frac{24 \, A a^{2}}{\tan \left (d x + c\right )^{\frac{5}{2}}} - \frac{120 \,{\left (A a^{2} - 2 \, B a b - A b^{2}\right )}}{\sqrt{\tan \left (d x + c\right )}} + \frac{40 \,{\left (B a^{2} + 2 \, A a b\right )}}{\tan \left (d x + c\right )^{\frac{3}{2}}}}{60 \, 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 )}^{2} \cot \left (d x + c\right )^{\frac{7}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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