Optimal. Leaf size=155 \[ -\frac{(a+b \tan (c+d x))^n \left (\frac{b \tan (c+d x)}{a}+1\right )^{-n} F_1\left (-\frac{1}{2};1,-n;\frac{1}{2};-i \tan (c+d x),-\frac{b \tan (c+d x)}{a}\right )}{d \sqrt{\tan (c+d x)}}-\frac{(a+b \tan (c+d x))^n \left (\frac{b \tan (c+d x)}{a}+1\right )^{-n} F_1\left (-\frac{1}{2};1,-n;\frac{1}{2};i \tan (c+d x),-\frac{b \tan (c+d x)}{a}\right )}{d \sqrt{\tan (c+d x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.224894, antiderivative size = 155, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used = {3575, 912, 130, 511, 510} \[ -\frac{(a+b \tan (c+d x))^n \left (\frac{b \tan (c+d x)}{a}+1\right )^{-n} F_1\left (-\frac{1}{2};1,-n;\frac{1}{2};-i \tan (c+d x),-\frac{b \tan (c+d x)}{a}\right )}{d \sqrt{\tan (c+d x)}}-\frac{(a+b \tan (c+d x))^n \left (\frac{b \tan (c+d x)}{a}+1\right )^{-n} F_1\left (-\frac{1}{2};1,-n;\frac{1}{2};i \tan (c+d x),-\frac{b \tan (c+d x)}{a}\right )}{d \sqrt{\tan (c+d x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3575
Rule 912
Rule 130
Rule 511
Rule 510
Rubi steps
\begin{align*} \int \frac{(a+b \tan (c+d x))^n}{\tan ^{\frac{3}{2}}(c+d x)} \, dx &=\frac{\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{\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{i \operatorname{Subst}\left (\int \frac{(a+b x)^n}{(i-x) x^{3/2}} \, dx,x,\tan (c+d x)\right )}{2 d}+\frac{i \operatorname{Subst}\left (\int \frac{(a+b x)^n}{x^{3/2} (i+x)} \, dx,x,\tan (c+d x)\right )}{2 d}\\ &=\frac{i \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{i \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 (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 (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 ) (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{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 = 0.764215, size = 0, normalized size = 0. \[ \int \frac{(a+b \tan (c+d x))^n}{\tan ^{\frac{3}{2}}(c+d x)} \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.213, size = 0, normalized size = 0. \begin{align*} \int{ \left ( a+b\tan \left ( dx+c \right ) \right ) ^{n} \left ( \tan \left ( dx+c \right ) \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\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.
[In]
[Out]
________________________________________________________________________________________
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 )}^{n}}{\tan \left (d x + c\right )^{\frac{3}{2}}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\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.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\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.
[In]
[Out]