Optimal. Leaf size=315 \[ -\frac{b \sqrt{\cot (c+d x)}}{d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}+\frac{\left (a^2-2 a b-b^2\right ) \log \left (\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right )}{2 \sqrt{2} d \left (a^2+b^2\right )^2}-\frac{\left (a^2-2 a b-b^2\right ) \log \left (\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right )}{2 \sqrt{2} d \left (a^2+b^2\right )^2}+\frac{\sqrt{b} \left (3 a^2-b^2\right ) \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{\cot (c+d x)}}{\sqrt{b}}\right )}{\sqrt{a} d \left (a^2+b^2\right )^2}+\frac{\left (a^2+2 a b-b^2\right ) \tan ^{-1}\left (1-\sqrt{2} \sqrt{\cot (c+d x)}\right )}{\sqrt{2} d \left (a^2+b^2\right )^2}-\frac{\left (a^2+2 a b-b^2\right ) \tan ^{-1}\left (\sqrt{2} \sqrt{\cot (c+d x)}+1\right )}{\sqrt{2} d \left (a^2+b^2\right )^2} \]
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Rubi [A] time = 0.508131, antiderivative size = 315, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 13, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.565, Rules used = {3673, 3567, 3653, 3534, 1168, 1162, 617, 204, 1165, 628, 3634, 63, 205} \[ -\frac{b \sqrt{\cot (c+d x)}}{d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}+\frac{\left (a^2-2 a b-b^2\right ) \log \left (\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right )}{2 \sqrt{2} d \left (a^2+b^2\right )^2}-\frac{\left (a^2-2 a b-b^2\right ) \log \left (\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right )}{2 \sqrt{2} d \left (a^2+b^2\right )^2}+\frac{\sqrt{b} \left (3 a^2-b^2\right ) \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{\cot (c+d x)}}{\sqrt{b}}\right )}{\sqrt{a} d \left (a^2+b^2\right )^2}+\frac{\left (a^2+2 a b-b^2\right ) \tan ^{-1}\left (1-\sqrt{2} \sqrt{\cot (c+d x)}\right )}{\sqrt{2} d \left (a^2+b^2\right )^2}-\frac{\left (a^2+2 a b-b^2\right ) \tan ^{-1}\left (\sqrt{2} \sqrt{\cot (c+d x)}+1\right )}{\sqrt{2} d \left (a^2+b^2\right )^2} \]
Antiderivative was successfully verified.
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Rule 3673
Rule 3567
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{1}{\sqrt{\cot (c+d x)} (a+b \tan (c+d x))^2} \, dx &=\int \frac{\cot ^{\frac{3}{2}}(c+d x)}{(b+a \cot (c+d x))^2} \, dx\\ &=-\frac{b \sqrt{\cot (c+d x)}}{\left (a^2+b^2\right ) d (b+a \cot (c+d x))}-\frac{\int \frac{\frac{b}{2}-a \cot (c+d x)-\frac{1}{2} b \cot ^2(c+d x)}{\sqrt{\cot (c+d x)} (b+a \cot (c+d x))} \, dx}{a^2+b^2}\\ &=-\frac{b \sqrt{\cot (c+d x)}}{\left (a^2+b^2\right ) d (b+a \cot (c+d x))}-\frac{\int \frac{-a^2+b^2-2 a b \cot (c+d x)}{\sqrt{\cot (c+d x)}} \, dx}{\left (a^2+b^2\right )^2}-\frac{\left (b \left (3 a^2-b^2\right )\right ) \int \frac{1+\cot ^2(c+d x)}{\sqrt{\cot (c+d x)} (b+a \cot (c+d x))} \, dx}{2 \left (a^2+b^2\right )^2}\\ &=-\frac{b \sqrt{\cot (c+d x)}}{\left (a^2+b^2\right ) d (b+a \cot (c+d x))}-\frac{2 \operatorname{Subst}\left (\int \frac{a^2-b^2+2 a b x^2}{1+x^4} \, dx,x,\sqrt{\cot (c+d x)}\right )}{\left (a^2+b^2\right )^2 d}-\frac{\left (b \left (3 a^2-b^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{-x} (b-a x)} \, dx,x,-\cot (c+d x)\right )}{2 \left (a^2+b^2\right )^2 d}\\ &=-\frac{b \sqrt{\cot (c+d x)}}{\left (a^2+b^2\right ) d (b+a \cot (c+d x))}+\frac{\left (b \left (3 a^2-b^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{b+a x^2} \, dx,x,\sqrt{\cot (c+d x)}\right )}{\left (a^2+b^2\right )^2 d}-\frac{\left (a^2-2 a b-b^2\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 (a^2+2 a b-b^2\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{b} \left (3 a^2-b^2\right ) \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{\cot (c+d x)}}{\sqrt{b}}\right )}{\sqrt{a} \left (a^2+b^2\right )^2 d}-\frac{b \sqrt{\cot (c+d x)}}{\left (a^2+b^2\right ) d (b+a \cot (c+d x))}+\frac{\left (a^2-2 a b-b^2\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^2-2 a b-b^2\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+2 a b-b^2\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 (a^2+2 a b-b^2\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{b} \left (3 a^2-b^2\right ) \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{\cot (c+d x)}}{\sqrt{b}}\right )}{\sqrt{a} \left (a^2+b^2\right )^2 d}-\frac{b \sqrt{\cot (c+d x)}}{\left (a^2+b^2\right ) d (b+a \cot (c+d x))}+\frac{\left (a^2-2 a b-b^2\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^2-2 a b-b^2\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+2 a b-b^2\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 (a^2+2 a b-b^2\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 (a^2+2 a b-b^2\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 (a^2+2 a b-b^2\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{b} \left (3 a^2-b^2\right ) \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{\cot (c+d x)}}{\sqrt{b}}\right )}{\sqrt{a} \left (a^2+b^2\right )^2 d}-\frac{b \sqrt{\cot (c+d x)}}{\left (a^2+b^2\right ) d (b+a \cot (c+d x))}+\frac{\left (a^2-2 a b-b^2\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^2-2 a b-b^2\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 [C] time = 4.56231, size = 301, normalized size = 0.96 \[ -\frac{\frac{24 a^2 \left (a^2+b^2\right ) \cot ^{\frac{5}{2}}(c+d x) \, _2F_1\left (2,\frac{5}{2};\frac{7}{2};-\frac{a \cot (c+d x)}{b}\right )}{b^2}-15 \left (a^2-b^2\right ) \left (8 \sqrt{\cot (c+d x)}+\sqrt{2} \log \left (\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right )-\sqrt{2} \log \left (\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right )+2 \sqrt{2} \tan ^{-1}\left (1-\sqrt{2} \sqrt{\cot (c+d x)}\right )-2 \sqrt{2} \tan ^{-1}\left (\sqrt{2} \sqrt{\cot (c+d x)}+1\right )\right )-240 b^2 \left (\sqrt{\cot (c+d x)}-\frac{\sqrt{b} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{\cot (c+d x)}}{\sqrt{b}}\right )}{\sqrt{a}}\right )+80 a b \cot ^{\frac{3}{2}}(c+d x) \left (\, _2F_1\left (\frac{3}{4},1;\frac{7}{4};-\cot ^2(c+d x)\right )-1\right )+80 a b \cot ^{\frac{3}{2}}(c+d x)}{60 d \left (a^2+b^2\right )^2} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.527, size = 20269, normalized size = 64.4 \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 [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{1}{{\left (b \tan \left (d x + c\right ) + a\right )}^{2} \sqrt{\cot \left (d x + c\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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