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

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Rubi [A]  time = 0.01, antiderivative size = 31, normalized size of antiderivative = 1.00, 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*}

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Mathematica [B]  time = 0.04, 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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fricas [A]  time = 0.49, size = 28, normalized size = 0.90 \[ \frac {\sqrt {\frac {1}{2} i \, a e^{\left (-d x - c\right )}} {\left (2 \, e^{\left (d x + c\right )} + 2 i\right )}}{d} \]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {i \, a \sinh \left (d x + c\right ) + a}\,{d x} \]

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)

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maple [B]  time = 0.13, size = 89, normalized size = 2.87 \[ \frac {i \sqrt {2}\, \sqrt {a \left (i {\mathrm e}^{2 d x +2 c}-i+2 \,{\mathrm e}^{d x +c}\right ) {\mathrm e}^{-d x -c}}\, \left ({\mathrm e}^{d x +c}+i\right ) \left ({\mathrm e}^{d x +c}-i\right )}{\left (i {\mathrm e}^{2 d x +2 c}-i+2 \,{\mathrm e}^{d x +c}\right ) d} \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {i \, a \sinh \left (d x + c\right ) + a}\,{d x} \]

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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mupad [B]  time = 0.75, size = 53, normalized size = 1.71 \[ \frac {\sqrt {2}\,\left ({\mathrm {e}}^{c+d\,x}+1{}\mathrm {i}\right )\,\sqrt {a\,{\mathrm {e}}^{-c-d\,x}\,{\left ({\mathrm {e}}^{c+d\,x}-\mathrm {i}\right )}^2\,1{}\mathrm {i}}}{d\,\left ({\mathrm {e}}^{c+d\,x}-\mathrm {i}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[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))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {i a \sinh {\left (c + d x \right )} + a}\, dx \]

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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