3.1.59 \(\int \frac {1}{(1+i \sinh (c+d x))^4} \, dx\) [59]

Optimal. Leaf size=117 \[ \frac {i \cosh (c+d x)}{7 d (1+i \sinh (c+d x))^4}+\frac {3 i \cosh (c+d x)}{35 d (1+i \sinh (c+d x))^3}+\frac {2 i \cosh (c+d x)}{35 d (1+i \sinh (c+d x))^2}+\frac {2 i \cosh (c+d x)}{35 d (1+i \sinh (c+d x))} \]

[Out]

1/7*I*cosh(d*x+c)/d/(1+I*sinh(d*x+c))^4+3/35*I*cosh(d*x+c)/d/(1+I*sinh(d*x+c))^3+2/35*I*cosh(d*x+c)/d/(1+I*sin
h(d*x+c))^2+2/35*I*cosh(d*x+c)/d/(1+I*sinh(d*x+c))

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Rubi [A]
time = 0.04, antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2729, 2727} \begin {gather*} \frac {2 i \cosh (c+d x)}{35 d (1+i \sinh (c+d x))}+\frac {2 i \cosh (c+d x)}{35 d (1+i \sinh (c+d x))^2}+\frac {3 i \cosh (c+d x)}{35 d (1+i \sinh (c+d x))^3}+\frac {i \cosh (c+d x)}{7 d (1+i \sinh (c+d x))^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 + I*Sinh[c + d*x])^(-4),x]

[Out]

((I/7)*Cosh[c + d*x])/(d*(1 + I*Sinh[c + d*x])^4) + (((3*I)/35)*Cosh[c + d*x])/(d*(1 + I*Sinh[c + d*x])^3) + (
((2*I)/35)*Cosh[c + d*x])/(d*(1 + I*Sinh[c + d*x])^2) + (((2*I)/35)*Cosh[c + d*x])/(d*(1 + I*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 2729

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]

Rubi steps

\begin {align*} \int \frac {1}{(1+i \sinh (c+d x))^4} \, dx &=\frac {i \cosh (c+d x)}{7 d (1+i \sinh (c+d x))^4}+\frac {3}{7} \int \frac {1}{(1+i \sinh (c+d x))^3} \, dx\\ &=\frac {i \cosh (c+d x)}{7 d (1+i \sinh (c+d x))^4}+\frac {3 i \cosh (c+d x)}{35 d (1+i \sinh (c+d x))^3}+\frac {6}{35} \int \frac {1}{(1+i \sinh (c+d x))^2} \, dx\\ &=\frac {i \cosh (c+d x)}{7 d (1+i \sinh (c+d x))^4}+\frac {3 i \cosh (c+d x)}{35 d (1+i \sinh (c+d x))^3}+\frac {2 i \cosh (c+d x)}{35 d (1+i \sinh (c+d x))^2}+\frac {2}{35} \int \frac {1}{1+i \sinh (c+d x)} \, dx\\ &=\frac {i \cosh (c+d x)}{7 d (1+i \sinh (c+d x))^4}+\frac {3 i \cosh (c+d x)}{35 d (1+i \sinh (c+d x))^3}+\frac {2 i \cosh (c+d x)}{35 d (1+i \sinh (c+d x))^2}+\frac {2 i \cosh (c+d x)}{35 d (1+i \sinh (c+d x))}\\ \end {align*}

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Mathematica [A]
time = 0.12, size = 87, normalized size = 0.74 \begin {gather*} \frac {21 i \cosh \left (\frac {3}{2} (c+d x)\right )-i \cosh \left (\frac {7}{2} (c+d x)\right )+35 \sinh \left (\frac {1}{2} (c+d x)\right )-7 \sinh \left (\frac {5}{2} (c+d x)\right )}{70 d \left (\cosh \left (\frac {1}{2} (c+d x)\right )+i \sinh \left (\frac {1}{2} (c+d x)\right )\right )^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 + I*Sinh[c + d*x])^(-4),x]

[Out]

((21*I)*Cosh[(3*(c + d*x))/2] - I*Cosh[(7*(c + d*x))/2] + 35*Sinh[(c + d*x)/2] - 7*Sinh[(5*(c + d*x))/2])/(70*
d*(Cosh[(c + d*x)/2] + I*Sinh[(c + d*x)/2])^7)

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Maple [A]
time = 1.41, size = 121, normalized size = 1.03

method	result	size
risch	\(-\frac {4 \left (-7 \,{\mathrm e}^{d x +c}-21 i {\mathrm e}^{2 d x +2 c}+35 \,{\mathrm e}^{3 d x +3 c}+i\right )}{35 \left ({\mathrm e}^{d x +c}-i\right )^{7} d}\)	\(51\)
derivativedivides	\(\frac {-\frac {16 i}{\left (-i+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}-\frac {16}{7 \left (-i+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}+\frac {6 i}{\left (-i+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\frac {2}{-i+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {72}{5 \left (-i+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}-\frac {12}{\left (-i+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+\frac {8 i}{\left (-i+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}}{d}\)	\(121\)
default	\(\frac {-\frac {16 i}{\left (-i+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}-\frac {16}{7 \left (-i+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}+\frac {6 i}{\left (-i+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\frac {2}{-i+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {72}{5 \left (-i+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}-\frac {12}{\left (-i+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+\frac {8 i}{\left (-i+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}}{d}\)	\(121\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(1+I*sinh(d*x+c))^4,x,method=_RETURNVERBOSE)

[Out]

1/d*(-16*I/(-I+tanh(1/2*d*x+1/2*c))^4-16/7/(-I+tanh(1/2*d*x+1/2*c))^7+6*I/(-I+tanh(1/2*d*x+1/2*c))^2+2/(-I+tan
h(1/2*d*x+1/2*c))+72/5/(-I+tanh(1/2*d*x+1/2*c))^5-12/(-I+tanh(1/2*d*x+1/2*c))^3+8*I/(-I+tanh(1/2*d*x+1/2*c))^6
)

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Maxima [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 372 vs. \(2 (93) = 186\).
time = 0.29, size = 372, normalized size = 3.18 \begin {gather*} \frac {4 \, e^{\left (-d x - c\right )}}{5 \, d {\left (7 \, e^{\left (-d x - c\right )} - 21 i \, e^{\left (-2 \, d x - 2 \, c\right )} - 35 \, e^{\left (-3 \, d x - 3 \, c\right )} + 35 i \, e^{\left (-4 \, d x - 4 \, c\right )} + 21 \, e^{\left (-5 \, d x - 5 \, c\right )} - 7 i \, e^{\left (-6 \, d x - 6 \, c\right )} - e^{\left (-7 \, d x - 7 \, c\right )} + i\right )}} - \frac {12 i \, e^{\left (-2 \, d x - 2 \, c\right )}}{5 \, d {\left (7 \, e^{\left (-d x - c\right )} - 21 i \, e^{\left (-2 \, d x - 2 \, c\right )} - 35 \, e^{\left (-3 \, d x - 3 \, c\right )} + 35 i \, e^{\left (-4 \, d x - 4 \, c\right )} + 21 \, e^{\left (-5 \, d x - 5 \, c\right )} - 7 i \, e^{\left (-6 \, d x - 6 \, c\right )} - e^{\left (-7 \, d x - 7 \, c\right )} + i\right )}} - \frac {4 \, e^{\left (-3 \, d x - 3 \, c\right )}}{d {\left (7 \, e^{\left (-d x - c\right )} - 21 i \, e^{\left (-2 \, d x - 2 \, c\right )} - 35 \, e^{\left (-3 \, d x - 3 \, c\right )} + 35 i \, e^{\left (-4 \, d x - 4 \, c\right )} + 21 \, e^{\left (-5 \, d x - 5 \, c\right )} - 7 i \, e^{\left (-6 \, d x - 6 \, c\right )} - e^{\left (-7 \, d x - 7 \, c\right )} + i\right )}} + \frac {4 i}{35 \, d {\left (7 \, e^{\left (-d x - c\right )} - 21 i \, e^{\left (-2 \, d x - 2 \, c\right )} - 35 \, e^{\left (-3 \, d x - 3 \, c\right )} + 35 i \, e^{\left (-4 \, d x - 4 \, c\right )} + 21 \, e^{\left (-5 \, d x - 5 \, c\right )} - 7 i \, e^{\left (-6 \, d x - 6 \, c\right )} - e^{\left (-7 \, d x - 7 \, c\right )} + i\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1+I*sinh(d*x+c))^4,x, algorithm="maxima")

[Out]

4/5*e^(-d*x - c)/(d*(7*e^(-d*x - c) - 21*I*e^(-2*d*x - 2*c) - 35*e^(-3*d*x - 3*c) + 35*I*e^(-4*d*x - 4*c) + 21
*e^(-5*d*x - 5*c) - 7*I*e^(-6*d*x - 6*c) - e^(-7*d*x - 7*c) + I)) - 12/5*I*e^(-2*d*x - 2*c)/(d*(7*e^(-d*x - c)
- 21*I*e^(-2*d*x - 2*c) - 35*e^(-3*d*x - 3*c) + 35*I*e^(-4*d*x - 4*c) + 21*e^(-5*d*x - 5*c) - 7*I*e^(-6*d*x -
6*c) - e^(-7*d*x - 7*c) + I)) - 4*e^(-3*d*x - 3*c)/(d*(7*e^(-d*x - c) - 21*I*e^(-2*d*x - 2*c) - 35*e^(-3*d*x
- 3*c) + 35*I*e^(-4*d*x - 4*c) + 21*e^(-5*d*x - 5*c) - 7*I*e^(-6*d*x - 6*c) - e^(-7*d*x - 7*c) + I)) + 4/35*I/
(d*(7*e^(-d*x - c) - 21*I*e^(-2*d*x - 2*c) - 35*e^(-3*d*x - 3*c) + 35*I*e^(-4*d*x - 4*c) + 21*e^(-5*d*x - 5*c)
- 7*I*e^(-6*d*x - 6*c) - e^(-7*d*x - 7*c) + I))

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Fricas [A]
time = 0.35, size = 120, normalized size = 1.03 \begin {gather*} -\frac {4 \, {\left (35 \, e^{\left (3 \, d x + 3 \, c\right )} - 21 i \, e^{\left (2 \, d x + 2 \, c\right )} - 7 \, e^{\left (d x + c\right )} + i\right )}}{35 \, {\left (d e^{\left (7 \, d x + 7 \, c\right )} - 7 i \, d e^{\left (6 \, d x + 6 \, c\right )} - 21 \, d e^{\left (5 \, d x + 5 \, c\right )} + 35 i \, d e^{\left (4 \, d x + 4 \, c\right )} + 35 \, d e^{\left (3 \, d x + 3 \, c\right )} - 21 i \, d e^{\left (2 \, d x + 2 \, c\right )} - 7 \, d e^{\left (d x + c\right )} + i \, d\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1+I*sinh(d*x+c))^4,x, algorithm="fricas")

[Out]

-4/35*(35*e^(3*d*x + 3*c) - 21*I*e^(2*d*x + 2*c) - 7*e^(d*x + c) + I)/(d*e^(7*d*x + 7*c) - 7*I*d*e^(6*d*x + 6*
c) - 21*d*e^(5*d*x + 5*c) + 35*I*d*e^(4*d*x + 4*c) + 35*d*e^(3*d*x + 3*c) - 21*I*d*e^(2*d*x + 2*c) - 7*d*e^(d*
x + c) + I*d)

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Sympy [A]
time = 0.27, size = 155, normalized size = 1.32 \begin {gather*} \frac {- 140 e^{3 c} e^{3 d x} + 84 i e^{2 c} e^{2 d x} + 28 e^{c} e^{d x} - 4 i}{35 d e^{7 c} e^{7 d x} - 245 i d e^{6 c} e^{6 d x} - 735 d e^{5 c} e^{5 d x} + 1225 i d e^{4 c} e^{4 d x} + 1225 d e^{3 c} e^{3 d x} - 735 i d e^{2 c} e^{2 d x} - 245 d e^{c} e^{d x} + 35 i d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1+I*sinh(d*x+c))**4,x)

[Out]

(-140*exp(3*c)*exp(3*d*x) + 84*I*exp(2*c)*exp(2*d*x) + 28*exp(c)*exp(d*x) - 4*I)/(35*d*exp(7*c)*exp(7*d*x) - 2
45*I*d*exp(6*c)*exp(6*d*x) - 735*d*exp(5*c)*exp(5*d*x) + 1225*I*d*exp(4*c)*exp(4*d*x) + 1225*d*exp(3*c)*exp(3*
d*x) - 735*I*d*exp(2*c)*exp(2*d*x) - 245*d*exp(c)*exp(d*x) + 35*I*d)

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Giac [A]
time = 0.42, size = 47, normalized size = 0.40 \begin {gather*} -\frac {4 \, {\left (35 \, e^{\left (3 \, d x + 3 \, c\right )} - 21 i \, e^{\left (2 \, d x + 2 \, c\right )} - 7 \, e^{\left (d x + c\right )} + i\right )}}{35 \, d {\left (e^{\left (d x + c\right )} - i\right )}^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1+I*sinh(d*x+c))^4,x, algorithm="giac")

[Out]

-4/35*(35*e^(3*d*x + 3*c) - 21*I*e^(2*d*x + 2*c) - 7*e^(d*x + c) + I)/(d*(e^(d*x + c) - I)^7)

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Mupad [B]
time = 0.96, size = 53, normalized size = 0.45 \begin {gather*} -\frac {\left (7\,{\mathrm {e}}^{c+d\,x}+{\mathrm {e}}^{2\,c+2\,d\,x}\,21{}\mathrm {i}-35\,{\mathrm {e}}^{3\,c+3\,d\,x}-\mathrm {i}\right )\,4{}\mathrm {i}}{35\,d\,{\left (1+{\mathrm {e}}^{c+d\,x}\,1{}\mathrm {i}\right )}^7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(sinh(c + d*x)*1i + 1)^4,x)

[Out]

-((7*exp(c + d*x) + exp(2*c + 2*d*x)*21i - 35*exp(3*c + 3*d*x) - 1i)*4i)/(35*d*(exp(c + d*x)*1i + 1)^7)

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