Optimal. Leaf size=338 \[ \frac{4 b^2 \left (15 a^2 b^2+4 a^4+8 b^4\right )}{3 a^4 d \left (a^2+b^2\right )^2 \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}+\frac{2 b^2 \left (7 a^2+8 b^2\right )}{3 a^3 d \left (a^2+b^2\right ) \sqrt{\cot (c+d x)} (a+b \tan (c+d x))^{3/2}}+\frac{4 b \sqrt{\cot (c+d x)}}{a^2 d (a+b \tan (c+d x))^{3/2}}-\frac{2 \cot ^{\frac{3}{2}}(c+d x)}{3 a d (a+b \tan (c+d x))^{3/2}}+\frac{\sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left (\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{d (-b+i a)^{5/2}}+\frac{\sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left (\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{d (b+i a)^{5/2}} \]
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Rubi [A] time = 1.2764, antiderivative size = 338, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.36, Rules used = {4241, 3569, 3649, 3650, 3616, 3615, 93, 203, 206} \[ \frac{4 b^2 \left (15 a^2 b^2+4 a^4+8 b^4\right )}{3 a^4 d \left (a^2+b^2\right )^2 \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}+\frac{2 b^2 \left (7 a^2+8 b^2\right )}{3 a^3 d \left (a^2+b^2\right ) \sqrt{\cot (c+d x)} (a+b \tan (c+d x))^{3/2}}+\frac{4 b \sqrt{\cot (c+d x)}}{a^2 d (a+b \tan (c+d x))^{3/2}}-\frac{2 \cot ^{\frac{3}{2}}(c+d x)}{3 a d (a+b \tan (c+d x))^{3/2}}+\frac{\sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left (\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{d (-b+i a)^{5/2}}+\frac{\sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left (\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{d (b+i a)^{5/2}} \]
Antiderivative was successfully verified.
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Rule 4241
Rule 3569
Rule 3649
Rule 3650
Rule 3616
Rule 3615
Rule 93
Rule 203
Rule 206
Rubi steps
\begin{align*} \int \frac{\cot ^{\frac{5}{2}}(c+d x)}{(a+b \tan (c+d x))^{5/2}} \, dx &=\left (\sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \int \frac{1}{\tan ^{\frac{5}{2}}(c+d x) (a+b \tan (c+d x))^{5/2}} \, dx\\ &=-\frac{2 \cot ^{\frac{3}{2}}(c+d x)}{3 a d (a+b \tan (c+d x))^{3/2}}-\frac{\left (2 \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \int \frac{3 b+\frac{3}{2} a \tan (c+d x)+3 b \tan ^2(c+d x)}{\tan ^{\frac{3}{2}}(c+d x) (a+b \tan (c+d x))^{5/2}} \, dx}{3 a}\\ &=\frac{4 b \sqrt{\cot (c+d x)}}{a^2 d (a+b \tan (c+d x))^{3/2}}-\frac{2 \cot ^{\frac{3}{2}}(c+d x)}{3 a d (a+b \tan (c+d x))^{3/2}}+\frac{\left (4 \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \int \frac{-\frac{3}{4} \left (a^2-8 b^2\right )+6 b^2 \tan ^2(c+d x)}{\sqrt{\tan (c+d x)} (a+b \tan (c+d x))^{5/2}} \, dx}{3 a^2}\\ &=\frac{2 b^2 \left (7 a^2+8 b^2\right )}{3 a^3 \left (a^2+b^2\right ) d \sqrt{\cot (c+d x)} (a+b \tan (c+d x))^{3/2}}+\frac{4 b \sqrt{\cot (c+d x)}}{a^2 d (a+b \tan (c+d x))^{3/2}}-\frac{2 \cot ^{\frac{3}{2}}(c+d x)}{3 a d (a+b \tan (c+d x))^{3/2}}+\frac{\left (8 \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \int \frac{-\frac{3}{8} \left (3 a^4-14 a^2 b^2-16 b^4\right )+\frac{9}{8} a^3 b \tan (c+d x)+\frac{3}{4} b^2 \left (7 a^2+8 b^2\right ) \tan ^2(c+d x)}{\sqrt{\tan (c+d x)} (a+b \tan (c+d x))^{3/2}} \, dx}{9 a^3 \left (a^2+b^2\right )}\\ &=\frac{2 b^2 \left (7 a^2+8 b^2\right )}{3 a^3 \left (a^2+b^2\right ) d \sqrt{\cot (c+d x)} (a+b \tan (c+d x))^{3/2}}+\frac{4 b \sqrt{\cot (c+d x)}}{a^2 d (a+b \tan (c+d x))^{3/2}}-\frac{2 \cot ^{\frac{3}{2}}(c+d x)}{3 a d (a+b \tan (c+d x))^{3/2}}+\frac{4 b^2 \left (4 a^4+15 a^2 b^2+8 b^4\right )}{3 a^4 \left (a^2+b^2\right )^2 d \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}+\frac{\left (16 \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \int \frac{-\frac{9}{16} a^4 \left (a^2-b^2\right )+\frac{9}{8} a^5 b \tan (c+d x)}{\sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)}} \, dx}{9 a^4 \left (a^2+b^2\right )^2}\\ &=\frac{2 b^2 \left (7 a^2+8 b^2\right )}{3 a^3 \left (a^2+b^2\right ) d \sqrt{\cot (c+d x)} (a+b \tan (c+d x))^{3/2}}+\frac{4 b \sqrt{\cot (c+d x)}}{a^2 d (a+b \tan (c+d x))^{3/2}}-\frac{2 \cot ^{\frac{3}{2}}(c+d x)}{3 a d (a+b \tan (c+d x))^{3/2}}+\frac{4 b^2 \left (4 a^4+15 a^2 b^2+8 b^4\right )}{3 a^4 \left (a^2+b^2\right )^2 d \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}-\frac{\left ((a-i b)^2 \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \int \frac{1-i \tan (c+d x)}{\sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)}} \, dx}{2 \left (a^2+b^2\right )^2}-\frac{\left ((a+i b)^2 \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \int \frac{1+i \tan (c+d x)}{\sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)}} \, dx}{2 \left (a^2+b^2\right )^2}\\ &=\frac{2 b^2 \left (7 a^2+8 b^2\right )}{3 a^3 \left (a^2+b^2\right ) d \sqrt{\cot (c+d x)} (a+b \tan (c+d x))^{3/2}}+\frac{4 b \sqrt{\cot (c+d x)}}{a^2 d (a+b \tan (c+d x))^{3/2}}-\frac{2 \cot ^{\frac{3}{2}}(c+d x)}{3 a d (a+b \tan (c+d x))^{3/2}}+\frac{4 b^2 \left (4 a^4+15 a^2 b^2+8 b^4\right )}{3 a^4 \left (a^2+b^2\right )^2 d \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}-\frac{\left ((a-i b)^2 \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \operatorname{Subst}\left (\int \frac{1}{(1+i x) \sqrt{x} \sqrt{a+b x}} \, dx,x,\tan (c+d x)\right )}{2 \left (a^2+b^2\right )^2 d}-\frac{\left ((a+i b)^2 \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \operatorname{Subst}\left (\int \frac{1}{(1-i x) \sqrt{x} \sqrt{a+b x}} \, dx,x,\tan (c+d x)\right )}{2 \left (a^2+b^2\right )^2 d}\\ &=\frac{2 b^2 \left (7 a^2+8 b^2\right )}{3 a^3 \left (a^2+b^2\right ) d \sqrt{\cot (c+d x)} (a+b \tan (c+d x))^{3/2}}+\frac{4 b \sqrt{\cot (c+d x)}}{a^2 d (a+b \tan (c+d x))^{3/2}}-\frac{2 \cot ^{\frac{3}{2}}(c+d x)}{3 a d (a+b \tan (c+d x))^{3/2}}+\frac{4 b^2 \left (4 a^4+15 a^2 b^2+8 b^4\right )}{3 a^4 \left (a^2+b^2\right )^2 d \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}-\frac{\left ((a-i b)^2 \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \operatorname{Subst}\left (\int \frac{1}{1-(-i a+b) x^2} \, dx,x,\frac{\sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{\left (a^2+b^2\right )^2 d}-\frac{\left ((a+i b)^2 \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \operatorname{Subst}\left (\int \frac{1}{1-(i a+b) x^2} \, dx,x,\frac{\sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{\left (a^2+b^2\right )^2 d}\\ &=\frac{\tan ^{-1}\left (\frac{\sqrt{i a-b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right ) \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}}{(i a-b)^{5/2} d}+\frac{\tanh ^{-1}\left (\frac{\sqrt{i a+b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right ) \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}}{(i a+b)^{5/2} d}+\frac{2 b^2 \left (7 a^2+8 b^2\right )}{3 a^3 \left (a^2+b^2\right ) d \sqrt{\cot (c+d x)} (a+b \tan (c+d x))^{3/2}}+\frac{4 b \sqrt{\cot (c+d x)}}{a^2 d (a+b \tan (c+d x))^{3/2}}-\frac{2 \cot ^{\frac{3}{2}}(c+d x)}{3 a d (a+b \tan (c+d x))^{3/2}}+\frac{4 b^2 \left (4 a^4+15 a^2 b^2+8 b^4\right )}{3 a^4 \left (a^2+b^2\right )^2 d \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}\\ \end{align*}
Mathematica [A] time = 6.28074, size = 504, normalized size = 1.49 \[ \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \left (-\frac{2}{3 a d \tan ^{\frac{3}{2}}(c+d x) (a+b \tan (c+d x))^{3/2}}-\frac{2 \left (-\frac{6 b}{a d \sqrt{\tan (c+d x)} (a+b \tan (c+d x))^{3/2}}-\frac{3 \left (\frac{32 b^2 \sqrt{\tan (c+d x)}}{3 a^2 \sqrt{a+b \tan (c+d x)}}-\frac{\frac{3 \sqrt [4]{-1} a^2 \tanh ^{-1}\left (\frac{\sqrt [4]{-1} \sqrt{a-i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{(a-i b)^{3/2}}+\frac{b (5 a-2 i b) \sqrt{\tan (c+d x)}}{(a-i b) \sqrt{a+b \tan (c+d x)}}}{3 (b+i a)}+\frac{\frac{3 \sqrt [4]{-1} a^2 \tanh ^{-1}\left (\frac{\sqrt [4]{-1} \sqrt{-a-i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{(-a-i b)^{3/2}}+\frac{b (5 a+2 i b) \sqrt{\tan (c+d x)}}{(a+i b) \sqrt{a+b \tan (c+d x)}}}{3 (-b+i a)}+\frac{16 b^2 \sqrt{\tan (c+d x)}}{3 a (a+b \tan (c+d x))^{3/2}}+\frac{a b \sqrt{\tan (c+d x)}}{3 (-b+i a) (a+b \tan (c+d x))^{3/2}}-\frac{a b \sqrt{\tan (c+d x)}}{3 (b+i a) (a+b \tan (c+d x))^{3/2}}\right )}{2 a d}\right )}{3 a}\right ) \]
Antiderivative was successfully verified.
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Maple [C] time = 1.939, size = 41237, normalized size = 122. \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] 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 )}^{\frac{5}{2}}}\,{d x} \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 )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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