Optimal. Leaf size=52 \[ \frac {i \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {a} \cosh (c+d x)}{\sqrt {2} \sqrt {a+i a \sinh (c+d x)}}\right )}{\sqrt {a} d} \]
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Rubi [A] time = 0.02, antiderivative size = 52, 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 = {2649, 206} \[ \frac {i \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {a} \cosh (c+d x)}{\sqrt {2} \sqrt {a+i a \sinh (c+d x)}}\right )}{\sqrt {a} d} \]
Antiderivative was successfully verified.
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Rule 206
Rule 2649
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {a+i a \sinh (c+d x)}} \, dx &=\frac {(2 i) \operatorname {Subst}\left (\int \frac {1}{2 a-x^2} \, dx,x,\frac {a \cosh (c+d x)}{\sqrt {a+i a \sinh (c+d x)}}\right )}{d}\\ &=\frac {i \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {a} \cosh (c+d x)}{\sqrt {2} \sqrt {a+i a \sinh (c+d x)}}\right )}{\sqrt {a} d}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 84, normalized size = 1.62 \[ \frac {(2+2 i) \sqrt [4]{-1} \tan ^{-1}\left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt [4]{-1} \left (1-i \tanh \left (\frac {1}{4} (c+d x)\right )\right )\right ) \left (\sinh \left (\frac {1}{2} (c+d x)\right )-i \cosh \left (\frac {1}{2} (c+d x)\right )\right )}{d \sqrt {a+i a \sinh (c+d x)}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.67, size = 93, normalized size = 1.79 \[ i \, \sqrt {2} \sqrt {\frac {1}{a d^{2}}} \log \left (\frac {1}{2} \, \sqrt {2} a d \sqrt {\frac {1}{a d^{2}}} + \sqrt {\frac {1}{2} i \, a e^{\left (-d x - c\right )}}\right ) - i \, \sqrt {2} \sqrt {\frac {1}{a d^{2}}} \log \left (-\frac {1}{2} \, \sqrt {2} a d \sqrt {\frac {1}{a d^{2}}} + \sqrt {\frac {1}{2} i \, a e^{\left (-d x - c\right )}}\right ) \]
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 \[ \int \frac {1}{\sqrt {i \, a \sinh \left (d x + c\right ) + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.54, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {a +i a \sinh \left (d x +c \right )}}\, dx \]
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 \[ \int \frac {1}{\sqrt {i \, a \sinh \left (d x + c\right ) + a}}\,{d x} \]
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 \[ \int \frac {1}{\sqrt {a+a\,\mathrm {sinh}\left (c+d\,x\right )\,1{}\mathrm {i}}} \,d x \]
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 \[ \int \frac {1}{\sqrt {i a \sinh {\left (c + d x \right )} + a}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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