Optimal. Leaf size=126 \[ -\frac{b \cot (c+d x) \sqrt{a+b \cot ^2(c+d x)}}{2 d}-\frac{(a-b)^{3/2} \tan ^{-1}\left (\frac{\sqrt{a-b} \cot (c+d x)}{\sqrt{a+b \cot ^2(c+d x)}}\right )}{d}-\frac{\sqrt{b} (3 a-2 b) \tanh ^{-1}\left (\frac{\sqrt{b} \cot (c+d x)}{\sqrt{a+b \cot ^2(c+d x)}}\right )}{2 d} \]
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Rubi [A] time = 0.0978133, antiderivative size = 126, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.438, Rules used = {3661, 416, 523, 217, 206, 377, 203} \[ -\frac{b \cot (c+d x) \sqrt{a+b \cot ^2(c+d x)}}{2 d}-\frac{(a-b)^{3/2} \tan ^{-1}\left (\frac{\sqrt{a-b} \cot (c+d x)}{\sqrt{a+b \cot ^2(c+d x)}}\right )}{d}-\frac{\sqrt{b} (3 a-2 b) \tanh ^{-1}\left (\frac{\sqrt{b} \cot (c+d x)}{\sqrt{a+b \cot ^2(c+d x)}}\right )}{2 d} \]
Antiderivative was successfully verified.
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Rule 3661
Rule 416
Rule 523
Rule 217
Rule 206
Rule 377
Rule 203
Rubi steps
\begin{align*} \int \left (a+b \cot ^2(c+d x)\right )^{3/2} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{\left (a+b x^2\right )^{3/2}}{1+x^2} \, dx,x,\cot (c+d x)\right )}{d}\\ &=-\frac{b \cot (c+d x) \sqrt{a+b \cot ^2(c+d x)}}{2 d}-\frac{\operatorname{Subst}\left (\int \frac{a (2 a-b)+(3 a-2 b) b x^2}{\left (1+x^2\right ) \sqrt{a+b x^2}} \, dx,x,\cot (c+d x)\right )}{2 d}\\ &=-\frac{b \cot (c+d x) \sqrt{a+b \cot ^2(c+d x)}}{2 d}-\frac{(a-b)^2 \operatorname{Subst}\left (\int \frac{1}{\left (1+x^2\right ) \sqrt{a+b x^2}} \, dx,x,\cot (c+d x)\right )}{d}-\frac{((3 a-2 b) b) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x^2}} \, dx,x,\cot (c+d x)\right )}{2 d}\\ &=-\frac{b \cot (c+d x) \sqrt{a+b \cot ^2(c+d x)}}{2 d}-\frac{(a-b)^2 \operatorname{Subst}\left (\int \frac{1}{1-(-a+b) x^2} \, dx,x,\frac{\cot (c+d x)}{\sqrt{a+b \cot ^2(c+d x)}}\right )}{d}-\frac{((3 a-2 b) b) \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{\cot (c+d x)}{\sqrt{a+b \cot ^2(c+d x)}}\right )}{2 d}\\ &=-\frac{(a-b)^{3/2} \tan ^{-1}\left (\frac{\sqrt{a-b} \cot (c+d x)}{\sqrt{a+b \cot ^2(c+d x)}}\right )}{d}-\frac{(3 a-2 b) \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} \cot (c+d x)}{\sqrt{a+b \cot ^2(c+d x)}}\right )}{2 d}-\frac{b \cot (c+d x) \sqrt{a+b \cot ^2(c+d x)}}{2 d}\\ \end{align*}
Mathematica [C] time = 1.38656, size = 234, normalized size = 1.86 \[ \frac{-b \cot (c+d x) \sqrt{a+b \cot ^2(c+d x)}+i (a-b)^{3/2} \log \left (-\frac{4 i \left (\sqrt{a-b} \sqrt{a+b \cot ^2(c+d x)}+a-i b \cot (c+d x)\right )}{(a-b)^{5/2} (\cot (c+d x)+i)}\right )-i (a-b)^{3/2} \log \left (\frac{4 i \left (\sqrt{a-b} \sqrt{a+b \cot ^2(c+d x)}+a+i b \cot (c+d x)\right )}{(a-b)^{5/2} (\cot (c+d x)-i)}\right )+\sqrt{b} (2 b-3 a) \log \left (\sqrt{b} \sqrt{a+b \cot ^2(c+d x)}+b \cot (c+d x)\right )}{2 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.02, size = 298, normalized size = 2.4 \begin{align*} -{\frac{b\cot \left ( dx+c \right ) }{2\,d}\sqrt{a+b \left ( \cot \left ( dx+c \right ) \right ) ^{2}}}-{\frac{3\,a}{2\,d}\sqrt{b}\ln \left ( \cot \left ( dx+c \right ) \sqrt{b}+\sqrt{a+b \left ( \cot \left ( dx+c \right ) \right ) ^{2}} \right ) }+{\frac{1}{d}{b}^{{\frac{3}{2}}}\ln \left ( \cot \left ( dx+c \right ) \sqrt{b}+\sqrt{a+b \left ( \cot \left ( dx+c \right ) \right ) ^{2}} \right ) }-{\frac{1}{d \left ( a-b \right ) }\sqrt{{b}^{4} \left ( a-b \right ) }\arctan \left ({ \left ( a-b \right ){b}^{2}\cot \left ( dx+c \right ){\frac{1}{\sqrt{{b}^{4} \left ( a-b \right ) }}}{\frac{1}{\sqrt{a+b \left ( \cot \left ( dx+c \right ) \right ) ^{2}}}}} \right ) }+2\,{\frac{a\sqrt{{b}^{4} \left ( a-b \right ) }}{db \left ( a-b \right ) }\arctan \left ({\frac{ \left ( a-b \right ){b}^{2}\cot \left ( dx+c \right ) }{\sqrt{{b}^{4} \left ( a-b \right ) }\sqrt{a+b \left ( \cot \left ( dx+c \right ) \right ) ^{2}}}} \right ) }-{\frac{{a}^{2}}{d{b}^{2} \left ( a-b \right ) }\sqrt{{b}^{4} \left ( a-b \right ) }\arctan \left ({ \left ( a-b \right ){b}^{2}\cot \left ( dx+c \right ){\frac{1}{\sqrt{{b}^{4} \left ( a-b \right ) }}}{\frac{1}{\sqrt{a+b \left ( \cot \left ( dx+c \right ) \right ) ^{2}}}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \cot \left (d x + c\right )^{2} + a\right )}^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.99124, size = 2558, normalized size = 20.3 \begin{align*} \text{result too large to display} \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 \left (a + b \cot ^{2}{\left (c + d x \right )}\right )^{\frac{3}{2}}\, 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{\left (b \cot \left (d x + c\right )^{2} + a\right )}^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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