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

Optimal. Leaf size=27 \[ \frac{A \cosh (x)}{1+i \sinh (x)}-B \log (-\sinh (x)+i) \]

[Out]

-(B*Log[I - Sinh[x]]) + (A*Cosh[x])/(1 + I*Sinh[x])

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Rubi [A]  time = 0.0865139, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235, Rules used = {4401, 2648, 2667, 31} \[ \frac{A \cosh (x)}{1+i \sinh (x)}-B \log (-\sinh (x)+i) \]

Antiderivative was successfully verified.

[In]

Int[(A + B*Cosh[x])/(I - Sinh[x]),x]

[Out]

-(B*Log[I - Sinh[x]]) + (A*Cosh[x])/(1 + I*Sinh[x])

Rule 4401

Int[u_, x_Symbol] :> With[{v = ExpandTrig[u, x]}, Int[v, x] /; SumQ[v]] /;  !InertTrigFreeQ[u]

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]

Rule 2667

Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(b^p*f), S
ubst[Int[(a + x)^(m + (p - 1)/2)*(a - x)^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x]
&& IntegerQ[(p - 1)/2] && EqQ[a^2 - b^2, 0] && (GeQ[p, -1] ||  !IntegerQ[m + 1/2])

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

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

Mathematica [B]  time = 0.0881459, size = 81, normalized size = 3. \[ -\frac{\left (\cosh \left (\frac{x}{2}\right )+i \sinh \left (\frac{x}{2}\right )\right ) \left (\sinh \left (\frac{x}{2}\right ) \left (2 A+2 i B \tan ^{-1}\left (\tanh \left (\frac{x}{2}\right )\right )+B \log (\cosh (x))\right )+B \cosh \left (\frac{x}{2}\right ) \left (2 \tan ^{-1}\left (\tanh \left (\frac{x}{2}\right )\right )-i \log (\cosh (x))\right )\right )}{\sinh (x)-i} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.037, size = 44, normalized size = 1.6 \begin{align*} B\ln \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) -{2\,iA \left ( \tanh \left ({\frac{x}{2}} \right ) -i \right ) ^{-1}}-2\,B\ln \left ( \tanh \left ( x/2 \right ) -i \right ) +B\ln \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.05507, size = 27, normalized size = 1. \begin{align*} -B \log \left (\sinh \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*cosh(x))/(I-sinh(x)),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 2.08174, size = 89, normalized size = 3.3 \begin{align*} \frac{B x e^{x} - i \, B x - 2 \,{\left (B e^{x} - i \, B\right )} \log \left (e^{x} - i\right ) + 2 \, A}{e^{x} - i} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.366195, size = 20, normalized size = 0.74 \begin{align*} \frac{2 A}{e^{x} - i} + B x - 2 B \log{\left (e^{x} - i \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*A/(exp(x) - I) + B*x - 2*B*log(exp(x) - I)

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Giac [A]  time = 1.14116, size = 28, normalized size = 1.04 \begin{align*} B x - 2 \, B \log \left (e^{x} - i\right ) + \frac{2 \, A}{e^{x} - i} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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