Optimal. Leaf size=398 \[ \frac{b^2 \cot ^{\frac{5}{2}}(c+d x)}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}-\frac{\left (2 a^2+5 b^2\right ) \cot ^{\frac{3}{2}}(c+d x)}{3 a^2 d \left (a^2+b^2\right )}+\frac{b \left (4 a^2+5 b^2\right ) \sqrt{\cot (c+d x)}}{a^3 d \left (a^2+b^2\right )}+\frac{\left (a^2+2 a b-b^2\right ) \log \left (\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right )}{2 \sqrt{2} d \left (a^2+b^2\right )^2}-\frac{\left (a^2+2 a b-b^2\right ) \log \left (\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right )}{2 \sqrt{2} d \left (a^2+b^2\right )^2}-\frac{b^{7/2} \left (9 a^2+5 b^2\right ) \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{\cot (c+d x)}}{\sqrt{b}}\right )}{a^{7/2} d \left (a^2+b^2\right )^2}-\frac{\left (a^2-2 a b-b^2\right ) \tan ^{-1}\left (1-\sqrt{2} \sqrt{\cot (c+d x)}\right )}{\sqrt{2} d \left (a^2+b^2\right )^2}+\frac{\left (a^2-2 a b-b^2\right ) \tan ^{-1}\left (\sqrt{2} \sqrt{\cot (c+d x)}+1\right )}{\sqrt{2} d \left (a^2+b^2\right )^2} \]
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Rubi [A] time = 1.09746, antiderivative size = 398, normalized size of antiderivative = 1., number of steps used = 18, number of rules used = 14, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.609, Rules used = {3673, 3565, 3647, 3653, 3534, 1168, 1162, 617, 204, 1165, 628, 3634, 63, 205} \[ \frac{b^2 \cot ^{\frac{5}{2}}(c+d x)}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}-\frac{\left (2 a^2+5 b^2\right ) \cot ^{\frac{3}{2}}(c+d x)}{3 a^2 d \left (a^2+b^2\right )}+\frac{b \left (4 a^2+5 b^2\right ) \sqrt{\cot (c+d x)}}{a^3 d \left (a^2+b^2\right )}+\frac{\left (a^2+2 a b-b^2\right ) \log \left (\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right )}{2 \sqrt{2} d \left (a^2+b^2\right )^2}-\frac{\left (a^2+2 a b-b^2\right ) \log \left (\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right )}{2 \sqrt{2} d \left (a^2+b^2\right )^2}-\frac{b^{7/2} \left (9 a^2+5 b^2\right ) \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{\cot (c+d x)}}{\sqrt{b}}\right )}{a^{7/2} d \left (a^2+b^2\right )^2}-\frac{\left (a^2-2 a b-b^2\right ) \tan ^{-1}\left (1-\sqrt{2} \sqrt{\cot (c+d x)}\right )}{\sqrt{2} d \left (a^2+b^2\right )^2}+\frac{\left (a^2-2 a b-b^2\right ) \tan ^{-1}\left (\sqrt{2} \sqrt{\cot (c+d x)}+1\right )}{\sqrt{2} d \left (a^2+b^2\right )^2} \]
Antiderivative was successfully verified.
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Rule 3673
Rule 3565
Rule 3647
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{\cot ^{\frac{5}{2}}(c+d x)}{(a+b \tan (c+d x))^2} \, dx &=\int \frac{\cot ^{\frac{9}{2}}(c+d x)}{(b+a \cot (c+d x))^2} \, dx\\ &=\frac{b^2 \cot ^{\frac{5}{2}}(c+d x)}{a \left (a^2+b^2\right ) d (b+a \cot (c+d x))}-\frac{\int \frac{\cot ^{\frac{3}{2}}(c+d x) \left (-\frac{5 b^2}{2}+a b \cot (c+d x)-\frac{1}{2} \left (2 a^2+5 b^2\right ) \cot ^2(c+d x)\right )}{b+a \cot (c+d x)} \, dx}{a \left (a^2+b^2\right )}\\ &=-\frac{\left (2 a^2+5 b^2\right ) \cot ^{\frac{3}{2}}(c+d x)}{3 a^2 \left (a^2+b^2\right ) d}+\frac{b^2 \cot ^{\frac{5}{2}}(c+d x)}{a \left (a^2+b^2\right ) d (b+a \cot (c+d x))}+\frac{2 \int \frac{\sqrt{\cot (c+d x)} \left (-\frac{3}{4} b \left (2 a^2+5 b^2\right )-\frac{3}{2} a^3 \cot (c+d x)-\frac{3}{4} b \left (4 a^2+5 b^2\right ) \cot ^2(c+d x)\right )}{b+a \cot (c+d x)} \, dx}{3 a^2 \left (a^2+b^2\right )}\\ &=\frac{b \left (4 a^2+5 b^2\right ) \sqrt{\cot (c+d x)}}{a^3 \left (a^2+b^2\right ) d}-\frac{\left (2 a^2+5 b^2\right ) \cot ^{\frac{3}{2}}(c+d x)}{3 a^2 \left (a^2+b^2\right ) d}+\frac{b^2 \cot ^{\frac{5}{2}}(c+d x)}{a \left (a^2+b^2\right ) d (b+a \cot (c+d x))}-\frac{4 \int \frac{-\frac{3}{8} b^2 \left (4 a^2+5 b^2\right )-\frac{3}{4} a^3 b \cot (c+d x)+\frac{3}{8} \left (2 a^4-4 a^2 b^2-5 b^4\right ) \cot ^2(c+d x)}{\sqrt{\cot (c+d x)} (b+a \cot (c+d x))} \, dx}{3 a^3 \left (a^2+b^2\right )}\\ &=\frac{b \left (4 a^2+5 b^2\right ) \sqrt{\cot (c+d x)}}{a^3 \left (a^2+b^2\right ) d}-\frac{\left (2 a^2+5 b^2\right ) \cot ^{\frac{3}{2}}(c+d x)}{3 a^2 \left (a^2+b^2\right ) d}+\frac{b^2 \cot ^{\frac{5}{2}}(c+d x)}{a \left (a^2+b^2\right ) d (b+a \cot (c+d x))}-\frac{4 \int \frac{-\frac{3 a^4 b}{2}+\frac{3}{4} a^3 \left (a^2-b^2\right ) \cot (c+d x)}{\sqrt{\cot (c+d x)}} \, dx}{3 a^3 \left (a^2+b^2\right )^2}+\frac{\left (b^4 \left (9 a^2+5 b^2\right )\right ) \int \frac{1+\cot ^2(c+d x)}{\sqrt{\cot (c+d x)} (b+a \cot (c+d x))} \, dx}{2 a^3 \left (a^2+b^2\right )^2}\\ &=\frac{b \left (4 a^2+5 b^2\right ) \sqrt{\cot (c+d x)}}{a^3 \left (a^2+b^2\right ) d}-\frac{\left (2 a^2+5 b^2\right ) \cot ^{\frac{3}{2}}(c+d x)}{3 a^2 \left (a^2+b^2\right ) d}+\frac{b^2 \cot ^{\frac{5}{2}}(c+d x)}{a \left (a^2+b^2\right ) d (b+a \cot (c+d x))}-\frac{8 \operatorname{Subst}\left (\int \frac{\frac{3 a^4 b}{2}-\frac{3}{4} a^3 \left (a^2-b^2\right ) x^2}{1+x^4} \, dx,x,\sqrt{\cot (c+d x)}\right )}{3 a^3 \left (a^2+b^2\right )^2 d}+\frac{\left (b^4 \left (9 a^2+5 b^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{-x} (b-a x)} \, dx,x,-\cot (c+d x)\right )}{2 a^3 \left (a^2+b^2\right )^2 d}\\ &=\frac{b \left (4 a^2+5 b^2\right ) \sqrt{\cot (c+d x)}}{a^3 \left (a^2+b^2\right ) d}-\frac{\left (2 a^2+5 b^2\right ) \cot ^{\frac{3}{2}}(c+d x)}{3 a^2 \left (a^2+b^2\right ) d}+\frac{b^2 \cot ^{\frac{5}{2}}(c+d x)}{a \left (a^2+b^2\right ) d (b+a \cot (c+d x))}+\frac{\left (a^2-2 a b-b^2\right ) \operatorname{Subst}\left (\int \frac{1+x^2}{1+x^4} \, dx,x,\sqrt{\cot (c+d x)}\right )}{\left (a^2+b^2\right )^2 d}-\frac{\left (a^2+2 a b-b^2\right ) \operatorname{Subst}\left (\int \frac{1-x^2}{1+x^4} \, dx,x,\sqrt{\cot (c+d x)}\right )}{\left (a^2+b^2\right )^2 d}-\frac{\left (b^4 \left (9 a^2+5 b^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{b+a x^2} \, dx,x,\sqrt{\cot (c+d x)}\right )}{a^3 \left (a^2+b^2\right )^2 d}\\ &=-\frac{b^{7/2} \left (9 a^2+5 b^2\right ) \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{\cot (c+d x)}}{\sqrt{b}}\right )}{a^{7/2} \left (a^2+b^2\right )^2 d}+\frac{b \left (4 a^2+5 b^2\right ) \sqrt{\cot (c+d x)}}{a^3 \left (a^2+b^2\right ) d}-\frac{\left (2 a^2+5 b^2\right ) \cot ^{\frac{3}{2}}(c+d x)}{3 a^2 \left (a^2+b^2\right ) d}+\frac{b^2 \cot ^{\frac{5}{2}}(c+d x)}{a \left (a^2+b^2\right ) d (b+a \cot (c+d x))}+\frac{\left (a^2-2 a b-b^2\right ) \operatorname{Subst}\left (\int \frac{1}{1-\sqrt{2} x+x^2} \, dx,x,\sqrt{\cot (c+d x)}\right )}{2 \left (a^2+b^2\right )^2 d}+\frac{\left (a^2-2 a b-b^2\right ) \operatorname{Subst}\left (\int \frac{1}{1+\sqrt{2} x+x^2} \, dx,x,\sqrt{\cot (c+d x)}\right )}{2 \left (a^2+b^2\right )^2 d}+\frac{\left (a^2+2 a b-b^2\right ) \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} \left (a^2+b^2\right )^2 d}+\frac{\left (a^2+2 a b-b^2\right ) \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} \left (a^2+b^2\right )^2 d}\\ &=-\frac{b^{7/2} \left (9 a^2+5 b^2\right ) \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{\cot (c+d x)}}{\sqrt{b}}\right )}{a^{7/2} \left (a^2+b^2\right )^2 d}+\frac{b \left (4 a^2+5 b^2\right ) \sqrt{\cot (c+d x)}}{a^3 \left (a^2+b^2\right ) d}-\frac{\left (2 a^2+5 b^2\right ) \cot ^{\frac{3}{2}}(c+d x)}{3 a^2 \left (a^2+b^2\right ) d}+\frac{b^2 \cot ^{\frac{5}{2}}(c+d x)}{a \left (a^2+b^2\right ) d (b+a \cot (c+d x))}+\frac{\left (a^2+2 a b-b^2\right ) \log \left (1-\sqrt{2} \sqrt{\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt{2} \left (a^2+b^2\right )^2 d}-\frac{\left (a^2+2 a b-b^2\right ) \log \left (1+\sqrt{2} \sqrt{\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt{2} \left (a^2+b^2\right )^2 d}+\frac{\left (a^2-2 a b-b^2\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\sqrt{2} \sqrt{\cot (c+d x)}\right )}{\sqrt{2} \left (a^2+b^2\right )^2 d}-\frac{\left (a^2-2 a b-b^2\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\sqrt{2} \sqrt{\cot (c+d x)}\right )}{\sqrt{2} \left (a^2+b^2\right )^2 d}\\ &=-\frac{\left (a^2-2 a b-b^2\right ) \tan ^{-1}\left (1-\sqrt{2} \sqrt{\cot (c+d x)}\right )}{\sqrt{2} \left (a^2+b^2\right )^2 d}+\frac{\left (a^2-2 a b-b^2\right ) \tan ^{-1}\left (1+\sqrt{2} \sqrt{\cot (c+d x)}\right )}{\sqrt{2} \left (a^2+b^2\right )^2 d}-\frac{b^{7/2} \left (9 a^2+5 b^2\right ) \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{\cot (c+d x)}}{\sqrt{b}}\right )}{a^{7/2} \left (a^2+b^2\right )^2 d}+\frac{b \left (4 a^2+5 b^2\right ) \sqrt{\cot (c+d x)}}{a^3 \left (a^2+b^2\right ) d}-\frac{\left (2 a^2+5 b^2\right ) \cot ^{\frac{3}{2}}(c+d x)}{3 a^2 \left (a^2+b^2\right ) d}+\frac{b^2 \cot ^{\frac{5}{2}}(c+d x)}{a \left (a^2+b^2\right ) d (b+a \cot (c+d x))}+\frac{\left (a^2+2 a b-b^2\right ) \log \left (1-\sqrt{2} \sqrt{\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt{2} \left (a^2+b^2\right )^2 d}-\frac{\left (a^2+2 a b-b^2\right ) \log \left (1+\sqrt{2} \sqrt{\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt{2} \left (a^2+b^2\right )^2 d}\\ \end{align*}
Mathematica [C] time = 6.16817, size = 455, normalized size = 1.14 \[ -\frac{\frac{2 a^2 \cot ^{\frac{11}{2}}(c+d x) \, _2F_1\left (2,\frac{11}{2};\frac{13}{2};-\frac{a \cot (c+d x)}{b}\right )}{11 b^2 \left (a^2+b^2\right )}+\frac{2 \left (a^2-b^2\right ) \left (-7 \cot ^{\frac{3}{2}}(c+d x) \, _2F_1\left (\frac{3}{4},1;\frac{7}{4};-\cot ^2(c+d x)\right )-3 \cot ^{\frac{7}{2}}(c+d x)+7 \cot ^{\frac{3}{2}}(c+d x)\right )}{21 \left (a^2+b^2\right )^2}+\frac{4 a b \cot ^{\frac{9}{2}}(c+d x)}{9 \left (a^2+b^2\right )^2}-\frac{4 b^2 \left (15 \cot ^{\frac{7}{2}}(c+d x)-7 b \left (\frac{3 \cot ^{\frac{5}{2}}(c+d x)}{a}-\frac{5 b \left (\frac{\cot ^{\frac{3}{2}}(c+d x)}{a}-\frac{3 b \left (\frac{\sqrt{\cot (c+d x)}}{a}-\frac{\sqrt{b} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{\cot (c+d x)}}{\sqrt{b}}\right )}{a^{3/2}}\right )}{a}\right )}{a}\right )\right )}{105 \left (a^2+b^2\right )^2}-\frac{a b \left (40 \cot ^{\frac{9}{2}}(c+d x)-72 \cot ^{\frac{5}{2}}(c+d x)+360 \sqrt{\cot (c+d x)}+45 \left (\sqrt{2} \log \left (\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right )-\sqrt{2} \log \left (\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right )+2 \left (\sqrt{2} \tan ^{-1}\left (1-\sqrt{2} \sqrt{\cot (c+d x)}\right )-\sqrt{2} \tan ^{-1}\left (\sqrt{2} \sqrt{\cot (c+d x)}+1\right )\right )\right )\right )}{90 \left (a^2+b^2\right )^2}}{d} \]
Antiderivative was successfully verified.
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Maple [C] time = 1.048, size = 37365, normalized size = 93.9 \begin{align*} \text{output 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{\cot \left (d x + c\right )^{\frac{5}{2}}}{{\left (b \tan \left (d x + c\right ) + a\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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