Optimal. Leaf size=408 \[ \frac{b \left (a^2+b^2\right ) \log \left (-\sqrt{2} \sqrt{\sqrt{a^2+b^2}+a} \sqrt{a+b \cot (c+d x)}+\sqrt{a^2+b^2}+a+b \cot (c+d x)\right )}{2 \sqrt{2} d \sqrt{\sqrt{a^2+b^2}+a}}-\frac{b \left (a^2+b^2\right ) \log \left (\sqrt{2} \sqrt{\sqrt{a^2+b^2}+a} \sqrt{a+b \cot (c+d x)}+\sqrt{a^2+b^2}+a+b \cot (c+d x)\right )}{2 \sqrt{2} d \sqrt{\sqrt{a^2+b^2}+a}}+\frac{b \left (a^2+b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{a^2+b^2}+a}-\sqrt{2} \sqrt{a+b \cot (c+d x)}}{\sqrt{a-\sqrt{a^2+b^2}}}\right )}{\sqrt{2} d \sqrt{a-\sqrt{a^2+b^2}}}-\frac{b \left (a^2+b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{a^2+b^2}+a}+\sqrt{2} \sqrt{a+b \cot (c+d x)}}{\sqrt{a-\sqrt{a^2+b^2}}}\right )}{\sqrt{2} d \sqrt{a-\sqrt{a^2+b^2}}}-\frac{2 b (a+b \cot (c+d x))^{3/2}}{3 d} \]
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Rubi [A] time = 0.510874, antiderivative size = 408, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 9, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {3528, 12, 3485, 700, 1129, 634, 618, 206, 628} \[ \frac{b \left (a^2+b^2\right ) \log \left (-\sqrt{2} \sqrt{\sqrt{a^2+b^2}+a} \sqrt{a+b \cot (c+d x)}+\sqrt{a^2+b^2}+a+b \cot (c+d x)\right )}{2 \sqrt{2} d \sqrt{\sqrt{a^2+b^2}+a}}-\frac{b \left (a^2+b^2\right ) \log \left (\sqrt{2} \sqrt{\sqrt{a^2+b^2}+a} \sqrt{a+b \cot (c+d x)}+\sqrt{a^2+b^2}+a+b \cot (c+d x)\right )}{2 \sqrt{2} d \sqrt{\sqrt{a^2+b^2}+a}}+\frac{b \left (a^2+b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{a^2+b^2}+a}-\sqrt{2} \sqrt{a+b \cot (c+d x)}}{\sqrt{a-\sqrt{a^2+b^2}}}\right )}{\sqrt{2} d \sqrt{a-\sqrt{a^2+b^2}}}-\frac{b \left (a^2+b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{a^2+b^2}+a}+\sqrt{2} \sqrt{a+b \cot (c+d x)}}{\sqrt{a-\sqrt{a^2+b^2}}}\right )}{\sqrt{2} d \sqrt{a-\sqrt{a^2+b^2}}}-\frac{2 b (a+b \cot (c+d x))^{3/2}}{3 d} \]
Antiderivative was successfully verified.
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Rule 3528
Rule 12
Rule 3485
Rule 700
Rule 1129
Rule 634
Rule 618
Rule 206
Rule 628
Rubi steps
\begin{align*} \int (-a+b \cot (c+d x)) (a+b \cot (c+d x))^{3/2} \, dx &=-\frac{2 b (a+b \cot (c+d x))^{3/2}}{3 d}+\int \left (-a^2-b^2\right ) \sqrt{a+b \cot (c+d x)} \, dx\\ &=-\frac{2 b (a+b \cot (c+d x))^{3/2}}{3 d}+\left (-a^2-b^2\right ) \int \sqrt{a+b \cot (c+d x)} \, dx\\ &=-\frac{2 b (a+b \cot (c+d x))^{3/2}}{3 d}+\frac{\left (b \left (a^2+b^2\right )\right ) \operatorname{Subst}\left (\int \frac{\sqrt{a+x}}{b^2+x^2} \, dx,x,b \cot (c+d x)\right )}{d}\\ &=-\frac{2 b (a+b \cot (c+d x))^{3/2}}{3 d}+\frac{\left (2 b \left (a^2+b^2\right )\right ) \operatorname{Subst}\left (\int \frac{x^2}{a^2+b^2-2 a x^2+x^4} \, dx,x,\sqrt{a+b \cot (c+d x)}\right )}{d}\\ &=-\frac{2 b (a+b \cot (c+d x))^{3/2}}{3 d}+\frac{\left (b \left (a^2+b^2\right )\right ) \operatorname{Subst}\left (\int \frac{x}{\sqrt{a^2+b^2}-\sqrt{2} \sqrt{a+\sqrt{a^2+b^2}} x+x^2} \, dx,x,\sqrt{a+b \cot (c+d x)}\right )}{\sqrt{2} \sqrt{a+\sqrt{a^2+b^2}} d}-\frac{\left (b \left (a^2+b^2\right )\right ) \operatorname{Subst}\left (\int \frac{x}{\sqrt{a^2+b^2}+\sqrt{2} \sqrt{a+\sqrt{a^2+b^2}} x+x^2} \, dx,x,\sqrt{a+b \cot (c+d x)}\right )}{\sqrt{2} \sqrt{a+\sqrt{a^2+b^2}} d}\\ &=-\frac{2 b (a+b \cot (c+d x))^{3/2}}{3 d}+\frac{\left (b \left (a^2+b^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a^2+b^2}-\sqrt{2} \sqrt{a+\sqrt{a^2+b^2}} x+x^2} \, dx,x,\sqrt{a+b \cot (c+d x)}\right )}{2 d}+\frac{\left (b \left (a^2+b^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a^2+b^2}+\sqrt{2} \sqrt{a+\sqrt{a^2+b^2}} x+x^2} \, dx,x,\sqrt{a+b \cot (c+d x)}\right )}{2 d}+\frac{\left (b \left (a^2+b^2\right )\right ) \operatorname{Subst}\left (\int \frac{-\sqrt{2} \sqrt{a+\sqrt{a^2+b^2}}+2 x}{\sqrt{a^2+b^2}-\sqrt{2} \sqrt{a+\sqrt{a^2+b^2}} x+x^2} \, dx,x,\sqrt{a+b \cot (c+d x)}\right )}{2 \sqrt{2} \sqrt{a+\sqrt{a^2+b^2}} d}-\frac{\left (b \left (a^2+b^2\right )\right ) \operatorname{Subst}\left (\int \frac{\sqrt{2} \sqrt{a+\sqrt{a^2+b^2}}+2 x}{\sqrt{a^2+b^2}+\sqrt{2} \sqrt{a+\sqrt{a^2+b^2}} x+x^2} \, dx,x,\sqrt{a+b \cot (c+d x)}\right )}{2 \sqrt{2} \sqrt{a+\sqrt{a^2+b^2}} d}\\ &=-\frac{2 b (a+b \cot (c+d x))^{3/2}}{3 d}+\frac{b \left (a^2+b^2\right ) \log \left (a+\sqrt{a^2+b^2}+b \cot (c+d x)-\sqrt{2} \sqrt{a+\sqrt{a^2+b^2}} \sqrt{a+b \cot (c+d x)}\right )}{2 \sqrt{2} \sqrt{a+\sqrt{a^2+b^2}} d}-\frac{b \left (a^2+b^2\right ) \log \left (a+\sqrt{a^2+b^2}+b \cot (c+d x)+\sqrt{2} \sqrt{a+\sqrt{a^2+b^2}} \sqrt{a+b \cot (c+d x)}\right )}{2 \sqrt{2} \sqrt{a+\sqrt{a^2+b^2}} d}-\frac{\left (b \left (a^2+b^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{2 \left (a-\sqrt{a^2+b^2}\right )-x^2} \, dx,x,-\sqrt{2} \sqrt{a+\sqrt{a^2+b^2}}+2 \sqrt{a+b \cot (c+d x)}\right )}{d}-\frac{\left (b \left (a^2+b^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{2 \left (a-\sqrt{a^2+b^2}\right )-x^2} \, dx,x,\sqrt{2} \sqrt{a+\sqrt{a^2+b^2}}+2 \sqrt{a+b \cot (c+d x)}\right )}{d}\\ &=\frac{b \left (a^2+b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{a+\sqrt{a^2+b^2}}-\sqrt{2} \sqrt{a+b \cot (c+d x)}}{\sqrt{a-\sqrt{a^2+b^2}}}\right )}{\sqrt{2} \sqrt{a-\sqrt{a^2+b^2}} d}-\frac{b \left (a^2+b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{a+\sqrt{a^2+b^2}}+\sqrt{2} \sqrt{a+b \cot (c+d x)}}{\sqrt{a-\sqrt{a^2+b^2}}}\right )}{\sqrt{2} \sqrt{a-\sqrt{a^2+b^2}} d}-\frac{2 b (a+b \cot (c+d x))^{3/2}}{3 d}+\frac{b \left (a^2+b^2\right ) \log \left (a+\sqrt{a^2+b^2}+b \cot (c+d x)-\sqrt{2} \sqrt{a+\sqrt{a^2+b^2}} \sqrt{a+b \cot (c+d x)}\right )}{2 \sqrt{2} \sqrt{a+\sqrt{a^2+b^2}} d}-\frac{b \left (a^2+b^2\right ) \log \left (a+\sqrt{a^2+b^2}+b \cot (c+d x)+\sqrt{2} \sqrt{a+\sqrt{a^2+b^2}} \sqrt{a+b \cot (c+d x)}\right )}{2 \sqrt{2} \sqrt{a+\sqrt{a^2+b^2}} d}\\ \end{align*}
Mathematica [C] time = 1.79328, size = 178, normalized size = 0.44 \[ \frac{\sin ^2(c+d x) (b \cot (c+d x)-a) (a+b \cot (c+d x)) \left (3 i \sqrt{a-i b} \left (a^2+b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{a+b \cot (c+d x)}}{\sqrt{a-i b}}\right )-3 i \sqrt{a+i b} \left (a^2+b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{a+b \cot (c+d x)}}{\sqrt{a+i b}}\right )+2 b (a+b \cot (c+d x))^{3/2}\right )}{3 a^2 d \sin ^2(c+d x)-3 b^2 d \cos ^2(c+d x)} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.052, size = 986, normalized size = 2.4 \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 [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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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 a^{2} \sqrt{a + b \cot{\left (c + d x \right )}}\, dx - \int - b^{2} \sqrt{a + b \cot{\left (c + d x \right )}} \cot ^{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{\left (b \cot \left (d x + c\right ) + a\right )}^{\frac{3}{2}}{\left (b \cot \left (d x + c\right ) - a\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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