Integrand size = 12, antiderivative size = 212 \[ \int \frac {1}{(d \tan (a+b x))^{3/2}} \, dx=\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {d \tan (a+b x)}}{\sqrt {d}}\right )}{\sqrt {2} b d^{3/2}}-\frac {\arctan \left (1+\frac {\sqrt {2} \sqrt {d \tan (a+b x)}}{\sqrt {d}}\right )}{\sqrt {2} b d^{3/2}}-\frac {\log \left (\sqrt {d}+\sqrt {d} \tan (a+b x)-\sqrt {2} \sqrt {d \tan (a+b x)}\right )}{2 \sqrt {2} b d^{3/2}}+\frac {\log \left (\sqrt {d}+\sqrt {d} \tan (a+b x)+\sqrt {2} \sqrt {d \tan (a+b x)}\right )}{2 \sqrt {2} b d^{3/2}}-\frac {2}{b d \sqrt {d \tan (a+b x)}} \]
[Out]
Time = 0.16 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {3555, 3557, 335, 303, 1176, 631, 210, 1179, 642} \[ \int \frac {1}{(d \tan (a+b x))^{3/2}} \, dx=\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {d \tan (a+b x)}}{\sqrt {d}}\right )}{\sqrt {2} b d^{3/2}}-\frac {\arctan \left (\frac {\sqrt {2} \sqrt {d \tan (a+b x)}}{\sqrt {d}}+1\right )}{\sqrt {2} b d^{3/2}}-\frac {\log \left (\sqrt {d} \tan (a+b x)-\sqrt {2} \sqrt {d \tan (a+b x)}+\sqrt {d}\right )}{2 \sqrt {2} b d^{3/2}}+\frac {\log \left (\sqrt {d} \tan (a+b x)+\sqrt {2} \sqrt {d \tan (a+b x)}+\sqrt {d}\right )}{2 \sqrt {2} b d^{3/2}}-\frac {2}{b d \sqrt {d \tan (a+b x)}} \]
[In]
[Out]
Rule 210
Rule 303
Rule 335
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 3555
Rule 3557
Rubi steps \begin{align*} \text {integral}& = -\frac {2}{b d \sqrt {d \tan (a+b x)}}-\frac {\int \sqrt {d \tan (a+b x)} \, dx}{d^2} \\ & = -\frac {2}{b d \sqrt {d \tan (a+b x)}}-\frac {\text {Subst}\left (\int \frac {\sqrt {x}}{d^2+x^2} \, dx,x,d \tan (a+b x)\right )}{b d} \\ & = -\frac {2}{b d \sqrt {d \tan (a+b x)}}-\frac {2 \text {Subst}\left (\int \frac {x^2}{d^2+x^4} \, dx,x,\sqrt {d \tan (a+b x)}\right )}{b d} \\ & = -\frac {2}{b d \sqrt {d \tan (a+b x)}}+\frac {\text {Subst}\left (\int \frac {d-x^2}{d^2+x^4} \, dx,x,\sqrt {d \tan (a+b x)}\right )}{b d}-\frac {\text {Subst}\left (\int \frac {d+x^2}{d^2+x^4} \, dx,x,\sqrt {d \tan (a+b x)}\right )}{b d} \\ & = -\frac {2}{b d \sqrt {d \tan (a+b x)}}-\frac {\text {Subst}\left (\int \frac {\sqrt {2} \sqrt {d}+2 x}{-d-\sqrt {2} \sqrt {d} x-x^2} \, dx,x,\sqrt {d \tan (a+b x)}\right )}{2 \sqrt {2} b d^{3/2}}-\frac {\text {Subst}\left (\int \frac {\sqrt {2} \sqrt {d}-2 x}{-d+\sqrt {2} \sqrt {d} x-x^2} \, dx,x,\sqrt {d \tan (a+b x)}\right )}{2 \sqrt {2} b d^{3/2}}-\frac {\text {Subst}\left (\int \frac {1}{d-\sqrt {2} \sqrt {d} x+x^2} \, dx,x,\sqrt {d \tan (a+b x)}\right )}{2 b d}-\frac {\text {Subst}\left (\int \frac {1}{d+\sqrt {2} \sqrt {d} x+x^2} \, dx,x,\sqrt {d \tan (a+b x)}\right )}{2 b d} \\ & = -\frac {\log \left (\sqrt {d}+\sqrt {d} \tan (a+b x)-\sqrt {2} \sqrt {d \tan (a+b x)}\right )}{2 \sqrt {2} b d^{3/2}}+\frac {\log \left (\sqrt {d}+\sqrt {d} \tan (a+b x)+\sqrt {2} \sqrt {d \tan (a+b x)}\right )}{2 \sqrt {2} b d^{3/2}}-\frac {2}{b d \sqrt {d \tan (a+b x)}}-\frac {\text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt {d \tan (a+b x)}}{\sqrt {d}}\right )}{\sqrt {2} b d^{3/2}}+\frac {\text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt {d \tan (a+b x)}}{\sqrt {d}}\right )}{\sqrt {2} b d^{3/2}} \\ & = \frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {d \tan (a+b x)}}{\sqrt {d}}\right )}{\sqrt {2} b d^{3/2}}-\frac {\arctan \left (1+\frac {\sqrt {2} \sqrt {d \tan (a+b x)}}{\sqrt {d}}\right )}{\sqrt {2} b d^{3/2}}-\frac {\log \left (\sqrt {d}+\sqrt {d} \tan (a+b x)-\sqrt {2} \sqrt {d \tan (a+b x)}\right )}{2 \sqrt {2} b d^{3/2}}+\frac {\log \left (\sqrt {d}+\sqrt {d} \tan (a+b x)+\sqrt {2} \sqrt {d \tan (a+b x)}\right )}{2 \sqrt {2} b d^{3/2}}-\frac {2}{b d \sqrt {d \tan (a+b x)}} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.39 \[ \int \frac {1}{(d \tan (a+b x))^{3/2}} \, dx=\frac {-2-\arctan \left (\sqrt [4]{-\tan ^2(a+b x)}\right ) \sqrt [4]{-\tan ^2(a+b x)}+\text {arctanh}\left (\sqrt [4]{-\tan ^2(a+b x)}\right ) \sqrt [4]{-\tan ^2(a+b x)}}{b d \sqrt {d \tan (a+b x)}} \]
[In]
[Out]
Time = 0.16 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.74
method | result | size |
derivativedivides | \(\frac {2 d \left (-\frac {1}{d^{2} \sqrt {d \tan \left (b x +a \right )}}-\frac {\sqrt {2}\, \left (\ln \left (\frac {d \tan \left (b x +a \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (b x +a \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (b x +a \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (b x +a \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (b x +a \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (b x +a \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 d^{2} \left (d^{2}\right )^{\frac {1}{4}}}\right )}{b}\) | \(157\) |
default | \(\frac {2 d \left (-\frac {1}{d^{2} \sqrt {d \tan \left (b x +a \right )}}-\frac {\sqrt {2}\, \left (\ln \left (\frac {d \tan \left (b x +a \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (b x +a \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (b x +a \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (b x +a \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (b x +a \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (b x +a \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 d^{2} \left (d^{2}\right )^{\frac {1}{4}}}\right )}{b}\) | \(157\) |
[In]
[Out]
Result contains complex when optimal does not.
Time = 0.25 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.11 \[ \int \frac {1}{(d \tan (a+b x))^{3/2}} \, dx=-\frac {b d^{2} \left (-\frac {1}{b^{4} d^{6}}\right )^{\frac {1}{4}} \log \left (b^{3} d^{5} \left (-\frac {1}{b^{4} d^{6}}\right )^{\frac {3}{4}} + \sqrt {d \tan \left (b x + a\right )}\right ) \tan \left (b x + a\right ) - i \, b d^{2} \left (-\frac {1}{b^{4} d^{6}}\right )^{\frac {1}{4}} \log \left (i \, b^{3} d^{5} \left (-\frac {1}{b^{4} d^{6}}\right )^{\frac {3}{4}} + \sqrt {d \tan \left (b x + a\right )}\right ) \tan \left (b x + a\right ) + i \, b d^{2} \left (-\frac {1}{b^{4} d^{6}}\right )^{\frac {1}{4}} \log \left (-i \, b^{3} d^{5} \left (-\frac {1}{b^{4} d^{6}}\right )^{\frac {3}{4}} + \sqrt {d \tan \left (b x + a\right )}\right ) \tan \left (b x + a\right ) - b d^{2} \left (-\frac {1}{b^{4} d^{6}}\right )^{\frac {1}{4}} \log \left (-b^{3} d^{5} \left (-\frac {1}{b^{4} d^{6}}\right )^{\frac {3}{4}} + \sqrt {d \tan \left (b x + a\right )}\right ) \tan \left (b x + a\right ) + 4 \, \sqrt {d \tan \left (b x + a\right )}}{2 \, b d^{2} \tan \left (b x + a\right )} \]
[In]
[Out]
\[ \int \frac {1}{(d \tan (a+b x))^{3/2}} \, dx=\int \frac {1}{\left (d \tan {\left (a + b x \right )}\right )^{\frac {3}{2}}}\, dx \]
[In]
[Out]
none
Time = 0.58 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.79 \[ \int \frac {1}{(d \tan (a+b x))^{3/2}} \, dx=-\frac {\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {d} + 2 \, \sqrt {d \tan \left (b x + a\right )}\right )}}{2 \, \sqrt {d}}\right )}{\sqrt {d}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {d} - 2 \, \sqrt {d \tan \left (b x + a\right )}\right )}}{2 \, \sqrt {d}}\right )}{\sqrt {d}} - \frac {\sqrt {2} \log \left (d \tan \left (b x + a\right ) + \sqrt {2} \sqrt {d \tan \left (b x + a\right )} \sqrt {d} + d\right )}{\sqrt {d}} + \frac {\sqrt {2} \log \left (d \tan \left (b x + a\right ) - \sqrt {2} \sqrt {d \tan \left (b x + a\right )} \sqrt {d} + d\right )}{\sqrt {d}} + \frac {8}{\sqrt {d \tan \left (b x + a\right )}}}{4 \, b d} \]
[In]
[Out]
Timed out. \[ \int \frac {1}{(d \tan (a+b x))^{3/2}} \, dx=\text {Timed out} \]
[In]
[Out]
Time = 2.83 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.36 \[ \int \frac {1}{(d \tan (a+b x))^{3/2}} \, dx=\frac {{\left (-1\right )}^{1/4}\,\mathrm {atanh}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {d\,\mathrm {tan}\left (a+b\,x\right )}}{\sqrt {d}}\right )}{b\,d^{3/2}}-\frac {{\left (-1\right )}^{1/4}\,\mathrm {atan}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {d\,\mathrm {tan}\left (a+b\,x\right )}}{\sqrt {d}}\right )}{b\,d^{3/2}}-\frac {2}{b\,d\,\sqrt {d\,\mathrm {tan}\left (a+b\,x\right )}} \]
[In]
[Out]