Integrand size = 20, antiderivative size = 66 \[ \int \frac {A+B \sinh (x)}{\sqrt {a+i a \sinh (x)}} \, dx=\frac {\sqrt {2} (i A-B) \text {arctanh}\left (\frac {\sqrt {a} \cosh (x)}{\sqrt {2} \sqrt {a+i a \sinh (x)}}\right )}{\sqrt {a}}+\frac {2 B \cosh (x)}{\sqrt {a+i a \sinh (x)}} \] Output:
2^(1/2)*(I*A-B)*arctanh(1/2*a^(1/2)*cosh(x)*2^(1/2)/(a+I*a*sinh(x))^(1/2)) /a^(1/2)+2*B*cosh(x)/(a+I*a*sinh(x))^(1/2)
Time = 0.21 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.29 \[ \int \frac {A+B \sinh (x)}{\sqrt {a+i a \sinh (x)}} \, dx=\frac {2 \left (\cosh \left (\frac {x}{2}\right )+i \sinh \left (\frac {x}{2}\right )\right ) \left ((1+i) \sqrt [4]{-1} (-i A+B) \arctan \left (\frac {i+\tanh \left (\frac {x}{4}\right )}{\sqrt {2}}\right )+B \cosh \left (\frac {x}{2}\right )-i B \sinh \left (\frac {x}{2}\right )\right )}{\sqrt {a+i a \sinh (x)}} \] Input:
Integrate[(A + B*Sinh[x])/Sqrt[a + I*a*Sinh[x]],x]
Output:
(2*(Cosh[x/2] + I*Sinh[x/2])*((1 + I)*(-1)^(1/4)*((-I)*A + B)*ArcTan[(I + Tanh[x/4])/Sqrt[2]] + B*Cosh[x/2] - I*B*Sinh[x/2]))/Sqrt[a + I*a*Sinh[x]]
Time = 0.33 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.02, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3042, 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.
\(\displaystyle \int \frac {A+B \sinh (x)}{\sqrt {a+i a \sinh (x)}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {A-i B \sin (i x)}{\sqrt {a+a \sin (i x)}}dx\) |
\(\Big \downarrow \) 3230 |
\(\displaystyle (A+i B) \int \frac {1}{\sqrt {i \sinh (x) a+a}}dx+\frac {2 B \cosh (x)}{\sqrt {a+i a \sinh (x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle (A+i B) \int \frac {1}{\sqrt {\sin (i x) a+a}}dx+\frac {2 B \cosh (x)}{\sqrt {a+i a \sinh (x)}}\) |
\(\Big \downarrow \) 3128 |
\(\displaystyle 2 i (A+i B) \int \frac {1}{2 a-\frac {a^2 \cosh ^2(x)}{i \sinh (x) a+a}}d\frac {a \cosh (x)}{\sqrt {i \sinh (x) a+a}}+\frac {2 B \cosh (x)}{\sqrt {a+i a \sinh (x)}}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {i \sqrt {2} (A+i B) \text {arctanh}\left (\frac {\sqrt {a} \cosh (x)}{\sqrt {2} \sqrt {a+i a \sinh (x)}}\right )}{\sqrt {a}}+\frac {2 B \cosh (x)}{\sqrt {a+i a \sinh (x)}}\) |
Input:
Int[(A + B*Sinh[x])/Sqrt[a + I*a*Sinh[x]],x]
Output:
(I*Sqrt[2]*(A + I*B)*ArcTanh[(Sqrt[a]*Cosh[x])/(Sqrt[2]*Sqrt[a + I*a*Sinh[ x]])])/Sqrt[a] + (2*B*Cosh[x])/Sqrt[a + I*a*Sinh[x]]
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. 125 vs. \(2 (52 ) = 104\).
Time = 1.22 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.91
method | result | size |
risch | \(\frac {\left (-2 A -i B +B \,{\mathrm e}^{x}\right ) \left ({\mathrm e}^{x}-i\right ) \sqrt {2}\, {\mathrm e}^{-x}}{\sqrt {a \left (i {\mathrm e}^{2 x}+2 \,{\mathrm e}^{x}-i\right ) {\mathrm e}^{-x}}}+\frac {i \left (2 i A -2 B \right ) \left (-{\mathrm e}^{x}+i\right ) \left (a^{\frac {3}{2}}+\arctan \left (\frac {\sqrt {i a \,{\mathrm e}^{x}}}{\sqrt {a}}\right ) a \sqrt {i a \,{\mathrm e}^{x}}\right ) \sqrt {2}\, {\mathrm e}^{-x}}{a^{\frac {3}{2}} \sqrt {a \left (i {\mathrm e}^{2 x}+2 \,{\mathrm e}^{x}-i\right ) {\mathrm e}^{-x}}}\) | \(126\) |
Input:
int((A+B*sinh(x))/(a+I*a*sinh(x))^(1/2),x,method=_RETURNVERBOSE)
Output:
(-2*A-I*B+B*exp(x))*(exp(x)-I)*2^(1/2)/(a*(I*exp(x)^2+2*exp(x)-I)/exp(x))^ (1/2)/exp(x)+I*(2*I*A-2*B)*(-exp(x)+I)*(a^(3/2)+arctan((I*a*exp(x))^(1/2)/ a^(1/2))*a*(I*a*exp(x))^(1/2))/a^(3/2)*2^(1/2)/(a*(I*exp(x)^2+2*exp(x)-I)/ exp(x))^(1/2)/exp(x)
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 = 0.09 (sec) , antiderivative size = 188, normalized size of antiderivative = 2.85 \[ \int \frac {A+B \sinh (x)}{\sqrt {a+i a \sinh (x)}} \, dx=\frac {\sqrt {2} a \sqrt {-\frac {A^{2} + 2 i \, A B - B^{2}}{a}} \log \left (-\frac {2 \, {\left (\sqrt {2} a \sqrt {-\frac {A^{2} + 2 i \, A B - B^{2}}{a}} + 2 \, \sqrt {\frac {1}{2} i \, a e^{\left (-x\right )}} {\left (i \, A - B\right )}\right )}}{-4 i \, A + 4 \, B}\right ) - \sqrt {2} a \sqrt {-\frac {A^{2} + 2 i \, A B - B^{2}}{a}} \log \left (\frac {2 \, {\left (\sqrt {2} a \sqrt {-\frac {A^{2} + 2 i \, A B - B^{2}}{a}} - 2 \, \sqrt {\frac {1}{2} i \, a e^{\left (-x\right )}} {\left (i \, A - B\right )}\right )}}{-4 i \, A + 4 \, B}\right ) - 2 \, \sqrt {\frac {1}{2} i \, a e^{\left (-x\right )}} {\left (i \, B e^{x} - B\right )}}{a} \] Input:
integrate((A+B*sinh(x))/(a+I*a*sinh(x))^(1/2),x, algorithm="fricas")
Output:
(sqrt(2)*a*sqrt(-(A^2 + 2*I*A*B - B^2)/a)*log(-2*(sqrt(2)*a*sqrt(-(A^2 + 2 *I*A*B - B^2)/a) + 2*sqrt(1/2*I*a*e^(-x))*(I*A - B))/(-4*I*A + 4*B)) - sqr t(2)*a*sqrt(-(A^2 + 2*I*A*B - B^2)/a)*log(2*(sqrt(2)*a*sqrt(-(A^2 + 2*I*A* B - B^2)/a) - 2*sqrt(1/2*I*a*e^(-x))*(I*A - B))/(-4*I*A + 4*B)) - 2*sqrt(1 /2*I*a*e^(-x))*(I*B*e^x - B))/a
\[ \int \frac {A+B \sinh (x)}{\sqrt {a+i a \sinh (x)}} \, dx=\int \frac {A + B \sinh {\left (x \right )}}{\sqrt {i a \left (\sinh {\left (x \right )} - i\right )}}\, dx \] Input:
integrate((A+B*sinh(x))/(a+I*a*sinh(x))**(1/2),x)
Output:
Integral((A + B*sinh(x))/sqrt(I*a*(sinh(x) - I)), x)
\[ \int \frac {A+B \sinh (x)}{\sqrt {a+i a \sinh (x)}} \, dx=\int { \frac {B \sinh \left (x\right ) + A}{\sqrt {i \, a \sinh \left (x\right ) + a}} \,d x } \] Input:
integrate((A+B*sinh(x))/(a+I*a*sinh(x))^(1/2),x, algorithm="maxima")
Output:
integrate((B*sinh(x) + A)/sqrt(I*a*sinh(x) + a), x)
\[ \int \frac {A+B \sinh (x)}{\sqrt {a+i a \sinh (x)}} \, dx=\int { \frac {B \sinh \left (x\right ) + A}{\sqrt {i \, a \sinh \left (x\right ) + a}} \,d x } \] Input:
integrate((A+B*sinh(x))/(a+I*a*sinh(x))^(1/2),x, algorithm="giac")
Output:
integrate((B*sinh(x) + A)/sqrt(I*a*sinh(x) + a), x)
Timed out. \[ \int \frac {A+B \sinh (x)}{\sqrt {a+i a \sinh (x)}} \, dx=\int \frac {A+B\,\mathrm {sinh}\left (x\right )}{\sqrt {a+a\,\mathrm {sinh}\left (x\right )\,1{}\mathrm {i}}} \,d x \] Input:
int((A + B*sinh(x))/(a + a*sinh(x)*1i)^(1/2),x)
Output:
int((A + B*sinh(x))/(a + a*sinh(x)*1i)^(1/2), x)
\[ \int \frac {A+B \sinh (x)}{\sqrt {a+i a \sinh (x)}} \, dx=\frac {\sqrt {a}\, \left (\left (\int \frac {\sqrt {\sinh \left (x \right ) i +1}}{\sinh \left (x \right )^{2}+1}d x \right ) a -\left (\int \frac {\sqrt {\sinh \left (x \right ) i +1}\, \sinh \left (x \right )^{2}}{\sinh \left (x \right )^{2}+1}d x \right ) b i -\left (\int \frac {\sqrt {\sinh \left (x \right ) i +1}\, \sinh \left (x \right )}{\sinh \left (x \right )^{2}+1}d x \right ) a i +\left (\int \frac {\sqrt {\sinh \left (x \right ) i +1}\, \sinh \left (x \right )}{\sinh \left (x \right )^{2}+1}d x \right ) b \right )}{a} \] Input:
int((A+B*sinh(x))/(a+I*a*sinh(x))^(1/2),x)
Output:
(sqrt(a)*(int(sqrt(sinh(x)*i + 1)/(sinh(x)**2 + 1),x)*a - int((sqrt(sinh(x )*i + 1)*sinh(x)**2)/(sinh(x)**2 + 1),x)*b*i - int((sqrt(sinh(x)*i + 1)*si nh(x))/(sinh(x)**2 + 1),x)*a*i + int((sqrt(sinh(x)*i + 1)*sinh(x))/(sinh(x )**2 + 1),x)*b))/a