\(\int \sqrt {a+i a \sinh (c+d x)} \, dx\) [68]

Optimal result

Integrand size = 17, antiderivative size = 31 \[ \int \sqrt {a+i a \sinh (c+d x)} \, dx=\frac {2 i a \cosh (c+d x)}{d \sqrt {a+i a \sinh (c+d x)}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {2725} \[ \int \sqrt {a+i a \sinh (c+d x)} \, dx=\frac {2 i a \cosh (c+d x)}{d \sqrt {a+i a \sinh (c+d x)}} \]

[In]

Int[Sqrt[a + I*a*Sinh[c + d*x]],x]

[Out]

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

Rule 2725

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[-2*b*(Cos[c + d*x]/(d*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 a \cosh (c+d x)}{d \sqrt {a+i a \sinh (c+d x)}} \\ \end{align*}

Mathematica [B] (verified)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(74\) vs. \(2(31)=62\).

Time = 0.03 (sec) , antiderivative size = 74, normalized size of antiderivative = 2.39 \[ \int \sqrt {a+i a \sinh (c+d x)} \, dx=\frac {2 \left (i \cosh \left (\frac {1}{2} (c+d x)\right )+\sinh \left (\frac {1}{2} (c+d x)\right )\right ) \sqrt {a+i a \sinh (c+d x)}}{d \left (\cosh \left (\frac {1}{2} (c+d x)\right )+i \sinh \left (\frac {1}{2} (c+d x)\right )\right )} \]

[In]

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

[Out]

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

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 88 vs. \(2 (27 ) = 54\).

Time = 2.19 (sec) , antiderivative size = 89, normalized size of antiderivative = 2.87

method	result	size
risch	\(\frac {i \sqrt {2}\, \sqrt {a \left (i {\mathrm e}^{2 d x +2 c}+2 \,{\mathrm e}^{d x +c}-i\right ) {\mathrm e}^{-d x -c}}\, \left ({\mathrm e}^{d x +c}+i\right ) \left ({\mathrm e}^{d x +c}-i\right )}{\left (i {\mathrm e}^{2 d x +2 c}+2 \,{\mathrm e}^{d x +c}-i\right ) d}\)	\(89\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.87 \[ \int \sqrt {a+i a \sinh (c+d x)} \, dx=\frac {2 \, \sqrt {\frac {1}{2} i \, a e^{\left (-d x - c\right )}} {\left (e^{\left (d x + c\right )} + i\right )}}{d} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 1.53 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.71 \[ \int \sqrt {a+i a \sinh (c+d x)} \, dx=\frac {\sqrt {2}\,\left ({\mathrm {e}}^{c+d\,x}+1{}\mathrm {i}\right )\,\sqrt {a\,{\mathrm {e}}^{-c-d\,x}\,{\left ({\mathrm {e}}^{c+d\,x}-\mathrm {i}\right )}^2\,1{}\mathrm {i}}}{d\,\left ({\mathrm {e}}^{c+d\,x}-\mathrm {i}\right )} \]

[In]

int((a + a*sinh(c + d*x)*1i)^(1/2),x)

[Out]

(2^(1/2)*(exp(c + d*x) + 1i)*(a*exp(- c - d*x)*(exp(c + d*x) - 1i)^2*1i)^(1/2))/(d*(exp(c + d*x) - 1i))