\(\int \frac {\sinh ^4(x)}{(i+\sinh (x))^2} \, dx\) [48]

Optimal result

Integrand size = 13, antiderivative size = 58 \[ \int \frac {\sinh ^4(x)}{(i+\sinh (x))^2} \, dx=-\frac {7 x}{2}-\frac {16}{3} i \cosh (x)+\frac {7}{2} \cosh (x) \sinh (x)-\frac {\cosh (x) \sinh ^3(x)}{3 (i+\sinh (x))^2}-\frac {8 \cosh (x) \sinh ^2(x)}{3 (i+\sinh (x))} \]

[Out]

-7/2*x-16/3*I*cosh(x)+7/2*cosh(x)*sinh(x)-1/3*cosh(x)*sinh(x)^3/(I+sinh(x))^2-8/3*cosh(x)*sinh(x)^2/(I+sinh(x)
)

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {2844, 3056, 2813} \[ \int \frac {\sinh ^4(x)}{(i+\sinh (x))^2} \, dx=-\frac {7 x}{2}-\frac {16}{3} i \cosh (x)-\frac {\sinh ^3(x) \cosh (x)}{3 (\sinh (x)+i)^2}-\frac {8 \sinh ^2(x) \cosh (x)}{3 (\sinh (x)+i)}+\frac {7}{2} \sinh (x) \cosh (x) \]

[In]

Int[Sinh[x]^4/(I + Sinh[x])^2,x]

[Out]

(-7*x)/2 - ((16*I)/3)*Cosh[x] + (7*Cosh[x]*Sinh[x])/2 - (Cosh[x]*Sinh[x]^3)/(3*(I + Sinh[x])^2) - (8*Cosh[x]*S
inh[x]^2)/(3*(I + Sinh[x]))

Rule 2813

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(2*a*c +
b*d)*(x/2), x] + (-Simp[(b*c + a*d)*(Cos[e + f*x]/f), x] - Simp[b*d*Cos[e + f*x]*(Sin[e + f*x]/(2*f)), x]) /;
FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]

Rule 2844

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim
p[(b*c - a*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^(n - 1)/(a*f*(2*m + 1))), x] + Dist[1/
(a*b*(2*m + 1)), Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^(n - 2)*Simp[b*(c^2*(m + 1) + d^2*(n -
1)) + a*c*d*(m - n + 1) + d*(a*d*(m - n + 1) + b*c*(m + n))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e,
f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -1] && GtQ[n, 1] && (IntegersQ
[2*m, 2*n] || (IntegerQ[m] && EqQ[c, 0]))

Rule 3056

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_
.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(A*b - a*B)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x]
)^n/(a*f*(2*m + 1))), x] - Dist[1/(a*b*(2*m + 1)), Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^(n -
1)*Simp[A*(a*d*n - b*c*(m + 1)) - B*(a*c*m + b*d*n) - d*(a*B*(m - n) + A*b*(m + n + 1))*Sin[e + f*x], x], x],
x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ
[m, -2^(-1)] && GtQ[n, 0] && IntegerQ[2*m] && (IntegerQ[2*n] || EqQ[c, 0])

Rubi steps \begin{align*} \text {integral}& = -\frac {\cosh (x) \sinh ^3(x)}{3 (i+\sinh (x))^2}+\frac {1}{3} \int \frac {\sinh ^2(x) (-3 i+5 \sinh (x))}{i+\sinh (x)} \, dx \\ & = -\frac {\cosh (x) \sinh ^3(x)}{3 (i+\sinh (x))^2}-\frac {8 \cosh (x) \sinh ^2(x)}{3 (i+\sinh (x))}-\frac {1}{3} i \int (16+21 i \sinh (x)) \sinh (x) \, dx \\ & = -\frac {7 x}{2}-\frac {16}{3} i \cosh (x)+\frac {7}{2} \cosh (x) \sinh (x)-\frac {\cosh (x) \sinh ^3(x)}{3 (i+\sinh (x))^2}-\frac {8 \cosh (x) \sinh ^2(x)}{3 (i+\sinh (x))} \\ \end{align*}

Mathematica [B] (verified)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(147\) vs. \(2(58)=116\).

Time = 0.15 (sec) , antiderivative size = 147, normalized size of antiderivative = 2.53 \[ \int \frac {\sinh ^4(x)}{(i+\sinh (x))^2} \, dx=\frac {5 i \cosh (x)}{6 (1-i \sinh (x))^2}-\frac {31 i \cosh (x)}{6 (1-i \sinh (x))}-\frac {i \sqrt {2} \cosh (x) \sqrt {1+\frac {1}{2} (-1+i \sinh (x))}}{\sqrt {1+i \sinh (x)}}-\frac {7 i \arcsin \left (\frac {\sqrt {1-i \sinh (x)}}{\sqrt {2}}\right ) \cosh (x)}{\sqrt {1-i \sinh (x)} \sqrt {1+i \sinh (x)}}-\frac {\cosh (x) \sinh ^3(x)}{2 (1-i \sinh (x))^2} \]

[In]

Integrate[Sinh[x]^4/(I + Sinh[x])^2,x]

[Out]

(((5*I)/6)*Cosh[x])/(1 - I*Sinh[x])^2 - (((31*I)/6)*Cosh[x])/(1 - I*Sinh[x]) - (I*Sqrt[2]*Cosh[x]*Sqrt[1 + (-1
+ I*Sinh[x])/2])/Sqrt[1 + I*Sinh[x]] - ((7*I)*ArcSin[Sqrt[1 - I*Sinh[x]]/Sqrt[2]]*Cosh[x])/(Sqrt[1 - I*Sinh[x
]]*Sqrt[1 + I*Sinh[x]]) - (Cosh[x]*Sinh[x]^3)/(2*(1 - I*Sinh[x])^2)

Maple [A] (verified)

Time = 3.04 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.90

method	result	size
risch	\(-\frac {7 x}{2}+\frac {{\mathrm e}^{2 x}}{8}-i {\mathrm e}^{x}-i {\mathrm e}^{-x}-\frac {{\mathrm e}^{-2 x}}{8}-\frac {2 i \left (21 i {\mathrm e}^{x}+12 \,{\mathrm e}^{2 x}-11\right )}{3 \left ({\mathrm e}^{x}+i\right )^{3}}\)	\(52\)
default	\(\frac {\frac {1}{2}+2 i}{\tanh \left (\frac {x}{2}\right )-1}+\frac {1}{2 \left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}+\frac {7 \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{2}+\frac {2 i}{\left (\tanh \left (\frac {x}{2}\right )+i\right )^{2}}+\frac {4}{3 \left (\tanh \left (\frac {x}{2}\right )+i\right )^{3}}+\frac {6}{\tanh \left (\frac {x}{2}\right )+i}+\frac {\frac {1}{2}-2 i}{\tanh \left (\frac {x}{2}\right )+1}-\frac {1}{2 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}-\frac {7 \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{2}\)	\(98\)
parallelrisch	\(\frac {-993 i+420 \left (\cosh \left (2 x \right )+4 i \sinh \left (x \right )-3-i \sinh \left (2 x \right )+2 \cosh \left (x \right )\right ) \ln \left (1-\coth \left (x \right )+\operatorname {csch}\left (x \right )\right )+420 \left (-\cosh \left (2 x \right )-4 i \sinh \left (x \right )+3+i \sinh \left (2 x \right )-2 \cosh \left (x \right )\right ) \ln \left (\coth \left (x \right )-\operatorname {csch}\left (x \right )+1\right )+1612 i \cosh \left (x \right )-544 i \cosh \left (2 x \right )-60 i \cosh \left (3 x \right )-15 i \cosh \left (4 x \right )-90 \sinh \left (3 x \right )+15 \sinh \left (4 x \right )-714 \sinh \left (x \right )+966 \sinh \left (2 x \right )}{120 \cosh \left (2 x \right )+480 i \sinh \left (x \right )-360-120 i \sinh \left (2 x \right )+240 \cosh \left (x \right )}\)	\(146\)

[In]

int(sinh(x)^4/(I+sinh(x))^2,x,method=_RETURNVERBOSE)

[Out]

-7/2*x+1/8*exp(x)^2-I*exp(x)-I/exp(x)-1/8/exp(x)^2-2/3*I*(21*I*exp(x)+12*exp(x)^2-11)/(exp(x)+I)^3

Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 89 vs. \(2 (42) = 84\).

Time = 0.25 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.53 \[ \int \frac {\sinh ^4(x)}{(i+\sinh (x))^2} \, dx=-\frac {21 \, {\left (4 \, x - 3\right )} e^{\left (5 \, x\right )} + 21 \, {\left (12 i \, x + 7 i\right )} e^{\left (4 \, x\right )} - 3 \, {\left (84 \, x + 127\right )} e^{\left (3 \, x\right )} - {\left (84 i \, x + 239 i\right )} e^{\left (2 \, x\right )} - 3 \, e^{\left (7 \, x\right )} + 15 i \, e^{\left (6 \, x\right )} + 15 \, e^{x} - 3 i}{24 \, {\left (e^{\left (5 \, x\right )} + 3 i \, e^{\left (4 \, x\right )} - 3 \, e^{\left (3 \, x\right )} - i \, e^{\left (2 \, x\right )}\right )}} \]

[In]

integrate(sinh(x)^4/(I+sinh(x))^2,x, algorithm="fricas")

[Out]

-1/24*(21*(4*x - 3)*e^(5*x) + 21*(12*I*x + 7*I)*e^(4*x) - 3*(84*x + 127)*e^(3*x) - (84*I*x + 239*I)*e^(2*x) -
3*e^(7*x) + 15*I*e^(6*x) + 15*e^x - 3*I)/(e^(5*x) + 3*I*e^(4*x) - 3*e^(3*x) - I*e^(2*x))

Sympy [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.21 \[ \int \frac {\sinh ^4(x)}{(i+\sinh (x))^2} \, dx=- \frac {7 x}{2} + \frac {- 24 i e^{2 x} + 42 e^{x} + 22 i}{3 e^{3 x} + 9 i e^{2 x} - 9 e^{x} - 3 i} + \frac {e^{2 x}}{8} - i e^{x} - i e^{- x} - \frac {e^{- 2 x}}{8} \]

[In]

integrate(sinh(x)**4/(I+sinh(x))**2,x)

[Out]

-7*x/2 + (-24*I*exp(2*x) + 42*exp(x) + 22*I)/(3*exp(3*x) + 9*I*exp(2*x) - 9*exp(x) - 3*I) + exp(2*x)/8 - I*exp
(x) - I*exp(-x) - exp(-2*x)/8

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.22 \[ \int \frac {\sinh ^4(x)}{(i+\sinh (x))^2} \, dx=-\frac {7}{2} \, x + \frac {15 \, e^{\left (-x\right )} + 239 i \, e^{\left (-2 \, x\right )} - 405 \, e^{\left (-3 \, x\right )} - 216 i \, e^{\left (-4 \, x\right )} + 3 i}{8 \, {\left (3 i \, e^{\left (-2 \, x\right )} - 9 \, e^{\left (-3 \, x\right )} - 9 i \, e^{\left (-4 \, x\right )} + 3 \, e^{\left (-5 \, x\right )}\right )}} - i \, e^{\left (-x\right )} - \frac {1}{8} \, e^{\left (-2 \, x\right )} \]

[In]

integrate(sinh(x)^4/(I+sinh(x))^2,x, algorithm="maxima")

[Out]

-7/2*x + 1/8*(15*e^(-x) + 239*I*e^(-2*x) - 405*e^(-3*x) - 216*I*e^(-4*x) + 3*I)/(3*I*e^(-2*x) - 9*e^(-3*x) - 9
*I*e^(-4*x) + 3*e^(-5*x)) - I*e^(-x) - 1/8*e^(-2*x)

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.86 \[ \int \frac {\sinh ^4(x)}{(i+\sinh (x))^2} \, dx=-\frac {7}{2} \, x - \frac {{\left (216 i \, e^{\left (4 \, x\right )} - 405 \, e^{\left (3 \, x\right )} - 239 i \, e^{\left (2 \, x\right )} + 15 \, e^{x} - 3 i\right )} e^{\left (-2 \, x\right )}}{24 \, {\left (e^{x} + i\right )}^{3}} + \frac {1}{8} \, e^{\left (2 \, x\right )} - i \, e^{x} \]

[In]

integrate(sinh(x)^4/(I+sinh(x))^2,x, algorithm="giac")

[Out]

-7/2*x - 1/24*(216*I*e^(4*x) - 405*e^(3*x) - 239*I*e^(2*x) + 15*e^x - 3*I)*e^(-2*x)/(e^x + I)^3 + 1/8*e^(2*x)
- I*e^x

Mupad [B] (verification not implemented)

Time = 1.30 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.67 \[ \int \frac {\sinh ^4(x)}{(i+\sinh (x))^2} \, dx=\frac {{\mathrm {e}}^{2\,x}}{8}-{\mathrm {e}}^{-x}\,1{}\mathrm {i}-\frac {{\mathrm {e}}^{-2\,x}}{8}-\frac {7\,x}{2}-{\mathrm {e}}^x\,1{}\mathrm {i}-\frac {-2+\frac {{\mathrm {e}}^x\,8{}\mathrm {i}}{3}}{{\mathrm {e}}^{2\,x}-1+{\mathrm {e}}^x\,2{}\mathrm {i}}+\frac {4\,{\mathrm {e}}^x-\frac {{\mathrm {e}}^{2\,x}\,8{}\mathrm {i}}{3}+\frac {8}{3}{}\mathrm {i}}{{\mathrm {e}}^{2\,x}\,3{}\mathrm {i}+{\mathrm {e}}^{3\,x}-3\,{\mathrm {e}}^x-\mathrm {i}}-\frac {8{}\mathrm {i}}{3\,\left ({\mathrm {e}}^x+1{}\mathrm {i}\right )} \]

[In]

int(sinh(x)^4/(sinh(x) + 1i)^2,x)

[Out]

exp(2*x)/8 - exp(-x)*1i - exp(-2*x)/8 - (7*x)/2 - exp(x)*1i - ((exp(x)*8i)/3 - 2)/(exp(2*x) + exp(x)*2i - 1) +
(4*exp(x) - (exp(2*x)*8i)/3 + 8i/3)/(exp(2*x)*3i + exp(3*x) - 3*exp(x) - 1i) - 8i/(3*(exp(x) + 1i))