Optimal. Leaf size=184 \[ \frac{(a-b) \log \left (\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right )}{2 \sqrt{2} d}-\frac{(a-b) \log \left (\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right )}{2 \sqrt{2} d}-\frac{(a+b) \tan ^{-1}\left (1-\sqrt{2} \sqrt{\cot (c+d x)}\right )}{\sqrt{2} d}+\frac{(a+b) \tan ^{-1}\left (\sqrt{2} \sqrt{\cot (c+d x)}+1\right )}{\sqrt{2} d}+\frac{2 a}{d \sqrt{\cot (c+d x)}}+\frac{2 b}{3 d \cot ^{\frac{3}{2}}(c+d x)} \]
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Rubi [A] time = 0.154313, antiderivative size = 184, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 9, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {3673, 3529, 3534, 1168, 1162, 617, 204, 1165, 628} \[ \frac{(a-b) \log \left (\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right )}{2 \sqrt{2} d}-\frac{(a-b) \log \left (\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right )}{2 \sqrt{2} d}-\frac{(a+b) \tan ^{-1}\left (1-\sqrt{2} \sqrt{\cot (c+d x)}\right )}{\sqrt{2} d}+\frac{(a+b) \tan ^{-1}\left (\sqrt{2} \sqrt{\cot (c+d x)}+1\right )}{\sqrt{2} d}+\frac{2 a}{d \sqrt{\cot (c+d x)}}+\frac{2 b}{3 d \cot ^{\frac{3}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Rule 3673
Rule 3529
Rule 3534
Rule 1168
Rule 1162
Rule 617
Rule 204
Rule 1165
Rule 628
Rubi steps
\begin{align*} \int \frac{a+b \tan (c+d x)}{\cot ^{\frac{3}{2}}(c+d x)} \, dx &=\int \frac{b+a \cot (c+d x)}{\cot ^{\frac{5}{2}}(c+d x)} \, dx\\ &=\frac{2 b}{3 d \cot ^{\frac{3}{2}}(c+d x)}+\int \frac{a-b \cot (c+d x)}{\cot ^{\frac{3}{2}}(c+d x)} \, dx\\ &=\frac{2 b}{3 d \cot ^{\frac{3}{2}}(c+d x)}+\frac{2 a}{d \sqrt{\cot (c+d x)}}+\int \frac{-b-a \cot (c+d x)}{\sqrt{\cot (c+d x)}} \, dx\\ &=\frac{2 b}{3 d \cot ^{\frac{3}{2}}(c+d x)}+\frac{2 a}{d \sqrt{\cot (c+d x)}}+\frac{2 \operatorname{Subst}\left (\int \frac{b+a x^2}{1+x^4} \, dx,x,\sqrt{\cot (c+d x)}\right )}{d}\\ &=\frac{2 b}{3 d \cot ^{\frac{3}{2}}(c+d x)}+\frac{2 a}{d \sqrt{\cot (c+d x)}}-\frac{(a-b) \operatorname{Subst}\left (\int \frac{1-x^2}{1+x^4} \, dx,x,\sqrt{\cot (c+d x)}\right )}{d}+\frac{(a+b) \operatorname{Subst}\left (\int \frac{1+x^2}{1+x^4} \, dx,x,\sqrt{\cot (c+d x)}\right )}{d}\\ &=\frac{2 b}{3 d \cot ^{\frac{3}{2}}(c+d x)}+\frac{2 a}{d \sqrt{\cot (c+d x)}}+\frac{(a-b) \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{(a-b) \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{(a+b) \operatorname{Subst}\left (\int \frac{1}{1-\sqrt{2} x+x^2} \, dx,x,\sqrt{\cot (c+d x)}\right )}{2 d}+\frac{(a+b) \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}{3 d \cot ^{\frac{3}{2}}(c+d x)}+\frac{2 a}{d \sqrt{\cot (c+d x)}}+\frac{(a-b) \log \left (1-\sqrt{2} \sqrt{\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt{2} d}-\frac{(a-b) \log \left (1+\sqrt{2} \sqrt{\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt{2} d}+\frac{(a+b) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\sqrt{2} \sqrt{\cot (c+d x)}\right )}{\sqrt{2} d}-\frac{(a+b) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\sqrt{2} \sqrt{\cot (c+d x)}\right )}{\sqrt{2} d}\\ &=-\frac{(a+b) \tan ^{-1}\left (1-\sqrt{2} \sqrt{\cot (c+d x)}\right )}{\sqrt{2} d}+\frac{(a+b) \tan ^{-1}\left (1+\sqrt{2} \sqrt{\cot (c+d x)}\right )}{\sqrt{2} d}+\frac{2 b}{3 d \cot ^{\frac{3}{2}}(c+d x)}+\frac{2 a}{d \sqrt{\cot (c+d x)}}+\frac{(a-b) \log \left (1-\sqrt{2} \sqrt{\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt{2} d}-\frac{(a-b) \log \left (1+\sqrt{2} \sqrt{\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt{2} d}\\ \end{align*}
Mathematica [C] time = 0.560499, size = 194, normalized size = 1.05 \[ \frac{\sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \left (3 a \left (2 \sqrt{2} \tan ^{-1}\left (1-\sqrt{2} \sqrt{\tan (c+d x)}\right )-2 \sqrt{2} \tan ^{-1}\left (\sqrt{2} \sqrt{\tan (c+d x)}+1\right )+8 \sqrt{\tan (c+d x)}+\sqrt{2} \log \left (\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right )-\sqrt{2} \log \left (\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right )\right )-8 b \tan ^{\frac{3}{2}}(c+d x) \left (\, _2F_1\left (\frac{3}{4},1;\frac{7}{4};-\tan ^2(c+d x)\right )-1\right )\right )}{12 d} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.227, size = 1208, normalized size = 6.6 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.6999, size = 205, normalized size = 1.11 \begin{align*} \frac{6 \, \sqrt{2}{\left (a + b\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 (a + b\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 (a - b\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 (a - b\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 + \frac{3 \, a}{\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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{a + b \tan{\left (c + d x \right )}}{\cot ^{\frac{3}{2}}{\left (c + d x \right )}}\, dx \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}{\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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