Optimal. Leaf size=245 \[ -\frac{\tan ^{-1}\left (\frac{\sqrt [3]{b \tan (c+d x)}}{\sqrt [3]{b}}\right )}{b^{4/3} d}+\frac{\tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [3]{b \tan (c+d x)}}{\sqrt [3]{b}}\right )}{2 b^{4/3} d}-\frac{\tan ^{-1}\left (\frac{2 \sqrt [3]{b \tan (c+d x)}}{\sqrt [3]{b}}+\sqrt{3}\right )}{2 b^{4/3} d}-\frac{\sqrt{3} \log \left (b^{2/3}-\sqrt{3} \sqrt [3]{b} \sqrt [3]{b \tan (c+d x)}+(b \tan (c+d x))^{2/3}\right )}{4 b^{4/3} d}+\frac{\sqrt{3} \log \left (b^{2/3}+\sqrt{3} \sqrt [3]{b} \sqrt [3]{b \tan (c+d x)}+(b \tan (c+d x))^{2/3}\right )}{4 b^{4/3} d}-\frac{3}{b d \sqrt [3]{b \tan (c+d x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.43572, antiderivative size = 245, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 9, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.75, Rules used = {3474, 3476, 329, 295, 634, 618, 204, 628, 203} \[ -\frac{\tan ^{-1}\left (\frac{\sqrt [3]{b \tan (c+d x)}}{\sqrt [3]{b}}\right )}{b^{4/3} d}+\frac{\tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [3]{b \tan (c+d x)}}{\sqrt [3]{b}}\right )}{2 b^{4/3} d}-\frac{\tan ^{-1}\left (\frac{2 \sqrt [3]{b \tan (c+d x)}}{\sqrt [3]{b}}+\sqrt{3}\right )}{2 b^{4/3} d}-\frac{\sqrt{3} \log \left (b^{2/3}-\sqrt{3} \sqrt [3]{b} \sqrt [3]{b \tan (c+d x)}+(b \tan (c+d x))^{2/3}\right )}{4 b^{4/3} d}+\frac{\sqrt{3} \log \left (b^{2/3}+\sqrt{3} \sqrt [3]{b} \sqrt [3]{b \tan (c+d x)}+(b \tan (c+d x))^{2/3}\right )}{4 b^{4/3} d}-\frac{3}{b d \sqrt [3]{b \tan (c+d x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3474
Rule 3476
Rule 329
Rule 295
Rule 634
Rule 618
Rule 204
Rule 628
Rule 203
Rubi steps
\begin{align*} \int \frac{1}{(b \tan (c+d x))^{4/3}} \, dx &=-\frac{3}{b d \sqrt [3]{b \tan (c+d x)}}-\frac{\int (b \tan (c+d x))^{2/3} \, dx}{b^2}\\ &=-\frac{3}{b d \sqrt [3]{b \tan (c+d x)}}-\frac{\operatorname{Subst}\left (\int \frac{x^{2/3}}{b^2+x^2} \, dx,x,b \tan (c+d x)\right )}{b d}\\ &=-\frac{3}{b d \sqrt [3]{b \tan (c+d x)}}-\frac{3 \operatorname{Subst}\left (\int \frac{x^4}{b^2+x^6} \, dx,x,\sqrt [3]{b \tan (c+d x)}\right )}{b d}\\ &=-\frac{3}{b d \sqrt [3]{b \tan (c+d x)}}-\frac{\operatorname{Subst}\left (\int \frac{-\frac{\sqrt [3]{b}}{2}+\frac{\sqrt{3} x}{2}}{b^{2/3}-\sqrt{3} \sqrt [3]{b} x+x^2} \, dx,x,\sqrt [3]{b \tan (c+d x)}\right )}{b^{4/3} d}-\frac{\operatorname{Subst}\left (\int \frac{-\frac{\sqrt [3]{b}}{2}-\frac{\sqrt{3} x}{2}}{b^{2/3}+\sqrt{3} \sqrt [3]{b} x+x^2} \, dx,x,\sqrt [3]{b \tan (c+d x)}\right )}{b^{4/3} d}-\frac{\operatorname{Subst}\left (\int \frac{1}{b^{2/3}+x^2} \, dx,x,\sqrt [3]{b \tan (c+d x)}\right )}{b d}\\ &=-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{b \tan (c+d x)}}{\sqrt [3]{b}}\right )}{b^{4/3} d}-\frac{3}{b d \sqrt [3]{b \tan (c+d x)}}-\frac{\sqrt{3} \operatorname{Subst}\left (\int \frac{-\sqrt{3} \sqrt [3]{b}+2 x}{b^{2/3}-\sqrt{3} \sqrt [3]{b} x+x^2} \, dx,x,\sqrt [3]{b \tan (c+d x)}\right )}{4 b^{4/3} d}+\frac{\sqrt{3} \operatorname{Subst}\left (\int \frac{\sqrt{3} \sqrt [3]{b}+2 x}{b^{2/3}+\sqrt{3} \sqrt [3]{b} x+x^2} \, dx,x,\sqrt [3]{b \tan (c+d x)}\right )}{4 b^{4/3} d}-\frac{\operatorname{Subst}\left (\int \frac{1}{b^{2/3}-\sqrt{3} \sqrt [3]{b} x+x^2} \, dx,x,\sqrt [3]{b \tan (c+d x)}\right )}{4 b d}-\frac{\operatorname{Subst}\left (\int \frac{1}{b^{2/3}+\sqrt{3} \sqrt [3]{b} x+x^2} \, dx,x,\sqrt [3]{b \tan (c+d x)}\right )}{4 b d}\\ &=-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{b \tan (c+d x)}}{\sqrt [3]{b}}\right )}{b^{4/3} d}-\frac{\sqrt{3} \log \left (b^{2/3}-\sqrt{3} \sqrt [3]{b} \sqrt [3]{b \tan (c+d x)}+(b \tan (c+d x))^{2/3}\right )}{4 b^{4/3} d}+\frac{\sqrt{3} \log \left (b^{2/3}+\sqrt{3} \sqrt [3]{b} \sqrt [3]{b \tan (c+d x)}+(b \tan (c+d x))^{2/3}\right )}{4 b^{4/3} d}-\frac{3}{b d \sqrt [3]{b \tan (c+d x)}}-\frac{\operatorname{Subst}\left (\int \frac{1}{-\frac{1}{3}-x^2} \, dx,x,1-\frac{2 \sqrt [3]{b \tan (c+d x)}}{\sqrt{3} \sqrt [3]{b}}\right )}{2 \sqrt{3} b^{4/3} d}+\frac{\operatorname{Subst}\left (\int \frac{1}{-\frac{1}{3}-x^2} \, dx,x,1+\frac{2 \sqrt [3]{b \tan (c+d x)}}{\sqrt{3} \sqrt [3]{b}}\right )}{2 \sqrt{3} b^{4/3} d}\\ &=-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{b \tan (c+d x)}}{\sqrt [3]{b}}\right )}{b^{4/3} d}+\frac{\tan ^{-1}\left (\frac{1}{3} \left (3 \sqrt{3}-\frac{6 \sqrt [3]{b \tan (c+d x)}}{\sqrt [3]{b}}\right )\right )}{2 b^{4/3} d}-\frac{\tan ^{-1}\left (\frac{1}{3} \left (3 \sqrt{3}+\frac{6 \sqrt [3]{b \tan (c+d x)}}{\sqrt [3]{b}}\right )\right )}{2 b^{4/3} d}-\frac{\sqrt{3} \log \left (b^{2/3}-\sqrt{3} \sqrt [3]{b} \sqrt [3]{b \tan (c+d x)}+(b \tan (c+d x))^{2/3}\right )}{4 b^{4/3} d}+\frac{\sqrt{3} \log \left (b^{2/3}+\sqrt{3} \sqrt [3]{b} \sqrt [3]{b \tan (c+d x)}+(b \tan (c+d x))^{2/3}\right )}{4 b^{4/3} d}-\frac{3}{b d \sqrt [3]{b \tan (c+d x)}}\\ \end{align*}
Mathematica [C] time = 0.0590962, size = 38, normalized size = 0.16 \[ -\frac{3 \, _2F_1\left (-\frac{1}{6},1;\frac{5}{6};-\tan ^2(c+d x)\right )}{b d \sqrt [3]{b \tan (c+d x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.042, size = 227, normalized size = 0.9 \begin{align*}{\frac{\sqrt{3}}{4\,d{b}^{3}} \left ({b}^{2} \right ) ^{{\frac{5}{6}}}\ln \left ( \left ( b\tan \left ( dx+c \right ) \right ) ^{{\frac{2}{3}}}+\sqrt{3}\sqrt [6]{{b}^{2}}\sqrt [3]{b\tan \left ( dx+c \right ) }+\sqrt [3]{{b}^{2}} \right ) }-{\frac{1}{2\,bd}\arctan \left ( 2\,{\frac{\sqrt [3]{b\tan \left ( dx+c \right ) }}{\sqrt [6]{{b}^{2}}}}+\sqrt{3} \right ){\frac{1}{\sqrt [6]{{b}^{2}}}}}-{\frac{\sqrt{3}}{4\,d{b}^{3}} \left ({b}^{2} \right ) ^{{\frac{5}{6}}}\ln \left ( \left ( b\tan \left ( dx+c \right ) \right ) ^{{\frac{2}{3}}}-\sqrt{3}\sqrt [6]{{b}^{2}}\sqrt [3]{b\tan \left ( dx+c \right ) }+\sqrt [3]{{b}^{2}} \right ) }-{\frac{1}{2\,bd}\arctan \left ( 2\,{\frac{\sqrt [3]{b\tan \left ( dx+c \right ) }}{\sqrt [6]{{b}^{2}}}}-\sqrt{3} \right ){\frac{1}{\sqrt [6]{{b}^{2}}}}}-{\frac{1}{bd}\arctan \left ({\sqrt [3]{b\tan \left ( dx+c \right ) }{\frac{1}{\sqrt [6]{{b}^{2}}}}} \right ){\frac{1}{\sqrt [6]{{b}^{2}}}}}-3\,{\frac{1}{bd\sqrt [3]{b\tan \left ( dx+c \right ) }}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.74943, size = 1852, normalized size = 7.56 \begin{align*} \text{result too large to display} \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{1}{\left (b \tan{\left (c + d x \right )}\right )^{\frac{4}{3}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.58753, size = 306, normalized size = 1.25 \begin{align*} \frac{1}{4} \, b{\left (\frac{\sqrt{3}{\left | b \right |}^{\frac{5}{3}} \log \left (\sqrt{3} \left (b \tan \left (d x + c\right )\right )^{\frac{1}{3}}{\left | b \right |}^{\frac{1}{3}} + \left (b \tan \left (d x + c\right )\right )^{\frac{2}{3}} +{\left | b \right |}^{\frac{2}{3}}\right )}{b^{4} d} - \frac{\sqrt{3}{\left | b \right |}^{\frac{5}{3}} \log \left (-\sqrt{3} \left (b \tan \left (d x + c\right )\right )^{\frac{1}{3}}{\left | b \right |}^{\frac{1}{3}} + \left (b \tan \left (d x + c\right )\right )^{\frac{2}{3}} +{\left | b \right |}^{\frac{2}{3}}\right )}{b^{4} d} - \frac{2 \,{\left | b \right |}^{\frac{5}{3}} \arctan \left (\frac{\sqrt{3}{\left | b \right |}^{\frac{1}{3}} + 2 \, \left (b \tan \left (d x + c\right )\right )^{\frac{1}{3}}}{{\left | b \right |}^{\frac{1}{3}}}\right )}{b^{4} d} - \frac{2 \,{\left | b \right |}^{\frac{5}{3}} \arctan \left (-\frac{\sqrt{3}{\left | b \right |}^{\frac{1}{3}} - 2 \, \left (b \tan \left (d x + c\right )\right )^{\frac{1}{3}}}{{\left | b \right |}^{\frac{1}{3}}}\right )}{b^{4} d} - \frac{4 \,{\left | b \right |}^{\frac{5}{3}} \arctan \left (\frac{\left (b \tan \left (d x + c\right )\right )^{\frac{1}{3}}}{{\left | b \right |}^{\frac{1}{3}}}\right )}{b^{4} d} - \frac{12}{\left (b \tan \left (d x + c\right )\right )^{\frac{1}{3}} b^{2} d}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]