Optimal. Leaf size=316 \[ \frac {d \tanh ^{-1}\left (\frac {\sqrt {c+\sqrt {c^2+d^2}}-\sqrt {2} \sqrt {c+d x}}{\sqrt {c-\sqrt {c^2+d^2}}}\right )}{\sqrt {2} \sqrt {c-\sqrt {c^2+d^2}}}-\frac {d \tanh ^{-1}\left (\frac {\sqrt {c+\sqrt {c^2+d^2}}+\sqrt {2} \sqrt {c+d x}}{\sqrt {c-\sqrt {c^2+d^2}}}\right )}{\sqrt {2} \sqrt {c-\sqrt {c^2+d^2}}}+\frac {d \log \left (c+\sqrt {c^2+d^2}+d x-\sqrt {2} \sqrt {c+\sqrt {c^2+d^2}} \sqrt {c+d x}\right )}{2 \sqrt {2} \sqrt {c+\sqrt {c^2+d^2}}}-\frac {d \log \left (c+\sqrt {c^2+d^2}+d x+\sqrt {2} \sqrt {c+\sqrt {c^2+d^2}} \sqrt {c+d x}\right )}{2 \sqrt {2} \sqrt {c+\sqrt {c^2+d^2}}} \]
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Rubi [A]
time = 0.34, antiderivative size = 316, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 6, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {714, 1143,
648, 632, 212, 642} \begin {gather*} \frac {d \log \left (-\sqrt {2} \sqrt {\sqrt {c^2+d^2}+c} \sqrt {c+d x}+\sqrt {c^2+d^2}+c+d x\right )}{2 \sqrt {2} \sqrt {\sqrt {c^2+d^2}+c}}-\frac {d \log \left (\sqrt {2} \sqrt {\sqrt {c^2+d^2}+c} \sqrt {c+d x}+\sqrt {c^2+d^2}+c+d x\right )}{2 \sqrt {2} \sqrt {\sqrt {c^2+d^2}+c}}+\frac {d \tanh ^{-1}\left (\frac {\sqrt {\sqrt {c^2+d^2}+c}-\sqrt {2} \sqrt {c+d x}}{\sqrt {c-\sqrt {c^2+d^2}}}\right )}{\sqrt {2} \sqrt {c-\sqrt {c^2+d^2}}}-\frac {d \tanh ^{-1}\left (\frac {\sqrt {\sqrt {c^2+d^2}+c}+\sqrt {2} \sqrt {c+d x}}{\sqrt {c-\sqrt {c^2+d^2}}}\right )}{\sqrt {2} \sqrt {c-\sqrt {c^2+d^2}}} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 632
Rule 642
Rule 648
Rule 714
Rule 1143
Rubi steps
\begin {align*} \int \frac {\sqrt {c+d x}}{1+x^2} \, dx &=(2 d) \text {Subst}\left (\int \frac {x^2}{c^2+d^2-2 c x^2+x^4} \, dx,x,\sqrt {c+d x}\right )\\ &=\frac {d \text {Subst}\left (\int \frac {x}{\sqrt {c^2+d^2}-\sqrt {2} \sqrt {c+\sqrt {c^2+d^2}} x+x^2} \, dx,x,\sqrt {c+d x}\right )}{\sqrt {2} \sqrt {c+\sqrt {c^2+d^2}}}-\frac {d \text {Subst}\left (\int \frac {x}{\sqrt {c^2+d^2}+\sqrt {2} \sqrt {c+\sqrt {c^2+d^2}} x+x^2} \, dx,x,\sqrt {c+d x}\right )}{\sqrt {2} \sqrt {c+\sqrt {c^2+d^2}}}\\ &=\frac {1}{2} d \text {Subst}\left (\int \frac {1}{\sqrt {c^2+d^2}-\sqrt {2} \sqrt {c+\sqrt {c^2+d^2}} x+x^2} \, dx,x,\sqrt {c+d x}\right )+\frac {1}{2} d \text {Subst}\left (\int \frac {1}{\sqrt {c^2+d^2}+\sqrt {2} \sqrt {c+\sqrt {c^2+d^2}} x+x^2} \, dx,x,\sqrt {c+d x}\right )+\frac {d \text {Subst}\left (\int \frac {-\sqrt {2} \sqrt {c+\sqrt {c^2+d^2}}+2 x}{\sqrt {c^2+d^2}-\sqrt {2} \sqrt {c+\sqrt {c^2+d^2}} x+x^2} \, dx,x,\sqrt {c+d x}\right )}{2 \sqrt {2} \sqrt {c+\sqrt {c^2+d^2}}}-\frac {d \text {Subst}\left (\int \frac {\sqrt {2} \sqrt {c+\sqrt {c^2+d^2}}+2 x}{\sqrt {c^2+d^2}+\sqrt {2} \sqrt {c+\sqrt {c^2+d^2}} x+x^2} \, dx,x,\sqrt {c+d x}\right )}{2 \sqrt {2} \sqrt {c+\sqrt {c^2+d^2}}}\\ &=\frac {d \log \left (c+\sqrt {c^2+d^2}+d x-\sqrt {2} \sqrt {c+\sqrt {c^2+d^2}} \sqrt {c+d x}\right )}{2 \sqrt {2} \sqrt {c+\sqrt {c^2+d^2}}}-\frac {d \log \left (c+\sqrt {c^2+d^2}+d x+\sqrt {2} \sqrt {c+\sqrt {c^2+d^2}} \sqrt {c+d x}\right )}{2 \sqrt {2} \sqrt {c+\sqrt {c^2+d^2}}}-d \text {Subst}\left (\int \frac {1}{2 \left (c-\sqrt {c^2+d^2}\right )-x^2} \, dx,x,-\sqrt {2} \sqrt {c+\sqrt {c^2+d^2}}+2 \sqrt {c+d x}\right )-d \text {Subst}\left (\int \frac {1}{2 \left (c-\sqrt {c^2+d^2}\right )-x^2} \, dx,x,\sqrt {2} \sqrt {c+\sqrt {c^2+d^2}}+2 \sqrt {c+d x}\right )\\ &=\frac {d \tanh ^{-1}\left (\frac {\sqrt {c+\sqrt {c^2+d^2}}-\sqrt {2} \sqrt {c+d x}}{\sqrt {c-\sqrt {c^2+d^2}}}\right )}{\sqrt {2} \sqrt {c-\sqrt {c^2+d^2}}}-\frac {d \tanh ^{-1}\left (\frac {\sqrt {c+\sqrt {c^2+d^2}}+\sqrt {2} \sqrt {c+d x}}{\sqrt {c-\sqrt {c^2+d^2}}}\right )}{\sqrt {2} \sqrt {c-\sqrt {c^2+d^2}}}+\frac {d \log \left (c+\sqrt {c^2+d^2}+d x-\sqrt {2} \sqrt {c+\sqrt {c^2+d^2}} \sqrt {c+d x}\right )}{2 \sqrt {2} \sqrt {c+\sqrt {c^2+d^2}}}-\frac {d \log \left (c+\sqrt {c^2+d^2}+d x+\sqrt {2} \sqrt {c+\sqrt {c^2+d^2}} \sqrt {c+d x}\right )}{2 \sqrt {2} \sqrt {c+\sqrt {c^2+d^2}}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.11, size = 82, normalized size = 0.26 \begin {gather*} i \left (\sqrt {-c-i d} \tan ^{-1}\left (\frac {\sqrt {c+d x}}{\sqrt {-c-i d}}\right )-\sqrt {-c+i d} \tan ^{-1}\left (\frac {\sqrt {c+d x}}{\sqrt {-c+i d}}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.44, size = 330, normalized size = 1.04
method | result | size |
derivativedivides | \(2 d \left (\frac {\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, \left (\sqrt {c^{2}+d^{2}}-c \right ) \left (\frac {\ln \left (\sqrt {d x +c}\, \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}-d x -c -\sqrt {c^{2}+d^{2}}\right )}{2}-\frac {\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, \arctan \left (\frac {\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}-2 \sqrt {d x +c}}{\sqrt {2 \sqrt {c^{2}+d^{2}}-2 c}}\right )}{\sqrt {2 \sqrt {c^{2}+d^{2}}-2 c}}\right )}{4 d^{2}}-\frac {\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, \left (\sqrt {c^{2}+d^{2}}-c \right ) \left (\frac {\ln \left (d x +c +\sqrt {d x +c}\, \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}+\sqrt {c^{2}+d^{2}}\right )}{2}-\frac {\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, \arctan \left (\frac {2 \sqrt {d x +c}+\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}}{\sqrt {2 \sqrt {c^{2}+d^{2}}-2 c}}\right )}{\sqrt {2 \sqrt {c^{2}+d^{2}}-2 c}}\right )}{4 d^{2}}\right )\) | \(330\) |
default | \(2 d \left (\frac {\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, \left (\sqrt {c^{2}+d^{2}}-c \right ) \left (\frac {\ln \left (\sqrt {d x +c}\, \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}-d x -c -\sqrt {c^{2}+d^{2}}\right )}{2}-\frac {\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, \arctan \left (\frac {\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}-2 \sqrt {d x +c}}{\sqrt {2 \sqrt {c^{2}+d^{2}}-2 c}}\right )}{\sqrt {2 \sqrt {c^{2}+d^{2}}-2 c}}\right )}{4 d^{2}}-\frac {\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, \left (\sqrt {c^{2}+d^{2}}-c \right ) \left (\frac {\ln \left (d x +c +\sqrt {d x +c}\, \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}+\sqrt {c^{2}+d^{2}}\right )}{2}-\frac {\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, \arctan \left (\frac {2 \sqrt {d x +c}+\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}}{\sqrt {2 \sqrt {c^{2}+d^{2}}-2 c}}\right )}{\sqrt {2 \sqrt {c^{2}+d^{2}}-2 c}}\right )}{4 d^{2}}\right )\) | \(330\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \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. 925 vs.
\(2 (247) = 494\).
time = 3.26, size = 925, normalized size = 2.93 \begin {gather*} -\frac {4 \, \sqrt {2} {\left (c^{2} + d^{2}\right )}^{\frac {3}{4}} \sqrt {d^{2}} \sqrt {\frac {c^{2} + d^{2} + \sqrt {c^{2} + d^{2}} c}{d^{2}}} \arctan \left (-\frac {\sqrt {2} {\left (c^{2} + d^{2}\right )}^{\frac {3}{4}} \sqrt {d^{2}} \sqrt {d x + c} d \sqrt {\frac {c^{2} + d^{2} + \sqrt {c^{2} + d^{2}} c}{d^{2}}} - \sqrt {2} {\left (c^{2} + d^{2}\right )}^{\frac {3}{4}} \sqrt {d^{2}} \sqrt {\frac {\sqrt {2} {\left (c^{2} + d^{2}\right )}^{\frac {3}{4}} \sqrt {d x + c} d^{3} \sqrt {\frac {c^{2} + d^{2} + \sqrt {c^{2} + d^{2}} c}{d^{2}}} + c^{3} d^{2} + c d^{4} + {\left (c^{2} d^{3} + d^{5}\right )} x + {\left (c^{2} d^{2} + d^{4}\right )} \sqrt {c^{2} + d^{2}}}{c^{2} + d^{2}}} \sqrt {\frac {c^{2} + d^{2} + \sqrt {c^{2} + d^{2}} c}{d^{2}}} + {\left (c^{2} + d^{2}\right )}^{\frac {3}{2}} \sqrt {d^{2}} + {\left (c^{3} + c d^{2}\right )} \sqrt {d^{2}}}{c^{2} d^{2} + d^{4}}\right ) + 4 \, \sqrt {2} {\left (c^{2} + d^{2}\right )}^{\frac {3}{4}} \sqrt {d^{2}} \sqrt {\frac {c^{2} + d^{2} + \sqrt {c^{2} + d^{2}} c}{d^{2}}} \arctan \left (-\frac {\sqrt {2} {\left (c^{2} + d^{2}\right )}^{\frac {3}{4}} \sqrt {d^{2}} \sqrt {d x + c} d \sqrt {\frac {c^{2} + d^{2} + \sqrt {c^{2} + d^{2}} c}{d^{2}}} - \sqrt {2} {\left (c^{2} + d^{2}\right )}^{\frac {3}{4}} \sqrt {d^{2}} \sqrt {-\frac {\sqrt {2} {\left (c^{2} + d^{2}\right )}^{\frac {3}{4}} \sqrt {d x + c} d^{3} \sqrt {\frac {c^{2} + d^{2} + \sqrt {c^{2} + d^{2}} c}{d^{2}}} - c^{3} d^{2} - c d^{4} - {\left (c^{2} d^{3} + d^{5}\right )} x - {\left (c^{2} d^{2} + d^{4}\right )} \sqrt {c^{2} + d^{2}}}{c^{2} + d^{2}}} \sqrt {\frac {c^{2} + d^{2} + \sqrt {c^{2} + d^{2}} c}{d^{2}}} - {\left (c^{2} + d^{2}\right )}^{\frac {3}{2}} \sqrt {d^{2}} - {\left (c^{3} + c d^{2}\right )} \sqrt {d^{2}}}{c^{2} d^{2} + d^{4}}\right ) + \sqrt {2} {\left (c^{2} + d^{2} - \sqrt {c^{2} + d^{2}} c\right )} {\left (c^{2} + d^{2}\right )}^{\frac {1}{4}} \sqrt {\frac {c^{2} + d^{2} + \sqrt {c^{2} + d^{2}} c}{d^{2}}} \log \left (\frac {\sqrt {2} {\left (c^{2} + d^{2}\right )}^{\frac {3}{4}} \sqrt {d x + c} d^{3} \sqrt {\frac {c^{2} + d^{2} + \sqrt {c^{2} + d^{2}} c}{d^{2}}} + c^{3} d^{2} + c d^{4} + {\left (c^{2} d^{3} + d^{5}\right )} x + {\left (c^{2} d^{2} + d^{4}\right )} \sqrt {c^{2} + d^{2}}}{c^{2} + d^{2}}\right ) - \sqrt {2} {\left (c^{2} + d^{2} - \sqrt {c^{2} + d^{2}} c\right )} {\left (c^{2} + d^{2}\right )}^{\frac {1}{4}} \sqrt {\frac {c^{2} + d^{2} + \sqrt {c^{2} + d^{2}} c}{d^{2}}} \log \left (-\frac {\sqrt {2} {\left (c^{2} + d^{2}\right )}^{\frac {3}{4}} \sqrt {d x + c} d^{3} \sqrt {\frac {c^{2} + d^{2} + \sqrt {c^{2} + d^{2}} c}{d^{2}}} - c^{3} d^{2} - c d^{4} - {\left (c^{2} d^{3} + d^{5}\right )} x - {\left (c^{2} d^{2} + d^{4}\right )} \sqrt {c^{2} + d^{2}}}{c^{2} + d^{2}}\right )}{4 \, {\left (c^{2} + d^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 2.22, size = 53, normalized size = 0.17 \begin {gather*} 2 d \operatorname {RootSum} {\left (256 t^{4} d^{4} + 32 t^{2} c d^{2} + c^{2} + d^{2}, \left ( t \mapsto t \log {\left (64 t^{3} d^{2} + 4 t c + \sqrt {c + d x} \right )} \right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Mupad [B]
time = 0.16, size = 133, normalized size = 0.42 \begin {gather*} -\mathrm {atan}\left (\frac {2\,c\,\sqrt {-\frac {c}{4}-\frac {d\,1{}\mathrm {i}}{4}}\,\sqrt {c+d\,x}-d\,\sqrt {-\frac {c}{4}-\frac {d\,1{}\mathrm {i}}{4}}\,\sqrt {c+d\,x}\,2{}\mathrm {i}}{c^2+d^2}\right )\,\sqrt {-\frac {c}{4}-\frac {d\,1{}\mathrm {i}}{4}}\,2{}\mathrm {i}+\mathrm {atan}\left (\frac {2\,c\,\sqrt {-\frac {c}{4}+\frac {d\,1{}\mathrm {i}}{4}}\,\sqrt {c+d\,x}+d\,\sqrt {-\frac {c}{4}+\frac {d\,1{}\mathrm {i}}{4}}\,\sqrt {c+d\,x}\,2{}\mathrm {i}}{c^2+d^2}\right )\,\sqrt {-\frac {c}{4}+\frac {d\,1{}\mathrm {i}}{4}}\,2{}\mathrm {i} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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