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

Optimal. Leaf size=101 \[ -\frac{2 (3 A-4 i B) \cosh (x)}{105 (-\sinh (x)+i)}-\frac{2 (4 B+3 i A) \cosh (x)}{105 (-\sinh (x)+i)^2}+\frac{(3 A-4 i B) \cosh (x)}{35 (-\sinh (x)+i)^3}+\frac{(-B+i A) \cosh (x)}{7 (-\sinh (x)+i)^4} \]

[Out]

((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]))

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Rubi [A]  time = 0.0699008, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, 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 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)]

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{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*}

Mathematica [A]  time = 0.0487166, size = 63, normalized size = 0.62 \[ \frac{\cosh (x) \left ((6 A-8 i B) \sinh ^3(x)+(-32 B-24 i A) \sinh ^2(x)+(-39 A+52 i B) \sinh (x)+36 i A+13 B\right )}{105 (\sinh (x)-i)^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Cosh[x]*((36*I)*A + 13*B + (-39*A + (52*I)*B)*Sinh[x] + ((-24*I)*A - 32*B)*Sinh[x]^2 + (6*A - (8*I)*B)*Sinh[x
]^3))/(105*(-I + Sinh[x])^4)

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Maple [A]  time = 0.033, size = 128, normalized size = 1.3 \begin{align*} -{\frac{32\,iA-24\,B}{2} \left ( \tanh \left ({\frac{x}{2}} \right ) -i \right ) ^{-4}}-{\frac{16\,A+16\,iB}{7} \left ( \tanh \left ({\frac{x}{2}} \right ) -i \right ) ^{-7}}+2\,{\frac{A}{\tanh \left ( x/2 \right ) -i}}-{(-6\,iA+2\,B) \left ( \tanh \left ({\frac{x}{2}} \right ) -i \right ) ^{-2}}-{\frac{-72\,A-64\,iB}{5} \left ( \tanh \left ({\frac{x}{2}} \right ) -i \right ) ^{-5}}-{\frac{-24\,iA+24\,B}{3} \left ( \tanh \left ({\frac{x}{2}} \right ) -i \right ) ^{-6}}-{\frac{36\,A+20\,iB}{3} \left ( \tanh \left ({\frac{x}{2}} \right ) -i \right ) ^{-3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.50314, size = 632, normalized size = 6.26 \begin{align*} \text{result too large to display} \end{align*}

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) - 367
5*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) -
1225*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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Fricas [A]  time = 1.72348, size = 304, normalized size = 3.01 \begin{align*} -\frac{280 \, B e^{\left (4 \, x\right )} + 140 \,{\left (3 \, A - 2 i \, B\right )} e^{\left (3 \, x\right )} -{\left (252 i \, A + 336 \, B\right )} e^{\left (2 \, x\right )} - 28 \,{\left (3 \, A - 4 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} \end{align*}

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) + 140*(3*A - 2*I*B)*e^(3*x) - (252*I*A + 336*B)*e^(2*x) - 28*(3*A - 4*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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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]  time = 1.28593, size = 81, normalized size = 0.8 \begin{align*} -\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}} \end{align*}

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