Optimal. Leaf size=40 \[ \frac {2 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a} \coth (c+d x)}{\sqrt {a-i a \text {csch}(c+d x)}}\right )}{d} \]
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Rubi [A] time = 0.02, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {3774, 203} \[ \frac {2 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a} \coth (c+d x)}{\sqrt {a-i a \text {csch}(c+d x)}}\right )}{d} \]
Antiderivative was successfully verified.
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Rule 203
Rule 3774
Rubi steps
\begin {align*} \int \sqrt {a-i a \text {csch}(c+d x)} \, dx &=\frac {(2 i a) \operatorname {Subst}\left (\int \frac {1}{a+x^2} \, dx,x,-\frac {i a \coth (c+d x)}{\sqrt {a-i a \text {csch}(c+d x)}}\right )}{d}\\ &=\frac {2 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a} \coth (c+d x)}{\sqrt {a-i a \text {csch}(c+d x)}}\right )}{d}\\ \end {align*}
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Mathematica [A] time = 1.01, size = 80, normalized size = 2.00 \[ -\frac {2 (-1)^{3/4} \coth (c+d x) \sqrt {a-i a \text {csch}(c+d x)} \tan ^{-1}\left ((-1)^{3/4} \sqrt {\text {csch}(c+d x)-i}\right )}{d \sqrt {\text {csch}(c+d x)-i} (\text {csch}(c+d x)+i)} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.58, size = 386, normalized size = 9.65 \[ \frac {1}{2} \, \sqrt {\frac {a}{d^{2}}} \log \left (\frac {{\left (2 \, {\left (d e^{\left (2 \, d x + 2 \, c\right )} - d\right )} \sqrt {\frac {a}{e^{\left (2 \, d x + 2 \, c\right )} - 1}} \sqrt {\frac {a}{d^{2}}} + 2 \, a e^{\left (d x + c\right )} - 2 i \, a\right )} e^{\left (-d x - c\right )}}{d}\right ) - \frac {1}{2} \, \sqrt {\frac {a}{d^{2}}} \log \left (-\frac {{\left (2 \, {\left (d e^{\left (2 \, d x + 2 \, c\right )} - d\right )} \sqrt {\frac {a}{e^{\left (2 \, d x + 2 \, c\right )} - 1}} \sqrt {\frac {a}{d^{2}}} - 2 \, a e^{\left (d x + c\right )} + 2 i \, a\right )} e^{\left (-d x - c\right )}}{d}\right ) + \frac {1}{2} \, \sqrt {\frac {a}{d^{2}}} \log \left (\frac {{\left ({\left (2 \, a e^{\left (3 \, d x + 3 \, c\right )} + 4 i \, a e^{\left (2 \, d x + 2 \, c\right )} - 2 \, a e^{\left (d x + c\right )} - 4 i \, a\right )} \sqrt {\frac {a}{e^{\left (2 \, d x + 2 \, c\right )} - 1}} + 2 \, {\left (a d e^{\left (2 \, d x + 2 \, c\right )} + i \, a d e^{\left (d x + c\right )} - 2 \, a d\right )} \sqrt {\frac {a}{d^{2}}}\right )} e^{\left (-2 \, d x - 2 \, c\right )}}{d}\right ) - \frac {1}{2} \, \sqrt {\frac {a}{d^{2}}} \log \left (\frac {{\left ({\left (2 \, a e^{\left (3 \, d x + 3 \, c\right )} + 4 i \, a e^{\left (2 \, d x + 2 \, c\right )} - 2 \, a e^{\left (d x + c\right )} - 4 i \, a\right )} \sqrt {\frac {a}{e^{\left (2 \, d x + 2 \, c\right )} - 1}} - 2 \, {\left (a d e^{\left (2 \, d x + 2 \, c\right )} + i \, a d e^{\left (d x + c\right )} - 2 \, a d\right )} \sqrt {\frac {a}{d^{2}}}\right )} e^{\left (-2 \, d x - 2 \, c\right )}}{d}\right ) \]
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 \[ \int \sqrt {-i \, a \operatorname {csch}\left (d x + c\right ) + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 2.78, size = 0, normalized size = 0.00 \[ \int \sqrt {a -i a \,\mathrm {csch}\left (d x +c \right )}\, dx \]
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 \[ \int \sqrt {-i \, a \operatorname {csch}\left (d x + c\right ) + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \sqrt {a-\frac {a\,1{}\mathrm {i}}{\mathrm {sinh}\left (c+d\,x\right )}} \,d x \]
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 \[ \int \sqrt {- i a \operatorname {csch}{\left (c + d x \right )} + a}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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