\(\int \frac {A+B \sinh (x)}{(i+\sinh (x))^4} \, dx\) [118]

Optimal result

Integrand size = 15, antiderivative size = 91 \[ \int \frac {A+B \sinh (x)}{(i+\sinh (x))^4} \, dx=-\frac {(i A+B) \cosh (x)}{7 (i+\sinh (x))^4}-\frac {(3 A+4 i B) \cosh (x)}{35 (i+\sinh (x))^3}+\frac {2 (3 i A-4 B) \cosh (x)}{105 (i+\sinh (x))^2}+\frac {2 (3 A+4 i B) \cosh (x)}{105 (i+\sinh (x))} \]

[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))

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2829, 2729, 2727} \[ \int \frac {A+B \sinh (x)}{(i+\sinh (x))^4} \, dx=\frac {2 (3 A+4 i B) \cosh (x)}{105 (\sinh (x)+i)}+\frac {2 (-4 B+3 i A) \cosh (x)}{105 (\sinh (x)+i)^2}-\frac {(3 A+4 i B) \cosh (x)}{35 (\sinh (x)+i)^3}-\frac {(B+i A) \cosh (x)}{7 (\sinh (x)+i)^4} \]

[In]

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

[Out]

-1/7*((I*A + B)*Cosh[x])/(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 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]

Rule 2829

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*} \text {integral}& = -\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*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.74 \[ \int \frac {A+B \sinh (x)}{(i+\sinh (x))^4} \, dx=\frac {\cosh (x) \left (-36 i A+13 B-13 (3 A+4 i B) \sinh (x)+8 i (3 A+4 i B) \sinh ^2(x)+(6 A+8 i B) \sinh ^3(x)\right )}{105 (i+\sinh (x))^4} \]

[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)

Maple [A] (verified)

Time = 1.22 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.73

method	result	size
risch	\(-\frac {4 \left (4 B -28 i B \,{\mathrm e}^{x}+63 i A \,{\mathrm e}^{2 x}+70 i B \,{\mathrm e}^{3 x}-3 i A +105 A \,{\mathrm e}^{3 x}+70 B \,{\mathrm e}^{4 x}-84 B \,{\mathrm e}^{2 x}-21 A \,{\mathrm e}^{x}\right )}{105 \left ({\mathrm e}^{x}+i\right )^{7}}\)	\(66\)
default	\(-\frac {2 \left (-10 i B +18 A \right )}{3 \left (\tanh \left (\frac {x}{2}\right )+i\right )^{3}}-\frac {6 i A +2 B}{\left (\tanh \left (\frac {x}{2}\right )+i\right )^{2}}-\frac {-32 i A -24 B}{2 \left (\tanh \left (\frac {x}{2}\right )+i\right )^{4}}-\frac {24 i A +24 B}{3 \left (\tanh \left (\frac {x}{2}\right )+i\right )^{6}}+\frac {2 A}{\tanh \left (\frac {x}{2}\right )+i}-\frac {2 \left (32 i B -36 A \right )}{5 \left (\tanh \left (\frac {x}{2}\right )+i\right )^{5}}-\frac {2 \left (-8 i B +8 A \right )}{7 \left (\tanh \left (\frac {x}{2}\right )+i\right )^{7}}\)	\(128\)
parallelrisch	\(\frac {\left (1092 i A -476 B \right ) \cosh \left (2 x \right )+\left (-168 i A +14 B \right ) \cosh \left (3 x \right )+\left (-42 i A +21 B \right ) \cosh \left (4 x \right )+\left (-42 i B +336 A \right ) \sinh \left (2 x \right )+\left (152 i B +324 A \right ) \sinh \left (3 x \right )+\left (-5 i B -30 A \right ) \sinh \left (4 x \right )+\left (168 i A -14 B \right ) \cosh \left (x \right )+\left (-560 i B -2100 A \right ) \sinh \left (x \right )-1050 i A +455 B}{840 i \sinh \left (3 x \right )-5880 i \sinh \left (x \right )+1470 i \sinh \left (2 x \right )-105 i \sinh \left (4 x \right )-1470 \cosh \left (x \right )+630 \cosh \left (3 x \right )+105 \cosh \left (4 x \right )-2940 \cosh \left (2 x \right )+3675}\)	\(162\)

[In]

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

[Out]

-4/105*(4*B-28*I*B*exp(x)+63*I*A*exp(x)^2+70*I*B*exp(x)^3-3*I*A+105*A*exp(x)^3+70*B*exp(x)^4-84*B*exp(x)^2-21*
A*exp(x))/(exp(x)+I)^7

Fricas [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.04 \[ \int \frac {A+B \sinh (x)}{(i+\sinh (x))^4} \, dx=-\frac {4 \, {\left (70 \, B e^{\left (4 \, x\right )} + 35 \, {\left (3 \, A + 2 i \, B\right )} e^{\left (3 \, x\right )} + 21 \, {\left (3 i \, A - 4 \, B\right )} e^{\left (2 \, x\right )} - 7 \, {\left (3 \, A + 4 i \, B\right )} e^{x} - 3 i \, A + 4 \, B\right )}}{105 \, {\left (e^{\left (7 \, x\right )} + 7 i \, e^{\left (6 \, x\right )} - 21 \, e^{\left (5 \, x\right )} - 35 i \, e^{\left (4 \, x\right )} + 35 \, e^{\left (3 \, x\right )} + 21 i \, e^{\left (2 \, x\right )} - 7 \, e^{x} - i\right )}} \]

[In]

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

[Out]

-4/105*(70*B*e^(4*x) + 35*(3*A + 2*I*B)*e^(3*x) + 21*(3*I*A - 4*B)*e^(2*x) - 7*(3*A + 4*I*B)*e^x - 3*I*A + 4*B
)/(e^(7*x) + 7*I*e^(6*x) - 21*e^(5*x) - 35*I*e^(4*x) + 35*e^(3*x) + 21*I*e^(2*x) - 7*e^x - I)

Sympy [A] (verification not implemented)

Time = 0.45 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.21 \[ \int \frac {A+B \sinh (x)}{(i+\sinh (x))^4} \, dx=\frac {12 i A - 280 B e^{4 x} - 16 B + \left (- 420 A - 280 i B\right ) e^{3 x} + \left (84 A + 112 i B\right ) e^{x} + \left (- 252 i A + 336 B\right ) e^{2 x}}{105 e^{7 x} + 735 i e^{6 x} - 2205 e^{5 x} - 3675 i e^{4 x} + 3675 e^{3 x} + 2205 i e^{2 x} - 735 e^{x} - 105 i} \]

[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)

Maxima [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 469 vs. \(2 (67) = 134\).

Time = 0.21 (sec) , antiderivative size = 469, normalized size of antiderivative = 5.15 \[ \int \frac {A+B \sinh (x)}{(i+\sinh (x))^4} \, dx=\frac {4}{35} \, A {\left (\frac {7 \, e^{\left (-x\right )}}{7 \, e^{\left (-x\right )} + 21 i \, e^{\left (-2 \, x\right )} - 35 \, e^{\left (-3 \, x\right )} - 35 i \, e^{\left (-4 \, x\right )} + 21 \, e^{\left (-5 \, x\right )} + 7 i \, e^{\left (-6 \, x\right )} - e^{\left (-7 \, x\right )} - i} + \frac {21 i \, e^{\left (-2 \, x\right )}}{7 \, e^{\left (-x\right )} + 21 i \, e^{\left (-2 \, x\right )} - 35 \, e^{\left (-3 \, x\right )} - 35 i \, e^{\left (-4 \, x\right )} + 21 \, e^{\left (-5 \, x\right )} + 7 i \, e^{\left (-6 \, x\right )} - e^{\left (-7 \, x\right )} - i} - \frac {35 \, e^{\left (-3 \, x\right )}}{7 \, e^{\left (-x\right )} + 21 i \, e^{\left (-2 \, x\right )} - 35 \, e^{\left (-3 \, x\right )} - 35 i \, e^{\left (-4 \, x\right )} + 21 \, e^{\left (-5 \, x\right )} + 7 i \, e^{\left (-6 \, x\right )} - e^{\left (-7 \, x\right )} - i} - \frac {i}{7 \, e^{\left (-x\right )} + 21 i \, e^{\left (-2 \, x\right )} - 35 \, e^{\left (-3 \, x\right )} - 35 i \, e^{\left (-4 \, x\right )} + 21 \, e^{\left (-5 \, x\right )} + 7 i \, e^{\left (-6 \, x\right )} - e^{\left (-7 \, x\right )} - i}\right )} - \frac {8}{105} \, B {\left (-\frac {14 i \, e^{\left (-x\right )}}{7 \, e^{\left (-x\right )} + 21 i \, e^{\left (-2 \, x\right )} - 35 \, e^{\left (-3 \, x\right )} - 35 i \, e^{\left (-4 \, x\right )} + 21 \, e^{\left (-5 \, x\right )} + 7 i \, e^{\left (-6 \, x\right )} - e^{\left (-7 \, x\right )} - i} + \frac {42 \, e^{\left (-2 \, x\right )}}{7 \, e^{\left (-x\right )} + 21 i \, e^{\left (-2 \, x\right )} - 35 \, e^{\left (-3 \, x\right )} - 35 i \, e^{\left (-4 \, x\right )} + 21 \, e^{\left (-5 \, x\right )} + 7 i \, e^{\left (-6 \, x\right )} - e^{\left (-7 \, x\right )} - i} + \frac {35 i \, e^{\left (-3 \, x\right )}}{7 \, e^{\left (-x\right )} + 21 i \, e^{\left (-2 \, x\right )} - 35 \, e^{\left (-3 \, x\right )} - 35 i \, e^{\left (-4 \, x\right )} + 21 \, e^{\left (-5 \, x\right )} + 7 i \, e^{\left (-6 \, x\right )} - e^{\left (-7 \, x\right )} - i} - \frac {35 \, e^{\left (-4 \, x\right )}}{7 \, e^{\left (-x\right )} + 21 i \, e^{\left (-2 \, x\right )} - 35 \, e^{\left (-3 \, x\right )} - 35 i \, e^{\left (-4 \, x\right )} + 21 \, e^{\left (-5 \, x\right )} + 7 i \, e^{\left (-6 \, x\right )} - e^{\left (-7 \, x\right )} - i} - \frac {2}{7 \, e^{\left (-x\right )} + 21 i \, e^{\left (-2 \, x\right )} - 35 \, e^{\left (-3 \, x\right )} - 35 i \, e^{\left (-4 \, x\right )} + 21 \, e^{\left (-5 \, x\right )} + 7 i \, e^{\left (-6 \, x\right )} - e^{\left (-7 \, x\right )} - i}\right )} \]

[In]

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

[Out]

4/35*A*(7*e^(-x)/(7*e^(-x) + 21*I*e^(-2*x) - 35*e^(-3*x) - 35*I*e^(-4*x) + 21*e^(-5*x) + 7*I*e^(-6*x) - e^(-7*
x) - I) + 21*I*e^(-2*x)/(7*e^(-x) + 21*I*e^(-2*x) - 35*e^(-3*x) - 35*I*e^(-4*x) + 21*e^(-5*x) + 7*I*e^(-6*x) -
e^(-7*x) - I) - 35*e^(-3*x)/(7*e^(-x) + 21*I*e^(-2*x) - 35*e^(-3*x) - 35*I*e^(-4*x) + 21*e^(-5*x) + 7*I*e^(-6
*x) - e^(-7*x) - I) - I/(7*e^(-x) + 21*I*e^(-2*x) - 35*e^(-3*x) - 35*I*e^(-4*x) + 21*e^(-5*x) + 7*I*e^(-6*x) -
e^(-7*x) - I)) - 8/105*B*(-14*I*e^(-x)/(7*e^(-x) + 21*I*e^(-2*x) - 35*e^(-3*x) - 35*I*e^(-4*x) + 21*e^(-5*x)
+ 7*I*e^(-6*x) - e^(-7*x) - I) + 42*e^(-2*x)/(7*e^(-x) + 21*I*e^(-2*x) - 35*e^(-3*x) - 35*I*e^(-4*x) + 21*e^(-
5*x) + 7*I*e^(-6*x) - e^(-7*x) - I) + 35*I*e^(-3*x)/(7*e^(-x) + 21*I*e^(-2*x) - 35*e^(-3*x) - 35*I*e^(-4*x) +
21*e^(-5*x) + 7*I*e^(-6*x) - e^(-7*x) - I) - 35*e^(-4*x)/(7*e^(-x) + 21*I*e^(-2*x) - 35*e^(-3*x) - 35*I*e^(-4*
x) + 21*e^(-5*x) + 7*I*e^(-6*x) - e^(-7*x) - I) - 2/(7*e^(-x) + 21*I*e^(-2*x) - 35*e^(-3*x) - 35*I*e^(-4*x) +
21*e^(-5*x) + 7*I*e^(-6*x) - e^(-7*x) - I))

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.66 \[ \int \frac {A+B \sinh (x)}{(i+\sinh (x))^4} \, dx=-\frac {4 \, {\left (70 \, B e^{\left (4 \, x\right )} + 105 \, A e^{\left (3 \, x\right )} + 70 i \, B e^{\left (3 \, x\right )} + 63 i \, A e^{\left (2 \, x\right )} - 84 \, B e^{\left (2 \, x\right )} - 21 \, A e^{x} - 28 i \, B e^{x} - 3 i \, A + 4 \, B\right )}}{105 \, {\left (e^{x} + i\right )}^{7}} \]

[In]

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

[Out]

-4/105*(70*B*e^(4*x) + 105*A*e^(3*x) + 70*I*B*e^(3*x) + 63*I*A*e^(2*x) - 84*B*e^(2*x) - 21*A*e^x - 28*I*B*e^x
- 3*I*A + 4*B)/(e^x + I)^7

Mupad [B] (verification not implemented)

Time = 1.79 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.73 \[ \int \frac {A+B \sinh (x)}{(i+\sinh (x))^4} \, dx=-\frac {\frac {16\,B}{105}+4\,A\,{\mathrm {e}}^{3\,x}-{\mathrm {e}}^x\,\left (\frac {4\,A}{5}+\frac {B\,16{}\mathrm {i}}{15}\right )-\frac {16\,B\,{\mathrm {e}}^{2\,x}}{5}+\frac {8\,B\,{\mathrm {e}}^{4\,x}}{3}-\frac {A\,4{}\mathrm {i}}{35}+\frac {A\,{\mathrm {e}}^{2\,x}\,12{}\mathrm {i}}{5}+\frac {B\,{\mathrm {e}}^{3\,x}\,8{}\mathrm {i}}{3}}{{\left ({\mathrm {e}}^x+1{}\mathrm {i}\right )}^7} \]

[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