3.28 \(\int \frac {(c+d x)^2}{(a+a \coth (e+f x))^3} \, dx\)

Optimal. Leaf size=246 \[ \frac {d (c+d x) e^{-6 e-6 f x}}{144 a^3 f^2}-\frac {3 d (c+d x) e^{-4 e-4 f x}}{64 a^3 f^2}+\frac {3 d (c+d x) e^{-2 e-2 f x}}{16 a^3 f^2}+\frac {(c+d x)^2 e^{-6 e-6 f x}}{48 a^3 f}-\frac {3 (c+d x)^2 e^{-4 e-4 f x}}{32 a^3 f}+\frac {3 (c+d x)^2 e^{-2 e-2 f x}}{16 a^3 f}+\frac {(c+d x)^3}{24 a^3 d}+\frac {d^2 e^{-6 e-6 f x}}{864 a^3 f^3}-\frac {3 d^2 e^{-4 e-4 f x}}{256 a^3 f^3}+\frac {3 d^2 e^{-2 e-2 f x}}{32 a^3 f^3} \]

[Out]

1/864*d^2*exp(-6*f*x-6*e)/a^3/f^3-3/256*d^2*exp(-4*f*x-4*e)/a^3/f^3+3/32*d^2*exp(-2*f*x-2*e)/a^3/f^3+1/144*d*e
xp(-6*f*x-6*e)*(d*x+c)/a^3/f^2-3/64*d*exp(-4*f*x-4*e)*(d*x+c)/a^3/f^2+3/16*d*exp(-2*f*x-2*e)*(d*x+c)/a^3/f^2+1
/48*exp(-6*f*x-6*e)*(d*x+c)^2/a^3/f-3/32*exp(-4*f*x-4*e)*(d*x+c)^2/a^3/f+3/16*exp(-2*f*x-2*e)*(d*x+c)^2/a^3/f+
1/24*(d*x+c)^3/a^3/d

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Rubi [A]  time = 0.26, antiderivative size = 246, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {3729, 2176, 2194} \[ \frac {d (c+d x) e^{-6 e-6 f x}}{144 a^3 f^2}-\frac {3 d (c+d x) e^{-4 e-4 f x}}{64 a^3 f^2}+\frac {3 d (c+d x) e^{-2 e-2 f x}}{16 a^3 f^2}+\frac {(c+d x)^2 e^{-6 e-6 f x}}{48 a^3 f}-\frac {3 (c+d x)^2 e^{-4 e-4 f x}}{32 a^3 f}+\frac {3 (c+d x)^2 e^{-2 e-2 f x}}{16 a^3 f}+\frac {(c+d x)^3}{24 a^3 d}+\frac {d^2 e^{-6 e-6 f x}}{864 a^3 f^3}-\frac {3 d^2 e^{-4 e-4 f x}}{256 a^3 f^3}+\frac {3 d^2 e^{-2 e-2 f x}}{32 a^3 f^3} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^2/(a + a*Coth[e + f*x])^3,x]

[Out]

(d^2*E^(-6*e - 6*f*x))/(864*a^3*f^3) - (3*d^2*E^(-4*e - 4*f*x))/(256*a^3*f^3) + (3*d^2*E^(-2*e - 2*f*x))/(32*a
^3*f^3) + (d*E^(-6*e - 6*f*x)*(c + d*x))/(144*a^3*f^2) - (3*d*E^(-4*e - 4*f*x)*(c + d*x))/(64*a^3*f^2) + (3*d*
E^(-2*e - 2*f*x)*(c + d*x))/(16*a^3*f^2) + (E^(-6*e - 6*f*x)*(c + d*x)^2)/(48*a^3*f) - (3*E^(-4*e - 4*f*x)*(c
+ d*x)^2)/(32*a^3*f) + (3*E^(-2*e - 2*f*x)*(c + d*x)^2)/(16*a^3*f) + (c + d*x)^3/(24*a^3*d)

Rule 2176

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^m
*(b*F^(g*(e + f*x)))^n)/(f*g*n*Log[F]), x] - Dist[(d*m)/(f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*(b*F^(g*(e + f*x
)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && GtQ[m, 0] && IntegerQ[2*m] &&  !$UseGamma === True

Rule 2194

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rule 3729

Int[((c_.) + (d_.)*(x_))^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Int[ExpandIntegrand[(c
+ d*x)^m, (1/(2*a) + E^((2*a*(e + f*x))/b)/(2*a))^(-n), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[a^2
+ b^2, 0] && ILtQ[n, 0]

Rubi steps

\begin {align*} \int \frac {(c+d x)^2}{(a+a \coth (e+f x))^3} \, dx &=\int \left (\frac {(c+d x)^2}{8 a^3}-\frac {e^{-6 e-6 f x} (c+d x)^2}{8 a^3}+\frac {3 e^{-4 e-4 f x} (c+d x)^2}{8 a^3}-\frac {3 e^{-2 e-2 f x} (c+d x)^2}{8 a^3}\right ) \, dx\\ &=\frac {(c+d x)^3}{24 a^3 d}-\frac {\int e^{-6 e-6 f x} (c+d x)^2 \, dx}{8 a^3}+\frac {3 \int e^{-4 e-4 f x} (c+d x)^2 \, dx}{8 a^3}-\frac {3 \int e^{-2 e-2 f x} (c+d x)^2 \, dx}{8 a^3}\\ &=\frac {e^{-6 e-6 f x} (c+d x)^2}{48 a^3 f}-\frac {3 e^{-4 e-4 f x} (c+d x)^2}{32 a^3 f}+\frac {3 e^{-2 e-2 f x} (c+d x)^2}{16 a^3 f}+\frac {(c+d x)^3}{24 a^3 d}-\frac {d \int e^{-6 e-6 f x} (c+d x) \, dx}{24 a^3 f}+\frac {(3 d) \int e^{-4 e-4 f x} (c+d x) \, dx}{16 a^3 f}-\frac {(3 d) \int e^{-2 e-2 f x} (c+d x) \, dx}{8 a^3 f}\\ &=\frac {d e^{-6 e-6 f x} (c+d x)}{144 a^3 f^2}-\frac {3 d e^{-4 e-4 f x} (c+d x)}{64 a^3 f^2}+\frac {3 d e^{-2 e-2 f x} (c+d x)}{16 a^3 f^2}+\frac {e^{-6 e-6 f x} (c+d x)^2}{48 a^3 f}-\frac {3 e^{-4 e-4 f x} (c+d x)^2}{32 a^3 f}+\frac {3 e^{-2 e-2 f x} (c+d x)^2}{16 a^3 f}+\frac {(c+d x)^3}{24 a^3 d}-\frac {d^2 \int e^{-6 e-6 f x} \, dx}{144 a^3 f^2}+\frac {\left (3 d^2\right ) \int e^{-4 e-4 f x} \, dx}{64 a^3 f^2}-\frac {\left (3 d^2\right ) \int e^{-2 e-2 f x} \, dx}{16 a^3 f^2}\\ &=\frac {d^2 e^{-6 e-6 f x}}{864 a^3 f^3}-\frac {3 d^2 e^{-4 e-4 f x}}{256 a^3 f^3}+\frac {3 d^2 e^{-2 e-2 f x}}{32 a^3 f^3}+\frac {d e^{-6 e-6 f x} (c+d x)}{144 a^3 f^2}-\frac {3 d e^{-4 e-4 f x} (c+d x)}{64 a^3 f^2}+\frac {3 d e^{-2 e-2 f x} (c+d x)}{16 a^3 f^2}+\frac {e^{-6 e-6 f x} (c+d x)^2}{48 a^3 f}-\frac {3 e^{-4 e-4 f x} (c+d x)^2}{32 a^3 f}+\frac {3 e^{-2 e-2 f x} (c+d x)^2}{16 a^3 f}+\frac {(c+d x)^3}{24 a^3 d}\\ \end {align*}

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Mathematica [A]  time = 1.50, size = 371, normalized size = 1.51 \[ \frac {\text {csch}^3(e+f x) \left (81 \left (8 c^2 f^2+4 c d f (4 f x+3)+d^2 \left (8 f^2 x^2+12 f x+7\right )\right ) \cosh (e+f x)+8 \left (18 c^2 f^2 (6 f x+1)+6 c d f \left (18 f^2 x^2+6 f x+1\right )+d^2 \left (36 f^3 x^3+18 f^2 x^2+6 f x+1\right )\right ) \cosh (3 (e+f x))+864 c^2 f^3 x \sinh (3 (e+f x))+1944 c^2 f^2 \sinh (e+f x)-144 c^2 f^2 \sinh (3 (e+f x))+864 c d f^3 x^2 \sinh (3 (e+f x))+3888 c d f^2 x \sinh (e+f x)-288 c d f^2 x \sinh (3 (e+f x))+1620 c d f \sinh (e+f x)-48 c d f \sinh (3 (e+f x))+288 d^2 f^3 x^3 \sinh (3 (e+f x))+1944 d^2 f^2 x^2 \sinh (e+f x)-144 d^2 f^2 x^2 \sinh (3 (e+f x))+1620 d^2 f x \sinh (e+f x)-48 d^2 f x \sinh (3 (e+f x))+729 d^2 \sinh (e+f x)-8 d^2 \sinh (3 (e+f x))\right )}{6912 a^3 f^3 (\coth (e+f x)+1)^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^2/(a + a*Coth[e + f*x])^3,x]

[Out]

(Csch[e + f*x]^3*(81*(8*c^2*f^2 + 4*c*d*f*(3 + 4*f*x) + d^2*(7 + 12*f*x + 8*f^2*x^2))*Cosh[e + f*x] + 8*(18*c^
2*f^2*(1 + 6*f*x) + 6*c*d*f*(1 + 6*f*x + 18*f^2*x^2) + d^2*(1 + 6*f*x + 18*f^2*x^2 + 36*f^3*x^3))*Cosh[3*(e +
f*x)] + 729*d^2*Sinh[e + f*x] + 1620*c*d*f*Sinh[e + f*x] + 1944*c^2*f^2*Sinh[e + f*x] + 1620*d^2*f*x*Sinh[e +
f*x] + 3888*c*d*f^2*x*Sinh[e + f*x] + 1944*d^2*f^2*x^2*Sinh[e + f*x] - 8*d^2*Sinh[3*(e + f*x)] - 48*c*d*f*Sinh
[3*(e + f*x)] - 144*c^2*f^2*Sinh[3*(e + f*x)] - 48*d^2*f*x*Sinh[3*(e + f*x)] - 288*c*d*f^2*x*Sinh[3*(e + f*x)]
+ 864*c^2*f^3*x*Sinh[3*(e + f*x)] - 144*d^2*f^2*x^2*Sinh[3*(e + f*x)] + 864*c*d*f^3*x^2*Sinh[3*(e + f*x)] + 2
88*d^2*f^3*x^3*Sinh[3*(e + f*x)]))/(6912*a^3*f^3*(1 + Coth[e + f*x])^3)

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fricas [B]  time = 0.39, size = 532, normalized size = 2.16 \[ \frac {8 \, {\left (36 \, d^{2} f^{3} x^{3} + 18 \, c^{2} f^{2} + 6 \, c d f + 18 \, {\left (6 \, c d f^{3} + d^{2} f^{2}\right )} x^{2} + d^{2} + 6 \, {\left (18 \, c^{2} f^{3} + 6 \, c d f^{2} + d^{2} f\right )} x\right )} \cosh \left (f x + e\right )^{3} + 24 \, {\left (36 \, d^{2} f^{3} x^{3} + 18 \, c^{2} f^{2} + 6 \, c d f + 18 \, {\left (6 \, c d f^{3} + d^{2} f^{2}\right )} x^{2} + d^{2} + 6 \, {\left (18 \, c^{2} f^{3} + 6 \, c d f^{2} + d^{2} f\right )} x\right )} \cosh \left (f x + e\right ) \sinh \left (f x + e\right )^{2} + 8 \, {\left (36 \, d^{2} f^{3} x^{3} - 18 \, c^{2} f^{2} - 6 \, c d f + 18 \, {\left (6 \, c d f^{3} - d^{2} f^{2}\right )} x^{2} - d^{2} + 6 \, {\left (18 \, c^{2} f^{3} - 6 \, c d f^{2} - d^{2} f\right )} x\right )} \sinh \left (f x + e\right )^{3} + 81 \, {\left (8 \, d^{2} f^{2} x^{2} + 8 \, c^{2} f^{2} + 12 \, c d f + 7 \, d^{2} + 4 \, {\left (4 \, c d f^{2} + 3 \, d^{2} f\right )} x\right )} \cosh \left (f x + e\right ) + 3 \, {\left (648 \, d^{2} f^{2} x^{2} + 648 \, c^{2} f^{2} + 540 \, c d f + 8 \, {\left (36 \, d^{2} f^{3} x^{3} - 18 \, c^{2} f^{2} - 6 \, c d f + 18 \, {\left (6 \, c d f^{3} - d^{2} f^{2}\right )} x^{2} - d^{2} + 6 \, {\left (18 \, c^{2} f^{3} - 6 \, c d f^{2} - d^{2} f\right )} x\right )} \cosh \left (f x + e\right )^{2} + 243 \, d^{2} + 108 \, {\left (12 \, c d f^{2} + 5 \, d^{2} f\right )} x\right )} \sinh \left (f x + e\right )}{6912 \, {\left (a^{3} f^{3} \cosh \left (f x + e\right )^{3} + 3 \, a^{3} f^{3} \cosh \left (f x + e\right )^{2} \sinh \left (f x + e\right ) + 3 \, a^{3} f^{3} \cosh \left (f x + e\right ) \sinh \left (f x + e\right )^{2} + a^{3} f^{3} \sinh \left (f x + e\right )^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^2/(a+a*coth(f*x+e))^3,x, algorithm="fricas")

[Out]

1/6912*(8*(36*d^2*f^3*x^3 + 18*c^2*f^2 + 6*c*d*f + 18*(6*c*d*f^3 + d^2*f^2)*x^2 + d^2 + 6*(18*c^2*f^3 + 6*c*d*
f^2 + d^2*f)*x)*cosh(f*x + e)^3 + 24*(36*d^2*f^3*x^3 + 18*c^2*f^2 + 6*c*d*f + 18*(6*c*d*f^3 + d^2*f^2)*x^2 + d
^2 + 6*(18*c^2*f^3 + 6*c*d*f^2 + d^2*f)*x)*cosh(f*x + e)*sinh(f*x + e)^2 + 8*(36*d^2*f^3*x^3 - 18*c^2*f^2 - 6*
c*d*f + 18*(6*c*d*f^3 - d^2*f^2)*x^2 - d^2 + 6*(18*c^2*f^3 - 6*c*d*f^2 - d^2*f)*x)*sinh(f*x + e)^3 + 81*(8*d^2
*f^2*x^2 + 8*c^2*f^2 + 12*c*d*f + 7*d^2 + 4*(4*c*d*f^2 + 3*d^2*f)*x)*cosh(f*x + e) + 3*(648*d^2*f^2*x^2 + 648*
c^2*f^2 + 540*c*d*f + 8*(36*d^2*f^3*x^3 - 18*c^2*f^2 - 6*c*d*f + 18*(6*c*d*f^3 - d^2*f^2)*x^2 - d^2 + 6*(18*c^
2*f^3 - 6*c*d*f^2 - d^2*f)*x)*cosh(f*x + e)^2 + 243*d^2 + 108*(12*c*d*f^2 + 5*d^2*f)*x)*sinh(f*x + e))/(a^3*f^
3*cosh(f*x + e)^3 + 3*a^3*f^3*cosh(f*x + e)^2*sinh(f*x + e) + 3*a^3*f^3*cosh(f*x + e)*sinh(f*x + e)^2 + a^3*f^
3*sinh(f*x + e)^3)

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giac [A]  time = 0.16, size = 331, normalized size = 1.35 \[ \frac {{\left (288 \, d^{2} f^{3} x^{3} e^{\left (6 \, f x + 6 \, e\right )} + 864 \, c d f^{3} x^{2} e^{\left (6 \, f x + 6 \, e\right )} + 864 \, c^{2} f^{3} x e^{\left (6 \, f x + 6 \, e\right )} + 1296 \, d^{2} f^{2} x^{2} e^{\left (4 \, f x + 4 \, e\right )} - 648 \, d^{2} f^{2} x^{2} e^{\left (2 \, f x + 2 \, e\right )} + 144 \, d^{2} f^{2} x^{2} + 2592 \, c d f^{2} x e^{\left (4 \, f x + 4 \, e\right )} - 1296 \, c d f^{2} x e^{\left (2 \, f x + 2 \, e\right )} + 288 \, c d f^{2} x + 1296 \, c^{2} f^{2} e^{\left (4 \, f x + 4 \, e\right )} + 1296 \, d^{2} f x e^{\left (4 \, f x + 4 \, e\right )} - 648 \, c^{2} f^{2} e^{\left (2 \, f x + 2 \, e\right )} - 324 \, d^{2} f x e^{\left (2 \, f x + 2 \, e\right )} + 144 \, c^{2} f^{2} + 48 \, d^{2} f x + 1296 \, c d f e^{\left (4 \, f x + 4 \, e\right )} - 324 \, c d f e^{\left (2 \, f x + 2 \, e\right )} + 48 \, c d f + 648 \, d^{2} e^{\left (4 \, f x + 4 \, e\right )} - 81 \, d^{2} e^{\left (2 \, f x + 2 \, e\right )} + 8 \, d^{2}\right )} e^{\left (-6 \, f x - 6 \, e\right )}}{6912 \, a^{3} f^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^2/(a+a*coth(f*x+e))^3,x, algorithm="giac")

[Out]

1/6912*(288*d^2*f^3*x^3*e^(6*f*x + 6*e) + 864*c*d*f^3*x^2*e^(6*f*x + 6*e) + 864*c^2*f^3*x*e^(6*f*x + 6*e) + 12
96*d^2*f^2*x^2*e^(4*f*x + 4*e) - 648*d^2*f^2*x^2*e^(2*f*x + 2*e) + 144*d^2*f^2*x^2 + 2592*c*d*f^2*x*e^(4*f*x +
4*e) - 1296*c*d*f^2*x*e^(2*f*x + 2*e) + 288*c*d*f^2*x + 1296*c^2*f^2*e^(4*f*x + 4*e) + 1296*d^2*f*x*e^(4*f*x
+ 4*e) - 648*c^2*f^2*e^(2*f*x + 2*e) - 324*d^2*f*x*e^(2*f*x + 2*e) + 144*c^2*f^2 + 48*d^2*f*x + 1296*c*d*f*e^(
4*f*x + 4*e) - 324*c*d*f*e^(2*f*x + 2*e) + 48*c*d*f + 648*d^2*e^(4*f*x + 4*e) - 81*d^2*e^(2*f*x + 2*e) + 8*d^2
)*e^(-6*f*x - 6*e)/(a^3*f^3)

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maple [B]  time = 0.62, size = 1767, normalized size = 7.18 \[ \text {Expression too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^2/(a+a*coth(f*x+e))^3,x)

[Out]

1/a^3/f*(-4/f^2*d^2*(1/6*(f*x+e)^2*sinh(f*x+e)^3*cosh(f*x+e)^3-1/8*(f*x+e)^2*sinh(f*x+e)*cosh(f*x+e)^3+1/16*(f
*x+e)^2*cosh(f*x+e)*sinh(f*x+e)+1/48*(f*x+e)^3-1/18*(f*x+e)*sinh(f*x+e)^2*cosh(f*x+e)^4+13/144*(f*x+e)*cosh(f*
x+e)^4+1/108*cosh(f*x+e)^5*sinh(f*x+e)-43/1728*cosh(f*x+e)^3*sinh(f*x+e)-7/1152*cosh(f*x+e)*sinh(f*x+e)-7/1152
*f*x-7/1152*e-1/16*(f*x+e)*cosh(f*x+e)^2)+4/f^2*d^2*(1/6*(f*x+e)^2*sinh(f*x+e)^2*cosh(f*x+e)^4-1/12*(f*x+e)^2*
cosh(f*x+e)^4-1/18*(f*x+e)*sinh(f*x+e)*cosh(f*x+e)^5+1/18*(f*x+e)*sinh(f*x+e)*cosh(f*x+e)^3+1/12*(f*x+e)*cosh(
f*x+e)*sinh(f*x+e)+1/24*(f*x+e)^2+1/108*cosh(f*x+e)^6-1/72*cosh(f*x+e)^4-1/24*cosh(f*x+e)^2)+1/f^2*d^2*(1/4*(f
*x+e)^2*cosh(f*x+e)*sinh(f*x+e)^3-3/8*(f*x+e)^2*cosh(f*x+e)*sinh(f*x+e)+1/8*(f*x+e)^3-1/8*(f*x+e)*sinh(f*x+e)^
4+1/32*sinh(f*x+e)^3*cosh(f*x+e)-15/64*cosh(f*x+e)*sinh(f*x+e)-9/64*f*x-9/64*e+3/8*(f*x+e)*cosh(f*x+e)^2)-3/f^
2*d^2*(1/4*(f*x+e)^2*sinh(f*x+e)^4-1/8*(f*x+e)*cosh(f*x+e)*sinh(f*x+e)^3+3/16*(f*x+e)*cosh(f*x+e)*sinh(f*x+e)-
3/32*(f*x+e)^2+1/32*sinh(f*x+e)^4-3/32*cosh(f*x+e)^2)+8/f^2*d^2*e*(1/6*(f*x+e)*sinh(f*x+e)^3*cosh(f*x+e)^3-1/8
*(f*x+e)*sinh(f*x+e)*cosh(f*x+e)^3+1/16*(f*x+e)*cosh(f*x+e)*sinh(f*x+e)+1/32*(f*x+e)^2-1/36*sinh(f*x+e)^2*cosh
(f*x+e)^4+13/288*cosh(f*x+e)^4-1/32*cosh(f*x+e)^2)-8/f^2*d^2*e*(1/6*(f*x+e)*sinh(f*x+e)^2*cosh(f*x+e)^4-1/12*(
f*x+e)*cosh(f*x+e)^4-1/36*cosh(f*x+e)^5*sinh(f*x+e)+1/36*cosh(f*x+e)^3*sinh(f*x+e)+1/24*cosh(f*x+e)*sinh(f*x+e
)+1/24*f*x+1/24*e)-2/f^2*d^2*e*(1/4*(f*x+e)*cosh(f*x+e)*sinh(f*x+e)^3-3/8*(f*x+e)*cosh(f*x+e)*sinh(f*x+e)+3/16
*(f*x+e)^2-1/16*sinh(f*x+e)^4+3/16*cosh(f*x+e)^2)+6/f^2*d^2*e*(1/4*(f*x+e)*sinh(f*x+e)^4-1/16*sinh(f*x+e)^3*co
sh(f*x+e)+3/32*cosh(f*x+e)*sinh(f*x+e)-3/32*f*x-3/32*e)-4/f^2*d^2*e^2*(1/6*sinh(f*x+e)^3*cosh(f*x+e)^3-1/8*cos
h(f*x+e)^3*sinh(f*x+e)+1/16*cosh(f*x+e)*sinh(f*x+e)+1/16*f*x+1/16*e)+4/f^2*d^2*e^2*(1/6*sinh(f*x+e)^2*cosh(f*x
+e)^4-1/12*cosh(f*x+e)^4)+1/f^2*d^2*e^2*((1/4*sinh(f*x+e)^3-3/8*sinh(f*x+e))*cosh(f*x+e)+3/8*f*x+3/8*e)-3/4*d^
2*e^2/f^2*sinh(f*x+e)^4-8/f*c*d*(1/6*(f*x+e)*sinh(f*x+e)^3*cosh(f*x+e)^3-1/8*(f*x+e)*sinh(f*x+e)*cosh(f*x+e)^3
+1/16*(f*x+e)*cosh(f*x+e)*sinh(f*x+e)+1/32*(f*x+e)^2-1/36*sinh(f*x+e)^2*cosh(f*x+e)^4+13/288*cosh(f*x+e)^4-1/3
2*cosh(f*x+e)^2)+8/f*c*d*(1/6*(f*x+e)*sinh(f*x+e)^2*cosh(f*x+e)^4-1/12*(f*x+e)*cosh(f*x+e)^4-1/36*cosh(f*x+e)^
5*sinh(f*x+e)+1/36*cosh(f*x+e)^3*sinh(f*x+e)+1/24*cosh(f*x+e)*sinh(f*x+e)+1/24*f*x+1/24*e)+2/f*c*d*(1/4*(f*x+e
)*cosh(f*x+e)*sinh(f*x+e)^3-3/8*(f*x+e)*cosh(f*x+e)*sinh(f*x+e)+3/16*(f*x+e)^2-1/16*sinh(f*x+e)^4+3/16*cosh(f*
x+e)^2)-6/f*d*c*(1/4*(f*x+e)*sinh(f*x+e)^4-1/16*sinh(f*x+e)^3*cosh(f*x+e)+3/32*cosh(f*x+e)*sinh(f*x+e)-3/32*f*
x-3/32*e)+8/f*c*d*e*(1/6*sinh(f*x+e)^3*cosh(f*x+e)^3-1/8*cosh(f*x+e)^3*sinh(f*x+e)+1/16*cosh(f*x+e)*sinh(f*x+e
)+1/16*f*x+1/16*e)-8/f*c*d*e*(1/6*sinh(f*x+e)^2*cosh(f*x+e)^4-1/12*cosh(f*x+e)^4)-2/f*c*d*e*((1/4*sinh(f*x+e)^
3-3/8*sinh(f*x+e))*cosh(f*x+e)+3/8*f*x+3/8*e)+3/2*d*e/f*c*sinh(f*x+e)^4-4*c^2*(1/6*sinh(f*x+e)^3*cosh(f*x+e)^3
-1/8*cosh(f*x+e)^3*sinh(f*x+e)+1/16*cosh(f*x+e)*sinh(f*x+e)+1/16*f*x+1/16*e)+4*c^2*(1/6*sinh(f*x+e)^2*cosh(f*x
+e)^4-1/12*cosh(f*x+e)^4)+c^2*((1/4*sinh(f*x+e)^3-3/8*sinh(f*x+e))*cosh(f*x+e)+3/8*f*x+3/8*e)-3/4*c^2*sinh(f*x
+e)^4)

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maxima [A]  time = 2.08, size = 254, normalized size = 1.03 \[ \frac {1}{96} \, c^{2} {\left (\frac {12 \, {\left (f x + e\right )}}{a^{3} f} + \frac {18 \, e^{\left (-2 \, f x - 2 \, e\right )} - 9 \, e^{\left (-4 \, f x - 4 \, e\right )} + 2 \, e^{\left (-6 \, f x - 6 \, e\right )}}{a^{3} f}\right )} + \frac {{\left (72 \, f^{2} x^{2} e^{\left (6 \, e\right )} + 108 \, {\left (2 \, f x e^{\left (4 \, e\right )} + e^{\left (4 \, e\right )}\right )} e^{\left (-2 \, f x\right )} - 27 \, {\left (4 \, f x e^{\left (2 \, e\right )} + e^{\left (2 \, e\right )}\right )} e^{\left (-4 \, f x\right )} + 4 \, {\left (6 \, f x + 1\right )} e^{\left (-6 \, f x\right )}\right )} c d e^{\left (-6 \, e\right )}}{576 \, a^{3} f^{2}} + \frac {{\left (288 \, f^{3} x^{3} e^{\left (6 \, e\right )} + 648 \, {\left (2 \, f^{2} x^{2} e^{\left (4 \, e\right )} + 2 \, f x e^{\left (4 \, e\right )} + e^{\left (4 \, e\right )}\right )} e^{\left (-2 \, f x\right )} - 81 \, {\left (8 \, f^{2} x^{2} e^{\left (2 \, e\right )} + 4 \, f x e^{\left (2 \, e\right )} + e^{\left (2 \, e\right )}\right )} e^{\left (-4 \, f x\right )} + 8 \, {\left (18 \, f^{2} x^{2} + 6 \, f x + 1\right )} e^{\left (-6 \, f x\right )}\right )} d^{2} e^{\left (-6 \, e\right )}}{6912 \, a^{3} f^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^2/(a+a*coth(f*x+e))^3,x, algorithm="maxima")

[Out]

1/96*c^2*(12*(f*x + e)/(a^3*f) + (18*e^(-2*f*x - 2*e) - 9*e^(-4*f*x - 4*e) + 2*e^(-6*f*x - 6*e))/(a^3*f)) + 1/
576*(72*f^2*x^2*e^(6*e) + 108*(2*f*x*e^(4*e) + e^(4*e))*e^(-2*f*x) - 27*(4*f*x*e^(2*e) + e^(2*e))*e^(-4*f*x) +
4*(6*f*x + 1)*e^(-6*f*x))*c*d*e^(-6*e)/(a^3*f^2) + 1/6912*(288*f^3*x^3*e^(6*e) + 648*(2*f^2*x^2*e^(4*e) + 2*f
*x*e^(4*e) + e^(4*e))*e^(-2*f*x) - 81*(8*f^2*x^2*e^(2*e) + 4*f*x*e^(2*e) + e^(2*e))*e^(-4*f*x) + 8*(18*f^2*x^2
+ 6*f*x + 1)*e^(-6*f*x))*d^2*e^(-6*e)/(a^3*f^3)

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mupad [B]  time = 1.34, size = 234, normalized size = 0.95 \[ {\mathrm {e}}^{-6\,e-6\,f\,x}\,\left (\frac {18\,c^2\,f^2+6\,c\,d\,f+d^2}{864\,a^3\,f^3}+\frac {d^2\,x^2}{48\,a^3\,f}+\frac {d\,x\,\left (d+6\,c\,f\right )}{144\,a^3\,f^2}\right )+{\mathrm {e}}^{-2\,e-2\,f\,x}\,\left (\frac {6\,c^2\,f^2+6\,c\,d\,f+3\,d^2}{32\,a^3\,f^3}+\frac {3\,d^2\,x^2}{16\,a^3\,f}+\frac {3\,d\,x\,\left (d+2\,c\,f\right )}{16\,a^3\,f^2}\right )-{\mathrm {e}}^{-4\,e-4\,f\,x}\,\left (\frac {24\,c^2\,f^2+12\,c\,d\,f+3\,d^2}{256\,a^3\,f^3}+\frac {3\,d^2\,x^2}{32\,a^3\,f}+\frac {3\,d\,x\,\left (d+4\,c\,f\right )}{64\,a^3\,f^2}\right )+\frac {c^2\,x}{8\,a^3}+\frac {d^2\,x^3}{24\,a^3}+\frac {c\,d\,x^2}{8\,a^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x)^2/(a + a*coth(e + f*x))^3,x)

[Out]

exp(- 6*e - 6*f*x)*((d^2 + 18*c^2*f^2 + 6*c*d*f)/(864*a^3*f^3) + (d^2*x^2)/(48*a^3*f) + (d*x*(d + 6*c*f))/(144
*a^3*f^2)) + exp(- 2*e - 2*f*x)*((3*d^2 + 6*c^2*f^2 + 6*c*d*f)/(32*a^3*f^3) + (3*d^2*x^2)/(16*a^3*f) + (3*d*x*
(d + 2*c*f))/(16*a^3*f^2)) - exp(- 4*e - 4*f*x)*((3*d^2 + 24*c^2*f^2 + 12*c*d*f)/(256*a^3*f^3) + (3*d^2*x^2)/(
32*a^3*f) + (3*d*x*(d + 4*c*f))/(64*a^3*f^2)) + (c^2*x)/(8*a^3) + (d^2*x^3)/(24*a^3) + (c*d*x^2)/(8*a^3)

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sympy [A]  time = 3.14, size = 2443, normalized size = 9.93 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)**2/(a+a*coth(f*x+e))**3,x)

[Out]

Piecewise((216*c**2*f**3*x*tanh(e + f*x)**3/(1728*a**3*f**3*tanh(e + f*x)**3 + 5184*a**3*f**3*tanh(e + f*x)**2
+ 5184*a**3*f**3*tanh(e + f*x) + 1728*a**3*f**3) + 648*c**2*f**3*x*tanh(e + f*x)**2/(1728*a**3*f**3*tanh(e +
f*x)**3 + 5184*a**3*f**3*tanh(e + f*x)**2 + 5184*a**3*f**3*tanh(e + f*x) + 1728*a**3*f**3) + 648*c**2*f**3*x*t
anh(e + f*x)/(1728*a**3*f**3*tanh(e + f*x)**3 + 5184*a**3*f**3*tanh(e + f*x)**2 + 5184*a**3*f**3*tanh(e + f*x)
+ 1728*a**3*f**3) + 216*c**2*f**3*x/(1728*a**3*f**3*tanh(e + f*x)**3 + 5184*a**3*f**3*tanh(e + f*x)**2 + 5184
*a**3*f**3*tanh(e + f*x) + 1728*a**3*f**3) + 1512*c**2*f**2*tanh(e + f*x)**2/(1728*a**3*f**3*tanh(e + f*x)**3
+ 5184*a**3*f**3*tanh(e + f*x)**2 + 5184*a**3*f**3*tanh(e + f*x) + 1728*a**3*f**3) + 1944*c**2*f**2*tanh(e + f
*x)/(1728*a**3*f**3*tanh(e + f*x)**3 + 5184*a**3*f**3*tanh(e + f*x)**2 + 5184*a**3*f**3*tanh(e + f*x) + 1728*a
**3*f**3) + 720*c**2*f**2/(1728*a**3*f**3*tanh(e + f*x)**3 + 5184*a**3*f**3*tanh(e + f*x)**2 + 5184*a**3*f**3*
tanh(e + f*x) + 1728*a**3*f**3) + 216*c*d*f**3*x**2*tanh(e + f*x)**3/(1728*a**3*f**3*tanh(e + f*x)**3 + 5184*a
**3*f**3*tanh(e + f*x)**2 + 5184*a**3*f**3*tanh(e + f*x) + 1728*a**3*f**3) + 648*c*d*f**3*x**2*tanh(e + f*x)**
2/(1728*a**3*f**3*tanh(e + f*x)**3 + 5184*a**3*f**3*tanh(e + f*x)**2 + 5184*a**3*f**3*tanh(e + f*x) + 1728*a**
3*f**3) + 648*c*d*f**3*x**2*tanh(e + f*x)/(1728*a**3*f**3*tanh(e + f*x)**3 + 5184*a**3*f**3*tanh(e + f*x)**2 +
5184*a**3*f**3*tanh(e + f*x) + 1728*a**3*f**3) + 216*c*d*f**3*x**2/(1728*a**3*f**3*tanh(e + f*x)**3 + 5184*a*
*3*f**3*tanh(e + f*x)**2 + 5184*a**3*f**3*tanh(e + f*x) + 1728*a**3*f**3) - 1044*c*d*f**2*x*tanh(e + f*x)**3/(
1728*a**3*f**3*tanh(e + f*x)**3 + 5184*a**3*f**3*tanh(e + f*x)**2 + 5184*a**3*f**3*tanh(e + f*x) + 1728*a**3*f
**3) - 108*c*d*f**2*x*tanh(e + f*x)**2/(1728*a**3*f**3*tanh(e + f*x)**3 + 5184*a**3*f**3*tanh(e + f*x)**2 + 51
84*a**3*f**3*tanh(e + f*x) + 1728*a**3*f**3) + 756*c*d*f**2*x*tanh(e + f*x)/(1728*a**3*f**3*tanh(e + f*x)**3 +
5184*a**3*f**3*tanh(e + f*x)**2 + 5184*a**3*f**3*tanh(e + f*x) + 1728*a**3*f**3) + 396*c*d*f**2*x/(1728*a**3*
f**3*tanh(e + f*x)**3 + 5184*a**3*f**3*tanh(e + f*x)**2 + 5184*a**3*f**3*tanh(e + f*x) + 1728*a**3*f**3) + 104
4*c*d*f*tanh(e + f*x)**2/(1728*a**3*f**3*tanh(e + f*x)**3 + 5184*a**3*f**3*tanh(e + f*x)**2 + 5184*a**3*f**3*t
anh(e + f*x) + 1728*a**3*f**3) + 1620*c*d*f*tanh(e + f*x)/(1728*a**3*f**3*tanh(e + f*x)**3 + 5184*a**3*f**3*ta
nh(e + f*x)**2 + 5184*a**3*f**3*tanh(e + f*x) + 1728*a**3*f**3) + 672*c*d*f/(1728*a**3*f**3*tanh(e + f*x)**3 +
5184*a**3*f**3*tanh(e + f*x)**2 + 5184*a**3*f**3*tanh(e + f*x) + 1728*a**3*f**3) + 72*d**2*f**3*x**3*tanh(e +
f*x)**3/(1728*a**3*f**3*tanh(e + f*x)**3 + 5184*a**3*f**3*tanh(e + f*x)**2 + 5184*a**3*f**3*tanh(e + f*x) + 1
728*a**3*f**3) + 216*d**2*f**3*x**3*tanh(e + f*x)**2/(1728*a**3*f**3*tanh(e + f*x)**3 + 5184*a**3*f**3*tanh(e
+ f*x)**2 + 5184*a**3*f**3*tanh(e + f*x) + 1728*a**3*f**3) + 216*d**2*f**3*x**3*tanh(e + f*x)/(1728*a**3*f**3*
tanh(e + f*x)**3 + 5184*a**3*f**3*tanh(e + f*x)**2 + 5184*a**3*f**3*tanh(e + f*x) + 1728*a**3*f**3) + 72*d**2*
f**3*x**3/(1728*a**3*f**3*tanh(e + f*x)**3 + 5184*a**3*f**3*tanh(e + f*x)**2 + 5184*a**3*f**3*tanh(e + f*x) +
1728*a**3*f**3) - 522*d**2*f**2*x**2*tanh(e + f*x)**3/(1728*a**3*f**3*tanh(e + f*x)**3 + 5184*a**3*f**3*tanh(e
+ f*x)**2 + 5184*a**3*f**3*tanh(e + f*x) + 1728*a**3*f**3) - 54*d**2*f**2*x**2*tanh(e + f*x)**2/(1728*a**3*f*
*3*tanh(e + f*x)**3 + 5184*a**3*f**3*tanh(e + f*x)**2 + 5184*a**3*f**3*tanh(e + f*x) + 1728*a**3*f**3) + 378*d
**2*f**2*x**2*tanh(e + f*x)/(1728*a**3*f**3*tanh(e + f*x)**3 + 5184*a**3*f**3*tanh(e + f*x)**2 + 5184*a**3*f**
3*tanh(e + f*x) + 1728*a**3*f**3) + 198*d**2*f**2*x**2/(1728*a**3*f**3*tanh(e + f*x)**3 + 5184*a**3*f**3*tanh(
e + f*x)**2 + 5184*a**3*f**3*tanh(e + f*x) + 1728*a**3*f**3) - 417*d**2*f*x*tanh(e + f*x)**3/(1728*a**3*f**3*t
anh(e + f*x)**3 + 5184*a**3*f**3*tanh(e + f*x)**2 + 5184*a**3*f**3*tanh(e + f*x) + 1728*a**3*f**3) - 207*d**2*
f*x*tanh(e + f*x)**2/(1728*a**3*f**3*tanh(e + f*x)**3 + 5184*a**3*f**3*tanh(e + f*x)**2 + 5184*a**3*f**3*tanh(
e + f*x) + 1728*a**3*f**3) + 369*d**2*f*x*tanh(e + f*x)/(1728*a**3*f**3*tanh(e + f*x)**3 + 5184*a**3*f**3*tanh
(e + f*x)**2 + 5184*a**3*f**3*tanh(e + f*x) + 1728*a**3*f**3) + 255*d**2*f*x/(1728*a**3*f**3*tanh(e + f*x)**3
+ 5184*a**3*f**3*tanh(e + f*x)**2 + 5184*a**3*f**3*tanh(e + f*x) + 1728*a**3*f**3) + 417*d**2*tanh(e + f*x)**2
/(1728*a**3*f**3*tanh(e + f*x)**3 + 5184*a**3*f**3*tanh(e + f*x)**2 + 5184*a**3*f**3*tanh(e + f*x) + 1728*a**3
*f**3) + 729*d**2*tanh(e + f*x)/(1728*a**3*f**3*tanh(e + f*x)**3 + 5184*a**3*f**3*tanh(e + f*x)**2 + 5184*a**3
*f**3*tanh(e + f*x) + 1728*a**3*f**3) + 328*d**2/(1728*a**3*f**3*tanh(e + f*x)**3 + 5184*a**3*f**3*tanh(e + f*
x)**2 + 5184*a**3*f**3*tanh(e + f*x) + 1728*a**3*f**3), Ne(f, 0)), ((c**2*x + c*d*x**2 + d**2*x**3/3)/(a*coth(
e) + a)**3, True))

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