3.18 \(\int \sqrt{b \sinh (c+d x)} \, dx\)

Optimal. Leaf size=56 \[ -\frac{2 i E\left (\left .\frac{1}{2} \left (i c+i d x-\frac{\pi }{2}\right )\right |2\right ) \sqrt{b \sinh (c+d x)}}{d \sqrt{i \sinh (c+d x)}} \]

[Out]

((-2*I)*EllipticE[(I*c - Pi/2 + I*d*x)/2, 2]*Sqrt[b*Sinh[c + d*x]])/(d*Sqrt[I*Sinh[c + d*x]])

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Rubi [A]  time = 0.0215649, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {2640, 2639} \[ -\frac{2 i E\left (\left .\frac{1}{2} \left (i c+i d x-\frac{\pi }{2}\right )\right |2\right ) \sqrt{b \sinh (c+d x)}}{d \sqrt{i \sinh (c+d x)}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[b*Sinh[c + d*x]],x]

[Out]

((-2*I)*EllipticE[(I*c - Pi/2 + I*d*x)/2, 2]*Sqrt[b*Sinh[c + d*x]])/(d*Sqrt[I*Sinh[c + d*x]])

Rule 2640

Int[Sqrt[(b_)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[Sqrt[b*Sin[c + d*x]]/Sqrt[Sin[c + d*x]], Int[Sqrt[Si
n[c + d*x]], x], x] /; FreeQ[{b, c, d}, x]

Rule 2639

Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticE[(1*(c - Pi/2 + d*x))/2, 2])/d, x] /; FreeQ[{
c, d}, x]

Rubi steps

\begin{align*} \int \sqrt{b \sinh (c+d x)} \, dx &=\frac{\sqrt{b \sinh (c+d x)} \int \sqrt{i \sinh (c+d x)} \, dx}{\sqrt{i \sinh (c+d x)}}\\ &=-\frac{2 i E\left (\left .\frac{1}{2} \left (i c-\frac{\pi }{2}+i d x\right )\right |2\right ) \sqrt{b \sinh (c+d x)}}{d \sqrt{i \sinh (c+d x)}}\\ \end{align*}

Mathematica [A]  time = 0.0450161, size = 52, normalized size = 0.93 \[ \frac{2 i E\left (\left .\frac{1}{4} (-2 i c-2 i d x+\pi )\right |2\right ) \sqrt{b \sinh (c+d x)}}{d \sqrt{i \sinh (c+d x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[b*Sinh[c + d*x]],x]

[Out]

((2*I)*EllipticE[((-2*I)*c + Pi - (2*I)*d*x)/4, 2]*Sqrt[b*Sinh[c + d*x]])/(d*Sqrt[I*Sinh[c + d*x]])

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Maple [A]  time = 0.043, size = 111, normalized size = 2. \begin{align*}{\frac{b\sqrt{2}}{d\cosh \left ( dx+c \right ) }\sqrt{-i \left ( \sinh \left ( dx+c \right ) +i \right ) }\sqrt{-i \left ( i-\sinh \left ( dx+c \right ) \right ) }\sqrt{i\sinh \left ( dx+c \right ) } \left ( 2\,{\it EllipticE} \left ( \sqrt{1-i\sinh \left ( dx+c \right ) },1/2\,\sqrt{2} \right ) -{\it EllipticF} \left ( \sqrt{1-i\sinh \left ( dx+c \right ) },{\frac{\sqrt{2}}{2}} \right ) \right ){\frac{1}{\sqrt{b\sinh \left ( dx+c \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b*(-I*(sinh(d*x+c)+I))^(1/2)*2^(1/2)*(-I*(I-sinh(d*x+c)))^(1/2)*(I*sinh(d*x+c))^(1/2)*(2*EllipticE((1-I*sinh(d
*x+c))^(1/2),1/2*2^(1/2))-EllipticF((1-I*sinh(d*x+c))^(1/2),1/2*2^(1/2)))/cosh(d*x+c)/(b*sinh(d*x+c))^(1/2)/d

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(b*sinh(d*x + c)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\sqrt{b \sinh \left (d x + c\right )}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(b*sinh(d*x + c)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(b*sinh(c + d*x)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(b*sinh(d*x + c)), x)