Optimal. Leaf size=169 \[ -\frac{(A+i B) (a+b \tan (c+d x))^n \left (\frac{b \tan (c+d x)}{a}+1\right )^{-n} F_1\left (-\frac{1}{2};1,-n;\frac{1}{2};-i \tan (c+d x),-\frac{b \tan (c+d x)}{a}\right )}{d \sqrt{\tan (c+d x)}}-\frac{(A-i B) (a+b \tan (c+d x))^n \left (\frac{b \tan (c+d x)}{a}+1\right )^{-n} F_1\left (-\frac{1}{2};1,-n;\frac{1}{2};i \tan (c+d x),-\frac{b \tan (c+d x)}{a}\right )}{d \sqrt{\tan (c+d x)}} \]
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Rubi [A] time = 0.375159, antiderivative size = 169, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.152, Rules used = {3603, 3602, 130, 511, 510} \[ -\frac{(A+i B) (a+b \tan (c+d x))^n \left (\frac{b \tan (c+d x)}{a}+1\right )^{-n} F_1\left (-\frac{1}{2};1,-n;\frac{1}{2};-i \tan (c+d x),-\frac{b \tan (c+d x)}{a}\right )}{d \sqrt{\tan (c+d x)}}-\frac{(A-i B) (a+b \tan (c+d x))^n \left (\frac{b \tan (c+d x)}{a}+1\right )^{-n} F_1\left (-\frac{1}{2};1,-n;\frac{1}{2};i \tan (c+d x),-\frac{b \tan (c+d x)}{a}\right )}{d \sqrt{\tan (c+d x)}} \]
Antiderivative was successfully verified.
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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))}{\tan ^{\frac{3}{2}}(c+d x)} \, dx &=\frac{1}{2} (A-i B) \int \frac{(1+i \tan (c+d x)) (a+b \tan (c+d x))^n}{\tan ^{\frac{3}{2}}(c+d x)} \, dx+\frac{1}{2} (A+i B) \int \frac{(1-i \tan (c+d x)) (a+b \tan (c+d x))^n}{\tan ^{\frac{3}{2}}(c+d x)} \, dx\\ &=\frac{(A-i B) \operatorname{Subst}\left (\int \frac{(a+b x)^n}{(1-i x) x^{3/2}} \, dx,x,\tan (c+d x)\right )}{2 d}+\frac{(A+i B) \operatorname{Subst}\left (\int \frac{(a+b x)^n}{(1+i x) x^{3/2}} \, dx,x,\tan (c+d x)\right )}{2 d}\\ &=\frac{(A-i B) \operatorname{Subst}\left (\int \frac{\left (a+b x^2\right )^n}{x^2 \left (1-i x^2\right )} \, dx,x,\sqrt{\tan (c+d x)}\right )}{d}+\frac{(A+i B) \operatorname{Subst}\left (\int \frac{\left (a+b x^2\right )^n}{x^2 \left (1+i x^2\right )} \, dx,x,\sqrt{\tan (c+d x)}\right )}{d}\\ &=\frac{\left ((A-i B) (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 (1-i x^2\right )} \, dx,x,\sqrt{\tan (c+d x)}\right )}{d}+\frac{\left ((A+i B) (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 (1+i x^2\right )} \, dx,x,\sqrt{\tan (c+d x)}\right )}{d}\\ &=-\frac{(A+i B) F_1\left (-\frac{1}{2};1,-n;\frac{1}{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}}{d \sqrt{\tan (c+d x)}}-\frac{(A-i B) F_1\left (-\frac{1}{2};1,-n;\frac{1}{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}}{d \sqrt{\tan (c+d x)}}\\ \end{align*}
Mathematica [F] time = 2.3891, size = 0, normalized size = 0. \[ \int \frac{(a+b \tan (c+d x))^n (A+B \tan (c+d x))}{\tan ^{\frac{3}{2}}(c+d x)} \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.373, 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 ) \left ( \tan \left ( dx+c \right ) \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [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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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}}{\tan \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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (A + B \tan{\left (c + d x \right )}\right ) \left (a + b \tan{\left (c + d x \right )}\right )^{n}}{\tan ^{\frac{3}{2}}{\left (c + d x \right )}}\, dx \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}}{\tan \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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