Optimal. Leaf size=192 \[ -\frac {\sqrt {b} \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt {b \tan (c+d x)}}{\sqrt {b}}\right )}{\sqrt {2} d}+\frac {\sqrt {b} \text {ArcTan}\left (1+\frac {\sqrt {2} \sqrt {b \tan (c+d x)}}{\sqrt {b}}\right )}{\sqrt {2} d}+\frac {\sqrt {b} \log \left (\sqrt {b}+\sqrt {b} \tan (c+d x)-\sqrt {2} \sqrt {b \tan (c+d x)}\right )}{2 \sqrt {2} d}-\frac {\sqrt {b} \log \left (\sqrt {b}+\sqrt {b} \tan (c+d x)+\sqrt {2} \sqrt {b \tan (c+d x)}\right )}{2 \sqrt {2} d} \]
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Rubi [A]
time = 0.09, antiderivative size = 192, normalized size of antiderivative = 1.00, number of steps
used = 11, number of rules used = 8, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {3557, 335,
303, 1176, 631, 210, 1179, 642} \begin {gather*} -\frac {\sqrt {b} \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt {b \tan (c+d x)}}{\sqrt {b}}\right )}{\sqrt {2} d}+\frac {\sqrt {b} \text {ArcTan}\left (\frac {\sqrt {2} \sqrt {b \tan (c+d x)}}{\sqrt {b}}+1\right )}{\sqrt {2} d}+\frac {\sqrt {b} \log \left (\sqrt {b} \tan (c+d x)-\sqrt {2} \sqrt {b \tan (c+d x)}+\sqrt {b}\right )}{2 \sqrt {2} d}-\frac {\sqrt {b} \log \left (\sqrt {b} \tan (c+d x)+\sqrt {2} \sqrt {b \tan (c+d x)}+\sqrt {b}\right )}{2 \sqrt {2} d} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 303
Rule 335
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 3557
Rubi steps
\begin {align*} \int \sqrt {b \tan (c+d x)} \, dx &=\frac {b \text {Subst}\left (\int \frac {\sqrt {x}}{b^2+x^2} \, dx,x,b \tan (c+d x)\right )}{d}\\ &=\frac {(2 b) \text {Subst}\left (\int \frac {x^2}{b^2+x^4} \, dx,x,\sqrt {b \tan (c+d x)}\right )}{d}\\ &=-\frac {b \text {Subst}\left (\int \frac {b-x^2}{b^2+x^4} \, dx,x,\sqrt {b \tan (c+d x)}\right )}{d}+\frac {b \text {Subst}\left (\int \frac {b+x^2}{b^2+x^4} \, dx,x,\sqrt {b \tan (c+d x)}\right )}{d}\\ &=\frac {\sqrt {b} \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} d}+\frac {\sqrt {b} \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} d}+\frac {b \text {Subst}\left (\int \frac {1}{b-\sqrt {2} \sqrt {b} x+x^2} \, dx,x,\sqrt {b \tan (c+d x)}\right )}{2 d}+\frac {b \text {Subst}\left (\int \frac {1}{b+\sqrt {2} \sqrt {b} x+x^2} \, dx,x,\sqrt {b \tan (c+d x)}\right )}{2 d}\\ &=\frac {\sqrt {b} \log \left (\sqrt {b}+\sqrt {b} \tan (c+d x)-\sqrt {2} \sqrt {b \tan (c+d x)}\right )}{2 \sqrt {2} d}-\frac {\sqrt {b} \log \left (\sqrt {b}+\sqrt {b} \tan (c+d x)+\sqrt {2} \sqrt {b \tan (c+d x)}\right )}{2 \sqrt {2} d}+\frac {\sqrt {b} \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} d}-\frac {\sqrt {b} \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} d}\\ &=-\frac {\sqrt {b} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {b \tan (c+d x)}}{\sqrt {b}}\right )}{\sqrt {2} d}+\frac {\sqrt {b} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt {b \tan (c+d x)}}{\sqrt {b}}\right )}{\sqrt {2} d}+\frac {\sqrt {b} \log \left (\sqrt {b}+\sqrt {b} \tan (c+d x)-\sqrt {2} \sqrt {b \tan (c+d x)}\right )}{2 \sqrt {2} d}-\frac {\sqrt {b} \log \left (\sqrt {b}+\sqrt {b} \tan (c+d x)+\sqrt {2} \sqrt {b \tan (c+d x)}\right )}{2 \sqrt {2} d}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 0.04, size = 40, normalized size = 0.21 \begin {gather*} \frac {2 \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};-\tan ^2(c+d x)\right ) (b \tan (c+d x))^{3/2}}{3 b d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 136, normalized size = 0.71
method | result | size |
derivativedivides | \(\frac {b \sqrt {2}\, \left (\ln \left (\frac {b \tan \left (d x +c \right )-\left (b^{2}\right )^{\frac {1}{4}} \sqrt {b \tan \left (d x +c \right )}\, \sqrt {2}+\sqrt {b^{2}}}{b \tan \left (d x +c \right )+\left (b^{2}\right )^{\frac {1}{4}} \sqrt {b \tan \left (d x +c \right )}\, \sqrt {2}+\sqrt {b^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {b \tan \left (d x +c \right )}}{\left (b^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {b \tan \left (d x +c \right )}}{\left (b^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 d \left (b^{2}\right )^{\frac {1}{4}}}\) | \(136\) |
default | \(\frac {b \sqrt {2}\, \left (\ln \left (\frac {b \tan \left (d x +c \right )-\left (b^{2}\right )^{\frac {1}{4}} \sqrt {b \tan \left (d x +c \right )}\, \sqrt {2}+\sqrt {b^{2}}}{b \tan \left (d x +c \right )+\left (b^{2}\right )^{\frac {1}{4}} \sqrt {b \tan \left (d x +c \right )}\, \sqrt {2}+\sqrt {b^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {b \tan \left (d x +c \right )}}{\left (b^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {b \tan \left (d x +c \right )}}{\left (b^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 d \left (b^{2}\right )^{\frac {1}{4}}}\) | \(136\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 153, normalized size = 0.80 \begin {gather*} \frac {b {\left (\frac {2 \, \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 )}{\sqrt {b}} + \frac {2 \, \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 )}{\sqrt {b}} - \frac {\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 )}{\sqrt {b}} + \frac {\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 )}{\sqrt {b}}\right )}}{4 \, 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. 519 vs.
\(2 (145) = 290\).
time = 0.39, size = 519, normalized size = 2.70 \begin {gather*} -\sqrt {2} \left (\frac {b^{2}}{d^{4}}\right )^{\frac {1}{4}} \arctan \left (-\frac {\sqrt {2} b d \sqrt {\frac {b \sin \left (d x + c\right )}{\cos \left (d x + c\right )}} \left (\frac {b^{2}}{d^{4}}\right )^{\frac {1}{4}} - \sqrt {2} d \sqrt {\frac {\sqrt {2} b d^{3} \sqrt {\frac {b \sin \left (d x + c\right )}{\cos \left (d x + c\right )}} \left (\frac {b^{2}}{d^{4}}\right )^{\frac {3}{4}} \cos \left (d x + c\right ) + b^{2} d^{2} \sqrt {\frac {b^{2}}{d^{4}}} \cos \left (d x + c\right ) + b^{3} \sin \left (d x + c\right )}{\cos \left (d x + c\right )}} \left (\frac {b^{2}}{d^{4}}\right )^{\frac {1}{4}} + b^{2}}{b^{2}}\right ) - \sqrt {2} \left (\frac {b^{2}}{d^{4}}\right )^{\frac {1}{4}} \arctan \left (-\frac {\sqrt {2} b d \sqrt {\frac {b \sin \left (d x + c\right )}{\cos \left (d x + c\right )}} \left (\frac {b^{2}}{d^{4}}\right )^{\frac {1}{4}} - \sqrt {2} d \sqrt {-\frac {\sqrt {2} b d^{3} \sqrt {\frac {b \sin \left (d x + c\right )}{\cos \left (d x + c\right )}} \left (\frac {b^{2}}{d^{4}}\right )^{\frac {3}{4}} \cos \left (d x + c\right ) - b^{2} d^{2} \sqrt {\frac {b^{2}}{d^{4}}} \cos \left (d x + c\right ) - b^{3} \sin \left (d x + c\right )}{\cos \left (d x + c\right )}} \left (\frac {b^{2}}{d^{4}}\right )^{\frac {1}{4}} - b^{2}}{b^{2}}\right ) - \frac {1}{4} \, \sqrt {2} \left (\frac {b^{2}}{d^{4}}\right )^{\frac {1}{4}} \log \left (\frac {\sqrt {2} b d^{3} \sqrt {\frac {b \sin \left (d x + c\right )}{\cos \left (d x + c\right )}} \left (\frac {b^{2}}{d^{4}}\right )^{\frac {3}{4}} \cos \left (d x + c\right ) + b^{2} d^{2} \sqrt {\frac {b^{2}}{d^{4}}} \cos \left (d x + c\right ) + b^{3} \sin \left (d x + c\right )}{\cos \left (d x + c\right )}\right ) + \frac {1}{4} \, \sqrt {2} \left (\frac {b^{2}}{d^{4}}\right )^{\frac {1}{4}} \log \left (-\frac {\sqrt {2} b d^{3} \sqrt {\frac {b \sin \left (d x + c\right )}{\cos \left (d x + c\right )}} \left (\frac {b^{2}}{d^{4}}\right )^{\frac {3}{4}} \cos \left (d x + c\right ) - b^{2} d^{2} \sqrt {\frac {b^{2}}{d^{4}}} \cos \left (d x + c\right ) - b^{3} \sin \left (d x + c\right )}{\cos \left (d x + c\right )}\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 \sqrt {b \tan {\left (c + d x \right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.47, size = 176, normalized size = 0.92 \begin {gather*} \frac {\frac {2 \, \sqrt {2} {\left | b \right |}^{\frac {3}{2}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {{\left | b \right |}} + 2 \, \sqrt {b \tan \left (d x + c\right )}\right )}}{2 \, \sqrt {{\left | b \right |}}}\right )}{d} + \frac {2 \, \sqrt {2} {\left | b \right |}^{\frac {3}{2}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {{\left | b \right |}} - 2 \, \sqrt {b \tan \left (d x + c\right )}\right )}}{2 \, \sqrt {{\left | b \right |}}}\right )}{d} - \frac {\sqrt {2} {\left | b \right |}^{\frac {3}{2}} \log \left (b \tan \left (d x + c\right ) + \sqrt {2} \sqrt {b \tan \left (d x + c\right )} \sqrt {{\left | b \right |}} + {\left | b \right |}\right )}{d} + \frac {\sqrt {2} {\left | b \right |}^{\frac {3}{2}} \log \left (b \tan \left (d x + c\right ) - \sqrt {2} \sqrt {b \tan \left (d x + c\right )} \sqrt {{\left | b \right |}} + {\left | b \right |}\right )}{d}}{4 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.62, size = 49, normalized size = 0.26 \begin {gather*} \frac {{\left (-1\right )}^{1/4}\,\sqrt {b}\,\left (\mathrm {atan}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {b\,\mathrm {tan}\left (c+d\,x\right )}}{\sqrt {b}}\right )-\mathrm {atanh}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {b\,\mathrm {tan}\left (c+d\,x\right )}}{\sqrt {b}}\right )\right )}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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