Optimal. Leaf size=213 \[ -\frac{2 a \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{3 d}+\frac{(-b+i a)^{3/2} \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left (\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{d}-\frac{8 b \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}{3 d}+\frac{(b+i a)^{3/2} \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left (\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{d} \]
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Rubi [A] time = 0.725429, antiderivative size = 213, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.32, Rules used = {4241, 3567, 3649, 3616, 3615, 93, 203, 206} \[ -\frac{2 a \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{3 d}+\frac{(-b+i a)^{3/2} \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left (\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{d}-\frac{8 b \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}{3 d}+\frac{(b+i a)^{3/2} \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left (\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{d} \]
Antiderivative was successfully verified.
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Rule 4241
Rule 3567
Rule 3649
Rule 3616
Rule 3615
Rule 93
Rule 203
Rule 206
Rubi steps
\begin{align*} \int \cot ^{\frac{5}{2}}(c+d x) (a+b \tan (c+d x))^{3/2} \, dx &=\left (\sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \int \frac{(a+b \tan (c+d x))^{3/2}}{\tan ^{\frac{5}{2}}(c+d x)} \, dx\\ &=-\frac{2 a \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{3 d}-\frac{1}{3} \left (2 \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \int \frac{-2 a b+\frac{3}{2} \left (a^2-b^2\right ) \tan (c+d x)+a b \tan ^2(c+d x)}{\tan ^{\frac{3}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}} \, dx\\ &=-\frac{8 b \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}{3 d}-\frac{2 a \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{3 d}+\frac{\left (4 \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \int \frac{-\frac{3}{4} a \left (a^2-b^2\right )-\frac{3}{2} a^2 b \tan (c+d x)}{\sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)}} \, dx}{3 a}\\ &=-\frac{8 b \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}{3 d}-\frac{2 a \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{3 d}-\frac{1}{2} \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-\frac{1}{2} \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\\ &=-\frac{8 b \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}{3 d}-\frac{2 a \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{3 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 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 d}\\ &=-\frac{8 b \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}{3 d}-\frac{2 a \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{3 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 )}{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 )}{d}\\ &=\frac{(i a-b)^{3/2} \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)^{3/2} \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{8 b \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}{3 d}-\frac{2 a \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{3 d}\\ \end{align*}
Mathematica [A] time = 0.991805, size = 189, normalized size = 0.89 \[ \frac{\cot ^{\frac{3}{2}}(c+d x) \left (-2 \sqrt{a+b \tan (c+d x)} (a+4 b \tan (c+d x))+3 (-1)^{3/4} (-a-i b)^{3/2} \tan ^{\frac{3}{2}}(c+d x) \tanh ^{-1}\left (\frac{\sqrt [4]{-1} \sqrt{-a-i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )+3 (-1)^{3/4} (a-i b)^{3/2} \tan ^{\frac{3}{2}}(c+d x) \tanh ^{-1}\left (\frac{\sqrt [4]{-1} \sqrt{a-i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )\right )}{3 d} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.633, size = 12573, normalized size = 59. \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{3}{2}} \cot \left (d x + c\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*} \mathit{sage}_{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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