Optimal. Leaf size=173 \[ \frac{(A+i B) (a+b \tan (c+d x))^n \left (\frac{b \tan (c+d x)}{a}+1\right )^{-n} F_1\left (\frac{3}{2};1,-n;\frac{5}{2};-i \tan (c+d x),-\frac{b \tan (c+d x)}{a}\right )}{3 d \cot ^{\frac{3}{2}}(c+d x)}+\frac{(A-i B) (a+b \tan (c+d x))^n \left (\frac{b \tan (c+d x)}{a}+1\right )^{-n} F_1\left (\frac{3}{2};1,-n;\frac{5}{2};i \tan (c+d x),-\frac{b \tan (c+d x)}{a}\right )}{3 d \cot ^{\frac{3}{2}}(c+d x)} \]
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Rubi [A] time = 0.463324, antiderivative size = 173, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {4241, 3603, 3602, 130, 511, 510} \[ \frac{(A+i B) (a+b \tan (c+d x))^n \left (\frac{b \tan (c+d x)}{a}+1\right )^{-n} F_1\left (\frac{3}{2};1,-n;\frac{5}{2};-i \tan (c+d x),-\frac{b \tan (c+d x)}{a}\right )}{3 d \cot ^{\frac{3}{2}}(c+d x)}+\frac{(A-i B) (a+b \tan (c+d x))^n \left (\frac{b \tan (c+d x)}{a}+1\right )^{-n} F_1\left (\frac{3}{2};1,-n;\frac{5}{2};i \tan (c+d x),-\frac{b \tan (c+d x)}{a}\right )}{3 d \cot ^{\frac{3}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Rule 4241
Rule 3603
Rule 3602
Rule 130
Rule 511
Rule 510
Rubi steps
\begin{align*} \int \frac{(a+b \tan (c+d x))^n (A+B \tan (c+d x))}{\sqrt{\cot (c+d x)}} \, dx &=\left (\sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \int \sqrt{\tan (c+d x)} (a+b \tan (c+d x))^n (A+B \tan (c+d x)) \, dx\\ &=\frac{1}{2} \left ((A-i B) \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \int (1+i \tan (c+d x)) \sqrt{\tan (c+d x)} (a+b \tan (c+d x))^n \, dx+\frac{1}{2} \left ((A+i B) \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \int (1-i \tan (c+d x)) \sqrt{\tan (c+d x)} (a+b \tan (c+d x))^n \, dx\\ &=\frac{\left ((A-i B) \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{x} (a+b x)^n}{1-i x} \, dx,x,\tan (c+d x)\right )}{2 d}+\frac{\left ((A+i B) \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{x} (a+b x)^n}{1+i x} \, dx,x,\tan (c+d x)\right )}{2 d}\\ &=\frac{\left ((A-i B) \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \operatorname{Subst}\left (\int \frac{x^2 \left (a+b x^2\right )^n}{1-i x^2} \, dx,x,\sqrt{\tan (c+d x)}\right )}{d}+\frac{\left ((A+i B) \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \operatorname{Subst}\left (\int \frac{x^2 \left (a+b x^2\right )^n}{1+i x^2} \, dx,x,\sqrt{\tan (c+d x)}\right )}{d}\\ &=\frac{\left ((A-i B) \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{x^2 \left (1+\frac{b x^2}{a}\right )^n}{1-i x^2} \, dx,x,\sqrt{\tan (c+d x)}\right )}{d}+\frac{\left ((A+i B) \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{x^2 \left (1+\frac{b x^2}{a}\right )^n}{1+i x^2} \, dx,x,\sqrt{\tan (c+d x)}\right )}{d}\\ &=\frac{(A+i B) F_1\left (\frac{3}{2};1,-n;\frac{5}{2};-i \tan (c+d x),-\frac{b \tan (c+d x)}{a}\right ) (a+b \tan (c+d x))^n \left (1+\frac{b \tan (c+d x)}{a}\right )^{-n}}{3 d \cot ^{\frac{3}{2}}(c+d x)}+\frac{(A-i B) F_1\left (\frac{3}{2};1,-n;\frac{5}{2};i \tan (c+d x),-\frac{b \tan (c+d x)}{a}\right ) (a+b \tan (c+d x))^n \left (1+\frac{b \tan (c+d x)}{a}\right )^{-n}}{3 d \cot ^{\frac{3}{2}}(c+d x)}\\ \end{align*}
Mathematica [F] time = 15.1959, size = 0, normalized size = 0. \[ \int \frac{(a+b \tan (c+d x))^n (A+B \tan (c+d x))}{\sqrt{\cot (c+d x)}} \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.438, size = 0, normalized size = 0. \begin{align*} \int{ \left ( a+b\tan \left ( dx+c \right ) \right ) ^{n} \left ( A+B\tan \left ( dx+c \right ) \right ){\frac{1}{\sqrt{\cot \left ( dx+c \right ) }}}}\, 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 \frac{{\left (B \tan \left (d x + c\right ) + A\right )}{\left (b \tan \left (d x + c\right ) + a\right )}^{n}}{\sqrt{\cot \left (d x + c\right )}}\,{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 (\frac{{\left (B \tan \left (d x + c\right ) + A\right )}{\left (b \tan \left (d x + c\right ) + a\right )}^{n}}{\sqrt{\cot \left (d x + c\right )}}, 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 \frac{{\left (B \tan \left (d x + c\right ) + A\right )}{\left (b \tan \left (d x + c\right ) + a\right )}^{n}}{\sqrt{\cot \left (d x + c\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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