Optimal. Leaf size=302 \[ \frac{(a+b) \log \left (\sqrt{e} \cot (c+d x)-\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right )}{2 \sqrt{2} d \sqrt{e} \left (a^2+b^2\right )}-\frac{(a+b) \log \left (\sqrt{e} \cot (c+d x)+\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right )}{2 \sqrt{2} d \sqrt{e} \left (a^2+b^2\right )}-\frac{2 b^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{e \cot (c+d x)}}{\sqrt{a} \sqrt{e}}\right )}{\sqrt{a} d \sqrt{e} \left (a^2+b^2\right )}+\frac{(a-b) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}\right )}{\sqrt{2} d \sqrt{e} \left (a^2+b^2\right )}-\frac{(a-b) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}+1\right )}{\sqrt{2} d \sqrt{e} \left (a^2+b^2\right )} \]
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Rubi [A] time = 0.369796, antiderivative size = 302, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 11, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.44, Rules used = {3574, 3534, 1168, 1162, 617, 204, 1165, 628, 3634, 63, 205} \[ \frac{(a+b) \log \left (\sqrt{e} \cot (c+d x)-\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right )}{2 \sqrt{2} d \sqrt{e} \left (a^2+b^2\right )}-\frac{(a+b) \log \left (\sqrt{e} \cot (c+d x)+\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right )}{2 \sqrt{2} d \sqrt{e} \left (a^2+b^2\right )}-\frac{2 b^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{e \cot (c+d x)}}{\sqrt{a} \sqrt{e}}\right )}{\sqrt{a} d \sqrt{e} \left (a^2+b^2\right )}+\frac{(a-b) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}\right )}{\sqrt{2} d \sqrt{e} \left (a^2+b^2\right )}-\frac{(a-b) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}+1\right )}{\sqrt{2} d \sqrt{e} \left (a^2+b^2\right )} \]
Antiderivative was successfully verified.
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Rule 3574
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{e \cot (c+d x)} (a+b \cot (c+d x))} \, dx &=\frac{\int \frac{a-b \cot (c+d x)}{\sqrt{e \cot (c+d x)}} \, dx}{a^2+b^2}+\frac{b^2 \int \frac{1+\cot ^2(c+d x)}{\sqrt{e \cot (c+d x)} (a+b \cot (c+d x))} \, dx}{a^2+b^2}\\ &=\frac{2 \operatorname{Subst}\left (\int \frac{-a e+b x^2}{e^2+x^4} \, dx,x,\sqrt{e \cot (c+d x)}\right )}{\left (a^2+b^2\right ) d}+\frac{b^2 \operatorname{Subst}\left (\int \frac{1}{\sqrt{-e x} (a-b x)} \, dx,x,-\cot (c+d x)\right )}{\left (a^2+b^2\right ) d}\\ &=-\frac{(a-b) \operatorname{Subst}\left (\int \frac{e+x^2}{e^2+x^4} \, dx,x,\sqrt{e \cot (c+d x)}\right )}{\left (a^2+b^2\right ) d}-\frac{(a+b) \operatorname{Subst}\left (\int \frac{e-x^2}{e^2+x^4} \, dx,x,\sqrt{e \cot (c+d x)}\right )}{\left (a^2+b^2\right ) d}-\frac{\left (2 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{a+\frac{b x^2}{e}} \, dx,x,\sqrt{e \cot (c+d x)}\right )}{\left (a^2+b^2\right ) d e}\\ &=-\frac{2 b^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{e \cot (c+d x)}}{\sqrt{a} \sqrt{e}}\right )}{\sqrt{a} \left (a^2+b^2\right ) d \sqrt{e}}-\frac{(a-b) \operatorname{Subst}\left (\int \frac{1}{e-\sqrt{2} \sqrt{e} x+x^2} \, dx,x,\sqrt{e \cot (c+d x)}\right )}{2 \left (a^2+b^2\right ) d}-\frac{(a-b) \operatorname{Subst}\left (\int \frac{1}{e+\sqrt{2} \sqrt{e} x+x^2} \, dx,x,\sqrt{e \cot (c+d x)}\right )}{2 \left (a^2+b^2\right ) d}+\frac{(a+b) \operatorname{Subst}\left (\int \frac{\sqrt{2} \sqrt{e}+2 x}{-e-\sqrt{2} \sqrt{e} x-x^2} \, dx,x,\sqrt{e \cot (c+d x)}\right )}{2 \sqrt{2} \left (a^2+b^2\right ) d \sqrt{e}}+\frac{(a+b) \operatorname{Subst}\left (\int \frac{\sqrt{2} \sqrt{e}-2 x}{-e+\sqrt{2} \sqrt{e} x-x^2} \, dx,x,\sqrt{e \cot (c+d x)}\right )}{2 \sqrt{2} \left (a^2+b^2\right ) d \sqrt{e}}\\ &=-\frac{2 b^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{e \cot (c+d x)}}{\sqrt{a} \sqrt{e}}\right )}{\sqrt{a} \left (a^2+b^2\right ) d \sqrt{e}}+\frac{(a+b) \log \left (\sqrt{e}+\sqrt{e} \cot (c+d x)-\sqrt{2} \sqrt{e \cot (c+d x)}\right )}{2 \sqrt{2} \left (a^2+b^2\right ) d \sqrt{e}}-\frac{(a+b) \log \left (\sqrt{e}+\sqrt{e} \cot (c+d x)+\sqrt{2} \sqrt{e \cot (c+d x)}\right )}{2 \sqrt{2} \left (a^2+b^2\right ) d \sqrt{e}}-\frac{(a-b) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}\right )}{\sqrt{2} \left (a^2+b^2\right ) d \sqrt{e}}+\frac{(a-b) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}\right )}{\sqrt{2} \left (a^2+b^2\right ) d \sqrt{e}}\\ &=-\frac{2 b^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{e \cot (c+d x)}}{\sqrt{a} \sqrt{e}}\right )}{\sqrt{a} \left (a^2+b^2\right ) d \sqrt{e}}+\frac{(a-b) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}\right )}{\sqrt{2} \left (a^2+b^2\right ) d \sqrt{e}}-\frac{(a-b) \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}\right )}{\sqrt{2} \left (a^2+b^2\right ) d \sqrt{e}}+\frac{(a+b) \log \left (\sqrt{e}+\sqrt{e} \cot (c+d x)-\sqrt{2} \sqrt{e \cot (c+d x)}\right )}{2 \sqrt{2} \left (a^2+b^2\right ) d \sqrt{e}}-\frac{(a+b) \log \left (\sqrt{e}+\sqrt{e} \cot (c+d x)+\sqrt{2} \sqrt{e \cot (c+d x)}\right )}{2 \sqrt{2} \left (a^2+b^2\right ) d \sqrt{e}}\\ \end{align*}
Mathematica [C] time = 0.244406, size = 248, normalized size = 0.82 \[ -\frac{\sqrt{\cot (c+d x)} \left (-\frac{2 b \cot ^{\frac{3}{2}}(c+d x) \text{Hypergeometric2F1}\left (\frac{3}{4},1,\frac{7}{4},-\cot ^2(c+d x)\right )}{3 \left (a^2+b^2\right )}+\frac{2 b^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{\cot (c+d x)}}{\sqrt{a}}\right )}{\sqrt{a} \left (a^2+b^2\right )}-\frac{a \left (2 \sqrt{2} \log \left (\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right )-2 \sqrt{2} \log \left (\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right )+4 \left (\sqrt{2} \tan ^{-1}\left (1-\sqrt{2} \sqrt{\cot (c+d x)}\right )-\sqrt{2} \tan ^{-1}\left (\sqrt{2} \sqrt{\cot (c+d x)}+1\right )\right )\right )}{8 \left (a^2+b^2\right )}\right )}{d \sqrt{e \cot (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 423, normalized size = 1.4 \begin{align*} -{\frac{a\sqrt{2}}{4\,de \left ({a}^{2}+{b}^{2} \right ) }\sqrt [4]{{e}^{2}}\ln \left ({ \left ( e\cot \left ( dx+c \right ) +\sqrt [4]{{e}^{2}}\sqrt{e\cot \left ( dx+c \right ) }\sqrt{2}+\sqrt{{e}^{2}} \right ) \left ( e\cot \left ( dx+c \right ) -\sqrt [4]{{e}^{2}}\sqrt{e\cot \left ( dx+c \right ) }\sqrt{2}+\sqrt{{e}^{2}} \right ) ^{-1}} \right ) }-{\frac{a\sqrt{2}}{2\,de \left ({a}^{2}+{b}^{2} \right ) }\sqrt [4]{{e}^{2}}\arctan \left ({\sqrt{2}\sqrt{e\cot \left ( dx+c \right ) }{\frac{1}{\sqrt [4]{{e}^{2}}}}}+1 \right ) }+{\frac{a\sqrt{2}}{2\,de \left ({a}^{2}+{b}^{2} \right ) }\sqrt [4]{{e}^{2}}\arctan \left ( -{\sqrt{2}\sqrt{e\cot \left ( dx+c \right ) }{\frac{1}{\sqrt [4]{{e}^{2}}}}}+1 \right ) }+{\frac{b\sqrt{2}}{4\,d \left ({a}^{2}+{b}^{2} \right ) }\ln \left ({ \left ( e\cot \left ( dx+c \right ) -\sqrt [4]{{e}^{2}}\sqrt{e\cot \left ( dx+c \right ) }\sqrt{2}+\sqrt{{e}^{2}} \right ) \left ( e\cot \left ( dx+c \right ) +\sqrt [4]{{e}^{2}}\sqrt{e\cot \left ( dx+c \right ) }\sqrt{2}+\sqrt{{e}^{2}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{e}^{2}}}}}+{\frac{b\sqrt{2}}{2\,d \left ({a}^{2}+{b}^{2} \right ) }\arctan \left ({\sqrt{2}\sqrt{e\cot \left ( dx+c \right ) }{\frac{1}{\sqrt [4]{{e}^{2}}}}}+1 \right ){\frac{1}{\sqrt [4]{{e}^{2}}}}}-{\frac{b\sqrt{2}}{2\,d \left ({a}^{2}+{b}^{2} \right ) }\arctan \left ( -{\sqrt{2}\sqrt{e\cot \left ( dx+c \right ) }{\frac{1}{\sqrt [4]{{e}^{2}}}}}+1 \right ){\frac{1}{\sqrt [4]{{e}^{2}}}}}-2\,{\frac{{b}^{2}}{d \left ({a}^{2}+{b}^{2} \right ) \sqrt{aeb}}\arctan \left ({\frac{\sqrt{e\cot \left ( dx+c \right ) }b}{\sqrt{aeb}}} \right ) } \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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{e \cot{\left (c + d x \right )}} \left (a + b \cot{\left (c + d x \right )}\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{1}{{\left (b \cot \left (d x + c\right ) + a\right )} \sqrt{e \cot \left (d x + c\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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