3.863 \(\int \sqrt{a+b \cosh (c+d x) \sinh (c+d x)} \, dx\)

Optimal. Leaf size=96 \[ -\frac{i \sqrt{2 a+b \sinh (2 c+2 d x)} E\left (\frac{1}{2} \left (2 i c+2 i d x-\frac{\pi }{2}\right )|\frac{2 b}{2 i a+b}\right )}{\sqrt{2} d \sqrt{\frac{2 a+b \sinh (2 c+2 d x)}{2 a-i b}}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0726453, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {2666, 2655, 2653} \[ -\frac{i \sqrt{2 a+b \sinh (2 c+2 d x)} E\left (\frac{1}{2} \left (2 i c+2 i d x-\frac{\pi }{2}\right )|\frac{2 b}{2 i a+b}\right )}{\sqrt{2} d \sqrt{\frac{2 a+b \sinh (2 c+2 d x)}{2 a-i b}}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + b*Cosh[c + d*x]*Sinh[c + d*x]],x]

[Out]

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

Rule 2666

Int[((a_) + cos[(c_.) + (d_.)*(x_)]*(b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Int[(a + (b*Sin[2*c + 2*
d*x])/2)^n, x] /; FreeQ[{a, b, c, d, n}, x]

Rule 2655

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

Rule 2653

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*Sqrt[a + b]*EllipticE[(1*(c - Pi/2 + d*x)
)/2, (2*b)/(a + b)])/d, x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]

Rubi steps

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

Mathematica [A]  time = 0.108877, size = 94, normalized size = 0.98 \[ \frac{(b+2 i a) \sqrt{\frac{2 a+b \sinh (2 (c+d x))}{2 a-i b}} E\left (\frac{1}{4} (-4 i c-4 i d x+\pi )|-\frac{2 i b}{2 a-i b}\right )}{d \sqrt{4 a+2 b \sinh (2 (c+d x))}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [B]  time = 0.396, size = 351, normalized size = 3.7 \begin{align*}{\frac{ib-2\,a}{b\cosh \left ( 2\,dx+2\,c \right ) d}\sqrt{-{\frac{2\,a+b\sinh \left ( 2\,dx+2\,c \right ) }{ib-2\,a}}}\sqrt{{\frac{ \left ( -\sinh \left ( 2\,dx+2\,c \right ) +i \right ) b}{ib+2\,a}}}\sqrt{{\frac{ \left ( i+\sinh \left ( 2\,dx+2\,c \right ) \right ) b}{ib-2\,a}}} \left ( i{\it EllipticE} \left ( \sqrt{-{\frac{2\,a+b\sinh \left ( 2\,dx+2\,c \right ) }{ib-2\,a}}},\sqrt{-{\frac{ib-2\,a}{ib+2\,a}}} \right ) b-i{\it EllipticF} \left ( \sqrt{-{\frac{2\,a+b\sinh \left ( 2\,dx+2\,c \right ) }{ib-2\,a}}},\sqrt{-{\frac{ib-2\,a}{ib+2\,a}}} \right ) b+2\,{\it EllipticE} \left ( \sqrt{-{\frac{2\,a+b\sinh \left ( 2\,dx+2\,c \right ) }{ib-2\,a}}},\sqrt{-{\frac{ib-2\,a}{ib+2\,a}}} \right ) a-2\,a{\it EllipticF} \left ( \sqrt{-{\frac{2\,a+b\sinh \left ( 2\,dx+2\,c \right ) }{ib-2\,a}}},\sqrt{-{\frac{ib-2\,a}{ib+2\,a}}} \right ) \right ){\frac{1}{\sqrt{4\,a+2\,b\sinh \left ( 2\,dx+2\,c \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/b*(I*b-2*a)*(-(2*a+b*sinh(2*d*x+2*c))/(I*b-2*a))^(1/2)*((-sinh(2*d*x+2*c)+I)*b/(I*b+2*a))^(1/2)*((I+sinh(2*d
*x+2*c))*b/(I*b-2*a))^(1/2)*(I*EllipticE((-(2*a+b*sinh(2*d*x+2*c))/(I*b-2*a))^(1/2),(-(I*b-2*a)/(I*b+2*a))^(1/
2))*b-I*EllipticF((-(2*a+b*sinh(2*d*x+2*c))/(I*b-2*a))^(1/2),(-(I*b-2*a)/(I*b+2*a))^(1/2))*b+2*EllipticE((-(2*
a+b*sinh(2*d*x+2*c))/(I*b-2*a))^(1/2),(-(I*b-2*a)/(I*b+2*a))^(1/2))*a-2*a*EllipticF((-(2*a+b*sinh(2*d*x+2*c))/
(I*b-2*a))^(1/2),(-(I*b-2*a)/(I*b+2*a))^(1/2)))/cosh(2*d*x+2*c)/(4*a+2*b*sinh(2*d*x+2*c))^(1/2)/d

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(b*cosh(d*x + c)*sinh(d*x + c) + a), x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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