Optimal. Leaf size=214 \[ \frac {\text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt {b \tan (c+d x)}}{\sqrt {b}}\right )}{\sqrt {2} b^{5/2} d}-\frac {\text {ArcTan}\left (1+\frac {\sqrt {2} \sqrt {b \tan (c+d x)}}{\sqrt {b}}\right )}{\sqrt {2} b^{5/2} d}+\frac {\log \left (\sqrt {b}+\sqrt {b} \tan (c+d x)-\sqrt {2} \sqrt {b \tan (c+d x)}\right )}{2 \sqrt {2} b^{5/2} d}-\frac {\log \left (\sqrt {b}+\sqrt {b} \tan (c+d x)+\sqrt {2} \sqrt {b \tan (c+d x)}\right )}{2 \sqrt {2} b^{5/2} d}-\frac {2}{3 b d (b \tan (c+d x))^{3/2}} \]
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Rubi [A]
time = 0.10, antiderivative size = 214, 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, 217, 1179, 642, 1176, 631, 210} \begin {gather*} \frac {\text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt {b \tan (c+d x)}}{\sqrt {b}}\right )}{\sqrt {2} b^{5/2} d}-\frac {\text {ArcTan}\left (\frac {\sqrt {2} \sqrt {b \tan (c+d x)}}{\sqrt {b}}+1\right )}{\sqrt {2} b^{5/2} d}+\frac {\log \left (\sqrt {b} \tan (c+d x)-\sqrt {2} \sqrt {b \tan (c+d x)}+\sqrt {b}\right )}{2 \sqrt {2} b^{5/2} d}-\frac {\log \left (\sqrt {b} \tan (c+d x)+\sqrt {2} \sqrt {b \tan (c+d x)}+\sqrt {b}\right )}{2 \sqrt {2} b^{5/2} d}-\frac {2}{3 b d (b \tan (c+d x))^{3/2}} \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 3555
Rule 3557
Rubi steps
\begin {align*} \int \frac {1}{(b \tan (c+d x))^{5/2}} \, dx &=-\frac {2}{3 b d (b \tan (c+d x))^{3/2}}-\frac {\int \frac {1}{\sqrt {b \tan (c+d x)}} \, dx}{b^2}\\ &=-\frac {2}{3 b d (b \tan (c+d x))^{3/2}}-\frac {\text {Subst}\left (\int \frac {1}{\sqrt {x} \left (b^2+x^2\right )} \, dx,x,b \tan (c+d x)\right )}{b d}\\ &=-\frac {2}{3 b d (b \tan (c+d x))^{3/2}}-\frac {2 \text {Subst}\left (\int \frac {1}{b^2+x^4} \, dx,x,\sqrt {b \tan (c+d x)}\right )}{b d}\\ &=-\frac {2}{3 b d (b \tan (c+d x))^{3/2}}-\frac {\text {Subst}\left (\int \frac {b-x^2}{b^2+x^4} \, dx,x,\sqrt {b \tan (c+d x)}\right )}{b^2 d}-\frac {\text {Subst}\left (\int \frac {b+x^2}{b^2+x^4} \, dx,x,\sqrt {b \tan (c+d x)}\right )}{b^2 d}\\ &=-\frac {2}{3 b d (b \tan (c+d x))^{3/2}}+\frac {\text {Subst}\left (\int \frac {\sqrt {2} \sqrt {b}+2 x}{-b-\sqrt {2} \sqrt {b} x-x^2} \, dx,x,\sqrt {b \tan (c+d x)}\right )}{2 \sqrt {2} b^{5/2} d}+\frac {\text {Subst}\left (\int \frac {\sqrt {2} \sqrt {b}-2 x}{-b+\sqrt {2} \sqrt {b} x-x^2} \, dx,x,\sqrt {b \tan (c+d x)}\right )}{2 \sqrt {2} b^{5/2} d}-\frac {\text {Subst}\left (\int \frac {1}{b-\sqrt {2} \sqrt {b} x+x^2} \, dx,x,\sqrt {b \tan (c+d x)}\right )}{2 b^2 d}-\frac {\text {Subst}\left (\int \frac {1}{b+\sqrt {2} \sqrt {b} x+x^2} \, dx,x,\sqrt {b \tan (c+d x)}\right )}{2 b^2 d}\\ &=\frac {\log \left (\sqrt {b}+\sqrt {b} \tan (c+d x)-\sqrt {2} \sqrt {b \tan (c+d x)}\right )}{2 \sqrt {2} b^{5/2} d}-\frac {\log \left (\sqrt {b}+\sqrt {b} \tan (c+d x)+\sqrt {2} \sqrt {b \tan (c+d x)}\right )}{2 \sqrt {2} b^{5/2} d}-\frac {2}{3 b d (b \tan (c+d x))^{3/2}}-\frac {\text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt {b \tan (c+d x)}}{\sqrt {b}}\right )}{\sqrt {2} b^{5/2} d}+\frac {\text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt {b \tan (c+d x)}}{\sqrt {b}}\right )}{\sqrt {2} b^{5/2} d}\\ &=\frac {\tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {b \tan (c+d x)}}{\sqrt {b}}\right )}{\sqrt {2} b^{5/2} d}-\frac {\tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt {b \tan (c+d x)}}{\sqrt {b}}\right )}{\sqrt {2} b^{5/2} d}+\frac {\log \left (\sqrt {b}+\sqrt {b} \tan (c+d x)-\sqrt {2} \sqrt {b \tan (c+d x)}\right )}{2 \sqrt {2} b^{5/2} d}-\frac {\log \left (\sqrt {b}+\sqrt {b} \tan (c+d x)+\sqrt {2} \sqrt {b \tan (c+d x)}\right )}{2 \sqrt {2} b^{5/2} d}-\frac {2}{3 b d (b \tan (c+d x))^{3/2}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 0.08, size = 40, normalized size = 0.19 \begin {gather*} -\frac {2 \, _2F_1\left (-\frac {3}{4},1;\frac {1}{4};-\tan ^2(c+d x)\right )}{3 b d (b \tan (c+d x))^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.05, size = 157, normalized size = 0.73
method | result | size |
derivativedivides | \(\frac {2 b \left (-\frac {1}{3 b^{2} \left (b \tan \left (d x +c \right )\right )^{\frac {3}{2}}}-\frac {\left (b^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {b \tan \left (d x +c \right )+\left (b^{2}\right )^{\frac {1}{4}} \sqrt {b \tan \left (d x +c \right )}\, \sqrt {2}+\sqrt {b^{2}}}{b \tan \left (d x +c \right )-\left (b^{2}\right )^{\frac {1}{4}} \sqrt {b \tan \left (d x +c \right )}\, \sqrt {2}+\sqrt {b^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {b \tan \left (d x +c \right )}}{\left (b^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {b \tan \left (d x +c \right )}}{\left (b^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 b^{4}}\right )}{d}\) | \(157\) |
default | \(\frac {2 b \left (-\frac {1}{3 b^{2} \left (b \tan \left (d x +c \right )\right )^{\frac {3}{2}}}-\frac {\left (b^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {b \tan \left (d x +c \right )+\left (b^{2}\right )^{\frac {1}{4}} \sqrt {b \tan \left (d x +c \right )}\, \sqrt {2}+\sqrt {b^{2}}}{b \tan \left (d x +c \right )-\left (b^{2}\right )^{\frac {1}{4}} \sqrt {b \tan \left (d x +c \right )}\, \sqrt {2}+\sqrt {b^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {b \tan \left (d x +c \right )}}{\left (b^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {b \tan \left (d x +c \right )}}{\left (b^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 b^{4}}\right )}{d}\) | \(157\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 168, normalized size = 0.79 \begin {gather*} -\frac {\frac {6 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {b} + 2 \, \sqrt {b \tan \left (d x + c\right )}\right )}}{2 \, \sqrt {b}}\right )}{b^{\frac {3}{2}}} + \frac {6 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {b} - 2 \, \sqrt {b \tan \left (d x + c\right )}\right )}}{2 \, \sqrt {b}}\right )}{b^{\frac {3}{2}}} + \frac {3 \, \sqrt {2} \log \left (b \tan \left (d x + c\right ) + \sqrt {2} \sqrt {b \tan \left (d x + c\right )} \sqrt {b} + b\right )}{b^{\frac {3}{2}}} - \frac {3 \, \sqrt {2} \log \left (b \tan \left (d x + c\right ) - \sqrt {2} \sqrt {b \tan \left (d x + c\right )} \sqrt {b} + b\right )}{b^{\frac {3}{2}}} + \frac {8}{\left (b \tan \left (d x + c\right )\right )^{\frac {3}{2}}}}{12 \, 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. 653 vs.
\(2 (163) = 326\).
time = 0.39, size = 653, normalized size = 3.05 \begin {gather*} \frac {8 \, \sqrt {\frac {b \sin \left (d x + c\right )}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )^{2} + 12 \, {\left (\sqrt {2} b^{3} d \cos \left (d x + c\right )^{2} - \sqrt {2} b^{3} d\right )} \left (\frac {1}{b^{10} d^{4}}\right )^{\frac {1}{4}} \arctan \left (-\sqrt {2} b^{7} d^{3} \sqrt {\frac {b \sin \left (d x + c\right )}{\cos \left (d x + c\right )}} \left (\frac {1}{b^{10} d^{4}}\right )^{\frac {3}{4}} + \sqrt {2} b^{7} d^{3} \sqrt {\frac {b^{6} d^{2} \sqrt {\frac {1}{b^{10} d^{4}}} \cos \left (d x + c\right ) + \sqrt {2} b^{3} d \sqrt {\frac {b \sin \left (d x + c\right )}{\cos \left (d x + c\right )}} \left (\frac {1}{b^{10} d^{4}}\right )^{\frac {1}{4}} \cos \left (d x + c\right ) + b \sin \left (d x + c\right )}{\cos \left (d x + c\right )}} \left (\frac {1}{b^{10} d^{4}}\right )^{\frac {3}{4}} - 1\right ) + 12 \, {\left (\sqrt {2} b^{3} d \cos \left (d x + c\right )^{2} - \sqrt {2} b^{3} d\right )} \left (\frac {1}{b^{10} d^{4}}\right )^{\frac {1}{4}} \arctan \left (-\sqrt {2} b^{7} d^{3} \sqrt {\frac {b \sin \left (d x + c\right )}{\cos \left (d x + c\right )}} \left (\frac {1}{b^{10} d^{4}}\right )^{\frac {3}{4}} + \sqrt {2} b^{7} d^{3} \sqrt {\frac {b^{6} d^{2} \sqrt {\frac {1}{b^{10} d^{4}}} \cos \left (d x + c\right ) - \sqrt {2} b^{3} d \sqrt {\frac {b \sin \left (d x + c\right )}{\cos \left (d x + c\right )}} \left (\frac {1}{b^{10} d^{4}}\right )^{\frac {1}{4}} \cos \left (d x + c\right ) + b \sin \left (d x + c\right )}{\cos \left (d x + c\right )}} \left (\frac {1}{b^{10} d^{4}}\right )^{\frac {3}{4}} + 1\right ) - 3 \, {\left (\sqrt {2} b^{3} d \cos \left (d x + c\right )^{2} - \sqrt {2} b^{3} d\right )} \left (\frac {1}{b^{10} d^{4}}\right )^{\frac {1}{4}} \log \left (\frac {b^{6} d^{2} \sqrt {\frac {1}{b^{10} d^{4}}} \cos \left (d x + c\right ) + \sqrt {2} b^{3} d \sqrt {\frac {b \sin \left (d x + c\right )}{\cos \left (d x + c\right )}} \left (\frac {1}{b^{10} d^{4}}\right )^{\frac {1}{4}} \cos \left (d x + c\right ) + b \sin \left (d x + c\right )}{\cos \left (d x + c\right )}\right ) + 3 \, {\left (\sqrt {2} b^{3} d \cos \left (d x + c\right )^{2} - \sqrt {2} b^{3} d\right )} \left (\frac {1}{b^{10} d^{4}}\right )^{\frac {1}{4}} \log \left (\frac {b^{6} d^{2} \sqrt {\frac {1}{b^{10} d^{4}}} \cos \left (d x + c\right ) - \sqrt {2} b^{3} d \sqrt {\frac {b \sin \left (d x + c\right )}{\cos \left (d x + c\right )}} \left (\frac {1}{b^{10} d^{4}}\right )^{\frac {1}{4}} \cos \left (d x + c\right ) + b \sin \left (d x + c\right )}{\cos \left (d x + c\right )}\right )}{12 \, {\left (b^{3} d \cos \left (d x + c\right )^{2} - b^{3} d\right )}} \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 \frac {1}{\left (b \tan {\left (c + d x \right )}\right )^{\frac {5}{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 = 3.07, size = 75, normalized size = 0.35 \begin {gather*} -\frac {2}{3\,b\,d\,{\left (b\,\mathrm {tan}\left (c+d\,x\right )\right )}^{3/2}}+\frac {{\left (-1\right )}^{1/4}\,\mathrm {atan}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {b\,\mathrm {tan}\left (c+d\,x\right )}}{\sqrt {b}}\right )\,1{}\mathrm {i}}{b^{5/2}\,d}+\frac {{\left (-1\right )}^{1/4}\,\mathrm {atanh}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {b\,\mathrm {tan}\left (c+d\,x\right )}}{\sqrt {b}}\right )\,1{}\mathrm {i}}{b^{5/2}\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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