Optimal. Leaf size=529 \[ \frac{(a+b) \left (a^2-4 a b+b^2\right ) \log \left (\sqrt{e} \cot (c+d x)-\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right )}{2 \sqrt{2} d e^{3/2} \left (a^2+b^2\right )^3}-\frac{(a+b) \left (a^2-4 a b+b^2\right ) \log \left (\sqrt{e} \cot (c+d x)+\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right )}{2 \sqrt{2} d e^{3/2} \left (a^2+b^2\right )^3}+\frac{b^{5/2} \left (46 a^2 b^2+63 a^4+15 b^4\right ) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{e \cot (c+d x)}}{\sqrt{a} \sqrt{e}}\right )}{4 a^{7/2} d e^{3/2} \left (a^2+b^2\right )^3}-\frac{(a-b) \left (a^2+4 a b+b^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}\right )}{\sqrt{2} d e^{3/2} \left (a^2+b^2\right )^3}+\frac{(a-b) \left (a^2+4 a b+b^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}+1\right )}{\sqrt{2} d e^{3/2} \left (a^2+b^2\right )^3}-\frac{b^2 \left (13 a^2+5 b^2\right )}{4 a^2 d e \left (a^2+b^2\right )^2 \sqrt{e \cot (c+d x)} (a+b \cot (c+d x))}-\frac{b^2}{2 a d e \left (a^2+b^2\right ) \sqrt{e \cot (c+d x)} (a+b \cot (c+d x))^2}+\frac{31 a^2 b^2+8 a^4+15 b^4}{4 a^3 d e \left (a^2+b^2\right )^2 \sqrt{e \cot (c+d x)}} \]
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Rubi [A] time = 1.65768, antiderivative size = 529, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 13, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.52, Rules used = {3569, 3649, 3653, 3534, 1168, 1162, 617, 204, 1165, 628, 3634, 63, 205} \[ \frac{(a+b) \left (a^2-4 a b+b^2\right ) \log \left (\sqrt{e} \cot (c+d x)-\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right )}{2 \sqrt{2} d e^{3/2} \left (a^2+b^2\right )^3}-\frac{(a+b) \left (a^2-4 a b+b^2\right ) \log \left (\sqrt{e} \cot (c+d x)+\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right )}{2 \sqrt{2} d e^{3/2} \left (a^2+b^2\right )^3}+\frac{b^{5/2} \left (46 a^2 b^2+63 a^4+15 b^4\right ) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{e \cot (c+d x)}}{\sqrt{a} \sqrt{e}}\right )}{4 a^{7/2} d e^{3/2} \left (a^2+b^2\right )^3}-\frac{(a-b) \left (a^2+4 a b+b^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}\right )}{\sqrt{2} d e^{3/2} \left (a^2+b^2\right )^3}+\frac{(a-b) \left (a^2+4 a b+b^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}+1\right )}{\sqrt{2} d e^{3/2} \left (a^2+b^2\right )^3}-\frac{b^2 \left (13 a^2+5 b^2\right )}{4 a^2 d e \left (a^2+b^2\right )^2 \sqrt{e \cot (c+d x)} (a+b \cot (c+d x))}-\frac{b^2}{2 a d e \left (a^2+b^2\right ) \sqrt{e \cot (c+d x)} (a+b \cot (c+d x))^2}+\frac{31 a^2 b^2+8 a^4+15 b^4}{4 a^3 d e \left (a^2+b^2\right )^2 \sqrt{e \cot (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 3569
Rule 3649
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}{(e \cot (c+d x))^{3/2} (a+b \cot (c+d x))^3} \, dx &=-\frac{b^2}{2 a \left (a^2+b^2\right ) d e \sqrt{e \cot (c+d x)} (a+b \cot (c+d x))^2}-\frac{\int \frac{-\frac{1}{2} \left (4 a^2+5 b^2\right ) e+2 a b e \cot (c+d x)-\frac{5}{2} b^2 e \cot ^2(c+d x)}{(e \cot (c+d x))^{3/2} (a+b \cot (c+d x))^2} \, dx}{2 a \left (a^2+b^2\right ) e}\\ &=-\frac{b^2}{2 a \left (a^2+b^2\right ) d e \sqrt{e \cot (c+d x)} (a+b \cot (c+d x))^2}-\frac{b^2 \left (13 a^2+5 b^2\right )}{4 a^2 \left (a^2+b^2\right )^2 d e \sqrt{e \cot (c+d x)} (a+b \cot (c+d x))}+\frac{\int \frac{\frac{1}{4} \left (8 a^4+31 a^2 b^2+15 b^4\right ) e^2-4 a^3 b e^2 \cot (c+d x)+\frac{3}{4} b^2 \left (13 a^2+5 b^2\right ) e^2 \cot ^2(c+d x)}{(e \cot (c+d x))^{3/2} (a+b \cot (c+d x))} \, dx}{2 a^2 \left (a^2+b^2\right )^2 e^2}\\ &=\frac{8 a^4+31 a^2 b^2+15 b^4}{4 a^3 \left (a^2+b^2\right )^2 d e \sqrt{e \cot (c+d x)}}-\frac{b^2}{2 a \left (a^2+b^2\right ) d e \sqrt{e \cot (c+d x)} (a+b \cot (c+d x))^2}-\frac{b^2 \left (13 a^2+5 b^2\right )}{4 a^2 \left (a^2+b^2\right )^2 d e \sqrt{e \cot (c+d x)} (a+b \cot (c+d x))}+\frac{\int \frac{-\frac{1}{8} b \left (24 a^4+31 a^2 b^2+15 b^4\right ) e^4-a^3 \left (a^2-b^2\right ) e^4 \cot (c+d x)-\frac{1}{8} b \left (8 a^4+31 a^2 b^2+15 b^4\right ) e^4 \cot ^2(c+d x)}{\sqrt{e \cot (c+d x)} (a+b \cot (c+d x))} \, dx}{a^3 \left (a^2+b^2\right )^2 e^5}\\ &=\frac{8 a^4+31 a^2 b^2+15 b^4}{4 a^3 \left (a^2+b^2\right )^2 d e \sqrt{e \cot (c+d x)}}-\frac{b^2}{2 a \left (a^2+b^2\right ) d e \sqrt{e \cot (c+d x)} (a+b \cot (c+d x))^2}-\frac{b^2 \left (13 a^2+5 b^2\right )}{4 a^2 \left (a^2+b^2\right )^2 d e \sqrt{e \cot (c+d x)} (a+b \cot (c+d x))}+\frac{\int \frac{-a^3 b \left (3 a^2-b^2\right ) e^4-a^4 \left (a^2-3 b^2\right ) e^4 \cot (c+d x)}{\sqrt{e \cot (c+d x)}} \, dx}{a^3 \left (a^2+b^2\right )^3 e^5}-\frac{\left (b^3 \left (63 a^4+46 a^2 b^2+15 b^4\right )\right ) \int \frac{1+\cot ^2(c+d x)}{\sqrt{e \cot (c+d x)} (a+b \cot (c+d x))} \, dx}{8 a^3 \left (a^2+b^2\right )^3 e}\\ &=\frac{8 a^4+31 a^2 b^2+15 b^4}{4 a^3 \left (a^2+b^2\right )^2 d e \sqrt{e \cot (c+d x)}}-\frac{b^2}{2 a \left (a^2+b^2\right ) d e \sqrt{e \cot (c+d x)} (a+b \cot (c+d x))^2}-\frac{b^2 \left (13 a^2+5 b^2\right )}{4 a^2 \left (a^2+b^2\right )^2 d e \sqrt{e \cot (c+d x)} (a+b \cot (c+d x))}+\frac{2 \operatorname{Subst}\left (\int \frac{a^3 b \left (3 a^2-b^2\right ) e^5+a^4 \left (a^2-3 b^2\right ) e^4 x^2}{e^2+x^4} \, dx,x,\sqrt{e \cot (c+d x)}\right )}{a^3 \left (a^2+b^2\right )^3 d e^5}-\frac{\left (b^3 \left (63 a^4+46 a^2 b^2+15 b^4\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{-e x} (a-b x)} \, dx,x,-\cot (c+d x)\right )}{8 a^3 \left (a^2+b^2\right )^3 d e}\\ &=\frac{8 a^4+31 a^2 b^2+15 b^4}{4 a^3 \left (a^2+b^2\right )^2 d e \sqrt{e \cot (c+d x)}}-\frac{b^2}{2 a \left (a^2+b^2\right ) d e \sqrt{e \cot (c+d x)} (a+b \cot (c+d x))^2}-\frac{b^2 \left (13 a^2+5 b^2\right )}{4 a^2 \left (a^2+b^2\right )^2 d e \sqrt{e \cot (c+d x)} (a+b \cot (c+d x))}+\frac{\left (b^3 \left (63 a^4+46 a^2 b^2+15 b^4\right )\right ) \operatorname{Subst}\left (\int \frac{1}{a+\frac{b x^2}{e}} \, dx,x,\sqrt{e \cot (c+d x)}\right )}{4 a^3 \left (a^2+b^2\right )^3 d e^2}-\frac{\left ((a+b) \left (a^2-4 a b+b^2\right )\right ) \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 )^3 d e}+\frac{\left ((a-b) \left (a^2+4 a b+b^2\right )\right ) \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 )^3 d e}\\ &=\frac{b^{5/2} \left (63 a^4+46 a^2 b^2+15 b^4\right ) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{e \cot (c+d x)}}{\sqrt{a} \sqrt{e}}\right )}{4 a^{7/2} \left (a^2+b^2\right )^3 d e^{3/2}}+\frac{8 a^4+31 a^2 b^2+15 b^4}{4 a^3 \left (a^2+b^2\right )^2 d e \sqrt{e \cot (c+d x)}}-\frac{b^2}{2 a \left (a^2+b^2\right ) d e \sqrt{e \cot (c+d x)} (a+b \cot (c+d x))^2}-\frac{b^2 \left (13 a^2+5 b^2\right )}{4 a^2 \left (a^2+b^2\right )^2 d e \sqrt{e \cot (c+d x)} (a+b \cot (c+d x))}+\frac{\left ((a+b) \left (a^2-4 a b+b^2\right )\right ) \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 )^3 d e^{3/2}}+\frac{\left ((a+b) \left (a^2-4 a b+b^2\right )\right ) \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 )^3 d e^{3/2}}+\frac{\left ((a-b) \left (a^2+4 a b+b^2\right )\right ) \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 )^3 d e}+\frac{\left ((a-b) \left (a^2+4 a b+b^2\right )\right ) \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 )^3 d e}\\ &=\frac{b^{5/2} \left (63 a^4+46 a^2 b^2+15 b^4\right ) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{e \cot (c+d x)}}{\sqrt{a} \sqrt{e}}\right )}{4 a^{7/2} \left (a^2+b^2\right )^3 d e^{3/2}}+\frac{8 a^4+31 a^2 b^2+15 b^4}{4 a^3 \left (a^2+b^2\right )^2 d e \sqrt{e \cot (c+d x)}}-\frac{b^2}{2 a \left (a^2+b^2\right ) d e \sqrt{e \cot (c+d x)} (a+b \cot (c+d x))^2}-\frac{b^2 \left (13 a^2+5 b^2\right )}{4 a^2 \left (a^2+b^2\right )^2 d e \sqrt{e \cot (c+d x)} (a+b \cot (c+d x))}+\frac{(a+b) \left (a^2-4 a b+b^2\right ) \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 )^3 d e^{3/2}}-\frac{(a+b) \left (a^2-4 a b+b^2\right ) \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 )^3 d e^{3/2}}+\frac{\left ((a-b) \left (a^2+4 a b+b^2\right )\right ) \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 )^3 d e^{3/2}}-\frac{\left ((a-b) \left (a^2+4 a b+b^2\right )\right ) \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 )^3 d e^{3/2}}\\ &=\frac{b^{5/2} \left (63 a^4+46 a^2 b^2+15 b^4\right ) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{e \cot (c+d x)}}{\sqrt{a} \sqrt{e}}\right )}{4 a^{7/2} \left (a^2+b^2\right )^3 d e^{3/2}}-\frac{(a-b) \left (a^2+4 a b+b^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}\right )}{\sqrt{2} \left (a^2+b^2\right )^3 d e^{3/2}}+\frac{(a-b) \left (a^2+4 a b+b^2\right ) \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}\right )}{\sqrt{2} \left (a^2+b^2\right )^3 d e^{3/2}}+\frac{8 a^4+31 a^2 b^2+15 b^4}{4 a^3 \left (a^2+b^2\right )^2 d e \sqrt{e \cot (c+d x)}}-\frac{b^2}{2 a \left (a^2+b^2\right ) d e \sqrt{e \cot (c+d x)} (a+b \cot (c+d x))^2}-\frac{b^2 \left (13 a^2+5 b^2\right )}{4 a^2 \left (a^2+b^2\right )^2 d e \sqrt{e \cot (c+d x)} (a+b \cot (c+d x))}+\frac{(a+b) \left (a^2-4 a b+b^2\right ) \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 )^3 d e^{3/2}}-\frac{(a+b) \left (a^2-4 a b+b^2\right ) \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 )^3 d e^{3/2}}\\ \end{align*}
Mathematica [C] time = 1.6798, size = 303, normalized size = 0.57 \[ -\frac{-8 a^4 \left (a^2-3 b^2\right ) \text{Hypergeometric2F1}\left (-\frac{1}{4},1,\frac{3}{4},-\cot ^2(c+d x)\right )-8 a^2 b^2 \left (3 a^2-b^2\right ) \text{Hypergeometric2F1}\left (-\frac{1}{2},1,\frac{1}{2},-\frac{b \cot (c+d x)}{a}\right )-16 a^2 b^2 \left (a^2+b^2\right ) \text{Hypergeometric2F1}\left (-\frac{1}{2},2,\frac{1}{2},-\frac{b \cot (c+d x)}{a}\right )-8 b^2 \left (a^2+b^2\right )^2 \text{Hypergeometric2F1}\left (-\frac{1}{2},3,\frac{1}{2},-\frac{b \cot (c+d x)}{a}\right )+\sqrt{2} a^3 b \left (3 a^2-b^2\right ) \sqrt{\cot (c+d x)} \left (\log \left (\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right )-\log \left (\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right )+2 \tan ^{-1}\left (1-\sqrt{2} \sqrt{\cot (c+d x)}\right )-2 \tan ^{-1}\left (\sqrt{2} \sqrt{\cot (c+d x)}+1\right )\right )}{4 a^3 d e \left (a^2+b^2\right )^3 \sqrt{e \cot (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.059, size = 1245, 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(-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 \cot \left (d x + c\right ) + a\right )}^{3} \left (e \cot \left (d x + c\right )\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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