Optimal. Leaf size=91 \[ \frac {2 \tanh ^{-1}\left (\frac {\sqrt {a} \coth (c+d x)}{\sqrt {a-i a \text {csch}(c+d x)}}\right )}{\sqrt {a} d}-\frac {\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {a} \coth (c+d x)}{\sqrt {2} \sqrt {a-i a \text {csch}(c+d x)}}\right )}{\sqrt {a} d} \]
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Rubi [A] time = 0.08, antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {3776, 3774, 203, 3795} \[ \frac {2 \tanh ^{-1}\left (\frac {\sqrt {a} \coth (c+d x)}{\sqrt {a-i a \text {csch}(c+d x)}}\right )}{\sqrt {a} d}-\frac {\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {a} \coth (c+d x)}{\sqrt {2} \sqrt {a-i a \text {csch}(c+d x)}}\right )}{\sqrt {a} d} \]
Antiderivative was successfully verified.
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Rule 203
Rule 3774
Rule 3776
Rule 3795
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {a-i a \text {csch}(c+d x)}} \, dx &=i \int \frac {\text {csch}(c+d x)}{\sqrt {a-i a \text {csch}(c+d x)}} \, dx+\frac {\int \sqrt {a-i a \text {csch}(c+d x)} \, dx}{a}\\ &=\frac {(2 i) \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 i) \operatorname {Subst}\left (\int \frac {1}{2 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 \tanh ^{-1}\left (\frac {\sqrt {a} \coth (c+d x)}{\sqrt {a-i a \text {csch}(c+d x)}}\right )}{\sqrt {a} d}-\frac {\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {a} \coth (c+d x)}{\sqrt {2} \sqrt {a-i a \text {csch}(c+d x)}}\right )}{\sqrt {a} d}\\ \end {align*}
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Mathematica [A] time = 1.10, size = 117, normalized size = 1.29 \[ \frac {\sqrt {a} \coth (c+d x) \left (2 \tan ^{-1}\left (\frac {\sqrt {-i a (\text {csch}(c+d x)-i)}}{\sqrt {a}}\right )-\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {-i a (\text {csch}(c+d x)-i)}}{\sqrt {2} \sqrt {a}}\right )\right )}{d \sqrt {a (-1-i \text {csch}(c+d x))} \sqrt {a-i a \text {csch}(c+d x)}} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.28, size = 561, normalized size = 6.16 \[ -\frac {1}{2} \, \sqrt {2} \sqrt {\frac {1}{a d^{2}}} \log \left ({\left (2 \, \sqrt {2} {\left (a d e^{\left (2 \, d x + 2 \, c\right )} - a d\right )} \sqrt {\frac {a}{e^{\left (2 \, d x + 2 \, c\right )} - 1}} \sqrt {\frac {1}{a d^{2}}} + 2 \, a e^{\left (d x + c\right )} + 2 i \, a\right )} e^{\left (-d x - c\right )}\right ) + \frac {1}{2} \, \sqrt {2} \sqrt {\frac {1}{a d^{2}}} \log \left (-{\left (2 \, \sqrt {2} {\left (a d e^{\left (2 \, d x + 2 \, c\right )} - a d\right )} \sqrt {\frac {a}{e^{\left (2 \, d x + 2 \, c\right )} - 1}} \sqrt {\frac {1}{a d^{2}}} - 2 \, a e^{\left (d x + c\right )} - 2 i \, a\right )} e^{\left (-d x - c\right )}\right ) + \frac {1}{2} \, \sqrt {\frac {1}{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 {1}{a d^{2}}} + 2 \, e^{\left (d x + c\right )} - 2 i\right )} e^{\left (-d x - c\right )}}{d}\right ) - \frac {1}{2} \, \sqrt {\frac {1}{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 {1}{a d^{2}}} - 2 \, e^{\left (d x + c\right )} + 2 i\right )} e^{\left (-d x - c\right )}}{d}\right ) + \frac {1}{2} \, \sqrt {\frac {1}{a d^{2}}} \log \left (\frac {{\left ({\left (2 \, a d e^{\left (2 \, d x + 2 \, c\right )} + 2 i \, a d e^{\left (d x + c\right )} - 4 \, a d\right )} \sqrt {\frac {1}{a d^{2}}} + \sqrt {\frac {a}{e^{\left (2 \, d x + 2 \, c\right )} - 1}} {\left (2 \, e^{\left (3 \, d x + 3 \, c\right )} + 4 i \, e^{\left (2 \, d x + 2 \, c\right )} - 2 \, e^{\left (d x + c\right )} - 4 i\right )}\right )} e^{\left (-2 \, d x - 2 \, c\right )}}{d}\right ) - \frac {1}{2} \, \sqrt {\frac {1}{a d^{2}}} \log \left (-\frac {{\left ({\left (2 \, a d e^{\left (2 \, d x + 2 \, c\right )} + 2 i \, a d e^{\left (d x + c\right )} - 4 \, a d\right )} \sqrt {\frac {1}{a d^{2}}} - \sqrt {\frac {a}{e^{\left (2 \, d x + 2 \, c\right )} - 1}} {\left (2 \, e^{\left (3 \, d x + 3 \, c\right )} + 4 i \, e^{\left (2 \, d x + 2 \, c\right )} - 2 \, e^{\left (d x + c\right )} - 4 i\right )}\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 \[ \int \frac {1}{\sqrt {-i \, a \operatorname {csch}\left (d x + c\right ) + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.88, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {a -i a \,\mathrm {csch}\left (d x +c \right )}}\, dx \]
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 \[ \int \frac {1}{\sqrt {-i \, a \operatorname {csch}\left (d x + c\right ) + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{\sqrt {a-\frac {a\,1{}\mathrm {i}}{\mathrm {sinh}\left (c+d\,x\right )}}} \,d x \]
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 \[ \int \frac {1}{\sqrt {- i a \operatorname {csch}{\left (c + d x \right )} + a}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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