Optimal. Leaf size=212 \[ \frac {\text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt {d \tan (a+b x)}}{\sqrt {d}}\right )}{\sqrt {2} b d^{3/2}}-\frac {\text {ArcTan}\left (1+\frac {\sqrt {2} \sqrt {d \tan (a+b x)}}{\sqrt {d}}\right )}{\sqrt {2} b d^{3/2}}-\frac {\log \left (\sqrt {d}+\sqrt {d} \tan (a+b x)-\sqrt {2} \sqrt {d \tan (a+b x)}\right )}{2 \sqrt {2} b d^{3/2}}+\frac {\log \left (\sqrt {d}+\sqrt {d} \tan (a+b x)+\sqrt {2} \sqrt {d \tan (a+b x)}\right )}{2 \sqrt {2} b d^{3/2}}-\frac {2}{b d \sqrt {d \tan (a+b x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.10, antiderivative size = 212, normalized size of antiderivative = 1.00, number of steps
used = 12, number of rules used = 9, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {3555, 3557,
335, 303, 1176, 631, 210, 1179, 642} \begin {gather*} \frac {\text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt {d \tan (a+b x)}}{\sqrt {d}}\right )}{\sqrt {2} b d^{3/2}}-\frac {\text {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)}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 210
Rule 303
Rule 335
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 3555
Rule 3557
Rubi steps
\begin {align*} \int \frac {1}{(d \tan (a+b x))^{3/2}} \, dx &=-\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 {\tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {d \tan (a+b x)}}{\sqrt {d}}\right )}{\sqrt {2} b d^{3/2}}-\frac {\tan ^{-1}\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*}
________________________________________________________________________________________
Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 0.04, size = 38, normalized size = 0.18 \begin {gather*} -\frac {2 \, _2F_1\left (-\frac {1}{4},1;\frac {3}{4};-\tan ^2(a+b x)\right )}{b d \sqrt {d \tan (a+b x)}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.10, size = 157, normalized size = 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 | \(\frac {2 d \left (-\frac {1}{d^{2} \sqrt {d \tan \left (b x +a \right )}}-\frac {\sqrt {2}\, \left (\ln \left (\frac {d \tan \left (b x +a \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (b x +a \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (b x +a \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (b x +a \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (b x +a \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (b x +a \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 d^{2} \left (d^{2}\right )^{\frac {1}{4}}}\right )}{b}\) | \(157\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.49, size = 167, normalized size = 0.79 \begin {gather*} -\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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 652 vs.
\(2 (163) = 326\).
time = 0.39, size = 652, normalized size = 3.08 \begin {gather*} \frac {8 \, \sqrt {\frac {d \sin \left (b x + a\right )}{\cos \left (b x + a\right )}} \cos \left (b x + a\right ) \sin \left (b x + a\right ) + 4 \, {\left (\sqrt {2} b d^{2} \cos \left (b x + a\right )^{2} - \sqrt {2} b d^{2}\right )} \left (\frac {1}{b^{4} d^{6}}\right )^{\frac {1}{4}} \arctan \left (-\sqrt {2} b d \sqrt {\frac {d \sin \left (b x + a\right )}{\cos \left (b x + a\right )}} \left (\frac {1}{b^{4} d^{6}}\right )^{\frac {1}{4}} + \sqrt {2} b d \sqrt {\frac {\sqrt {2} b^{3} d^{5} \sqrt {\frac {d \sin \left (b x + a\right )}{\cos \left (b x + a\right )}} \left (\frac {1}{b^{4} d^{6}}\right )^{\frac {3}{4}} \cos \left (b x + a\right ) + b^{2} d^{4} \sqrt {\frac {1}{b^{4} d^{6}}} \cos \left (b x + a\right ) + d \sin \left (b x + a\right )}{\cos \left (b x + a\right )}} \left (\frac {1}{b^{4} d^{6}}\right )^{\frac {1}{4}} - 1\right ) + 4 \, {\left (\sqrt {2} b d^{2} \cos \left (b x + a\right )^{2} - \sqrt {2} b d^{2}\right )} \left (\frac {1}{b^{4} d^{6}}\right )^{\frac {1}{4}} \arctan \left (-\sqrt {2} b d \sqrt {\frac {d \sin \left (b x + a\right )}{\cos \left (b x + a\right )}} \left (\frac {1}{b^{4} d^{6}}\right )^{\frac {1}{4}} + \sqrt {2} b d \sqrt {-\frac {\sqrt {2} b^{3} d^{5} \sqrt {\frac {d \sin \left (b x + a\right )}{\cos \left (b x + a\right )}} \left (\frac {1}{b^{4} d^{6}}\right )^{\frac {3}{4}} \cos \left (b x + a\right ) - b^{2} d^{4} \sqrt {\frac {1}{b^{4} d^{6}}} \cos \left (b x + a\right ) - d \sin \left (b x + a\right )}{\cos \left (b x + a\right )}} \left (\frac {1}{b^{4} d^{6}}\right )^{\frac {1}{4}} + 1\right ) + {\left (\sqrt {2} b d^{2} \cos \left (b x + a\right )^{2} - \sqrt {2} b d^{2}\right )} \left (\frac {1}{b^{4} d^{6}}\right )^{\frac {1}{4}} \log \left (\frac {\sqrt {2} b^{3} d^{5} \sqrt {\frac {d \sin \left (b x + a\right )}{\cos \left (b x + a\right )}} \left (\frac {1}{b^{4} d^{6}}\right )^{\frac {3}{4}} \cos \left (b x + a\right ) + b^{2} d^{4} \sqrt {\frac {1}{b^{4} d^{6}}} \cos \left (b x + a\right ) + d \sin \left (b x + a\right )}{\cos \left (b x + a\right )}\right ) - {\left (\sqrt {2} b d^{2} \cos \left (b x + a\right )^{2} - \sqrt {2} b d^{2}\right )} \left (\frac {1}{b^{4} d^{6}}\right )^{\frac {1}{4}} \log \left (-\frac {\sqrt {2} b^{3} d^{5} \sqrt {\frac {d \sin \left (b x + a\right )}{\cos \left (b x + a\right )}} \left (\frac {1}{b^{4} d^{6}}\right )^{\frac {3}{4}} \cos \left (b x + a\right ) - b^{2} d^{4} \sqrt {\frac {1}{b^{4} d^{6}}} \cos \left (b x + a\right ) - d \sin \left (b x + a\right )}{\cos \left (b x + a\right )}\right )}{4 \, {\left (b d^{2} \cos \left (b x + a\right )^{2} - b d^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (d \tan {\left (a + b x \right )}\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 2.69, size = 76, normalized size = 0.36 \begin {gather*} \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 )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________