\(\int \sqrt {a+i a \sinh (x)} (A+B \sinh (x)) \, dx\) [114]

Optimal result

Integrand size = 20, antiderivative size = 48 \[ \int \sqrt {a+i a \sinh (x)} (A+B \sinh (x)) \, dx=\frac {2 a (3 i A+B) \cosh (x)}{3 \sqrt {a+i a \sinh (x)}}+\frac {2}{3} B \cosh (x) \sqrt {a+i a \sinh (x)} \]

[Out]

2/3*a*(3*I*A+B)*cosh(x)/(a+I*a*sinh(x))^(1/2)+2/3*B*cosh(x)*(a+I*a*sinh(x))^(1/2)

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2830, 2725} \[ \int \sqrt {a+i a \sinh (x)} (A+B \sinh (x)) \, dx=\frac {2 a (B+3 i A) \cosh (x)}{3 \sqrt {a+i a \sinh (x)}}+\frac {2}{3} B \cosh (x) \sqrt {a+i a \sinh (x)} \]

[In]

Int[Sqrt[a + I*a*Sinh[x]]*(A + B*Sinh[x]),x]

[Out]

(2*a*((3*I)*A + B)*Cosh[x])/(3*Sqrt[a + I*a*Sinh[x]]) + (2*B*Cosh[x]*Sqrt[a + I*a*Sinh[x]])/3

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]

Rule 2830

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-d
)*Cos[e + f*x]*((a + b*Sin[e + f*x])^m/(f*(m + 1))), x] + Dist[(a*d*m + b*c*(m + 1))/(b*(m + 1)), Int[(a + b*S
in[e + f*x])^m, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] &&  !LtQ[m
, -2^(-1)]

Rubi steps \begin{align*} \text {integral}& = \frac {2}{3} B \cosh (x) \sqrt {a+i a \sinh (x)}+\frac {1}{3} (3 A-i B) \int \sqrt {a+i a \sinh (x)} \, dx \\ & = \frac {2 a (3 i A+B) \cosh (x)}{3 \sqrt {a+i a \sinh (x)}}+\frac {2}{3} B \cosh (x) \sqrt {a+i a \sinh (x)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.38 \[ \int \sqrt {a+i a \sinh (x)} (A+B \sinh (x)) \, dx=\frac {2 \left (i \cosh \left (\frac {x}{2}\right )+\sinh \left (\frac {x}{2}\right )\right ) \sqrt {a+i a \sinh (x)} (3 A-2 i B+B \sinh (x))}{3 \left (\cosh \left (\frac {x}{2}\right )+i \sinh \left (\frac {x}{2}\right )\right )} \]

[In]

Integrate[Sqrt[a + I*a*Sinh[x]]*(A + B*Sinh[x]),x]

[Out]

(2*(I*Cosh[x/2] + Sinh[x/2])*Sqrt[a + I*a*Sinh[x]]*(3*A - (2*I)*B + B*Sinh[x]))/(3*(Cosh[x/2] + I*Sinh[x/2]))

Maple [F]

\[\int \sqrt {a +i a \sinh \left (x \right )}\, \left (A +B \sinh \left (x \right )\right )d x\]

[In]

int((a+I*a*sinh(x))^(1/2)*(A+B*sinh(x)),x)

[Out]

int((a+I*a*sinh(x))^(1/2)*(A+B*sinh(x)),x)

Fricas [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.02 \[ \int \sqrt {a+i a \sinh (x)} (A+B \sinh (x)) \, dx=\frac {1}{3} \, {\left (B e^{\left (3 \, x\right )} + 3 \, {\left (2 \, A - i \, B\right )} e^{\left (2 \, x\right )} - 3 \, {\left (-2 i \, A - B\right )} e^{x} - i \, B\right )} \sqrt {\frac {1}{2} i \, a e^{\left (-x\right )}} e^{\left (-x\right )} \]

[In]

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

[Out]

1/3*(B*e^(3*x) + 3*(2*A - I*B)*e^(2*x) - 3*(-2*I*A - B)*e^x - I*B)*sqrt(1/2*I*a*e^(-x))*e^(-x)

Sympy [F]

\[ \int \sqrt {a+i a \sinh (x)} (A+B \sinh (x)) \, dx=\int \sqrt {i a \left (\sinh {\left (x \right )} - i\right )} \left (A + B \sinh {\left (x \right )}\right )\, dx \]

[In]

integrate((a+I*a*sinh(x))**(1/2)*(A+B*sinh(x)),x)

[Out]

Integral(sqrt(I*a*(sinh(x) - I))*(A + B*sinh(x)), x)

Maxima [F]

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

[In]

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

[Out]

integrate((B*sinh(x) + A)*sqrt(I*a*sinh(x) + a), x)

Giac [F]

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

[In]

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

[Out]

integrate((B*sinh(x) + A)*sqrt(I*a*sinh(x) + a), x)

Mupad [F(-1)]

Timed out. \[ \int \sqrt {a+i a \sinh (x)} (A+B \sinh (x)) \, dx=\int \left (A+B\,\mathrm {sinh}\left (x\right )\right )\,\sqrt {a+a\,\mathrm {sinh}\left (x\right )\,1{}\mathrm {i}} \,d x \]

[In]

int((A + B*sinh(x))*(a + a*sinh(x)*1i)^(1/2),x)

[Out]

int((A + B*sinh(x))*(a + a*sinh(x)*1i)^(1/2), x)