3.57 \(\int \frac{1}{(1+i \sinh (c+d x))^2} \, dx\)

Optimal. Leaf size=59 \[ \frac{i \cosh (c+d x)}{3 d (1+i \sinh (c+d x))}+\frac{i \cosh (c+d x)}{3 d (1+i \sinh (c+d x))^2} \]

[Out]

((I/3)*Cosh[c + d*x])/(d*(1 + I*Sinh[c + d*x])^2) + ((I/3)*Cosh[c + d*x])/(d*(1 + I*Sinh[c + d*x]))

________________________________________________________________________________________

Rubi [A]  time = 0.0263944, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {2650, 2648} \[ \frac{i \cosh (c+d x)}{3 d (1+i \sinh (c+d x))}+\frac{i \cosh (c+d x)}{3 d (1+i \sinh (c+d x))^2} \]

Antiderivative was successfully verified.

[In]

Int[(1 + I*Sinh[c + d*x])^(-2),x]

[Out]

((I/3)*Cosh[c + d*x])/(d*(1 + I*Sinh[c + d*x])^2) + ((I/3)*Cosh[c + d*x])/(d*(1 + I*Sinh[c + d*x]))

Rule 2650

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

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{1}{(1+i \sinh (c+d x))^2} \, dx &=\frac{i \cosh (c+d x)}{3 d (1+i \sinh (c+d x))^2}+\frac{1}{3} \int \frac{1}{1+i \sinh (c+d x)} \, dx\\ &=\frac{i \cosh (c+d x)}{3 d (1+i \sinh (c+d x))^2}+\frac{i \cosh (c+d x)}{3 d (1+i \sinh (c+d x))}\\ \end{align*}

Mathematica [A]  time = 0.0996662, size = 61, normalized size = 1.03 \[ \frac{-4 \sinh (c+d x)+\sinh (2 (c+d x))-4 i \cosh (c+d x)-i \cosh (2 (c+d x))+3 i}{6 d (\sinh (c+d x)-i)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + I*Sinh[c + d*x])^(-2),x]

[Out]

(3*I - (4*I)*Cosh[c + d*x] - I*Cosh[2*(c + d*x)] - 4*Sinh[c + d*x] + Sinh[2*(c + d*x)])/(6*d*(-I + Sinh[c + d*
x])^2)

________________________________________________________________________________________

Maple [A]  time = 0.035, size = 55, normalized size = 0.9 \begin{align*}{\frac{1}{d} \left ( 2\, \left ( -i+\tanh \left ( 1/2\,dx+c/2 \right ) \right ) ^{-1}+{2\,i \left ( -i+\tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-2}}-{\frac{4}{3} \left ( -i+\tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-3}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(1+I*sinh(d*x+c))^2,x)

[Out]

1/d*(2/(-I+tanh(1/2*d*x+1/2*c))+2*I/(-I+tanh(1/2*d*x+1/2*c))^2-4/3/(-I+tanh(1/2*d*x+1/2*c))^3)

________________________________________________________________________________________

Maxima [B]  time = 1.14607, size = 127, normalized size = 2.15 \begin{align*} \frac{6 \, e^{\left (-d x - c\right )}}{d{\left (9 \, e^{\left (-d x - c\right )} - 9 i \, e^{\left (-2 \, d x - 2 \, c\right )} - 3 \, e^{\left (-3 \, d x - 3 \, c\right )} + 3 i\right )}} + \frac{2 i}{d{\left (9 \, e^{\left (-d x - c\right )} - 9 i \, e^{\left (-2 \, d x - 2 \, c\right )} - 3 \, e^{\left (-3 \, d x - 3 \, c\right )} + 3 i\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1+I*sinh(d*x+c))^2,x, algorithm="maxima")

[Out]

6*e^(-d*x - c)/(d*(9*e^(-d*x - c) - 9*I*e^(-2*d*x - 2*c) - 3*e^(-3*d*x - 3*c) + 3*I)) + 2*I/(d*(9*e^(-d*x - c)
- 9*I*e^(-2*d*x - 2*c) - 3*e^(-3*d*x - 3*c) + 3*I))

________________________________________________________________________________________

Fricas [A]  time = 1.99318, size = 128, normalized size = 2.17 \begin{align*} \frac{6 \, e^{\left (d x + c\right )} - 2 i}{3 \, d e^{\left (3 \, d x + 3 \, c\right )} - 9 i \, d e^{\left (2 \, d x + 2 \, c\right )} - 9 \, d e^{\left (d x + c\right )} + 3 i \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1+I*sinh(d*x+c))^2,x, algorithm="fricas")

[Out]

(6*e^(d*x + c) - 2*I)/(3*d*e^(3*d*x + 3*c) - 9*I*d*e^(2*d*x + 2*c) - 9*d*e^(d*x + c) + 3*I*d)

________________________________________________________________________________________

Sympy [A]  time = 0.964546, size = 63, normalized size = 1.07 \begin{align*} \frac{\frac{2 e^{- 2 c} e^{d x}}{d} - \frac{2 i e^{- 3 c}}{3 d}}{e^{3 d x} - 3 i e^{- c} e^{2 d x} - 3 e^{- 2 c} e^{d x} + i e^{- 3 c}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1+I*sinh(d*x+c))**2,x)

[Out]

(2*exp(-2*c)*exp(d*x)/d - 2*I*exp(-3*c)/(3*d))/(exp(3*d*x) - 3*I*exp(-c)*exp(2*d*x) - 3*exp(-2*c)*exp(d*x) + I
*exp(-3*c))

________________________________________________________________________________________

Giac [A]  time = 1.33114, size = 34, normalized size = 0.58 \begin{align*} \frac{6 \, e^{\left (d x + c\right )} - 2 i}{3 \, d{\left (e^{\left (d x + c\right )} - i\right )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1+I*sinh(d*x+c))^2,x, algorithm="giac")

[Out]

1/3*(6*e^(d*x + c) - 2*I)/(d*(e^(d*x + c) - I)^3)