Optimal. Leaf size=358 \[ \frac{2 \left (21 a^2-25 b^2\right ) \cot ^{\frac{5}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{105 d}+\frac{2 b \left (231 a^2-5 b^2\right ) \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{315 a d}-\frac{2 \left (-483 a^2 b^2+315 a^4-10 b^4\right ) \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}{315 a^2 d}-\frac{2 a^2 \cot ^{\frac{9}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{9 d}-\frac{38 a b \cot ^{\frac{7}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{63 d}+\frac{(-b+i a)^{5/2} \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left (\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{d}-\frac{(b+i a)^{5/2} \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left (\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{d} \]
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Rubi [A] time = 1.6715, antiderivative size = 358, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 8, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.32, Rules used = {4241, 3565, 3649, 3616, 3615, 93, 203, 206} \[ \frac{2 \left (21 a^2-25 b^2\right ) \cot ^{\frac{5}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{105 d}+\frac{2 b \left (231 a^2-5 b^2\right ) \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{315 a d}-\frac{2 \left (-483 a^2 b^2+315 a^4-10 b^4\right ) \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}{315 a^2 d}-\frac{2 a^2 \cot ^{\frac{9}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{9 d}-\frac{38 a b \cot ^{\frac{7}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{63 d}+\frac{(-b+i a)^{5/2} \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left (\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{d}-\frac{(b+i a)^{5/2} \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left (\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{d} \]
Antiderivative was successfully verified.
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Rule 4241
Rule 3565
Rule 3649
Rule 3616
Rule 3615
Rule 93
Rule 203
Rule 206
Rubi steps
\begin{align*} \int \cot ^{\frac{11}{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{(a+b \tan (c+d x))^{5/2}}{\tan ^{\frac{11}{2}}(c+d x)} \, dx\\ &=-\frac{2 a^2 \cot ^{\frac{9}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{9 d}+\frac{1}{9} \left (2 \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \int \frac{\frac{19 a^2 b}{2}-\frac{9}{2} a \left (a^2-3 b^2\right ) \tan (c+d x)-\frac{1}{2} b \left (8 a^2-9 b^2\right ) \tan ^2(c+d x)}{\tan ^{\frac{9}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}} \, dx\\ &=-\frac{38 a b \cot ^{\frac{7}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{63 d}-\frac{2 a^2 \cot ^{\frac{9}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{9 d}-\frac{\left (4 \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \int \frac{\frac{3}{4} a^2 \left (21 a^2-25 b^2\right )+\frac{63}{4} a b \left (3 a^2-b^2\right ) \tan (c+d x)+\frac{57}{2} a^2 b^2 \tan ^2(c+d x)}{\tan ^{\frac{7}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}} \, dx}{63 a}\\ &=\frac{2 \left (21 a^2-25 b^2\right ) \cot ^{\frac{5}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{105 d}-\frac{38 a b \cot ^{\frac{7}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{63 d}-\frac{2 a^2 \cot ^{\frac{9}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{9 d}+\frac{\left (8 \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \int \frac{-\frac{3}{8} a^2 b \left (231 a^2-5 b^2\right )+\frac{315}{8} a^3 \left (a^2-3 b^2\right ) \tan (c+d x)+\frac{3}{2} a^2 b \left (21 a^2-25 b^2\right ) \tan ^2(c+d x)}{\tan ^{\frac{5}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}} \, dx}{315 a^2}\\ &=\frac{2 b \left (231 a^2-5 b^2\right ) \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{315 a d}+\frac{2 \left (21 a^2-25 b^2\right ) \cot ^{\frac{5}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{105 d}-\frac{38 a b \cot ^{\frac{7}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{63 d}-\frac{2 a^2 \cot ^{\frac{9}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{9 d}-\frac{\left (16 \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \int \frac{-\frac{3}{16} a^2 \left (315 a^4-483 a^2 b^2-10 b^4\right )-\frac{945}{16} a^3 b \left (3 a^2-b^2\right ) \tan (c+d x)-\frac{3}{8} a^2 b^2 \left (231 a^2-5 b^2\right ) \tan ^2(c+d x)}{\tan ^{\frac{3}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}} \, dx}{945 a^3}\\ &=-\frac{2 \left (315 a^4-483 a^2 b^2-10 b^4\right ) \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}{315 a^2 d}+\frac{2 b \left (231 a^2-5 b^2\right ) \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{315 a d}+\frac{2 \left (21 a^2-25 b^2\right ) \cot ^{\frac{5}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{105 d}-\frac{38 a b \cot ^{\frac{7}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{63 d}-\frac{2 a^2 \cot ^{\frac{9}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{9 d}+\frac{\left (32 \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \int \frac{\frac{945}{32} a^4 b \left (3 a^2-b^2\right )-\frac{945}{32} a^5 \left (a^2-3 b^2\right ) \tan (c+d x)}{\sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)}} \, dx}{945 a^4}\\ &=-\frac{2 \left (315 a^4-483 a^2 b^2-10 b^4\right ) \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}{315 a^2 d}+\frac{2 b \left (231 a^2-5 b^2\right ) \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{315 a d}+\frac{2 \left (21 a^2-25 b^2\right ) \cot ^{\frac{5}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{105 d}-\frac{38 a b \cot ^{\frac{7}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{63 d}-\frac{2 a^2 \cot ^{\frac{9}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{9 d}+\frac{1}{2} \left ((i a-b)^3 \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-\frac{1}{2} \left ((i a+b)^3 \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\\ &=-\frac{2 \left (315 a^4-483 a^2 b^2-10 b^4\right ) \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}{315 a^2 d}+\frac{2 b \left (231 a^2-5 b^2\right ) \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{315 a d}+\frac{2 \left (21 a^2-25 b^2\right ) \cot ^{\frac{5}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{105 d}-\frac{38 a b \cot ^{\frac{7}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{63 d}-\frac{2 a^2 \cot ^{\frac{9}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{9 d}+\frac{\left ((i a-b)^3 \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 d}-\frac{\left ((i a+b)^3 \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 d}\\ &=-\frac{2 \left (315 a^4-483 a^2 b^2-10 b^4\right ) \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}{315 a^2 d}+\frac{2 b \left (231 a^2-5 b^2\right ) \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{315 a d}+\frac{2 \left (21 a^2-25 b^2\right ) \cot ^{\frac{5}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{105 d}-\frac{38 a b \cot ^{\frac{7}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{63 d}-\frac{2 a^2 \cot ^{\frac{9}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{9 d}+\frac{\left ((i a-b)^3 \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 )}{d}-\frac{\left ((i a+b)^3 \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 )}{d}\\ &=\frac{(i a-b)^{5/2} \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)}}{d}-\frac{(i a+b)^{5/2} \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)}}{d}-\frac{2 \left (315 a^4-483 a^2 b^2-10 b^4\right ) \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}{315 a^2 d}+\frac{2 b \left (231 a^2-5 b^2\right ) \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{315 a d}+\frac{2 \left (21 a^2-25 b^2\right ) \cot ^{\frac{5}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{105 d}-\frac{38 a b \cot ^{\frac{7}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{63 d}-\frac{2 a^2 \cot ^{\frac{9}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{9 d}\\ \end{align*}
Mathematica [A] time = 5.22063, size = 329, normalized size = 0.92 \[ \frac{\cot ^{\frac{9}{2}}(c+d x) \left (\frac{1}{4} \sec ^4(c+d x) \sqrt{a+b \tan (c+d x)} \left (4 \left (-483 a^2 b^2+280 a^4-10 b^4\right ) \cos (2 (c+d x))+\left (558 a^2 b^2-413 a^4+10 b^4\right ) \cos (4 (c+d x))+1374 a^2 b^2+272 a^3 b \sin (2 (c+d x))-326 a^3 b \sin (4 (c+d x))-987 a^4-10 a b^3 \sin (2 (c+d x))+5 a b^3 \sin (4 (c+d x))+30 b^4\right )+315 \sqrt [4]{-1} a^2 (-a-i b)^{5/2} \tan ^{\frac{9}{2}}(c+d x) \tanh ^{-1}\left (\frac{\sqrt [4]{-1} \sqrt{-a-i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )+315 \sqrt [4]{-1} a^2 (a-i b)^{5/2} \tan ^{\frac{9}{2}}(c+d x) \tanh ^{-1}\left (\frac{\sqrt [4]{-1} \sqrt{a-i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )\right )}{315 a^2 d} \]
Antiderivative was successfully verified.
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Maple [C] time = 1.642, size = 49870, normalized size = 139.3 \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{\left (b \tan \left (d x + c\right ) + a\right )}^{\frac{5}{2}} \cot \left (d x + c\right )^{\frac{11}{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(-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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