3.119 \(\int \frac{A+B \sinh (x)}{i-\sinh (x)} \, dx\)

Optimal. Leaf size=27 \[ -B x+\frac{(-B+i A) \cosh (x)}{-\sinh (x)+i} \]

[Out]

-(B*x) + ((I*A - B)*Cosh[x])/(I - Sinh[x])

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Rubi [A]  time = 0.042611, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {2735, 2648} \[ -B x+\frac{(-B+i A) \cosh (x)}{-\sinh (x)+i} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*Sinh[x])/(I - Sinh[x]),x]

[Out]

-(B*x) + ((I*A - B)*Cosh[x])/(I - Sinh[x])

Rule 2735

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

Rule 2648

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

Rubi steps

\begin{align*} \int \frac{A+B \sinh (x)}{i-\sinh (x)} \, dx &=-B x+(A+i B) \int \frac{1}{i-\sinh (x)} \, dx\\ &=-B x+\frac{(i A-B) \cosh (x)}{i-\sinh (x)}\\ \end{align*}

Mathematica [B]  time = 0.0753079, size = 59, normalized size = 2.19 \[ \frac{\left (-\sinh \left (\frac{x}{2}\right )+i \cosh \left (\frac{x}{2}\right )\right ) \left (B x \cosh \left (\frac{x}{2}\right )+i \sinh \left (\frac{x}{2}\right ) (2 A+B (x+2 i))\right )}{\sinh (x)-i} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*Sinh[x])/(I - Sinh[x]),x]

[Out]

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

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Maple [A]  time = 0.03, size = 46, normalized size = 1.7 \begin{align*} -B\ln \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) +B\ln \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) +2\,{\frac{B}{\tanh \left ( x/2 \right ) -i}}-{2\,iA \left ( \tanh \left ({\frac{x}{2}} \right ) -i \right ) ^{-1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A+B*sinh(x))/(I-sinh(x)),x)

[Out]

-B*ln(tanh(1/2*x)+1)+B*ln(tanh(1/2*x)-1)+2/(tanh(1/2*x)-I)*B-2*I/(tanh(1/2*x)-I)*A

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Maxima [A]  time = 1.19698, size = 36, normalized size = 1.33 \begin{align*} -B{\left (x - \frac{2 i}{e^{\left (-x\right )} + i}\right )} + \frac{2 \, A}{e^{\left (-x\right )} + i} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*sinh(x))/(I-sinh(x)),x, algorithm="maxima")

[Out]

-B*(x - 2*I/(e^(-x) + I)) + 2*A/(e^(-x) + I)

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Fricas [A]  time = 1.7196, size = 59, normalized size = 2.19 \begin{align*} -\frac{B x e^{x} - i \, B x - 2 \, A - 2 i \, B}{e^{x} - i} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*sinh(x))/(I-sinh(x)),x, algorithm="fricas")

[Out]

-(B*x*e^x - I*B*x - 2*A - 2*I*B)/(e^x - I)

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Sympy [A]  time = 0.209746, size = 15, normalized size = 0.56 \begin{align*} - B x + \frac{2 A + 2 i B}{e^{x} - i} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*sinh(x))/(I-sinh(x)),x)

[Out]

-B*x + (2*A + 2*I*B)/(exp(x) - I)

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Giac [A]  time = 1.29128, size = 24, normalized size = 0.89 \begin{align*} -B x + \frac{2 \,{\left (A + i \, B\right )}}{e^{x} - i} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*sinh(x))/(I-sinh(x)),x, algorithm="giac")

[Out]

-B*x + 2*(A + I*B)/(e^x - I)