3.190 \(\int \frac {\sinh (c+d x)}{a+i a \sinh (c+d x)} \, dx\)

Optimal. Leaf size=35 \[ -\frac {\cosh (c+d x)}{d (a+i a \sinh (c+d x))}-\frac {i x}{a} \]

[Out]

-I*x/a-cosh(d*x+c)/d/(a+I*a*sinh(d*x+c))

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Rubi [A]  time = 0.04, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {2735, 2648} \[ -\frac {\cosh (c+d x)}{d (a+i a \sinh (c+d x))}-\frac {i x}{a} \]

Antiderivative was successfully verified.

[In]

Int[Sinh[c + d*x]/(a + I*a*Sinh[c + d*x]),x]

[Out]

((-I)*x)/a - Cosh[c + d*x]/(d*(a + I*a*Sinh[c + d*x]))

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 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]

Rubi steps

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

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Mathematica [A]  time = 0.21, size = 61, normalized size = 1.74 \[ \frac {i \cosh (c+d x) \left (1-\frac {\sinh ^{-1}(\sinh (c+d x)) (\sinh (c+d x)-i)}{\sqrt {\cosh ^2(c+d x)}}\right )}{a d (\sinh (c+d x)-i)} \]

Antiderivative was successfully verified.

[In]

Integrate[Sinh[c + d*x]/(a + I*a*Sinh[c + d*x]),x]

[Out]

(I*Cosh[c + d*x]*(1 - (ArcSinh[Sinh[c + d*x]]*(-I + Sinh[c + d*x]))/Sqrt[Cosh[c + d*x]^2]))/(a*d*(-I + Sinh[c
+ d*x]))

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fricas [A]  time = 0.45, size = 33, normalized size = 0.94 \[ \frac {-i \, d x e^{\left (d x + c\right )} - d x - 2}{a d e^{\left (d x + c\right )} - i \, a d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(d*x+c)/(a+I*a*sinh(d*x+c)),x, algorithm="fricas")

[Out]

(-I*d*x*e^(d*x + c) - d*x - 2)/(a*d*e^(d*x + c) - I*a*d)

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giac [A]  time = 0.56, size = 33, normalized size = 0.94 \[ \frac {-\frac {2 i \, {\left (d x + c\right )}}{a} - \frac {4 i}{a {\left (i \, e^{\left (d x + c\right )} + 1\right )}}}{2 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(d*x+c)/(a+I*a*sinh(d*x+c)),x, algorithm="giac")

[Out]

1/2*(-2*I*(d*x + c)/a - 4*I/(a*(I*e^(d*x + c) + 1)))/d

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maple [A]  time = 0.05, size = 67, normalized size = 1.91 \[ \frac {i \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d a}-\frac {i \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d a}+\frac {2 i}{d a \left (-i+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(d*x+c)/(a+I*a*sinh(d*x+c)),x)

[Out]

I/d/a*ln(tanh(1/2*d*x+1/2*c)-1)-I/d/a*ln(tanh(1/2*d*x+1/2*c)+1)+2*I/d/a/(-I+tanh(1/2*d*x+1/2*c))

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maxima [A]  time = 0.31, size = 36, normalized size = 1.03 \[ -\frac {i \, {\left (d x + c\right )}}{a d} - \frac {2}{{\left (a e^{\left (-d x - c\right )} + i \, a\right )} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(d*x+c)/(a+I*a*sinh(d*x+c)),x, algorithm="maxima")

[Out]

-I*(d*x + c)/(a*d) - 2/((a*e^(-d*x - c) + I*a)*d)

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mupad [B]  time = 0.24, size = 27, normalized size = 0.77 \[ -\frac {x\,1{}\mathrm {i}}{a}-\frac {2}{a\,d\,\left ({\mathrm {e}}^{c+d\,x}-\mathrm {i}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(c + d*x)/(a + a*sinh(c + d*x)*1i),x)

[Out]

- (x*1i)/a - 2/(a*d*(exp(c + d*x) - 1i))

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sympy [A]  time = 0.17, size = 27, normalized size = 0.77 \[ \frac {2 e^{c}}{- i a d e^{c} - a d e^{- d x}} - \frac {i x}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(d*x+c)/(a+I*a*sinh(d*x+c)),x)

[Out]

2*exp(c)/(-I*a*d*exp(c) - a*d*exp(-d*x)) - I*x/a

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