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

Optimal. Leaf size=48 \[ \frac{2 a (B+3 i A) \cosh (x)}{3 \sqrt{a+i a \sinh (x)}}+\frac{2}{3} B \cosh (x) \sqrt{a+i a \sinh (x)} \]

[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

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Rubi [A]  time = 0.0538788, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {2751, 2646} \[ \frac{2 a (B+3 i A) \cosh (x)}{3 \sqrt{a+i a \sinh (x)}}+\frac{2}{3} B \cosh (x) \sqrt{a+i a \sinh (x)} \]

Antiderivative was successfully verified.

[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 2751

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*Sin
[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)]

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 (x)} (A+B \sinh (x)) \, dx &=\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]  time = 0.0739076, size = 66, normalized size = 1.38 \[ \frac{2 \sqrt{a+i a \sinh (x)} \left (\sinh \left (\frac{x}{2}\right )+i \cosh \left (\frac{x}{2}\right )\right ) (3 A+B \sinh (x)-2 i B)}{3 \left (\cosh \left (\frac{x}{2}\right )+i \sinh \left (\frac{x}{2}\right )\right )} \]

Antiderivative was successfully verified.

[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]))

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Maple [F]  time = 0.108, size = 0, normalized size = 0. \begin{align*} \int \sqrt{a+ia\sinh \left ( x \right ) } \left ( A+B\sinh \left ( x \right ) \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [B]  time = 1.72814, size = 189, normalized size = 3.94 \begin{align*} \frac{\sqrt{\frac{1}{2}}{\left (B e^{\left (3 \, x\right )} + 3 \,{\left (2 \, A - i \, B\right )} e^{\left (2 \, x\right )} +{\left (6 i \, A + 3 \, B\right )} e^{x} - i \, B\right )} \sqrt{i \, a e^{\left (2 \, x\right )} + 2 \, a e^{x} - i \, a} e^{\left (-\frac{1}{2} \, x\right )}}{3 \,{\left (e^{\left (2 \, x\right )} - i \, e^{x}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*sqrt(1/2)*(B*e^(3*x) + 3*(2*A - I*B)*e^(2*x) + (6*I*A + 3*B)*e^x - I*B)*sqrt(I*a*e^(2*x) + 2*a*e^x - I*a)*
e^(-1/2*x)/(e^(2*x) - I*e^x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{a \left (i \sinh{\left (x \right )} + 1\right )} \left (A + B \sinh{\left (x \right )}\right )\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(a*(I*sinh(x) + 1))*(A + B*sinh(x)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[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)