3.118 \(\int \frac {A+B \sinh (x)}{(i+\sinh (x))^4} \, dx\)

Optimal. Leaf size=91 \[ \frac {2 (3 A+4 i B) \cosh (x)}{105 (\sinh (x)+i)}+\frac {2 (-4 B+3 i A) \cosh (x)}{105 (\sinh (x)+i)^2}-\frac {(3 A+4 i B) \cosh (x)}{35 (\sinh (x)+i)^3}-\frac {(B+i A) \cosh (x)}{7 (\sinh (x)+i)^4} \]

[Out]

-1/7*(I*A+B)*cosh(x)/(I+sinh(x))^4-1/35*(3*A+4*I*B)*cosh(x)/(I+sinh(x))^3+2/105*(3*I*A-4*B)*cosh(x)/(I+sinh(x)
)^2+2/105*(3*A+4*I*B)*cosh(x)/(I+sinh(x))

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Rubi [A]  time = 0.07, antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2750, 2650, 2648} \[ \frac {2 (3 A+4 i B) \cosh (x)}{105 (\sinh (x)+i)}+\frac {2 (-4 B+3 i A) \cosh (x)}{105 (\sinh (x)+i)^2}-\frac {(3 A+4 i B) \cosh (x)}{35 (\sinh (x)+i)^3}-\frac {(B+i A) \cosh (x)}{7 (\sinh (x)+i)^4} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*Sinh[x])/(I + Sinh[x])^4,x]

[Out]

-((I*A + B)*Cosh[x])/(7*(I + Sinh[x])^4) - ((3*A + (4*I)*B)*Cosh[x])/(35*(I + Sinh[x])^3) + (2*((3*I)*A - 4*B)
*Cosh[x])/(105*(I + Sinh[x])^2) + (2*(3*A + (4*I)*B)*Cosh[x])/(105*(I + Sinh[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 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 2750

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

Rubi steps

\begin {align*} \int \frac {A+B \sinh (x)}{(i+\sinh (x))^4} \, dx &=-\frac {(i A+B) \cosh (x)}{7 (i+\sinh (x))^4}+\frac {1}{7} (-3 i A+4 B) \int \frac {1}{(i+\sinh (x))^3} \, dx\\ &=-\frac {(i A+B) \cosh (x)}{7 (i+\sinh (x))^4}-\frac {(3 A+4 i B) \cosh (x)}{35 (i+\sinh (x))^3}-\frac {1}{35} (2 (3 A+4 i B)) \int \frac {1}{(i+\sinh (x))^2} \, dx\\ &=-\frac {(i A+B) \cosh (x)}{7 (i+\sinh (x))^4}-\frac {(3 A+4 i B) \cosh (x)}{35 (i+\sinh (x))^3}+\frac {2 (3 i A-4 B) \cosh (x)}{105 (i+\sinh (x))^2}+\frac {1}{105} (2 (3 i A-4 B)) \int \frac {1}{i+\sinh (x)} \, dx\\ &=-\frac {(i A+B) \cosh (x)}{7 (i+\sinh (x))^4}-\frac {(3 A+4 i B) \cosh (x)}{35 (i+\sinh (x))^3}+\frac {2 (3 i A-4 B) \cosh (x)}{105 (i+\sinh (x))^2}+\frac {2 (3 A+4 i B) \cosh (x)}{105 (i+\sinh (x))}\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 67, normalized size = 0.74 \[ \frac {\cosh (x) \left ((6 A+8 i B) \sinh ^3(x)+8 i (3 A+4 i B) \sinh ^2(x)-13 (3 A+4 i B) \sinh (x)-36 i A+13 B\right )}{105 (\sinh (x)+i)^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*Sinh[x])/(I + Sinh[x])^4,x]

[Out]

(Cosh[x]*((-36*I)*A + 13*B - 13*(3*A + (4*I)*B)*Sinh[x] + (8*I)*(3*A + (4*I)*B)*Sinh[x]^2 + (6*A + (8*I)*B)*Si
nh[x]^3))/(105*(I + Sinh[x])^4)

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fricas [A]  time = 0.51, size = 96, normalized size = 1.05 \[ -\frac {280 \, B e^{\left (4 \, x\right )} + {\left (420 \, A + 280 i \, B\right )} e^{\left (3 \, x\right )} + 84 \, {\left (3 i \, A - 4 \, B\right )} e^{\left (2 \, x\right )} - {\left (84 \, A + 112 i \, B\right )} e^{x} - 12 i \, A + 16 \, B}{105 \, e^{\left (7 \, x\right )} + 735 i \, e^{\left (6 \, x\right )} - 2205 \, e^{\left (5 \, x\right )} - 3675 i \, e^{\left (4 \, x\right )} + 3675 \, e^{\left (3 \, x\right )} + 2205 i \, e^{\left (2 \, x\right )} - 735 \, e^{x} - 105 i} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*sinh(x))/(I+sinh(x))^4,x, algorithm="fricas")

[Out]

-(280*B*e^(4*x) + (420*A + 280*I*B)*e^(3*x) + 84*(3*I*A - 4*B)*e^(2*x) - (84*A + 112*I*B)*e^x - 12*I*A + 16*B)
/(105*e^(7*x) + 735*I*e^(6*x) - 2205*e^(5*x) - 3675*I*e^(4*x) + 3675*e^(3*x) + 2205*I*e^(2*x) - 735*e^x - 105*
I)

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giac [A]  time = 0.18, size = 60, normalized size = 0.66 \[ -\frac {280 \, B e^{\left (4 \, x\right )} + 420 \, A e^{\left (3 \, x\right )} + 280 i \, B e^{\left (3 \, x\right )} + 252 i \, A e^{\left (2 \, x\right )} - 336 \, B e^{\left (2 \, x\right )} - 84 \, A e^{x} - 112 i \, B e^{x} - 12 i \, A + 16 \, B}{105 \, {\left (e^{x} + i\right )}^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*sinh(x))/(I+sinh(x))^4,x, algorithm="giac")

[Out]

-1/105*(280*B*e^(4*x) + 420*A*e^(3*x) + 280*I*B*e^(3*x) + 252*I*A*e^(2*x) - 336*B*e^(2*x) - 84*A*e^x - 112*I*B
*e^x - 12*I*A + 16*B)/(e^x + I)^7

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maple [A]  time = 0.05, size = 128, normalized size = 1.41 \[ -\frac {24 i A +24 B}{3 \left (\tanh \left (\frac {x}{2}\right )+i\right )^{6}}+\frac {2 A}{\tanh \left (\frac {x}{2}\right )+i}-\frac {-32 i A -24 B}{2 \left (\tanh \left (\frac {x}{2}\right )+i\right )^{4}}-\frac {6 i A +2 B}{\left (\tanh \left (\frac {x}{2}\right )+i\right )^{2}}-\frac {2 \left (32 i B -36 A \right )}{5 \left (\tanh \left (\frac {x}{2}\right )+i\right )^{5}}-\frac {2 \left (-10 i B +18 A \right )}{3 \left (\tanh \left (\frac {x}{2}\right )+i\right )^{3}}-\frac {2 \left (-8 i B +8 A \right )}{7 \left (\tanh \left (\frac {x}{2}\right )+i\right )^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A+B*sinh(x))/(I+sinh(x))^4,x)

[Out]

-1/3*(24*I*A+24*B)/(tanh(1/2*x)+I)^6+2*A/(tanh(1/2*x)+I)-1/2*(-32*I*A-24*B)/(tanh(1/2*x)+I)^4-(6*I*A+2*B)/(tan
h(1/2*x)+I)^2-2/5*(-36*A+32*I*B)/(tanh(1/2*x)+I)^5-2/3*(18*A-10*I*B)/(tanh(1/2*x)+I)^3-2/7*(8*A-8*I*B)/(tanh(1
/2*x)+I)^7

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maxima [B]  time = 0.34, size = 468, normalized size = 5.14 \[ \frac {1}{2} \, B {\left (\frac {224 i \, e^{\left (-x\right )}}{735 \, e^{\left (-x\right )} + 2205 i \, e^{\left (-2 \, x\right )} - 3675 \, e^{\left (-3 \, x\right )} - 3675 i \, e^{\left (-4 \, x\right )} + 2205 \, e^{\left (-5 \, x\right )} + 735 i \, e^{\left (-6 \, x\right )} - 105 \, e^{\left (-7 \, x\right )} - 105 i} - \frac {672 \, e^{\left (-2 \, x\right )}}{735 \, e^{\left (-x\right )} + 2205 i \, e^{\left (-2 \, x\right )} - 3675 \, e^{\left (-3 \, x\right )} - 3675 i \, e^{\left (-4 \, x\right )} + 2205 \, e^{\left (-5 \, x\right )} + 735 i \, e^{\left (-6 \, x\right )} - 105 \, e^{\left (-7 \, x\right )} - 105 i} - \frac {560 i \, e^{\left (-3 \, x\right )}}{735 \, e^{\left (-x\right )} + 2205 i \, e^{\left (-2 \, x\right )} - 3675 \, e^{\left (-3 \, x\right )} - 3675 i \, e^{\left (-4 \, x\right )} + 2205 \, e^{\left (-5 \, x\right )} + 735 i \, e^{\left (-6 \, x\right )} - 105 \, e^{\left (-7 \, x\right )} - 105 i} + \frac {560 \, e^{\left (-4 \, x\right )}}{735 \, e^{\left (-x\right )} + 2205 i \, e^{\left (-2 \, x\right )} - 3675 \, e^{\left (-3 \, x\right )} - 3675 i \, e^{\left (-4 \, x\right )} + 2205 \, e^{\left (-5 \, x\right )} + 735 i \, e^{\left (-6 \, x\right )} - 105 \, e^{\left (-7 \, x\right )} - 105 i} + \frac {32}{735 \, e^{\left (-x\right )} + 2205 i \, e^{\left (-2 \, x\right )} - 3675 \, e^{\left (-3 \, x\right )} - 3675 i \, e^{\left (-4 \, x\right )} + 2205 \, e^{\left (-5 \, x\right )} + 735 i \, e^{\left (-6 \, x\right )} - 105 \, e^{\left (-7 \, x\right )} - 105 i}\right )} + A {\left (\frac {28 \, e^{\left (-x\right )}}{245 \, e^{\left (-x\right )} + 735 i \, e^{\left (-2 \, x\right )} - 1225 \, e^{\left (-3 \, x\right )} - 1225 i \, e^{\left (-4 \, x\right )} + 735 \, e^{\left (-5 \, x\right )} + 245 i \, e^{\left (-6 \, x\right )} - 35 \, e^{\left (-7 \, x\right )} - 35 i} + \frac {84 i \, e^{\left (-2 \, x\right )}}{245 \, e^{\left (-x\right )} + 735 i \, e^{\left (-2 \, x\right )} - 1225 \, e^{\left (-3 \, x\right )} - 1225 i \, e^{\left (-4 \, x\right )} + 735 \, e^{\left (-5 \, x\right )} + 245 i \, e^{\left (-6 \, x\right )} - 35 \, e^{\left (-7 \, x\right )} - 35 i} - \frac {140 \, e^{\left (-3 \, x\right )}}{245 \, e^{\left (-x\right )} + 735 i \, e^{\left (-2 \, x\right )} - 1225 \, e^{\left (-3 \, x\right )} - 1225 i \, e^{\left (-4 \, x\right )} + 735 \, e^{\left (-5 \, x\right )} + 245 i \, e^{\left (-6 \, x\right )} - 35 \, e^{\left (-7 \, x\right )} - 35 i} - \frac {4 i}{245 \, e^{\left (-x\right )} + 735 i \, e^{\left (-2 \, x\right )} - 1225 \, e^{\left (-3 \, x\right )} - 1225 i \, e^{\left (-4 \, x\right )} + 735 \, e^{\left (-5 \, x\right )} + 245 i \, e^{\left (-6 \, x\right )} - 35 \, e^{\left (-7 \, x\right )} - 35 i}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*sinh(x))/(I+sinh(x))^4,x, algorithm="maxima")

[Out]

1/2*B*(224*I*e^(-x)/(735*e^(-x) + 2205*I*e^(-2*x) - 3675*e^(-3*x) - 3675*I*e^(-4*x) + 2205*e^(-5*x) + 735*I*e^
(-6*x) - 105*e^(-7*x) - 105*I) - 672*e^(-2*x)/(735*e^(-x) + 2205*I*e^(-2*x) - 3675*e^(-3*x) - 3675*I*e^(-4*x)
+ 2205*e^(-5*x) + 735*I*e^(-6*x) - 105*e^(-7*x) - 105*I) - 560*I*e^(-3*x)/(735*e^(-x) + 2205*I*e^(-2*x) - 3675
*e^(-3*x) - 3675*I*e^(-4*x) + 2205*e^(-5*x) + 735*I*e^(-6*x) - 105*e^(-7*x) - 105*I) + 560*e^(-4*x)/(735*e^(-x
) + 2205*I*e^(-2*x) - 3675*e^(-3*x) - 3675*I*e^(-4*x) + 2205*e^(-5*x) + 735*I*e^(-6*x) - 105*e^(-7*x) - 105*I)
+ 32/(735*e^(-x) + 2205*I*e^(-2*x) - 3675*e^(-3*x) - 3675*I*e^(-4*x) + 2205*e^(-5*x) + 735*I*e^(-6*x) - 105*e
^(-7*x) - 105*I)) + A*(28*e^(-x)/(245*e^(-x) + 735*I*e^(-2*x) - 1225*e^(-3*x) - 1225*I*e^(-4*x) + 735*e^(-5*x)
+ 245*I*e^(-6*x) - 35*e^(-7*x) - 35*I) + 84*I*e^(-2*x)/(245*e^(-x) + 735*I*e^(-2*x) - 1225*e^(-3*x) - 1225*I*
e^(-4*x) + 735*e^(-5*x) + 245*I*e^(-6*x) - 35*e^(-7*x) - 35*I) - 140*e^(-3*x)/(245*e^(-x) + 735*I*e^(-2*x) - 1
225*e^(-3*x) - 1225*I*e^(-4*x) + 735*e^(-5*x) + 245*I*e^(-6*x) - 35*e^(-7*x) - 35*I) - 4*I/(245*e^(-x) + 735*I
*e^(-2*x) - 1225*e^(-3*x) - 1225*I*e^(-4*x) + 735*e^(-5*x) + 245*I*e^(-6*x) - 35*e^(-7*x) - 35*I))

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mupad [B]  time = 1.04, size = 66, normalized size = 0.73 \[ -\frac {\frac {16\,B}{105}+4\,A\,{\mathrm {e}}^{3\,x}-{\mathrm {e}}^x\,\left (\frac {4\,A}{5}+\frac {B\,16{}\mathrm {i}}{15}\right )-\frac {16\,B\,{\mathrm {e}}^{2\,x}}{5}+\frac {8\,B\,{\mathrm {e}}^{4\,x}}{3}-\frac {A\,4{}\mathrm {i}}{35}+\frac {A\,{\mathrm {e}}^{2\,x}\,12{}\mathrm {i}}{5}+\frac {B\,{\mathrm {e}}^{3\,x}\,8{}\mathrm {i}}{3}}{{\left ({\mathrm {e}}^x+1{}\mathrm {i}\right )}^7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*sinh(x))/(sinh(x) + 1i)^4,x)

[Out]

-((16*B)/105 - (A*4i)/35 + (A*exp(2*x)*12i)/5 + 4*A*exp(3*x) - exp(x)*((4*A)/5 + (B*16i)/15) - (16*B*exp(2*x))
/5 + (B*exp(3*x)*8i)/3 + (8*B*exp(4*x))/3)/(exp(x) + 1i)^7

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sympy [A]  time = 0.78, size = 110, normalized size = 1.21 \[ \frac {12 i A - 280 B e^{4 x} - 16 B + \left (- 420 A - 280 i B\right ) e^{3 x} + \left (84 A + 112 i B\right ) e^{x} + \left (- 252 i A + 336 B\right ) e^{2 x}}{105 e^{7 x} + 735 i e^{6 x} - 2205 e^{5 x} - 3675 i e^{4 x} + 3675 e^{3 x} + 2205 i e^{2 x} - 735 e^{x} - 105 i} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*sinh(x))/(I+sinh(x))**4,x)

[Out]

(12*I*A - 280*B*exp(4*x) - 16*B + (-420*A - 280*I*B)*exp(3*x) + (84*A + 112*I*B)*exp(x) + (-252*I*A + 336*B)*e
xp(2*x))/(105*exp(7*x) + 735*I*exp(6*x) - 2205*exp(5*x) - 3675*I*exp(4*x) + 3675*exp(3*x) + 2205*I*exp(2*x) -
735*exp(x) - 105*I)

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