Optimal. Leaf size=155 \[ -\frac{\sqrt{\cot (c+d x)} (a+b \tan (c+d x))^n \left (\frac{b \tan (c+d x)}{a}+1\right )^{-n} F_1\left (-\frac{1}{2};1,-n;\frac{1}{2};-i \tan (c+d x),-\frac{b \tan (c+d x)}{a}\right )}{d}-\frac{\sqrt{\cot (c+d x)} (a+b \tan (c+d x))^n \left (\frac{b \tan (c+d x)}{a}+1\right )^{-n} F_1\left (-\frac{1}{2};1,-n;\frac{1}{2};i \tan (c+d x),-\frac{b \tan (c+d x)}{a}\right )}{d} \]
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Rubi [A] time = 0.277467, antiderivative size = 155, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261, Rules used = {4241, 3575, 912, 130, 511, 510} \[ -\frac{\sqrt{\cot (c+d x)} (a+b \tan (c+d x))^n \left (\frac{b \tan (c+d x)}{a}+1\right )^{-n} F_1\left (-\frac{1}{2};1,-n;\frac{1}{2};-i \tan (c+d x),-\frac{b \tan (c+d x)}{a}\right )}{d}-\frac{\sqrt{\cot (c+d x)} (a+b \tan (c+d x))^n \left (\frac{b \tan (c+d x)}{a}+1\right )^{-n} F_1\left (-\frac{1}{2};1,-n;\frac{1}{2};i \tan (c+d x),-\frac{b \tan (c+d x)}{a}\right )}{d} \]
Antiderivative was successfully verified.
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Rule 4241
Rule 3575
Rule 912
Rule 130
Rule 511
Rule 510
Rubi steps
\begin{align*} \int \cot ^{\frac{3}{2}}(c+d x) (a+b \tan (c+d x))^n \, dx &=\left (\sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \int \frac{(a+b \tan (c+d x))^n}{\tan ^{\frac{3}{2}}(c+d x)} \, dx\\ &=\frac{\left (\sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \operatorname{Subst}\left (\int \frac{(a+b x)^n}{x^{3/2} \left (1+x^2\right )} \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac{\left (\sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \operatorname{Subst}\left (\int \left (\frac{i (a+b x)^n}{2 (i-x) x^{3/2}}+\frac{i (a+b x)^n}{2 x^{3/2} (i+x)}\right ) \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac{\left (i \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \operatorname{Subst}\left (\int \frac{(a+b x)^n}{(i-x) x^{3/2}} \, dx,x,\tan (c+d x)\right )}{2 d}+\frac{\left (i \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \operatorname{Subst}\left (\int \frac{(a+b x)^n}{x^{3/2} (i+x)} \, dx,x,\tan (c+d x)\right )}{2 d}\\ &=\frac{\left (i \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \operatorname{Subst}\left (\int \frac{\left (a+b x^2\right )^n}{x^2 \left (i-x^2\right )} \, dx,x,\sqrt{\tan (c+d x)}\right )}{d}+\frac{\left (i \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \operatorname{Subst}\left (\int \frac{\left (a+b x^2\right )^n}{x^2 \left (i+x^2\right )} \, dx,x,\sqrt{\tan (c+d x)}\right )}{d}\\ &=\frac{\left (i \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)} (a+b \tan (c+d x))^n \left (1+\frac{b \tan (c+d x)}{a}\right )^{-n}\right ) \operatorname{Subst}\left (\int \frac{\left (1+\frac{b x^2}{a}\right )^n}{x^2 \left (i-x^2\right )} \, dx,x,\sqrt{\tan (c+d x)}\right )}{d}+\frac{\left (i \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)} (a+b \tan (c+d x))^n \left (1+\frac{b \tan (c+d x)}{a}\right )^{-n}\right ) \operatorname{Subst}\left (\int \frac{\left (1+\frac{b x^2}{a}\right )^n}{x^2 \left (i+x^2\right )} \, dx,x,\sqrt{\tan (c+d x)}\right )}{d}\\ &=-\frac{F_1\left (-\frac{1}{2};1,-n;\frac{1}{2};-i \tan (c+d x),-\frac{b \tan (c+d x)}{a}\right ) \sqrt{\cot (c+d x)} (a+b \tan (c+d x))^n \left (1+\frac{b \tan (c+d x)}{a}\right )^{-n}}{d}-\frac{F_1\left (-\frac{1}{2};1,-n;\frac{1}{2};i \tan (c+d x),-\frac{b \tan (c+d x)}{a}\right ) \sqrt{\cot (c+d x)} (a+b \tan (c+d x))^n \left (1+\frac{b \tan (c+d x)}{a}\right )^{-n}}{d}\\ \end{align*}
Mathematica [F] time = 3.57057, size = 0, normalized size = 0. \[ \int \cot ^{\frac{3}{2}}(c+d x) (a+b \tan (c+d x))^n \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.26, size = 0, normalized size = 0. \begin{align*} \int \left ( \cot \left ( dx+c \right ) \right ) ^{{\frac{3}{2}}} \left ( a+b\tan \left ( dx+c \right ) \right ) ^{n}\, dx \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 )}^{n} \cot \left (d x + c\right )^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (b \tan \left (d x + c\right ) + a\right )}^{n} \cot \left (d x + c\right )^{\frac{3}{2}}, x\right ) \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{\left (b \tan \left (d x + c\right ) + a\right )}^{n} \cot \left (d x + c\right )^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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