Integrand size = 20, antiderivative size = 66 \[ \int \frac {A+B \sinh (x)}{\sqrt {a+i a \sinh (x)}} \, dx=\frac {\sqrt {2} (i A-B) \text {arctanh}\left (\frac {\sqrt {a} \cosh (x)}{\sqrt {2} \sqrt {a+i a \sinh (x)}}\right )}{\sqrt {a}}+\frac {2 B \cosh (x)}{\sqrt {a+i a \sinh (x)}} \]
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Time = 0.05 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {2830, 2728, 212} \[ \int \frac {A+B \sinh (x)}{\sqrt {a+i a \sinh (x)}} \, dx=\frac {\sqrt {2} (-B+i A) \text {arctanh}\left (\frac {\sqrt {a} \cosh (x)}{\sqrt {2} \sqrt {a+i a \sinh (x)}}\right )}{\sqrt {a}}+\frac {2 B \cosh (x)}{\sqrt {a+i a \sinh (x)}} \]
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Rule 212
Rule 2728
Rule 2830
Rubi steps \begin{align*} \text {integral}& = \frac {2 B \cosh (x)}{\sqrt {a+i a \sinh (x)}}+(A+i B) \int \frac {1}{\sqrt {a+i a \sinh (x)}} \, dx \\ & = \frac {2 B \cosh (x)}{\sqrt {a+i a \sinh (x)}}+(2 (i A-B)) \text {Subst}\left (\int \frac {1}{2 a-x^2} \, dx,x,\frac {a \cosh (x)}{\sqrt {a+i a \sinh (x)}}\right ) \\ & = \frac {\sqrt {2} (i A-B) \text {arctanh}\left (\frac {\sqrt {a} \cosh (x)}{\sqrt {2} \sqrt {a+i a \sinh (x)}}\right )}{\sqrt {a}}+\frac {2 B \cosh (x)}{\sqrt {a+i a \sinh (x)}} \\ \end{align*}
Time = 0.22 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.29 \[ \int \frac {A+B \sinh (x)}{\sqrt {a+i a \sinh (x)}} \, dx=\frac {2 \left (\cosh \left (\frac {x}{2}\right )+i \sinh \left (\frac {x}{2}\right )\right ) \left ((1+i) \sqrt [4]{-1} (-i A+B) \arctan \left (\frac {i+\tanh \left (\frac {x}{4}\right )}{\sqrt {2}}\right )+B \cosh \left (\frac {x}{2}\right )-i B \sinh \left (\frac {x}{2}\right )\right )}{\sqrt {a+i a \sinh (x)}} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 125 vs. \(2 (52 ) = 104\).
Time = 5.19 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.91
method | result | size |
risch | \(\frac {\left (-2 A -i B +B \,{\mathrm e}^{x}\right ) \left ({\mathrm e}^{x}-i\right ) \sqrt {2}\, {\mathrm e}^{-x}}{\sqrt {a \left (i {\mathrm e}^{2 x}+2 \,{\mathrm e}^{x}-i\right ) {\mathrm e}^{-x}}}+\frac {i \left (2 i A -2 B \right ) \left (-{\mathrm e}^{x}+i\right ) \left (a^{\frac {3}{2}}+\arctan \left (\frac {\sqrt {i a \,{\mathrm e}^{x}}}{\sqrt {a}}\right ) a \sqrt {i a \,{\mathrm e}^{x}}\right ) \sqrt {2}\, {\mathrm e}^{-x}}{a^{\frac {3}{2}} \sqrt {a \left (i {\mathrm e}^{2 x}+2 \,{\mathrm e}^{x}-i\right ) {\mathrm e}^{-x}}}\) | \(126\) |
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 188 vs. \(2 (49) = 98\).
Time = 0.32 (sec) , antiderivative size = 188, normalized size of antiderivative = 2.85 \[ \int \frac {A+B \sinh (x)}{\sqrt {a+i a \sinh (x)}} \, dx=\frac {\sqrt {2} a \sqrt {-\frac {A^{2} + 2 i \, A B - B^{2}}{a}} \log \left (-\frac {2 \, {\left (\sqrt {2} a \sqrt {-\frac {A^{2} + 2 i \, A B - B^{2}}{a}} + 2 \, \sqrt {\frac {1}{2} i \, a e^{\left (-x\right )}} {\left (i \, A - B\right )}\right )}}{-4 i \, A + 4 \, B}\right ) - \sqrt {2} a \sqrt {-\frac {A^{2} + 2 i \, A B - B^{2}}{a}} \log \left (\frac {2 \, {\left (\sqrt {2} a \sqrt {-\frac {A^{2} + 2 i \, A B - B^{2}}{a}} - 2 \, \sqrt {\frac {1}{2} i \, a e^{\left (-x\right )}} {\left (i \, A - B\right )}\right )}}{-4 i \, A + 4 \, B}\right ) - 2 \, \sqrt {\frac {1}{2} i \, a e^{\left (-x\right )}} {\left (i \, B e^{x} - B\right )}}{a} \]
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\[ \int \frac {A+B \sinh (x)}{\sqrt {a+i a \sinh (x)}} \, dx=\int \frac {A + B \sinh {\left (x \right )}}{\sqrt {i a \left (\sinh {\left (x \right )} - i\right )}}\, dx \]
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\[ \int \frac {A+B \sinh (x)}{\sqrt {a+i a \sinh (x)}} \, dx=\int { \frac {B \sinh \left (x\right ) + A}{\sqrt {i \, a \sinh \left (x\right ) + a}} \,d x } \]
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\[ \int \frac {A+B \sinh (x)}{\sqrt {a+i a \sinh (x)}} \, dx=\int { \frac {B \sinh \left (x\right ) + A}{\sqrt {i \, a \sinh \left (x\right ) + a}} \,d x } \]
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Timed out. \[ \int \frac {A+B \sinh (x)}{\sqrt {a+i a \sinh (x)}} \, dx=\int \frac {A+B\,\mathrm {sinh}\left (x\right )}{\sqrt {a+a\,\mathrm {sinh}\left (x\right )\,1{}\mathrm {i}}} \,d x \]
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