\(\int \frac {1}{\sqrt {a+i a \sinh (c+d x)}} \, dx\) [69]

Optimal result

Integrand size = 17, antiderivative size = 52 \[ \int \frac {1}{\sqrt {a+i a \sinh (c+d x)}} \, dx=\frac {i \sqrt {2} \text {arctanh}\left (\frac {\sqrt {a} \cosh (c+d x)}{\sqrt {2} \sqrt {a+i a \sinh (c+d x)}}\right )}{\sqrt {a} d} \]

[Out]

I*arctanh(1/2*cosh(d*x+c)*a^(1/2)*2^(1/2)/(a+I*a*sinh(d*x+c))^(1/2))*2^(1/2)/d/a^(1/2)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2728, 212} \[ \int \frac {1}{\sqrt {a+i a \sinh (c+d x)}} \, dx=\frac {i \sqrt {2} \text {arctanh}\left (\frac {\sqrt {a} \cosh (c+d x)}{\sqrt {2} \sqrt {a+i a \sinh (c+d x)}}\right )}{\sqrt {a} d} \]

[In]

Int[1/Sqrt[a + I*a*Sinh[c + d*x]],x]

[Out]

(I*Sqrt[2]*ArcTanh[(Sqrt[a]*Cosh[c + d*x])/(Sqrt[2]*Sqrt[a + I*a*Sinh[c + d*x]])])/(Sqrt[a]*d)

Rule 212

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] && (GtQ[a, 0] || LtQ[b, 0])

Rule 2728

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[-2/d, Subst[Int[1/(2*a - x^2), x], x, b*(C
os[c + d*x]/Sqrt[a + b*Sin[c + d*x]])], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {(2 i) \text {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} \text {arctanh}\left (\frac {\sqrt {a} \cosh (c+d x)}{\sqrt {2} \sqrt {a+i a \sinh (c+d x)}}\right )}{\sqrt {a} d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.62 \[ \int \frac {1}{\sqrt {a+i a \sinh (c+d x)}} \, dx=\frac {(2+2 i) \sqrt [4]{-1} \arctan \left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt [4]{-1} \left (1-i \tanh \left (\frac {1}{4} (c+d x)\right )\right )\right ) \left (-i \cosh \left (\frac {1}{2} (c+d x)\right )+\sinh \left (\frac {1}{2} (c+d x)\right )\right )}{d \sqrt {a+i a \sinh (c+d x)}} \]

[In]

Integrate[1/Sqrt[a + I*a*Sinh[c + d*x]],x]

[Out]

((2 + 2*I)*(-1)^(1/4)*ArcTan[(1/2 + I/2)*(-1)^(1/4)*(1 - I*Tanh[(c + d*x)/4])]*((-I)*Cosh[(c + d*x)/2] + Sinh[
(c + d*x)/2]))/(d*Sqrt[a + I*a*Sinh[c + d*x]])

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 159 vs. \(2 (41 ) = 82\).

Time = 6.70 (sec) , antiderivative size = 160, normalized size of antiderivative = 3.08

method	result	size
risch	\(-\frac {2 \left ({\mathrm e}^{d x +c}-i\right ) \sqrt {2}\, {\mathrm e}^{-d x -c}}{d \sqrt {a \left (i {\mathrm e}^{2 d x +2 c}+2 \,{\mathrm e}^{d x +c}-i\right ) {\mathrm e}^{-d x -c}}}-\frac {2 \left (-{\mathrm e}^{d x +c}+i\right ) \left (a^{\frac {3}{2}}+\arctan \left (\frac {\sqrt {i {\mathrm e}^{d x +c} a}}{\sqrt {a}}\right ) a \sqrt {i {\mathrm e}^{d x +c} a}\right ) \sqrt {2}\, {\mathrm e}^{-d x -c}}{d \,a^{\frac {3}{2}} \sqrt {a \left (i {\mathrm e}^{2 d x +2 c}+2 \,{\mathrm e}^{d x +c}-i\right ) {\mathrm e}^{-d x -c}}}\)	\(160\)

[In]

int(1/(a+I*a*sinh(d*x+c))^(1/2),x,method=_RETURNVERBOSE)

[Out]

-2/d*(exp(d*x+c)-I)*2^(1/2)/(a*(I*exp(d*x+c)^2+2*exp(d*x+c)-I)/exp(d*x+c))^(1/2)/exp(d*x+c)-2/d*(-exp(d*x+c)+I
)*(a^(3/2)+arctan((I*exp(d*x+c)*a)^(1/2)/a^(1/2))*a*(I*exp(d*x+c)*a)^(1/2))/a^(3/2)*2^(1/2)/(a*(I*exp(d*x+c)^2
+2*exp(d*x+c)-I)/exp(d*x+c))^(1/2)/exp(d*x+c)

Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 93 vs. \(2 (39) = 78\).

Time = 0.24 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.79 \[ \int \frac {1}{\sqrt {a+i a \sinh (c+d x)}} \, dx=i \, \sqrt {2} \sqrt {\frac {1}{a d^{2}}} \log \left (\frac {1}{2} \, \sqrt {2} a d \sqrt {\frac {1}{a d^{2}}} + \sqrt {\frac {1}{2} i \, a e^{\left (-d x - c\right )}}\right ) - i \, \sqrt {2} \sqrt {\frac {1}{a d^{2}}} \log \left (-\frac {1}{2} \, \sqrt {2} a d \sqrt {\frac {1}{a d^{2}}} + \sqrt {\frac {1}{2} i \, a e^{\left (-d x - c\right )}}\right ) \]

[In]

integrate(1/(a+I*a*sinh(d*x+c))^(1/2),x, algorithm="fricas")

[Out]

I*sqrt(2)*sqrt(1/(a*d^2))*log(1/2*sqrt(2)*a*d*sqrt(1/(a*d^2)) + sqrt(1/2*I*a*e^(-d*x - c))) - I*sqrt(2)*sqrt(1
/(a*d^2))*log(-1/2*sqrt(2)*a*d*sqrt(1/(a*d^2)) + sqrt(1/2*I*a*e^(-d*x - c)))

Sympy [F]

\[ \int \frac {1}{\sqrt {a+i a \sinh (c+d x)}} \, dx=\int \frac {1}{\sqrt {i a \sinh {\left (c + d x \right )} + a}}\, dx \]

[In]

integrate(1/(a+I*a*sinh(d*x+c))**(1/2),x)

[Out]

Integral(1/sqrt(I*a*sinh(c + d*x) + a), x)

Maxima [F]

\[ \int \frac {1}{\sqrt {a+i a \sinh (c+d x)}} \, dx=\int { \frac {1}{\sqrt {i \, a \sinh \left (d x + c\right ) + a}} \,d x } \]

[In]

integrate(1/(a+I*a*sinh(d*x+c))^(1/2),x, algorithm="maxima")

[Out]

integrate(1/sqrt(I*a*sinh(d*x + c) + a), x)

Giac [F]

\[ \int \frac {1}{\sqrt {a+i a \sinh (c+d x)}} \, dx=\int { \frac {1}{\sqrt {i \, a \sinh \left (d x + c\right ) + a}} \,d x } \]

[In]

integrate(1/(a+I*a*sinh(d*x+c))^(1/2),x, algorithm="giac")

[Out]

integrate(1/sqrt(I*a*sinh(d*x + c) + a), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {a+i a \sinh (c+d x)}} \, dx=\int \frac {1}{\sqrt {a+a\,\mathrm {sinh}\left (c+d\,x\right )\,1{}\mathrm {i}}} \,d x \]

[In]

int(1/(a + a*sinh(c + d*x)*1i)^(1/2),x)

[Out]

int(1/(a + a*sinh(c + d*x)*1i)^(1/2), x)