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

Optimal. Leaf size=31 \[ \frac{2 i a \cosh (c+d x)}{d \sqrt{a+i a \sinh (c+d x)}} \]

[Out]

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

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Rubi [A]  time = 0.0139036, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {2646} \[ \frac{2 i a \cosh (c+d x)}{d \sqrt{a+i a \sinh (c+d x)}} \]

Antiderivative was successfully verified.

[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 2646

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*} \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)}}\\ \end{align*}

Mathematica [B]  time = 0.041178, size = 74, normalized size = 2.39 \[ \frac{2 \sqrt{a+i a \sinh (c+d x)} \left (\sinh \left (\frac{1}{2} (c+d x)\right )+i \cosh \left (\frac{1}{2} (c+d x)\right )\right )}{d \left (\cosh \left (\frac{1}{2} (c+d x)\right )+i \sinh \left (\frac{1}{2} (c+d x)\right )\right )} \]

Antiderivative was successfully verified.

[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]))

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Maple [B]  time = 0.125, size = 89, normalized size = 2.9 \begin{align*}{\frac{i\sqrt{2} \left ({{\rm e}^{dx+c}}+i \right ) \left ({{\rm e}^{dx+c}}-i \right ) }{ \left ( i{{\rm e}^{2\,dx+2\,c}}-i+2\,{{\rm e}^{dx+c}} \right ) d}\sqrt{a \left ( i{{\rm e}^{2\,dx+2\,c}}-i+2\,{{\rm e}^{dx+c}} \right ){{\rm e}^{-dx-c}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{i \, a \sinh \left (d x + c\right ) + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [B]  time = 2.00736, size = 170, normalized size = 5.48 \begin{align*} \frac{\sqrt{\frac{1}{2}} \sqrt{i \, a e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a e^{\left (d x + c\right )} - i \, a}{\left (2 \, e^{\left (d x + c\right )} + 2 i\right )} e^{\left (-\frac{1}{2} \, d x - \frac{1}{2} \, c\right )}}{d e^{\left (d x + c\right )} - i \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{i a \sinh{\left (c + d x \right )} + a}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{i \, a \sinh \left (d x + c\right ) + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)