3.265 \(\int \frac {(e+f x)^3 \cosh ^3(c+d x)}{a+i a \sinh (c+d x)} \, dx\)

Optimal. Leaf size=231 \[ -\frac {6 f^3 \cosh (c+d x)}{a d^4}+\frac {3 i f^3 \sinh (c+d x) \cosh (c+d x)}{8 a d^4}-\frac {3 i f^2 (e+f x) \sinh ^2(c+d x)}{4 a d^3}+\frac {6 f^2 (e+f x) \sinh (c+d x)}{a d^3}-\frac {3 f (e+f x)^2 \cosh (c+d x)}{a d^2}+\frac {3 i f (e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{4 a d^2}-\frac {i (e+f x)^3 \sinh ^2(c+d x)}{2 a d}+\frac {(e+f x)^3 \sinh (c+d x)}{a d}-\frac {3 i f^3 x}{8 a d^3}-\frac {i (e+f x)^3}{4 a d} \]

[Out]

-3/8*I*f^3*x/a/d^3-1/4*I*(f*x+e)^3/a/d-6*f^3*cosh(d*x+c)/a/d^4-3*f*(f*x+e)^2*cosh(d*x+c)/a/d^2+6*f^2*(f*x+e)*s
inh(d*x+c)/a/d^3+(f*x+e)^3*sinh(d*x+c)/a/d+3/8*I*f^3*cosh(d*x+c)*sinh(d*x+c)/a/d^4+3/4*I*f*(f*x+e)^2*cosh(d*x+
c)*sinh(d*x+c)/a/d^2-3/4*I*f^2*(f*x+e)*sinh(d*x+c)^2/a/d^3-1/2*I*(f*x+e)^3*sinh(d*x+c)^2/a/d

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Rubi [A]  time = 0.26, antiderivative size = 231, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.258, Rules used = {5563, 3296, 2638, 5446, 3311, 32, 2635, 8} \[ -\frac {3 i f^2 (e+f x) \sinh ^2(c+d x)}{4 a d^3}+\frac {6 f^2 (e+f x) \sinh (c+d x)}{a d^3}-\frac {3 f (e+f x)^2 \cosh (c+d x)}{a d^2}+\frac {3 i f (e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{4 a d^2}-\frac {6 f^3 \cosh (c+d x)}{a d^4}+\frac {3 i f^3 \sinh (c+d x) \cosh (c+d x)}{8 a d^4}-\frac {i (e+f x)^3 \sinh ^2(c+d x)}{2 a d}+\frac {(e+f x)^3 \sinh (c+d x)}{a d}-\frac {3 i f^3 x}{8 a d^3}-\frac {i (e+f x)^3}{4 a d} \]

Antiderivative was successfully verified.

[In]

Int[((e + f*x)^3*Cosh[c + d*x]^3)/(a + I*a*Sinh[c + d*x]),x]

[Out]

(((-3*I)/8)*f^3*x)/(a*d^3) - ((I/4)*(e + f*x)^3)/(a*d) - (6*f^3*Cosh[c + d*x])/(a*d^4) - (3*f*(e + f*x)^2*Cosh
[c + d*x])/(a*d^2) + (6*f^2*(e + f*x)*Sinh[c + d*x])/(a*d^3) + ((e + f*x)^3*Sinh[c + d*x])/(a*d) + (((3*I)/8)*
f^3*Cosh[c + d*x]*Sinh[c + d*x])/(a*d^4) + (((3*I)/4)*f*(e + f*x)^2*Cosh[c + d*x]*Sinh[c + d*x])/(a*d^2) - (((
3*I)/4)*f^2*(e + f*x)*Sinh[c + d*x]^2)/(a*d^3) - ((I/2)*(e + f*x)^3*Sinh[c + d*x]^2)/(a*d)

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rule 2635

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Sin[c + d*x])^(n - 1))/(d*n),
x] + Dist[(b^2*(n - 1))/n, Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integer
Q[2*n]

Rule 2638

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 3296

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> -Simp[((c + d*x)^m*Cos[e + f*x])/f, x] +
Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 3311

Int[((c_.) + (d_.)*(x_))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(d*m*(c + d*x)^(m - 1)*(
b*Sin[e + f*x])^n)/(f^2*n^2), x] + (Dist[(b^2*(n - 1))/n, Int[(c + d*x)^m*(b*Sin[e + f*x])^(n - 2), x], x] - D
ist[(d^2*m*(m - 1))/(f^2*n^2), Int[(c + d*x)^(m - 2)*(b*Sin[e + f*x])^n, x], x] - Simp[(b*(c + d*x)^m*Cos[e +
f*x]*(b*Sin[e + f*x])^(n - 1))/(f*n), x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1] && GtQ[m, 1]

Rule 5446

Int[Cosh[(a_.) + (b_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Simp[((c
+ d*x)^m*Sinh[a + b*x]^(n + 1))/(b*(n + 1)), x] - Dist[(d*m)/(b*(n + 1)), Int[(c + d*x)^(m - 1)*Sinh[a + b*x]^
(n + 1), x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && NeQ[n, -1]

Rule 5563

Int[(Cosh[(c_.) + (d_.)*(x_)]^(n_)*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sinh[(c_.) + (d_.)*(x_)]), x_Symb
ol] :> Dist[1/a, Int[(e + f*x)^m*Cosh[c + d*x]^(n - 2), x], x] + Dist[1/b, Int[(e + f*x)^m*Cosh[c + d*x]^(n -
2)*Sinh[c + d*x], x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[n, 1] && EqQ[a^2 + b^2, 0]

Rubi steps

\begin {align*} \int \frac {(e+f x)^3 \cosh ^3(c+d x)}{a+i a \sinh (c+d x)} \, dx &=-\frac {i \int (e+f x)^3 \cosh (c+d x) \sinh (c+d x) \, dx}{a}+\frac {\int (e+f x)^3 \cosh (c+d x) \, dx}{a}\\ &=\frac {(e+f x)^3 \sinh (c+d x)}{a d}-\frac {i (e+f x)^3 \sinh ^2(c+d x)}{2 a d}+\frac {(3 i f) \int (e+f x)^2 \sinh ^2(c+d x) \, dx}{2 a d}-\frac {(3 f) \int (e+f x)^2 \sinh (c+d x) \, dx}{a d}\\ &=-\frac {3 f (e+f x)^2 \cosh (c+d x)}{a d^2}+\frac {(e+f x)^3 \sinh (c+d x)}{a d}+\frac {3 i f (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{4 a d^2}-\frac {3 i f^2 (e+f x) \sinh ^2(c+d x)}{4 a d^3}-\frac {i (e+f x)^3 \sinh ^2(c+d x)}{2 a d}-\frac {(3 i f) \int (e+f x)^2 \, dx}{4 a d}+\frac {\left (6 f^2\right ) \int (e+f x) \cosh (c+d x) \, dx}{a d^2}+\frac {\left (3 i f^3\right ) \int \sinh ^2(c+d x) \, dx}{4 a d^3}\\ &=-\frac {i (e+f x)^3}{4 a d}-\frac {3 f (e+f x)^2 \cosh (c+d x)}{a d^2}+\frac {6 f^2 (e+f x) \sinh (c+d x)}{a d^3}+\frac {(e+f x)^3 \sinh (c+d x)}{a d}+\frac {3 i f^3 \cosh (c+d x) \sinh (c+d x)}{8 a d^4}+\frac {3 i f (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{4 a d^2}-\frac {3 i f^2 (e+f x) \sinh ^2(c+d x)}{4 a d^3}-\frac {i (e+f x)^3 \sinh ^2(c+d x)}{2 a d}-\frac {\left (3 i f^3\right ) \int 1 \, dx}{8 a d^3}-\frac {\left (6 f^3\right ) \int \sinh (c+d x) \, dx}{a d^3}\\ &=-\frac {3 i f^3 x}{8 a d^3}-\frac {i (e+f x)^3}{4 a d}-\frac {6 f^3 \cosh (c+d x)}{a d^4}-\frac {3 f (e+f x)^2 \cosh (c+d x)}{a d^2}+\frac {6 f^2 (e+f x) \sinh (c+d x)}{a d^3}+\frac {(e+f x)^3 \sinh (c+d x)}{a d}+\frac {3 i f^3 \cosh (c+d x) \sinh (c+d x)}{8 a d^4}+\frac {3 i f (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{4 a d^2}-\frac {3 i f^2 (e+f x) \sinh ^2(c+d x)}{4 a d^3}-\frac {i (e+f x)^3 \sinh ^2(c+d x)}{2 a d}\\ \end {align*}

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Mathematica [A]  time = 1.29, size = 134, normalized size = 0.58 \[ \frac {-96 f \cosh (c+d x) \left (d^2 (e+f x)^2+2 f^2\right )-4 i d (e+f x) \cosh (2 (c+d x)) \left (2 d^2 (e+f x)^2+3 f^2\right )+4 \sinh (c+d x) \left (8 d (e+f x) \left (d^2 (e+f x)^2+6 f^2\right )+3 i f \cosh (c+d x) \left (2 d^2 (e+f x)^2+f^2\right )\right )}{32 a d^4} \]

Antiderivative was successfully verified.

[In]

Integrate[((e + f*x)^3*Cosh[c + d*x]^3)/(a + I*a*Sinh[c + d*x]),x]

[Out]

(-96*f*(2*f^2 + d^2*(e + f*x)^2)*Cosh[c + d*x] - (4*I)*d*(e + f*x)*(3*f^2 + 2*d^2*(e + f*x)^2)*Cosh[2*(c + d*x
)] + 4*(8*d*(e + f*x)*(6*f^2 + d^2*(e + f*x)^2) + (3*I)*f*(f^2 + 2*d^2*(e + f*x)^2)*Cosh[c + d*x])*Sinh[c + d*
x])/(32*a*d^4)

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fricas [A]  time = 0.47, size = 401, normalized size = 1.74 \[ \frac {{\left (-4 i \, d^{3} f^{3} x^{3} - 4 i \, d^{3} e^{3} - 6 i \, d^{2} e^{2} f - 6 i \, d e f^{2} - 3 i \, f^{3} + {\left (-12 i \, d^{3} e f^{2} - 6 i \, d^{2} f^{3}\right )} x^{2} + {\left (-12 i \, d^{3} e^{2} f - 12 i \, d^{2} e f^{2} - 6 i \, d f^{3}\right )} x + {\left (-4 i \, d^{3} f^{3} x^{3} - 4 i \, d^{3} e^{3} + 6 i \, d^{2} e^{2} f - 6 i \, d e f^{2} + 3 i \, f^{3} + {\left (-12 i \, d^{3} e f^{2} + 6 i \, d^{2} f^{3}\right )} x^{2} + {\left (-12 i \, d^{3} e^{2} f + 12 i \, d^{2} e f^{2} - 6 i \, d f^{3}\right )} x\right )} e^{\left (4 \, d x + 4 \, c\right )} + 16 \, {\left (d^{3} f^{3} x^{3} + d^{3} e^{3} - 3 \, d^{2} e^{2} f + 6 \, d e f^{2} - 6 \, f^{3} + 3 \, {\left (d^{3} e f^{2} - d^{2} f^{3}\right )} x^{2} + 3 \, {\left (d^{3} e^{2} f - 2 \, d^{2} e f^{2} + 2 \, d f^{3}\right )} x\right )} e^{\left (3 \, d x + 3 \, c\right )} - 16 \, {\left (d^{3} f^{3} x^{3} + d^{3} e^{3} + 3 \, d^{2} e^{2} f + 6 \, d e f^{2} + 6 \, f^{3} + 3 \, {\left (d^{3} e f^{2} + d^{2} f^{3}\right )} x^{2} + 3 \, {\left (d^{3} e^{2} f + 2 \, d^{2} e f^{2} + 2 \, d f^{3}\right )} x\right )} e^{\left (d x + c\right )}\right )} e^{\left (-2 \, d x - 2 \, c\right )}}{32 \, a d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)^3*cosh(d*x+c)^3/(a+I*a*sinh(d*x+c)),x, algorithm="fricas")

[Out]

1/32*(-4*I*d^3*f^3*x^3 - 4*I*d^3*e^3 - 6*I*d^2*e^2*f - 6*I*d*e*f^2 - 3*I*f^3 + (-12*I*d^3*e*f^2 - 6*I*d^2*f^3)
*x^2 + (-12*I*d^3*e^2*f - 12*I*d^2*e*f^2 - 6*I*d*f^3)*x + (-4*I*d^3*f^3*x^3 - 4*I*d^3*e^3 + 6*I*d^2*e^2*f - 6*
I*d*e*f^2 + 3*I*f^3 + (-12*I*d^3*e*f^2 + 6*I*d^2*f^3)*x^2 + (-12*I*d^3*e^2*f + 12*I*d^2*e*f^2 - 6*I*d*f^3)*x)*
e^(4*d*x + 4*c) + 16*(d^3*f^3*x^3 + d^3*e^3 - 3*d^2*e^2*f + 6*d*e*f^2 - 6*f^3 + 3*(d^3*e*f^2 - d^2*f^3)*x^2 +
3*(d^3*e^2*f - 2*d^2*e*f^2 + 2*d*f^3)*x)*e^(3*d*x + 3*c) - 16*(d^3*f^3*x^3 + d^3*e^3 + 3*d^2*e^2*f + 6*d*e*f^2
+ 6*f^3 + 3*(d^3*e*f^2 + d^2*f^3)*x^2 + 3*(d^3*e^2*f + 2*d^2*e*f^2 + 2*d*f^3)*x)*e^(d*x + c))*e^(-2*d*x - 2*c
)/(a*d^4)

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giac [B]  time = 1.69, size = 1005, normalized size = 4.35 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)^3*cosh(d*x+c)^3/(a+I*a*sinh(d*x+c)),x, algorithm="giac")

[Out]

1/32*(-4*I*d^3*f^3*x^3*e^(5*d*x + 6*c) + 12*d^3*f^3*x^3*e^(4*d*x + 5*c) - 16*I*d^3*f^3*x^3*e^(3*d*x + 4*c) - 1
6*d^3*f^3*x^3*e^(2*d*x + 3*c) + 12*I*d^3*f^3*x^3*e^(d*x + 2*c) - 4*d^3*f^3*x^3*e^c - 12*I*d^3*f^2*x^2*e^(5*d*x
+ 6*c + 1) + 6*I*d^2*f^3*x^2*e^(5*d*x + 6*c) + 36*d^3*f^2*x^2*e^(4*d*x + 5*c + 1) - 42*d^2*f^3*x^2*e^(4*d*x +
5*c) - 48*I*d^3*f^2*x^2*e^(3*d*x + 4*c + 1) + 48*I*d^2*f^3*x^2*e^(3*d*x + 4*c) - 48*d^3*f^2*x^2*e^(2*d*x + 3*
c + 1) - 48*d^2*f^3*x^2*e^(2*d*x + 3*c) + 36*I*d^3*f^2*x^2*e^(d*x + 2*c + 1) + 42*I*d^2*f^3*x^2*e^(d*x + 2*c)
- 12*d^3*f^2*x^2*e^(c + 1) - 6*d^2*f^3*x^2*e^c - 12*I*d^3*f*x*e^(5*d*x + 6*c + 2) + 12*I*d^2*f^2*x*e^(5*d*x +
6*c + 1) - 6*I*d*f^3*x*e^(5*d*x + 6*c) + 36*d^3*f*x*e^(4*d*x + 5*c + 2) - 84*d^2*f^2*x*e^(4*d*x + 5*c + 1) + 9
0*d*f^3*x*e^(4*d*x + 5*c) - 48*I*d^3*f*x*e^(3*d*x + 4*c + 2) + 96*I*d^2*f^2*x*e^(3*d*x + 4*c + 1) - 96*I*d*f^3
*x*e^(3*d*x + 4*c) - 48*d^3*f*x*e^(2*d*x + 3*c + 2) - 96*d^2*f^2*x*e^(2*d*x + 3*c + 1) - 96*d*f^3*x*e^(2*d*x +
3*c) + 36*I*d^3*f*x*e^(d*x + 2*c + 2) + 84*I*d^2*f^2*x*e^(d*x + 2*c + 1) + 90*I*d*f^3*x*e^(d*x + 2*c) - 12*d^
3*f*x*e^(c + 2) - 12*d^2*f^2*x*e^(c + 1) - 6*d*f^3*x*e^c - 4*I*d^3*e^(5*d*x + 6*c + 3) + 6*I*d^2*f*e^(5*d*x +
6*c + 2) - 6*I*d*f^2*e^(5*d*x + 6*c + 1) + 3*I*f^3*e^(5*d*x + 6*c) + 12*d^3*e^(4*d*x + 5*c + 3) - 42*d^2*f*e^(
4*d*x + 5*c + 2) + 90*d*f^2*e^(4*d*x + 5*c + 1) - 93*f^3*e^(4*d*x + 5*c) - 16*I*d^3*e^(3*d*x + 4*c + 3) + 48*I
*d^2*f*e^(3*d*x + 4*c + 2) - 96*I*d*f^2*e^(3*d*x + 4*c + 1) + 96*I*f^3*e^(3*d*x + 4*c) - 16*d^3*e^(2*d*x + 3*c
+ 3) - 48*d^2*f*e^(2*d*x + 3*c + 2) - 96*d*f^2*e^(2*d*x + 3*c + 1) - 96*f^3*e^(2*d*x + 3*c) + 12*I*d^3*e^(d*x
+ 2*c + 3) + 42*I*d^2*f*e^(d*x + 2*c + 2) + 90*I*d*f^2*e^(d*x + 2*c + 1) + 93*I*f^3*e^(d*x + 2*c) - 4*d^3*e^(
c + 3) - 6*d^2*f*e^(c + 2) - 6*d*f^2*e^(c + 1) - 3*f^3*e^c)/(a*d^4*e^(3*d*x + 4*c) - I*a*d^4*e^(2*d*x + 3*c))

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maple [B]  time = 0.14, size = 726, normalized size = 3.14 \[ -\frac {\frac {3 i c^{2} f^{2} e d \left (\cosh ^{2}\left (d x +c \right )\right )}{2}-\frac {i f^{3} c^{3} \left (\cosh ^{2}\left (d x +c \right )\right )}{2}+3 i f \,e^{2} d^{2} \left (\frac {\left (d x +c \right ) \left (\cosh ^{2}\left (d x +c \right )\right )}{2}-\frac {\cosh \left (d x +c \right ) \sinh \left (d x +c \right )}{4}-\frac {d x}{4}-\frac {c}{4}\right )-3 i c \,f^{3} \left (\frac {\left (d x +c \right )^{2} \left (\cosh ^{2}\left (d x +c \right )\right )}{2}-\frac {\left (d x +c \right ) \cosh \left (d x +c \right ) \sinh \left (d x +c \right )}{2}-\frac {\left (d x +c \right )^{2}}{4}+\frac {\left (\cosh ^{2}\left (d x +c \right )\right )}{4}\right )+3 i c^{2} f^{3} \left (\frac {\left (d x +c \right ) \left (\cosh ^{2}\left (d x +c \right )\right )}{2}-\frac {\cosh \left (d x +c \right ) \sinh \left (d x +c \right )}{4}-\frac {d x}{4}-\frac {c}{4}\right )-6 i f^{2} e c d \left (\frac {\left (d x +c \right ) \left (\cosh ^{2}\left (d x +c \right )\right )}{2}-\frac {\cosh \left (d x +c \right ) \sinh \left (d x +c \right )}{4}-\frac {d x}{4}-\frac {c}{4}\right )+\frac {i e^{3} d^{3} \left (\cosh ^{2}\left (d x +c \right )\right )}{2}-\frac {3 i c f \,e^{2} d^{2} \left (\cosh ^{2}\left (d x +c \right )\right )}{2}+i f^{3} \left (\frac {\left (d x +c \right )^{3} \left (\cosh ^{2}\left (d x +c \right )\right )}{2}-\frac {3 \left (d x +c \right )^{2} \cosh \left (d x +c \right ) \sinh \left (d x +c \right )}{4}-\frac {\left (d x +c \right )^{3}}{4}+\frac {3 \left (d x +c \right ) \left (\cosh ^{2}\left (d x +c \right )\right )}{4}-\frac {3 \cosh \left (d x +c \right ) \sinh \left (d x +c \right )}{8}-\frac {3 d x}{8}-\frac {3 c}{8}\right )+3 i f^{2} e d \left (\frac {\left (d x +c \right )^{2} \left (\cosh ^{2}\left (d x +c \right )\right )}{2}-\frac {\left (d x +c \right ) \cosh \left (d x +c \right ) \sinh \left (d x +c \right )}{2}-\frac {\left (d x +c \right )^{2}}{4}+\frac {\left (\cosh ^{2}\left (d x +c \right )\right )}{4}\right )-f^{3} \left (\left (d x +c \right )^{3} \sinh \left (d x +c \right )-3 \left (d x +c \right )^{2} \cosh \left (d x +c \right )+6 \left (d x +c \right ) \sinh \left (d x +c \right )-6 \cosh \left (d x +c \right )\right )+3 c \,f^{3} \left (\left (d x +c \right )^{2} \sinh \left (d x +c \right )-2 \left (d x +c \right ) \cosh \left (d x +c \right )+2 \sinh \left (d x +c \right )\right )-3 d e \,f^{2} \left (\left (d x +c \right )^{2} \sinh \left (d x +c \right )-2 \left (d x +c \right ) \cosh \left (d x +c \right )+2 \sinh \left (d x +c \right )\right )-3 c^{2} f^{3} \left (\left (d x +c \right ) \sinh \left (d x +c \right )-\cosh \left (d x +c \right )\right )+6 c d e \,f^{2} \left (\left (d x +c \right ) \sinh \left (d x +c \right )-\cosh \left (d x +c \right )\right )-3 d^{2} e^{2} f \left (\left (d x +c \right ) \sinh \left (d x +c \right )-\cosh \left (d x +c \right )\right )+f^{3} c^{3} \sinh \left (d x +c \right )-3 c^{2} f^{2} e d \sinh \left (d x +c \right )+3 c f \,e^{2} d^{2} \sinh \left (d x +c \right )-e^{3} d^{3} \sinh \left (d x +c \right )}{d^{4} a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f*x+e)^3*cosh(d*x+c)^3/(a+I*a*sinh(d*x+c)),x)

[Out]

-1/d^4/a*(3/2*I*c^2*f^2*e*d*cosh(d*x+c)^2-1/2*I*f^3*c^3*cosh(d*x+c)^2+3*I*f*e^2*d^2*(1/2*(d*x+c)*cosh(d*x+c)^2
-1/4*cosh(d*x+c)*sinh(d*x+c)-1/4*d*x-1/4*c)-3*I*c*f^3*(1/2*(d*x+c)^2*cosh(d*x+c)^2-1/2*(d*x+c)*cosh(d*x+c)*sin
h(d*x+c)-1/4*(d*x+c)^2+1/4*cosh(d*x+c)^2)+3*I*c^2*f^3*(1/2*(d*x+c)*cosh(d*x+c)^2-1/4*cosh(d*x+c)*sinh(d*x+c)-1
/4*d*x-1/4*c)-6*I*f^2*e*c*d*(1/2*(d*x+c)*cosh(d*x+c)^2-1/4*cosh(d*x+c)*sinh(d*x+c)-1/4*d*x-1/4*c)+1/2*I*e^3*d^
3*cosh(d*x+c)^2-3/2*I*c*f*e^2*d^2*cosh(d*x+c)^2+I*f^3*(1/2*(d*x+c)^3*cosh(d*x+c)^2-3/4*(d*x+c)^2*cosh(d*x+c)*s
inh(d*x+c)-1/4*(d*x+c)^3+3/4*(d*x+c)*cosh(d*x+c)^2-3/8*cosh(d*x+c)*sinh(d*x+c)-3/8*d*x-3/8*c)+3*I*f^2*e*d*(1/2
*(d*x+c)^2*cosh(d*x+c)^2-1/2*(d*x+c)*cosh(d*x+c)*sinh(d*x+c)-1/4*(d*x+c)^2+1/4*cosh(d*x+c)^2)-f^3*((d*x+c)^3*s
inh(d*x+c)-3*(d*x+c)^2*cosh(d*x+c)+6*(d*x+c)*sinh(d*x+c)-6*cosh(d*x+c))+3*c*f^3*((d*x+c)^2*sinh(d*x+c)-2*(d*x+
c)*cosh(d*x+c)+2*sinh(d*x+c))-3*d*e*f^2*((d*x+c)^2*sinh(d*x+c)-2*(d*x+c)*cosh(d*x+c)+2*sinh(d*x+c))-3*c^2*f^3*
((d*x+c)*sinh(d*x+c)-cosh(d*x+c))+6*c*d*e*f^2*((d*x+c)*sinh(d*x+c)-cosh(d*x+c))-3*d^2*e^2*f*((d*x+c)*sinh(d*x+
c)-cosh(d*x+c))+f^3*c^3*sinh(d*x+c)-3*c^2*f^2*e*d*sinh(d*x+c)+3*c*f*e^2*d^2*sinh(d*x+c)-e^3*d^3*sinh(d*x+c))

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)^3*cosh(d*x+c)^3/(a+I*a*sinh(d*x+c)),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is undefined.

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mupad [B]  time = 1.28, size = 449, normalized size = 1.94 \[ -{\mathrm {e}}^{c+d\,x}\,\left (\frac {-d^3\,e^3+3\,d^2\,e^2\,f-6\,d\,e\,f^2+6\,f^3}{2\,a\,d^4}-\frac {f^3\,x^3}{2\,a\,d}+\frac {3\,f^2\,x^2\,\left (f-d\,e\right )}{2\,a\,d^2}-\frac {3\,f\,x\,\left (d^2\,e^2-2\,d\,e\,f+2\,f^2\right )}{2\,a\,d^3}\right )-{\mathrm {e}}^{-2\,c-2\,d\,x}\,\left (\frac {\left (4\,d^3\,e^3+6\,d^2\,e^2\,f+6\,d\,e\,f^2+3\,f^3\right )\,1{}\mathrm {i}}{32\,a\,d^4}+\frac {f^3\,x^3\,1{}\mathrm {i}}{8\,a\,d}+\frac {f\,x\,\left (2\,d^2\,e^2+2\,d\,e\,f+f^2\right )\,3{}\mathrm {i}}{16\,a\,d^3}+\frac {f^2\,x^2\,\left (f+2\,d\,e\right )\,3{}\mathrm {i}}{16\,a\,d^2}\right )+{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (\frac {\left (-4\,d^3\,e^3+6\,d^2\,e^2\,f-6\,d\,e\,f^2+3\,f^3\right )\,1{}\mathrm {i}}{32\,a\,d^4}-\frac {f^3\,x^3\,1{}\mathrm {i}}{8\,a\,d}-\frac {f\,x\,\left (2\,d^2\,e^2-2\,d\,e\,f+f^2\right )\,3{}\mathrm {i}}{16\,a\,d^3}+\frac {f^2\,x^2\,\left (f-2\,d\,e\right )\,3{}\mathrm {i}}{16\,a\,d^2}\right )-{\mathrm {e}}^{-c-d\,x}\,\left (\frac {d^3\,e^3+3\,d^2\,e^2\,f+6\,d\,e\,f^2+6\,f^3}{2\,a\,d^4}+\frac {f^3\,x^3}{2\,a\,d}+\frac {3\,f^2\,x^2\,\left (f+d\,e\right )}{2\,a\,d^2}+\frac {3\,f\,x\,\left (d^2\,e^2+2\,d\,e\,f+2\,f^2\right )}{2\,a\,d^3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((cosh(c + d*x)^3*(e + f*x)^3)/(a + a*sinh(c + d*x)*1i),x)

[Out]

exp(2*c + 2*d*x)*(((3*f^3 - 4*d^3*e^3 + 6*d^2*e^2*f - 6*d*e*f^2)*1i)/(32*a*d^4) - (f^3*x^3*1i)/(8*a*d) - (f*x*
(f^2 + 2*d^2*e^2 - 2*d*e*f)*3i)/(16*a*d^3) + (f^2*x^2*(f - 2*d*e)*3i)/(16*a*d^2)) - exp(- 2*c - 2*d*x)*(((3*f^
3 + 4*d^3*e^3 + 6*d^2*e^2*f + 6*d*e*f^2)*1i)/(32*a*d^4) + (f^3*x^3*1i)/(8*a*d) + (f*x*(f^2 + 2*d^2*e^2 + 2*d*e
*f)*3i)/(16*a*d^3) + (f^2*x^2*(f + 2*d*e)*3i)/(16*a*d^2)) - exp(c + d*x)*((6*f^3 - d^3*e^3 + 3*d^2*e^2*f - 6*d
*e*f^2)/(2*a*d^4) - (f^3*x^3)/(2*a*d) + (3*f^2*x^2*(f - d*e))/(2*a*d^2) - (3*f*x*(2*f^2 + d^2*e^2 - 2*d*e*f))/
(2*a*d^3)) - exp(- c - d*x)*((6*f^3 + d^3*e^3 + 3*d^2*e^2*f + 6*d*e*f^2)/(2*a*d^4) + (f^3*x^3)/(2*a*d) + (3*f^
2*x^2*(f + d*e))/(2*a*d^2) + (3*f*x*(2*f^2 + d^2*e^2 + 2*d*e*f))/(2*a*d^3))

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sympy [A]  time = 1.18, size = 1042, normalized size = 4.51 \[ \begin {cases} \frac {\left (\left (- 2048 a^{3} d^{15} e^{3} e^{2 c} - 6144 a^{3} d^{15} e^{2} f x e^{2 c} - 6144 a^{3} d^{15} e f^{2} x^{2} e^{2 c} - 2048 a^{3} d^{15} f^{3} x^{3} e^{2 c} - 6144 a^{3} d^{14} e^{2} f e^{2 c} - 12288 a^{3} d^{14} e f^{2} x e^{2 c} - 6144 a^{3} d^{14} f^{3} x^{2} e^{2 c} - 12288 a^{3} d^{13} e f^{2} e^{2 c} - 12288 a^{3} d^{13} f^{3} x e^{2 c} - 12288 a^{3} d^{12} f^{3} e^{2 c}\right ) e^{- d x} + \left (2048 a^{3} d^{15} e^{3} e^{4 c} + 6144 a^{3} d^{15} e^{2} f x e^{4 c} + 6144 a^{3} d^{15} e f^{2} x^{2} e^{4 c} + 2048 a^{3} d^{15} f^{3} x^{3} e^{4 c} - 6144 a^{3} d^{14} e^{2} f e^{4 c} - 12288 a^{3} d^{14} e f^{2} x e^{4 c} - 6144 a^{3} d^{14} f^{3} x^{2} e^{4 c} + 12288 a^{3} d^{13} e f^{2} e^{4 c} + 12288 a^{3} d^{13} f^{3} x e^{4 c} - 12288 a^{3} d^{12} f^{3} e^{4 c}\right ) e^{d x} + \left (- 512 i a^{3} d^{15} e^{3} e^{c} - 1536 i a^{3} d^{15} e^{2} f x e^{c} - 1536 i a^{3} d^{15} e f^{2} x^{2} e^{c} - 512 i a^{3} d^{15} f^{3} x^{3} e^{c} - 768 i a^{3} d^{14} e^{2} f e^{c} - 1536 i a^{3} d^{14} e f^{2} x e^{c} - 768 i a^{3} d^{14} f^{3} x^{2} e^{c} - 768 i a^{3} d^{13} e f^{2} e^{c} - 768 i a^{3} d^{13} f^{3} x e^{c} - 384 i a^{3} d^{12} f^{3} e^{c}\right ) e^{- 2 d x} + \left (- 512 i a^{3} d^{15} e^{3} e^{5 c} - 1536 i a^{3} d^{15} e^{2} f x e^{5 c} - 1536 i a^{3} d^{15} e f^{2} x^{2} e^{5 c} - 512 i a^{3} d^{15} f^{3} x^{3} e^{5 c} + 768 i a^{3} d^{14} e^{2} f e^{5 c} + 1536 i a^{3} d^{14} e f^{2} x e^{5 c} + 768 i a^{3} d^{14} f^{3} x^{2} e^{5 c} - 768 i a^{3} d^{13} e f^{2} e^{5 c} - 768 i a^{3} d^{13} f^{3} x e^{5 c} + 384 i a^{3} d^{12} f^{3} e^{5 c}\right ) e^{2 d x}\right ) e^{- 3 c}}{4096 a^{4} d^{16}} & \text {for}\: 4096 a^{4} d^{16} e^{3 c} \neq 0 \\\frac {x^{4} \left (- i f^{3} e^{4 c} + 2 f^{3} e^{3 c} + 2 f^{3} e^{c} + i f^{3}\right ) e^{- 2 c}}{16 a} + \frac {x^{3} \left (- i e f^{2} e^{4 c} + 2 e f^{2} e^{3 c} + 2 e f^{2} e^{c} + i e f^{2}\right ) e^{- 2 c}}{4 a} + \frac {x^{2} \left (- 3 i e^{2} f e^{4 c} + 6 e^{2} f e^{3 c} + 6 e^{2} f e^{c} + 3 i e^{2} f\right ) e^{- 2 c}}{8 a} + \frac {x \left (- i e^{3} e^{4 c} + 2 e^{3} e^{3 c} + 2 e^{3} e^{c} + i e^{3}\right ) e^{- 2 c}}{4 a} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)**3*cosh(d*x+c)**3/(a+I*a*sinh(d*x+c)),x)

[Out]

Piecewise((((-2048*a**3*d**15*e**3*exp(2*c) - 6144*a**3*d**15*e**2*f*x*exp(2*c) - 6144*a**3*d**15*e*f**2*x**2*
exp(2*c) - 2048*a**3*d**15*f**3*x**3*exp(2*c) - 6144*a**3*d**14*e**2*f*exp(2*c) - 12288*a**3*d**14*e*f**2*x*ex
p(2*c) - 6144*a**3*d**14*f**3*x**2*exp(2*c) - 12288*a**3*d**13*e*f**2*exp(2*c) - 12288*a**3*d**13*f**3*x*exp(2
*c) - 12288*a**3*d**12*f**3*exp(2*c))*exp(-d*x) + (2048*a**3*d**15*e**3*exp(4*c) + 6144*a**3*d**15*e**2*f*x*ex
p(4*c) + 6144*a**3*d**15*e*f**2*x**2*exp(4*c) + 2048*a**3*d**15*f**3*x**3*exp(4*c) - 6144*a**3*d**14*e**2*f*ex
p(4*c) - 12288*a**3*d**14*e*f**2*x*exp(4*c) - 6144*a**3*d**14*f**3*x**2*exp(4*c) + 12288*a**3*d**13*e*f**2*exp
(4*c) + 12288*a**3*d**13*f**3*x*exp(4*c) - 12288*a**3*d**12*f**3*exp(4*c))*exp(d*x) + (-512*I*a**3*d**15*e**3*
exp(c) - 1536*I*a**3*d**15*e**2*f*x*exp(c) - 1536*I*a**3*d**15*e*f**2*x**2*exp(c) - 512*I*a**3*d**15*f**3*x**3
*exp(c) - 768*I*a**3*d**14*e**2*f*exp(c) - 1536*I*a**3*d**14*e*f**2*x*exp(c) - 768*I*a**3*d**14*f**3*x**2*exp(
c) - 768*I*a**3*d**13*e*f**2*exp(c) - 768*I*a**3*d**13*f**3*x*exp(c) - 384*I*a**3*d**12*f**3*exp(c))*exp(-2*d*
x) + (-512*I*a**3*d**15*e**3*exp(5*c) - 1536*I*a**3*d**15*e**2*f*x*exp(5*c) - 1536*I*a**3*d**15*e*f**2*x**2*ex
p(5*c) - 512*I*a**3*d**15*f**3*x**3*exp(5*c) + 768*I*a**3*d**14*e**2*f*exp(5*c) + 1536*I*a**3*d**14*e*f**2*x*e
xp(5*c) + 768*I*a**3*d**14*f**3*x**2*exp(5*c) - 768*I*a**3*d**13*e*f**2*exp(5*c) - 768*I*a**3*d**13*f**3*x*exp
(5*c) + 384*I*a**3*d**12*f**3*exp(5*c))*exp(2*d*x))*exp(-3*c)/(4096*a**4*d**16), Ne(4096*a**4*d**16*exp(3*c),
0)), (x**4*(-I*f**3*exp(4*c) + 2*f**3*exp(3*c) + 2*f**3*exp(c) + I*f**3)*exp(-2*c)/(16*a) + x**3*(-I*e*f**2*ex
p(4*c) + 2*e*f**2*exp(3*c) + 2*e*f**2*exp(c) + I*e*f**2)*exp(-2*c)/(4*a) + x**2*(-3*I*e**2*f*exp(4*c) + 6*e**2
*f*exp(3*c) + 6*e**2*f*exp(c) + 3*I*e**2*f)*exp(-2*c)/(8*a) + x*(-I*e**3*exp(4*c) + 2*e**3*exp(3*c) + 2*e**3*e
xp(c) + I*e**3)*exp(-2*c)/(4*a), True))

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