Integrand size = 16, antiderivative size = 57 \[ \int \frac {\sinh (x)}{\sqrt {a-i a \sinh (x)}} \, dx=-\frac {\sqrt {2} \text {arctanh}\left (\frac {\sqrt {a} \cosh (x)}{\sqrt {2} \sqrt {a-i a \sinh (x)}}\right )}{\sqrt {a}}+\frac {2 \cosh (x)}{\sqrt {a-i a \sinh (x)}} \] Output:
-2^(1/2)*arctanh(1/2*a^(1/2)*cosh(x)*2^(1/2)/(a-I*a*sinh(x))^(1/2))/a^(1/2 )+2*cosh(x)/(a-I*a*sinh(x))^(1/2)
Time = 0.24 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.33 \[ \int \frac {\sinh (x)}{\sqrt {a-i a \sinh (x)}} \, dx=\frac {2 \left (\cosh \left (\frac {x}{2}\right )-i \sinh \left (\frac {x}{2}\right )\right ) \left (\cosh \left (\frac {x}{2}\right )+i \left ((1+i) (-1)^{3/4} \arctan \left (\frac {-i+\tanh \left (\frac {x}{4}\right )}{\sqrt {2}}\right )+\sinh \left (\frac {x}{2}\right )\right )\right )}{\sqrt {a-i a \sinh (x)}} \] Input:
Integrate[Sinh[x]/Sqrt[a - I*a*Sinh[x]],x]
Output:
(2*(Cosh[x/2] - I*Sinh[x/2])*(Cosh[x/2] + I*((1 + I)*(-1)^(3/4)*ArcTan[(-I + Tanh[x/4])/Sqrt[2]] + Sinh[x/2])))/Sqrt[a - I*a*Sinh[x]]
Time = 0.29 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.14, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {3042, 26, 3230, 3042, 3128, 219}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\sinh (x)}{\sqrt {a-i a \sinh (x)}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int -\frac {i \sin (i x)}{\sqrt {a-a \sin (i x)}}dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -i \int \frac {\sin (i x)}{\sqrt {a-a \sin (i x)}}dx\) |
\(\Big \downarrow \) 3230 |
\(\displaystyle -i \left (\int \frac {1}{\sqrt {a-i a \sinh (x)}}dx+\frac {2 i \cosh (x)}{\sqrt {a-i a \sinh (x)}}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -i \left (\int \frac {1}{\sqrt {a-a \sin (i x)}}dx+\frac {2 i \cosh (x)}{\sqrt {a-i a \sinh (x)}}\right )\) |
\(\Big \downarrow \) 3128 |
\(\displaystyle -i \left (2 i \int \frac {1}{2 a-\frac {a^2 \cosh ^2(x)}{a-i a \sinh (x)}}d\left (-\frac {a \cosh (x)}{\sqrt {a-i a \sinh (x)}}\right )+\frac {2 i \cosh (x)}{\sqrt {a-i a \sinh (x)}}\right )\) |
\(\Big \downarrow \) 219 |
\(\displaystyle -i \left (\frac {2 i \cosh (x)}{\sqrt {a-i a \sinh (x)}}-\frac {i \sqrt {2} \text {arctanh}\left (\frac {\sqrt {a} \cosh (x)}{\sqrt {2} \sqrt {a-i a \sinh (x)}}\right )}{\sqrt {a}}\right )\) |
Input:
Int[Sinh[x]/Sqrt[a - I*a*Sinh[x]],x]
Output:
(-I)*(((-I)*Sqrt[2]*ArcTanh[(Sqrt[a]*Cosh[x])/(Sqrt[2]*Sqrt[a - I*a*Sinh[x ]])])/Sqrt[a] + ((2*I)*Cosh[x])/Sqrt[a - I*a*Sinh[x]])
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[-2/d Subst[Int[1/(2*a - x^2), x], x, b*(Cos[c + d*x]/Sqrt[a + b*Sin[c + d*x]])], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-d)*Cos[e + f*x]*((a + b*Sin[e + f*x])^m/( f*(m + 1))), x] + Simp[(a*d*m + b*c*(m + 1))/(b*(m + 1)) Int[(a + b*Sin[e + f*x])^m, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && !LtQ[m, -2^(-1)]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 107 vs. \(2 (44 ) = 88\).
Time = 0.78 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.89
method | result | size |
risch | \(\frac {\left ({\mathrm e}^{x}+i\right )^{2} \sqrt {2}\, {\mathrm e}^{-x}}{\sqrt {-a \left (i {\mathrm e}^{2 x}-i-2 \,{\mathrm e}^{x}\right ) {\mathrm e}^{-x}}}-\frac {2 i \left ({\mathrm e}^{x}+i\right ) \left (\arctan \left (\frac {\sqrt {-i a \,{\mathrm e}^{x}}}{\sqrt {a}}\right ) a \sqrt {-i a \,{\mathrm e}^{x}}+a^{\frac {3}{2}}\right ) \sqrt {2}\, {\mathrm e}^{-x}}{a^{\frac {3}{2}} \sqrt {-a \left (i {\mathrm e}^{2 x}-i-2 \,{\mathrm e}^{x}\right ) {\mathrm e}^{-x}}}\) | \(108\) |
Input:
int(sinh(x)/(a-I*a*sinh(x))^(1/2),x,method=_RETURNVERBOSE)
Output:
(exp(x)+I)^2*2^(1/2)/(-a*(I*exp(x)^2-I-2*exp(x))/exp(x))^(1/2)/exp(x)-2*I* (exp(x)+I)*(arctan((-I*a*exp(x))^(1/2)/a^(1/2))*a*(-I*a*exp(x))^(1/2)+a^(3 /2))/a^(3/2)*2^(1/2)/(-a*(I*exp(x)^2-I-2*exp(x))/exp(x))^(1/2)/exp(x)
Time = 0.08 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.33 \[ \int \frac {\sinh (x)}{\sqrt {a-i a \sinh (x)}} \, dx=-\frac {\sqrt {2} \sqrt {a} \log \left (\frac {1}{2} \, \sqrt {2} \sqrt {a} + \sqrt {-\frac {1}{2} i \, a e^{\left (-x\right )}}\right ) - \sqrt {2} \sqrt {a} \log \left (-\frac {1}{2} \, \sqrt {2} \sqrt {a} + \sqrt {-\frac {1}{2} i \, a e^{\left (-x\right )}}\right ) + 2 \, \sqrt {-\frac {1}{2} i \, a e^{\left (-x\right )}} {\left (-i \, e^{x} - 1\right )}}{a} \] Input:
integrate(sinh(x)/(a-I*a*sinh(x))^(1/2),x, algorithm="fricas")
Output:
-(sqrt(2)*sqrt(a)*log(1/2*sqrt(2)*sqrt(a) + sqrt(-1/2*I*a*e^(-x))) - sqrt( 2)*sqrt(a)*log(-1/2*sqrt(2)*sqrt(a) + sqrt(-1/2*I*a*e^(-x))) + 2*sqrt(-1/2 *I*a*e^(-x))*(-I*e^x - 1))/a
\[ \int \frac {\sinh (x)}{\sqrt {a-i a \sinh (x)}} \, dx=\int \frac {\sinh {\left (x \right )}}{\sqrt {- i a \left (\sinh {\left (x \right )} + i\right )}}\, dx \] Input:
integrate(sinh(x)/(a-I*a*sinh(x))**(1/2),x)
Output:
Integral(sinh(x)/sqrt(-I*a*(sinh(x) + I)), x)
\[ \int \frac {\sinh (x)}{\sqrt {a-i a \sinh (x)}} \, dx=\int { \frac {\sinh \left (x\right )}{\sqrt {-i \, a \sinh \left (x\right ) + a}} \,d x } \] Input:
integrate(sinh(x)/(a-I*a*sinh(x))^(1/2),x, algorithm="maxima")
Output:
integrate(sinh(x)/sqrt(-I*a*sinh(x) + a), x)
\[ \int \frac {\sinh (x)}{\sqrt {a-i a \sinh (x)}} \, dx=\int { \frac {\sinh \left (x\right )}{\sqrt {-i \, a \sinh \left (x\right ) + a}} \,d x } \] Input:
integrate(sinh(x)/(a-I*a*sinh(x))^(1/2),x, algorithm="giac")
Output:
integrate(sinh(x)/sqrt(-I*a*sinh(x) + a), x)
Timed out. \[ \int \frac {\sinh (x)}{\sqrt {a-i a \sinh (x)}} \, dx=\int \frac {\mathrm {sinh}\left (x\right )}{\sqrt {a-a\,\mathrm {sinh}\left (x\right )\,1{}\mathrm {i}}} \,d x \] Input:
int(sinh(x)/(a - a*sinh(x)*1i)^(1/2),x)
Output:
int(sinh(x)/(a - a*sinh(x)*1i)^(1/2), x)
\[ \int \frac {\sinh (x)}{\sqrt {a-i a \sinh (x)}} \, dx=\frac {\sqrt {a}\, \left (\left (\int \frac {\sqrt {-\sinh \left (x \right ) i +1}\, \sinh \left (x \right )^{2}}{\sinh \left (x \right )^{2}+1}d x \right ) i +\int \frac {\sqrt {-\sinh \left (x \right ) i +1}\, \sinh \left (x \right )}{\sinh \left (x \right )^{2}+1}d x \right )}{a} \] Input:
int(sinh(x)/(a-I*a*sinh(x))^(1/2),x)
Output:
(sqrt(a)*(int((sqrt( - sinh(x)*i + 1)*sinh(x)**2)/(sinh(x)**2 + 1),x)*i + int((sqrt( - sinh(x)*i + 1)*sinh(x))/(sinh(x)**2 + 1),x)))/a