Optimal. Leaf size=210 \[ \frac {d^{3/2} \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt {d \tan (a+b x)}}{\sqrt {d}}\right )}{\sqrt {2} b}-\frac {d^{3/2} \text {ArcTan}\left (1+\frac {\sqrt {2} \sqrt {d \tan (a+b x)}}{\sqrt {d}}\right )}{\sqrt {2} b}+\frac {d^{3/2} \log \left (\sqrt {d}+\sqrt {d} \tan (a+b x)-\sqrt {2} \sqrt {d \tan (a+b x)}\right )}{2 \sqrt {2} b}-\frac {d^{3/2} \log \left (\sqrt {d}+\sqrt {d} \tan (a+b x)+\sqrt {2} \sqrt {d \tan (a+b x)}\right )}{2 \sqrt {2} b}+\frac {2 d \sqrt {d \tan (a+b x)}}{b} \]
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Rubi [A]
time = 0.10, antiderivative size = 210, 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 = {3554, 3557,
335, 217, 1179, 642, 1176, 631, 210} \begin {gather*} \frac {d^{3/2} \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt {d \tan (a+b x)}}{\sqrt {d}}\right )}{\sqrt {2} b}-\frac {d^{3/2} \text {ArcTan}\left (\frac {\sqrt {2} \sqrt {d \tan (a+b x)}}{\sqrt {d}}+1\right )}{\sqrt {2} b}+\frac {d^{3/2} \log \left (\sqrt {d} \tan (a+b x)-\sqrt {2} \sqrt {d \tan (a+b x)}+\sqrt {d}\right )}{2 \sqrt {2} b}-\frac {d^{3/2} \log \left (\sqrt {d} \tan (a+b x)+\sqrt {2} \sqrt {d \tan (a+b x)}+\sqrt {d}\right )}{2 \sqrt {2} b}+\frac {2 d \sqrt {d \tan (a+b x)}}{b} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 217
Rule 335
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 3554
Rule 3557
Rubi steps
\begin {align*} \int (d \tan (a+b x))^{3/2} \, dx &=\frac {2 d \sqrt {d \tan (a+b x)}}{b}-d^2 \int \frac {1}{\sqrt {d \tan (a+b x)}} \, dx\\ &=\frac {2 d \sqrt {d \tan (a+b x)}}{b}-\frac {d^3 \text {Subst}\left (\int \frac {1}{\sqrt {x} \left (d^2+x^2\right )} \, dx,x,d \tan (a+b x)\right )}{b}\\ &=\frac {2 d \sqrt {d \tan (a+b x)}}{b}-\frac {\left (2 d^3\right ) \text {Subst}\left (\int \frac {1}{d^2+x^4} \, dx,x,\sqrt {d \tan (a+b x)}\right )}{b}\\ &=\frac {2 d \sqrt {d \tan (a+b x)}}{b}-\frac {d^2 \text {Subst}\left (\int \frac {d-x^2}{d^2+x^4} \, dx,x,\sqrt {d \tan (a+b x)}\right )}{b}-\frac {d^2 \text {Subst}\left (\int \frac {d+x^2}{d^2+x^4} \, dx,x,\sqrt {d \tan (a+b x)}\right )}{b}\\ &=\frac {2 d \sqrt {d \tan (a+b x)}}{b}+\frac {d^{3/2} \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}+\frac {d^{3/2} \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}-\frac {d^2 \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}-\frac {d^2 \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}\\ &=\frac {d^{3/2} \log \left (\sqrt {d}+\sqrt {d} \tan (a+b x)-\sqrt {2} \sqrt {d \tan (a+b x)}\right )}{2 \sqrt {2} b}-\frac {d^{3/2} \log \left (\sqrt {d}+\sqrt {d} \tan (a+b x)+\sqrt {2} \sqrt {d \tan (a+b x)}\right )}{2 \sqrt {2} b}+\frac {2 d \sqrt {d \tan (a+b x)}}{b}-\frac {d^{3/2} \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}+\frac {d^{3/2} \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}\\ &=\frac {d^{3/2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {d \tan (a+b x)}}{\sqrt {d}}\right )}{\sqrt {2} b}-\frac {d^{3/2} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt {d \tan (a+b x)}}{\sqrt {d}}\right )}{\sqrt {2} b}+\frac {d^{3/2} \log \left (\sqrt {d}+\sqrt {d} \tan (a+b x)-\sqrt {2} \sqrt {d \tan (a+b x)}\right )}{2 \sqrt {2} b}-\frac {d^{3/2} \log \left (\sqrt {d}+\sqrt {d} \tan (a+b x)+\sqrt {2} \sqrt {d \tan (a+b x)}\right )}{2 \sqrt {2} b}+\frac {2 d \sqrt {d \tan (a+b x)}}{b}\\ \end {align*}
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Mathematica [A]
time = 0.29, size = 159, normalized size = 0.76 \begin {gather*} \frac {\left (2 \sqrt {2} \text {ArcTan}\left (1-\sqrt {2} \sqrt {\tan (a+b x)}\right )-2 \sqrt {2} \text {ArcTan}\left (1+\sqrt {2} \sqrt {\tan (a+b x)}\right )+\sqrt {2} \log \left (1-\sqrt {2} \sqrt {\tan (a+b x)}+\tan (a+b x)\right )-\sqrt {2} \log \left (1+\sqrt {2} \sqrt {\tan (a+b x)}+\tan (a+b x)\right )+8 \sqrt {\tan (a+b x)}\right ) (d \tan (a+b x))^{3/2}}{4 b \tan ^{\frac {3}{2}}(a+b x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.16, size = 149, normalized size = 0.71
method | result | size |
derivativedivides | \(\frac {2 d \left (\sqrt {d \tan \left (b x +a \right )}-\frac {\left (d^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d \tan \left (b x +a \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (b x +a \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (b x +a \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (b x +a \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (b x +a \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (b x +a \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8}\right )}{b}\) | \(149\) |
default | \(\frac {2 d \left (\sqrt {d \tan \left (b x +a \right )}-\frac {\left (d^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d \tan \left (b x +a \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (b x +a \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (b x +a \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (b x +a \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (b x +a \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (b x +a \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8}\right )}{b}\) | \(149\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 170, normalized size = 0.81 \begin {gather*} -\frac {2 \, \sqrt {2} d^{\frac {5}{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 ) + 2 \, \sqrt {2} d^{\frac {5}{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 {2} d^{\frac {5}{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 {2} d^{\frac {5}{2}} \log \left (d \tan \left (b x + a\right ) - \sqrt {2} \sqrt {d \tan \left (b x + a\right )} \sqrt {d} + d\right ) - 8 \, \sqrt {d \tan \left (b x + a\right )} d^{2}}{4 \, b d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 533 vs.
\(2 (161) = 322\).
time = 0.47, size = 533, normalized size = 2.54 \begin {gather*} \frac {4 \, \sqrt {2} \left (\frac {d^{6}}{b^{4}}\right )^{\frac {1}{4}} b \arctan \left (-\frac {d^{6} + \sqrt {2} \left (\frac {d^{6}}{b^{4}}\right )^{\frac {3}{4}} b^{3} d \sqrt {\frac {d \sin \left (b x + a\right )}{\cos \left (b x + a\right )}} - \sqrt {2} \left (\frac {d^{6}}{b^{4}}\right )^{\frac {3}{4}} b^{3} \sqrt {\frac {\sqrt {2} \left (\frac {d^{6}}{b^{4}}\right )^{\frac {1}{4}} b d \sqrt {\frac {d \sin \left (b x + a\right )}{\cos \left (b x + a\right )}} \cos \left (b x + a\right ) + d^{3} \sin \left (b x + a\right ) + \sqrt {\frac {d^{6}}{b^{4}}} b^{2} \cos \left (b x + a\right )}{\cos \left (b x + a\right )}}}{d^{6}}\right ) + 4 \, \sqrt {2} \left (\frac {d^{6}}{b^{4}}\right )^{\frac {1}{4}} b \arctan \left (\frac {d^{6} - \sqrt {2} \left (\frac {d^{6}}{b^{4}}\right )^{\frac {3}{4}} b^{3} d \sqrt {\frac {d \sin \left (b x + a\right )}{\cos \left (b x + a\right )}} + \sqrt {2} \left (\frac {d^{6}}{b^{4}}\right )^{\frac {3}{4}} b^{3} \sqrt {-\frac {\sqrt {2} \left (\frac {d^{6}}{b^{4}}\right )^{\frac {1}{4}} b d \sqrt {\frac {d \sin \left (b x + a\right )}{\cos \left (b x + a\right )}} \cos \left (b x + a\right ) - d^{3} \sin \left (b x + a\right ) - \sqrt {\frac {d^{6}}{b^{4}}} b^{2} \cos \left (b x + a\right )}{\cos \left (b x + a\right )}}}{d^{6}}\right ) - \sqrt {2} \left (\frac {d^{6}}{b^{4}}\right )^{\frac {1}{4}} b \log \left (\frac {\sqrt {2} \left (\frac {d^{6}}{b^{4}}\right )^{\frac {1}{4}} b d \sqrt {\frac {d \sin \left (b x + a\right )}{\cos \left (b x + a\right )}} \cos \left (b x + a\right ) + d^{3} \sin \left (b x + a\right ) + \sqrt {\frac {d^{6}}{b^{4}}} b^{2} \cos \left (b x + a\right )}{\cos \left (b x + a\right )}\right ) + \sqrt {2} \left (\frac {d^{6}}{b^{4}}\right )^{\frac {1}{4}} b \log \left (-\frac {\sqrt {2} \left (\frac {d^{6}}{b^{4}}\right )^{\frac {1}{4}} b d \sqrt {\frac {d \sin \left (b x + a\right )}{\cos \left (b x + a\right )}} \cos \left (b x + a\right ) - d^{3} \sin \left (b x + a\right ) - \sqrt {\frac {d^{6}}{b^{4}}} b^{2} \cos \left (b x + a\right )}{\cos \left (b x + a\right )}\right ) + 8 \, d \sqrt {\frac {d \sin \left (b x + a\right )}{\cos \left (b x + a\right )}}}{4 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \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.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.65, size = 73, normalized size = 0.35 \begin {gather*} \frac {2\,d\,\sqrt {d\,\mathrm {tan}\left (a+b\,x\right )}}{b}+\frac {{\left (-1\right )}^{1/4}\,d^{3/2}\,\mathrm {atan}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {d\,\mathrm {tan}\left (a+b\,x\right )}}{\sqrt {d}}\right )\,1{}\mathrm {i}}{b}+\frac {{\left (-1\right )}^{1/4}\,d^{3/2}\,\mathrm {atanh}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {d\,\mathrm {tan}\left (a+b\,x\right )}}{\sqrt {d}}\right )\,1{}\mathrm {i}}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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