3.2.90 \(\int \frac {\sinh (c+d x)}{a+i a \sinh (c+d x)} \, dx\) [190]

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

[Out]

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

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Rubi [A]
time = 0.03, 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 = {2814, 2727} \begin {gather*} -\frac {\cosh (c+d x)}{d (a+i a \sinh (c+d x))}-\frac {i x}{a} \end {gather*}

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 2727

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 2814

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 [B] Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(84\) vs. \(2(35)=70\).
time = 0.10, size = 84, normalized size = 2.40 \begin {gather*} -\frac {\left (\cosh \left (\frac {1}{2} (c+d x)\right )+i \sinh \left (\frac {1}{2} (c+d x)\right )\right ) \left ((c+d x) \cosh \left (\frac {1}{2} (c+d x)\right )+i (2 i+c+d x) \sinh \left (\frac {1}{2} (c+d x)\right )\right )}{a d (-i+\sinh (c+d x))} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.88, size = 57, normalized size = 1.63

method	result	size
risch	\(-\frac {i x}{a}-\frac {2}{d a \left ({\mathrm e}^{d x +c}-i\right )}\)	\(28\)
derivativedivides	\(\frac {\frac {4 i}{-2 i+2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}+i \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-i \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{a d}\)	\(57\)
default	\(\frac {\frac {4 i}{-2 i+2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}+i \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-i \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{a d}\)	\(57\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/d/a*(1/2*I/(-I+tanh(1/2*d*x+1/2*c))+1/4*I*ln(tanh(1/2*d*x+1/2*c)-1)-1/4*I*ln(tanh(1/2*d*x+1/2*c)+1))

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Maxima [A]
time = 0.26, size = 36, normalized size = 1.03 \begin {gather*} -\frac {i \, {\left (d x + c\right )}}{a d} - \frac {2}{{\left (a e^{\left (-d x - c\right )} + i \, a\right )} d} \end {gather*}

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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Fricas [A]
time = 0.33, size = 33, normalized size = 0.94 \begin {gather*} \frac {-i \, d x e^{\left (d x + c\right )} - d x - 2}{a d e^{\left (d x + c\right )} - i \, a d} \end {gather*}

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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Sympy [A]
time = 0.08, size = 24, normalized size = 0.69 \begin {gather*} - \frac {2}{a d e^{c} e^{d x} - i a d} - \frac {i x}{a} \end {gather*}

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

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Giac [A]
time = 0.42, size = 33, normalized size = 0.94 \begin {gather*} -\frac {\frac {i \, {\left (d x + c\right )}}{a} + \frac {2 i}{a {\left (i \, e^{\left (d x + c\right )} + 1\right )}}}{d} \end {gather*}

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]

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

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

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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