Optimal. Leaf size=233 \[ -\frac{2 b \left (a^2+2 b^2\right )}{a^2 d \left (a^2+b^2\right ) \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}+\frac{\sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left (\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{d (-b+i a)^{3/2}}-\frac{2 \sqrt{\cot (c+d x)}}{a d \sqrt{a+b \tan (c+d x)}}-\frac{\sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left (\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{d (b+i a)^{3/2}} \]
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Rubi [A] time = 0.721442, antiderivative size = 233, 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, 3569, 3649, 3616, 3615, 93, 203, 206} \[ -\frac{2 b \left (a^2+2 b^2\right )}{a^2 d \left (a^2+b^2\right ) \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}+\frac{\sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left (\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{d (-b+i a)^{3/2}}-\frac{2 \sqrt{\cot (c+d x)}}{a d \sqrt{a+b \tan (c+d x)}}-\frac{\sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left (\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{d (b+i a)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 4241
Rule 3569
Rule 3649
Rule 3616
Rule 3615
Rule 93
Rule 203
Rule 206
Rubi steps
\begin{align*} \int \frac{\cot ^{\frac{3}{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{1}{\tan ^{\frac{3}{2}}(c+d x) (a+b \tan (c+d x))^{3/2}} \, dx\\ &=-\frac{2 \sqrt{\cot (c+d x)}}{a d \sqrt{a+b \tan (c+d x)}}-\frac{\left (2 \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \int \frac{b+\frac{1}{2} a \tan (c+d x)+b \tan ^2(c+d x)}{\sqrt{\tan (c+d x)} (a+b \tan (c+d x))^{3/2}} \, dx}{a}\\ &=-\frac{2 b \left (a^2+2 b^2\right )}{a^2 \left (a^2+b^2\right ) d \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}-\frac{2 \sqrt{\cot (c+d x)}}{a d \sqrt{a+b \tan (c+d x)}}-\frac{\left (4 \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \int \frac{\frac{a^2 b}{4}+\frac{1}{4} a^3 \tan (c+d x)}{\sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)}} \, dx}{a^2 \left (a^2+b^2\right )}\\ &=-\frac{2 b \left (a^2+2 b^2\right )}{a^2 \left (a^2+b^2\right ) d \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}-\frac{2 \sqrt{\cot (c+d x)}}{a d \sqrt{a+b \tan (c+d x)}}+\frac{\left ((i a-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 )}-\frac{\left ((i a+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 )}\\ &=-\frac{2 b \left (a^2+2 b^2\right )}{a^2 \left (a^2+b^2\right ) d \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}-\frac{2 \sqrt{\cot (c+d x)}}{a d \sqrt{a+b \tan (c+d x)}}+\frac{\left ((i a-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 ) d}-\frac{\left ((i a+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 ) d}\\ &=-\frac{2 b \left (a^2+2 b^2\right )}{a^2 \left (a^2+b^2\right ) d \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}-\frac{2 \sqrt{\cot (c+d x)}}{a d \sqrt{a+b \tan (c+d x)}}+\frac{\left ((i a-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 ) d}-\frac{\left ((i a+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 ) 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)^{3/2} 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)^{3/2} d}-\frac{2 b \left (a^2+2 b^2\right )}{a^2 \left (a^2+b^2\right ) d \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}-\frac{2 \sqrt{\cot (c+d x)}}{a d \sqrt{a+b \tan (c+d x)}}\\ \end{align*}
Mathematica [A] time = 2.88594, size = 228, normalized size = 0.98 \[ \frac{\sqrt{\cot (c+d x)} \left (\frac{-\frac{2 \left (b \left (a^2+2 b^2\right ) \tan (c+d x)+a \left (a^2+b^2\right )\right )}{\sqrt{a+b \tan (c+d x)}}+\frac{\sqrt [4]{-1} a^2 (a+i b) \sqrt{\tan (c+d x)} \tanh ^{-1}\left (\frac{\sqrt [4]{-1} \sqrt{a-i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{\sqrt{a-i b}}}{a \left (a^2+b^2\right )}+\frac{\sqrt [4]{-1} a \sqrt{\tan (c+d x)} \tanh ^{-1}\left (\frac{\sqrt [4]{-1} \sqrt{-a-i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{(-a-i b)^{3/2}}\right )}{a d} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.473, size = 9463, normalized size = 40.6 \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{3}{2}}}{{\left (b \tan \left (d x + c\right ) + a\right )}^{\frac{3}{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{3}{2}}}{{\left (b \tan \left (d x + c\right ) + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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