Optimal. Leaf size=399 \[ \frac{2 b \left (30 a^2 A b^3+8 a^4 A b-17 a^3 b^2 B-3 a^5 B-8 a b^4 B+16 A b^5\right )}{3 a^4 d \left (a^2+b^2\right )^2 \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}+\frac{2 b \left (7 a^2 A b-3 a^3 B-4 a b^2 B+8 A b^3\right )}{3 a^3 d \left (a^2+b^2\right ) \sqrt{\cot (c+d x)} (a+b \tan (c+d x))^{3/2}}+\frac{2 (2 A b-a B) \sqrt{\cot (c+d x)}}{a^2 d (a+b \tan (c+d x))^{3/2}}+\frac{(A+i B) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left (\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{d (-b+i a)^{5/2}}+\frac{(A-i B) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left (\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{d (b+i a)^{5/2}}-\frac{2 A \cot ^{\frac{3}{2}}(c+d x)}{3 a d (a+b \tan (c+d x))^{3/2}} \]
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Rubi [A] time = 1.7451, antiderivative size = 399, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 8, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.229, Rules used = {4241, 3609, 3649, 3616, 3615, 93, 203, 206} \[ \frac{2 b \left (30 a^2 A b^3+8 a^4 A b-17 a^3 b^2 B-3 a^5 B-8 a b^4 B+16 A b^5\right )}{3 a^4 d \left (a^2+b^2\right )^2 \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}+\frac{2 b \left (7 a^2 A b-3 a^3 B-4 a b^2 B+8 A b^3\right )}{3 a^3 d \left (a^2+b^2\right ) \sqrt{\cot (c+d x)} (a+b \tan (c+d x))^{3/2}}+\frac{2 (2 A b-a B) \sqrt{\cot (c+d x)}}{a^2 d (a+b \tan (c+d x))^{3/2}}+\frac{(A+i B) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left (\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{d (-b+i a)^{5/2}}+\frac{(A-i B) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left (\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{d (b+i a)^{5/2}}-\frac{2 A \cot ^{\frac{3}{2}}(c+d x)}{3 a d (a+b \tan (c+d x))^{3/2}} \]
Antiderivative was successfully verified.
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Rule 4241
Rule 3609
Rule 3649
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))}{(a+b \tan (c+d x))^{5/2}} \, dx &=\left (\sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \int \frac{A+B \tan (c+d x)}{\tan ^{\frac{5}{2}}(c+d x) (a+b \tan (c+d x))^{5/2}} \, dx\\ &=-\frac{2 A \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{\frac{3}{2} (2 A b-a B)+\frac{3}{2} a A \tan (c+d x)+3 A 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{2 (2 A b-a B) \sqrt{\cot (c+d x)}}{a^2 d (a+b \tan (c+d x))^{3/2}}-\frac{2 A \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 A-8 A b^2+4 a b B\right )-\frac{3}{4} a^2 B \tan (c+d x)+3 b (2 A b-a B) \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 \left (7 a^2 A b+8 A b^3-3 a^3 B-4 a b^2 B\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{2 (2 A b-a B) \sqrt{\cot (c+d x)}}{a^2 d (a+b \tan (c+d x))^{3/2}}-\frac{2 A \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 A-14 a^2 A b^2-16 A b^4+9 a^3 b B+8 a b^3 B\right )+\frac{9}{8} a^3 (A b-a B) \tan (c+d x)+\frac{3}{4} b \left (7 a^2 A b+8 A b^3-3 a^3 B-4 a b^2 B\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 \left (7 a^2 A b+8 A b^3-3 a^3 B-4 a b^2 B\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{2 (2 A b-a B) \sqrt{\cot (c+d x)}}{a^2 d (a+b \tan (c+d x))^{3/2}}-\frac{2 A \cot ^{\frac{3}{2}}(c+d x)}{3 a d (a+b \tan (c+d x))^{3/2}}+\frac{2 b \left (8 a^4 A b+30 a^2 A b^3+16 A b^5-3 a^5 B-17 a^3 b^2 B-8 a b^4 B\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 A-A b^2+2 a b B\right )+\frac{9}{16} a^4 \left (2 a A b-a^2 B+b^2 B\right ) \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 \left (7 a^2 A b+8 A b^3-3 a^3 B-4 a b^2 B\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{2 (2 A b-a B) \sqrt{\cot (c+d x)}}{a^2 d (a+b \tan (c+d x))^{3/2}}-\frac{2 A \cot ^{\frac{3}{2}}(c+d x)}{3 a d (a+b \tan (c+d x))^{3/2}}+\frac{2 b \left (8 a^4 A b+30 a^2 A b^3+16 A b^5-3 a^5 B-17 a^3 b^2 B-8 a b^4 B\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 (A-i B) \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 (A+i B) \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 \left (7 a^2 A b+8 A b^3-3 a^3 B-4 a b^2 B\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{2 (2 A b-a B) \sqrt{\cot (c+d x)}}{a^2 d (a+b \tan (c+d x))^{3/2}}-\frac{2 A \cot ^{\frac{3}{2}}(c+d x)}{3 a d (a+b \tan (c+d x))^{3/2}}+\frac{2 b \left (8 a^4 A b+30 a^2 A b^3+16 A b^5-3 a^5 B-17 a^3 b^2 B-8 a b^4 B\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 (A-i B) \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 (A+i B) \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 \left (7 a^2 A b+8 A b^3-3 a^3 B-4 a b^2 B\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{2 (2 A b-a B) \sqrt{\cot (c+d x)}}{a^2 d (a+b \tan (c+d x))^{3/2}}-\frac{2 A \cot ^{\frac{3}{2}}(c+d x)}{3 a d (a+b \tan (c+d x))^{3/2}}+\frac{2 b \left (8 a^4 A b+30 a^2 A b^3+16 A b^5-3 a^5 B-17 a^3 b^2 B-8 a b^4 B\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 (A-i B) \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 (A+i B) \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{(A+i B) \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{(A-i B) \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 \left (7 a^2 A b+8 A b^3-3 a^3 B-4 a b^2 B\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{2 (2 A b-a B) \sqrt{\cot (c+d x)}}{a^2 d (a+b \tan (c+d x))^{3/2}}-\frac{2 A \cot ^{\frac{3}{2}}(c+d x)}{3 a d (a+b \tan (c+d x))^{3/2}}+\frac{2 b \left (8 a^4 A b+30 a^2 A b^3+16 A b^5-3 a^5 B-17 a^3 b^2 B-8 a b^4 B\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 = 3.91508, size = 385, normalized size = 0.96 \[ \frac{\sqrt{\cot (c+d x)} \left (\frac{6 b \left (7 a^2 A b-3 a^3 B-4 a b^2 B+8 A b^3\right ) \tan (c+d x)}{a^2 \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}+\frac{\sqrt{\tan (c+d x)} \left (\frac{6 b \left (30 a^2 A b^3+8 a^4 A b-17 a^3 b^2 B-3 a^5 B-8 a b^4 B+16 A b^5\right ) \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}+9 (-1)^{3/4} a^4 \left (\frac{(a-i b)^2 (A+i B) \tanh ^{-1}\left (\frac{\sqrt [4]{-1} \sqrt{-a-i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{\sqrt{-a-i b}}+\frac{(a+i b)^2 (A-i B) \tanh ^{-1}\left (\frac{\sqrt [4]{-1} \sqrt{a-i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{\sqrt{a-i b}}\right )\right )}{a^3 \left (a^2+b^2\right )^2}+\frac{6 (6 A b-3 a B)}{a (a+b \tan (c+d x))^{3/2}}-\frac{6 A \cot (c+d x)}{(a+b \tan (c+d x))^{3/2}}\right )}{9 a d} \]
Antiderivative was successfully verified.
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Maple [C] time = 3.816, size = 80979, normalized size = 203. \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{{\left (B \tan \left (d x + c\right ) + A\right )} \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{{\left (B \tan \left (d x + c\right ) + A\right )} \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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