Optimal. Leaf size=392 \[ \frac{a (A b-a B) \sqrt{\cot (c+d x)}}{b d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}+\frac{\left (a^2 (A-B)+2 a b (A+B)-b^2 (A-B)\right ) \log \left (\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right )}{2 \sqrt{2} d \left (a^2+b^2\right )^2}-\frac{\left (a^2 (A-B)+2 a b (A+B)-b^2 (A-B)\right ) \log \left (\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right )}{2 \sqrt{2} d \left (a^2+b^2\right )^2}+\frac{\left (a^2 (-(A+B))+2 a b (A-B)+b^2 (A+B)\right ) \tan ^{-1}\left (1-\sqrt{2} \sqrt{\cot (c+d x)}\right )}{\sqrt{2} d \left (a^2+b^2\right )^2}-\frac{\left (a^2 (-(A+B))+2 a b (A-B)+b^2 (A+B)\right ) \tan ^{-1}\left (\sqrt{2} \sqrt{\cot (c+d x)}+1\right )}{\sqrt{2} d \left (a^2+b^2\right )^2}-\frac{\sqrt{a} \left (a^2 A b+a^3 B+5 a b^2 B-3 A b^3\right ) \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{\cot (c+d x)}}{\sqrt{b}}\right )}{b^{3/2} d \left (a^2+b^2\right )^2} \]
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Rubi [A] time = 0.916095, antiderivative size = 392, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 13, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.394, Rules used = {3581, 3609, 3653, 3534, 1168, 1162, 617, 204, 1165, 628, 3634, 63, 205} \[ \frac{a (A b-a B) \sqrt{\cot (c+d x)}}{b d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}+\frac{\left (a^2 (A-B)+2 a b (A+B)-b^2 (A-B)\right ) \log \left (\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right )}{2 \sqrt{2} d \left (a^2+b^2\right )^2}-\frac{\left (a^2 (A-B)+2 a b (A+B)-b^2 (A-B)\right ) \log \left (\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right )}{2 \sqrt{2} d \left (a^2+b^2\right )^2}+\frac{\left (a^2 (-(A+B))+2 a b (A-B)+b^2 (A+B)\right ) \tan ^{-1}\left (1-\sqrt{2} \sqrt{\cot (c+d x)}\right )}{\sqrt{2} d \left (a^2+b^2\right )^2}-\frac{\left (a^2 (-(A+B))+2 a b (A-B)+b^2 (A+B)\right ) \tan ^{-1}\left (\sqrt{2} \sqrt{\cot (c+d x)}+1\right )}{\sqrt{2} d \left (a^2+b^2\right )^2}-\frac{\sqrt{a} \left (a^2 A b+a^3 B+5 a b^2 B-3 A b^3\right ) \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{\cot (c+d x)}}{\sqrt{b}}\right )}{b^{3/2} d \left (a^2+b^2\right )^2} \]
Antiderivative was successfully verified.
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Rule 3581
Rule 3609
Rule 3653
Rule 3534
Rule 1168
Rule 1162
Rule 617
Rule 204
Rule 1165
Rule 628
Rule 3634
Rule 63
Rule 205
Rubi steps
\begin{align*} \int \frac{A+B \tan (c+d x)}{\cot ^{\frac{3}{2}}(c+d x) (a+b \tan (c+d x))^2} \, dx &=\int \frac{B+A \cot (c+d x)}{\sqrt{\cot (c+d x)} (b+a \cot (c+d x))^2} \, dx\\ &=\frac{a (A b-a B) \sqrt{\cot (c+d x)}}{b \left (a^2+b^2\right ) d (b+a \cot (c+d x))}-\frac{\int \frac{\frac{1}{2} \left (-a A b-a^2 B-2 b^2 B\right )-b (A b-a B) \cot (c+d x)+\frac{1}{2} a (A b-a B) \cot ^2(c+d x)}{\sqrt{\cot (c+d x)} (b+a \cot (c+d x))} \, dx}{b \left (a^2+b^2\right )}\\ &=\frac{a (A b-a B) \sqrt{\cot (c+d x)}}{b \left (a^2+b^2\right ) d (b+a \cot (c+d x))}-\frac{\int \frac{-b \left (2 a A b-a^2 B+b^2 B\right )+b \left (a^2 A-A b^2+2 a b B\right ) \cot (c+d x)}{\sqrt{\cot (c+d x)}} \, dx}{b \left (a^2+b^2\right )^2}+\frac{\left (a \left (a^2 A b-3 A b^3+a^3 B+5 a b^2 B\right )\right ) \int \frac{1+\cot ^2(c+d x)}{\sqrt{\cot (c+d x)} (b+a \cot (c+d x))} \, dx}{2 b \left (a^2+b^2\right )^2}\\ &=\frac{a (A b-a B) \sqrt{\cot (c+d x)}}{b \left (a^2+b^2\right ) d (b+a \cot (c+d x))}-\frac{2 \operatorname{Subst}\left (\int \frac{b \left (2 a A b-a^2 B+b^2 B\right )-b \left (a^2 A-A b^2+2 a b B\right ) x^2}{1+x^4} \, dx,x,\sqrt{\cot (c+d x)}\right )}{b \left (a^2+b^2\right )^2 d}+\frac{\left (a \left (a^2 A b-3 A b^3+a^3 B+5 a b^2 B\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{-x} (b-a x)} \, dx,x,-\cot (c+d x)\right )}{2 b \left (a^2+b^2\right )^2 d}\\ &=\frac{a (A b-a B) \sqrt{\cot (c+d x)}}{b \left (a^2+b^2\right ) d (b+a \cot (c+d x))}-\frac{\left (a \left (a^2 A b-3 A b^3+a^3 B+5 a b^2 B\right )\right ) \operatorname{Subst}\left (\int \frac{1}{b+a x^2} \, dx,x,\sqrt{\cot (c+d x)}\right )}{b \left (a^2+b^2\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-x^2}{1+x^4} \, dx,x,\sqrt{\cot (c+d x)}\right )}{\left (a^2+b^2\right )^2 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 )}{\left (a^2+b^2\right )^2 d}\\ &=-\frac{\sqrt{a} \left (a^2 A b-3 A b^3+a^3 B+5 a b^2 B\right ) \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{\cot (c+d x)}}{\sqrt{b}}\right )}{b^{3/2} \left (a^2+b^2\right )^2 d}+\frac{a (A b-a B) \sqrt{\cot (c+d x)}}{b \left (a^2+b^2\right ) d (b+a \cot (c+d x))}+\frac{\left (a^2 (A-B)-b^2 (A-B)+2 a b (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} \left (a^2+b^2\right )^2 d}+\frac{\left (a^2 (A-B)-b^2 (A-B)+2 a b (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} \left (a^2+b^2\right )^2 d}-\frac{\left (2 a b (A-B)-a^2 (A+B)+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 \left (a^2+b^2\right )^2 d}-\frac{\left (2 a b (A-B)-a^2 (A+B)+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 \left (a^2+b^2\right )^2 d}\\ &=-\frac{\sqrt{a} \left (a^2 A b-3 A b^3+a^3 B+5 a b^2 B\right ) \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{\cot (c+d x)}}{\sqrt{b}}\right )}{b^{3/2} \left (a^2+b^2\right )^2 d}+\frac{a (A b-a B) \sqrt{\cot (c+d x)}}{b \left (a^2+b^2\right ) d (b+a \cot (c+d x))}+\frac{\left (a^2 (A-B)-b^2 (A-B)+2 a b (A+B)\right ) \log \left (1-\sqrt{2} \sqrt{\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt{2} \left (a^2+b^2\right )^2 d}-\frac{\left (a^2 (A-B)-b^2 (A-B)+2 a b (A+B)\right ) \log \left (1+\sqrt{2} \sqrt{\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt{2} \left (a^2+b^2\right )^2 d}-\frac{\left (2 a b (A-B)-a^2 (A+B)+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} \left (a^2+b^2\right )^2 d}+\frac{\left (2 a b (A-B)-a^2 (A+B)+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} \left (a^2+b^2\right )^2 d}\\ &=\frac{\left (2 a b (A-B)-a^2 (A+B)+b^2 (A+B)\right ) \tan ^{-1}\left (1-\sqrt{2} \sqrt{\cot (c+d x)}\right )}{\sqrt{2} \left (a^2+b^2\right )^2 d}-\frac{\left (2 a b (A-B)-a^2 (A+B)+b^2 (A+B)\right ) \tan ^{-1}\left (1+\sqrt{2} \sqrt{\cot (c+d x)}\right )}{\sqrt{2} \left (a^2+b^2\right )^2 d}-\frac{\sqrt{a} \left (a^2 A b-3 A b^3+a^3 B+5 a b^2 B\right ) \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{\cot (c+d x)}}{\sqrt{b}}\right )}{b^{3/2} \left (a^2+b^2\right )^2 d}+\frac{a (A b-a B) \sqrt{\cot (c+d x)}}{b \left (a^2+b^2\right ) d (b+a \cot (c+d x))}+\frac{\left (a^2 (A-B)-b^2 (A-B)+2 a b (A+B)\right ) \log \left (1-\sqrt{2} \sqrt{\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt{2} \left (a^2+b^2\right )^2 d}-\frac{\left (a^2 (A-B)-b^2 (A-B)+2 a b (A+B)\right ) \log \left (1+\sqrt{2} \sqrt{\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt{2} \left (a^2+b^2\right )^2 d}\\ \end{align*}
Mathematica [A] time = 2.552, size = 342, normalized size = 0.87 \[ \frac{\sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \left (\frac{4 \sqrt{a} \left (a^2+b^2\right ) (A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right )}{b^{3/2}}-2 \sqrt{2} \left (a^2 (-(A+B))+2 a b (A-B)+b^2 (A+B)\right ) \left (\tan ^{-1}\left (1-\sqrt{2} \sqrt{\tan (c+d x)}\right )-\tan ^{-1}\left (\sqrt{2} \sqrt{\tan (c+d x)}+1\right )\right )+\frac{8 \sqrt{a} \left (a B \left (a^2+3 b^2\right )-2 A b^3\right ) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right )}{b^{3/2}}+\frac{4 a \left (a^2+b^2\right ) (A b-a B) \sqrt{\tan (c+d x)}}{b (a+b \tan (c+d x))}+\sqrt{2} \left (a^2 (A-B)+2 a b (A+B)+b^2 (B-A)\right ) \left (\log \left (\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right )-\log \left (\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right )\right )\right )}{4 d \left (a^2+b^2\right )^2} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.675, size = 40734, normalized size = 103.9 \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 [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \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 \frac{B \tan \left (d x + c\right ) + A}{{\left (b \tan \left (d x + c\right ) + a\right )}^{2} \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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