3.1.22 \(\int \frac {(c+d x)^3}{(a+a \coth (e+f x))^2} \, dx\) [22]

Optimal. Leaf size=230 \[ -\frac {3 d^3 e^{-4 e-4 f x}}{512 a^2 f^4}+\frac {3 d^3 e^{-2 e-2 f x}}{16 a^2 f^4}-\frac {3 d^2 e^{-4 e-4 f x} (c+d x)}{128 a^2 f^3}+\frac {3 d^2 e^{-2 e-2 f x} (c+d x)}{8 a^2 f^3}-\frac {3 d e^{-4 e-4 f x} (c+d x)^2}{64 a^2 f^2}+\frac {3 d e^{-2 e-2 f x} (c+d x)^2}{8 a^2 f^2}-\frac {e^{-4 e-4 f x} (c+d x)^3}{16 a^2 f}+\frac {e^{-2 e-2 f x} (c+d x)^3}{4 a^2 f}+\frac {(c+d x)^4}{16 a^2 d} \]

[Out]

-3/512*d^3*exp(-4*f*x-4*e)/a^2/f^4+3/16*d^3*exp(-2*f*x-2*e)/a^2/f^4-3/128*d^2*exp(-4*f*x-4*e)*(d*x+c)/a^2/f^3+
3/8*d^2*exp(-2*f*x-2*e)*(d*x+c)/a^2/f^3-3/64*d*exp(-4*f*x-4*e)*(d*x+c)^2/a^2/f^2+3/8*d*exp(-2*f*x-2*e)*(d*x+c)
^2/a^2/f^2-1/16*exp(-4*f*x-4*e)*(d*x+c)^3/a^2/f+1/4*exp(-2*f*x-2*e)*(d*x+c)^3/a^2/f+1/16*(d*x+c)^4/a^2/d

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Rubi [A]
time = 0.19, antiderivative size = 230, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {3810, 2207, 2225} \begin {gather*} -\frac {3 d^2 (c+d x) e^{-4 e-4 f x}}{128 a^2 f^3}+\frac {3 d^2 (c+d x) e^{-2 e-2 f x}}{8 a^2 f^3}-\frac {3 d (c+d x)^2 e^{-4 e-4 f x}}{64 a^2 f^2}+\frac {3 d (c+d x)^2 e^{-2 e-2 f x}}{8 a^2 f^2}-\frac {(c+d x)^3 e^{-4 e-4 f x}}{16 a^2 f}+\frac {(c+d x)^3 e^{-2 e-2 f x}}{4 a^2 f}+\frac {(c+d x)^4}{16 a^2 d}-\frac {3 d^3 e^{-4 e-4 f x}}{512 a^2 f^4}+\frac {3 d^3 e^{-2 e-2 f x}}{16 a^2 f^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*d^3*E^(-4*e - 4*f*x))/(512*a^2*f^4) + (3*d^3*E^(-2*e - 2*f*x))/(16*a^2*f^4) - (3*d^2*E^(-4*e - 4*f*x)*(c +
d*x))/(128*a^2*f^3) + (3*d^2*E^(-2*e - 2*f*x)*(c + d*x))/(8*a^2*f^3) - (3*d*E^(-4*e - 4*f*x)*(c + d*x)^2)/(64
*a^2*f^2) + (3*d*E^(-2*e - 2*f*x)*(c + d*x)^2)/(8*a^2*f^2) - (E^(-4*e - 4*f*x)*(c + d*x)^3)/(16*a^2*f) + (E^(-
2*e - 2*f*x)*(c + d*x)^3)/(4*a^2*f) + (c + d*x)^4/(16*a^2*d)

Rule 2207

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] &&  !TrueQ[$UseGamma]

Rule 2225

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 3810

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/b)*(e + f*x))/(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)^3}{(a+a \coth (e+f x))^2} \, dx &=\int \left (\frac {(c+d x)^3}{4 a^2}+\frac {e^{-4 e-4 f x} (c+d x)^3}{4 a^2}-\frac {e^{-2 e-2 f x} (c+d x)^3}{2 a^2}\right ) \, dx\\ &=\frac {(c+d x)^4}{16 a^2 d}+\frac {\int e^{-4 e-4 f x} (c+d x)^3 \, dx}{4 a^2}-\frac {\int e^{-2 e-2 f x} (c+d x)^3 \, dx}{2 a^2}\\ &=-\frac {e^{-4 e-4 f x} (c+d x)^3}{16 a^2 f}+\frac {e^{-2 e-2 f x} (c+d x)^3}{4 a^2 f}+\frac {(c+d x)^4}{16 a^2 d}+\frac {(3 d) \int e^{-4 e-4 f x} (c+d x)^2 \, dx}{16 a^2 f}-\frac {(3 d) \int e^{-2 e-2 f x} (c+d x)^2 \, dx}{4 a^2 f}\\ &=-\frac {3 d e^{-4 e-4 f x} (c+d x)^2}{64 a^2 f^2}+\frac {3 d e^{-2 e-2 f x} (c+d x)^2}{8 a^2 f^2}-\frac {e^{-4 e-4 f x} (c+d x)^3}{16 a^2 f}+\frac {e^{-2 e-2 f x} (c+d x)^3}{4 a^2 f}+\frac {(c+d x)^4}{16 a^2 d}+\frac {\left (3 d^2\right ) \int e^{-4 e-4 f x} (c+d x) \, dx}{32 a^2 f^2}-\frac {\left (3 d^2\right ) \int e^{-2 e-2 f x} (c+d x) \, dx}{4 a^2 f^2}\\ &=-\frac {3 d^2 e^{-4 e-4 f x} (c+d x)}{128 a^2 f^3}+\frac {3 d^2 e^{-2 e-2 f x} (c+d x)}{8 a^2 f^3}-\frac {3 d e^{-4 e-4 f x} (c+d x)^2}{64 a^2 f^2}+\frac {3 d e^{-2 e-2 f x} (c+d x)^2}{8 a^2 f^2}-\frac {e^{-4 e-4 f x} (c+d x)^3}{16 a^2 f}+\frac {e^{-2 e-2 f x} (c+d x)^3}{4 a^2 f}+\frac {(c+d x)^4}{16 a^2 d}+\frac {\left (3 d^3\right ) \int e^{-4 e-4 f x} \, dx}{128 a^2 f^3}-\frac {\left (3 d^3\right ) \int e^{-2 e-2 f x} \, dx}{8 a^2 f^3}\\ &=-\frac {3 d^3 e^{-4 e-4 f x}}{512 a^2 f^4}+\frac {3 d^3 e^{-2 e-2 f x}}{16 a^2 f^4}-\frac {3 d^2 e^{-4 e-4 f x} (c+d x)}{128 a^2 f^3}+\frac {3 d^2 e^{-2 e-2 f x} (c+d x)}{8 a^2 f^3}-\frac {3 d e^{-4 e-4 f x} (c+d x)^2}{64 a^2 f^2}+\frac {3 d e^{-2 e-2 f x} (c+d x)^2}{8 a^2 f^2}-\frac {e^{-4 e-4 f x} (c+d x)^3}{16 a^2 f}+\frac {e^{-2 e-2 f x} (c+d x)^3}{4 a^2 f}+\frac {(c+d x)^4}{16 a^2 d}\\ \end {align*}

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Mathematica [A]
time = 0.52, size = 420, normalized size = 1.83 \begin {gather*} \frac {\text {csch}^2(e+f x) (\cosh (f x)+\sinh (f x))^2 \left (\left (4 c^3 f^3+6 c^2 d f^2 (1+2 f x)+6 c d^2 f \left (1+2 f x+2 f^2 x^2\right )+d^3 \left (3+6 f x+6 f^2 x^2+4 f^3 x^3\right )\right ) \cosh (2 f x)+\frac {1}{32} \left (32 c^3 f^3+24 c^2 d f^2 (1+4 f x)+12 c d^2 f \left (1+4 f x+8 f^2 x^2\right )+d^3 \left (3+12 f x+24 f^2 x^2+32 f^3 x^3\right )\right ) \cosh (4 f x) (-\cosh (2 e)+\sinh (2 e))+f^4 x \left (4 c^3+6 c^2 d x+4 c d^2 x^2+d^3 x^3\right ) (\cosh (2 e)+\sinh (2 e))-\left (4 c^3 f^3+6 c^2 d f^2 (1+2 f x)+6 c d^2 f \left (1+2 f x+2 f^2 x^2\right )+d^3 \left (3+6 f x+6 f^2 x^2+4 f^3 x^3\right )\right ) \sinh (2 f x)+\frac {1}{32} \left (32 c^3 f^3+24 c^2 d f^2 (1+4 f x)+12 c d^2 f \left (1+4 f x+8 f^2 x^2\right )+d^3 \left (3+12 f x+24 f^2 x^2+32 f^3 x^3\right )\right ) (\cosh (2 e)-\sinh (2 e)) \sinh (4 f x)\right )}{16 a^2 f^4 (1+\coth (e+f x))^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Csch[e + f*x]^2*(Cosh[f*x] + Sinh[f*x])^2*((4*c^3*f^3 + 6*c^2*d*f^2*(1 + 2*f*x) + 6*c*d^2*f*(1 + 2*f*x + 2*f^
2*x^2) + d^3*(3 + 6*f*x + 6*f^2*x^2 + 4*f^3*x^3))*Cosh[2*f*x] + ((32*c^3*f^3 + 24*c^2*d*f^2*(1 + 4*f*x) + 12*c
*d^2*f*(1 + 4*f*x + 8*f^2*x^2) + d^3*(3 + 12*f*x + 24*f^2*x^2 + 32*f^3*x^3))*Cosh[4*f*x]*(-Cosh[2*e] + Sinh[2*
e]))/32 + f^4*x*(4*c^3 + 6*c^2*d*x + 4*c*d^2*x^2 + d^3*x^3)*(Cosh[2*e] + Sinh[2*e]) - (4*c^3*f^3 + 6*c^2*d*f^2
*(1 + 2*f*x) + 6*c*d^2*f*(1 + 2*f*x + 2*f^2*x^2) + d^3*(3 + 6*f*x + 6*f^2*x^2 + 4*f^3*x^3))*Sinh[2*f*x] + ((32
*c^3*f^3 + 24*c^2*d*f^2*(1 + 4*f*x) + 12*c*d^2*f*(1 + 4*f*x + 8*f^2*x^2) + d^3*(3 + 12*f*x + 24*f^2*x^2 + 32*f
^3*x^3))*(Cosh[2*e] - Sinh[2*e])*Sinh[4*f*x])/32))/(16*a^2*f^4*(1 + Coth[e + f*x])^2)

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Maple [A]
time = 2.97, size = 273, normalized size = 1.19

method	result	size
risch	\(\frac {d^{3} x^{4}}{16 a^{2}}+\frac {d^{2} c \,x^{3}}{4 a^{2}}+\frac {3 d \,c^{2} x^{2}}{8 a^{2}}+\frac {c^{3} x}{4 a^{2}}+\frac {c^{4}}{16 a^{2} d}+\frac {\left (4 d^{3} x^{3} f^{3}+12 c \,d^{2} f^{3} x^{2}+12 c^{2} d \,f^{3} x +6 d^{3} f^{2} x^{2}+4 c^{3} f^{3}+12 c \,d^{2} f^{2} x +6 c^{2} d \,f^{2}+6 d^{3} f x +6 c \,d^{2} f +3 d^{3}\right ) {\mathrm e}^{-2 f x -2 e}}{16 a^{2} f^{4}}-\frac {\left (32 d^{3} x^{3} f^{3}+96 c \,d^{2} f^{3} x^{2}+96 c^{2} d \,f^{3} x +24 d^{3} f^{2} x^{2}+32 c^{3} f^{3}+48 c \,d^{2} f^{2} x +24 c^{2} d \,f^{2}+12 d^{3} f x +12 c \,d^{2} f +3 d^{3}\right ) {\mathrm e}^{-4 f x -4 e}}{512 a^{2} f^{4}}\)	\(273\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16/a^2*d^3*x^4+1/4/a^2*d^2*c*x^3+3/8/a^2*d*c^2*x^2+1/4/a^2*c^3*x+1/16/a^2/d*c^4+1/16*(4*d^3*f^3*x^3+12*c*d^2
*f^3*x^2+12*c^2*d*f^3*x+6*d^3*f^2*x^2+4*c^3*f^3+12*c*d^2*f^2*x+6*c^2*d*f^2+6*d^3*f*x+6*c*d^2*f+3*d^3)/a^2/f^4*
exp(-2*f*x-2*e)-1/512*(32*d^3*f^3*x^3+96*c*d^2*f^3*x^2+96*c^2*d*f^3*x+24*d^3*f^2*x^2+32*c^3*f^3+48*c*d^2*f^2*x
+24*c^2*d*f^2+12*d^3*f*x+12*c*d^2*f+3*d^3)/a^2/f^4*exp(-4*f*x-4*e)

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Maxima [A]
time = 0.56, size = 315, normalized size = 1.37 \begin {gather*} \frac {1}{16} \, c^{3} {\left (\frac {4 \, {\left (f x + e\right )}}{a^{2} f} + \frac {4 \, e^{\left (-2 \, f x - 2 \, e\right )} - e^{\left (-4 \, f x - 4 \, e\right )}}{a^{2} f}\right )} + \frac {3 \, {\left (8 \, f^{2} x^{2} e^{\left (4 \, e\right )} + 8 \, {\left (2 \, f x e^{\left (2 \, e\right )} + e^{\left (2 \, e\right )}\right )} e^{\left (-2 \, f x\right )} - {\left (4 \, f x + 1\right )} e^{\left (-4 \, f x\right )}\right )} c^{2} d e^{\left (-4 \, e\right )}}{64 \, a^{2} f^{2}} + \frac {{\left (32 \, f^{3} x^{3} e^{\left (4 \, e\right )} + 48 \, {\left (2 \, f^{2} x^{2} e^{\left (2 \, e\right )} + 2 \, f x e^{\left (2 \, e\right )} + e^{\left (2 \, e\right )}\right )} e^{\left (-2 \, f x\right )} - 3 \, {\left (8 \, f^{2} x^{2} + 4 \, f x + 1\right )} e^{\left (-4 \, f x\right )}\right )} c d^{2} e^{\left (-4 \, e\right )}}{128 \, a^{2} f^{3}} + \frac {{\left (32 \, f^{4} x^{4} e^{\left (4 \, e\right )} + 32 \, {\left (4 \, f^{3} x^{3} e^{\left (2 \, e\right )} + 6 \, f^{2} x^{2} e^{\left (2 \, e\right )} + 6 \, f x e^{\left (2 \, e\right )} + 3 \, e^{\left (2 \, e\right )}\right )} e^{\left (-2 \, f x\right )} - {\left (32 \, f^{3} x^{3} + 24 \, f^{2} x^{2} + 12 \, f x + 3\right )} e^{\left (-4 \, f x\right )}\right )} d^{3} e^{\left (-4 \, e\right )}}{512 \, a^{2} f^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*c^3*(4*(f*x + e)/(a^2*f) + (4*e^(-2*f*x - 2*e) - e^(-4*f*x - 4*e))/(a^2*f)) + 3/64*(8*f^2*x^2*e^(4*e) + 8
*(2*f*x*e^(2*e) + e^(2*e))*e^(-2*f*x) - (4*f*x + 1)*e^(-4*f*x))*c^2*d*e^(-4*e)/(a^2*f^2) + 1/128*(32*f^3*x^3*e
^(4*e) + 48*(2*f^2*x^2*e^(2*e) + 2*f*x*e^(2*e) + e^(2*e))*e^(-2*f*x) - 3*(8*f^2*x^2 + 4*f*x + 1)*e^(-4*f*x))*c
*d^2*e^(-4*e)/(a^2*f^3) + 1/512*(32*f^4*x^4*e^(4*e) + 32*(4*f^3*x^3*e^(2*e) + 6*f^2*x^2*e^(2*e) + 6*f*x*e^(2*e
) + 3*e^(2*e))*e^(-2*f*x) - (32*f^3*x^3 + 24*f^2*x^2 + 12*f*x + 3)*e^(-4*f*x))*d^3*e^(-4*e)/(a^2*f^4)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 595 vs. \(2 (212) = 424\).
time = 0.38, size = 595, normalized size = 2.59 \begin {gather*} \frac {128 \, d^{3} f^{3} x^{3} + 128 \, c^{3} f^{3} + 192 \, c^{2} d f^{2} + 192 \, c d^{2} f + 96 \, d^{3} + 192 \, {\left (2 \, c d^{2} f^{3} + d^{3} f^{2}\right )} x^{2} + {\left (32 \, d^{3} f^{4} x^{4} - 32 \, c^{3} f^{3} - 24 \, c^{2} d f^{2} - 12 \, c d^{2} f + 32 \, {\left (4 \, c d^{2} f^{4} - d^{3} f^{3}\right )} x^{3} - 3 \, d^{3} + 24 \, {\left (8 \, c^{2} d f^{4} - 4 \, c d^{2} f^{3} - d^{3} f^{2}\right )} x^{2} + 4 \, {\left (32 \, c^{3} f^{4} - 24 \, c^{2} d f^{3} - 12 \, c d^{2} f^{2} - 3 \, d^{3} f\right )} x\right )} \cosh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right )^{2} + 2 \, {\left (32 \, d^{3} f^{4} x^{4} + 32 \, c^{3} f^{3} + 24 \, c^{2} d f^{2} + 12 \, c d^{2} f + 32 \, {\left (4 \, c d^{2} f^{4} + d^{3} f^{3}\right )} x^{3} + 3 \, d^{3} + 24 \, {\left (8 \, c^{2} d f^{4} + 4 \, c d^{2} f^{3} + d^{3} f^{2}\right )} x^{2} + 4 \, {\left (32 \, c^{3} f^{4} + 24 \, c^{2} d f^{3} + 12 \, c d^{2} f^{2} + 3 \, d^{3} f\right )} x\right )} \cosh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) \sinh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) + {\left (32 \, d^{3} f^{4} x^{4} - 32 \, c^{3} f^{3} - 24 \, c^{2} d f^{2} - 12 \, c d^{2} f + 32 \, {\left (4 \, c d^{2} f^{4} - d^{3} f^{3}\right )} x^{3} - 3 \, d^{3} + 24 \, {\left (8 \, c^{2} d f^{4} - 4 \, c d^{2} f^{3} - d^{3} f^{2}\right )} x^{2} + 4 \, {\left (32 \, c^{3} f^{4} - 24 \, c^{2} d f^{3} - 12 \, c d^{2} f^{2} - 3 \, d^{3} f\right )} x\right )} \sinh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right )^{2} + 192 \, {\left (2 \, c^{2} d f^{3} + 2 \, c d^{2} f^{2} + d^{3} f\right )} x}{512 \, {\left (a^{2} f^{4} \cosh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right )^{2} + 2 \, a^{2} f^{4} \cosh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) \sinh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) + a^{2} f^{4} \sinh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right )^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/512*(128*d^3*f^3*x^3 + 128*c^3*f^3 + 192*c^2*d*f^2 + 192*c*d^2*f + 96*d^3 + 192*(2*c*d^2*f^3 + d^3*f^2)*x^2
+ (32*d^3*f^4*x^4 - 32*c^3*f^3 - 24*c^2*d*f^2 - 12*c*d^2*f + 32*(4*c*d^2*f^4 - d^3*f^3)*x^3 - 3*d^3 + 24*(8*c^
2*d*f^4 - 4*c*d^2*f^3 - d^3*f^2)*x^2 + 4*(32*c^3*f^4 - 24*c^2*d*f^3 - 12*c*d^2*f^2 - 3*d^3*f)*x)*cosh(f*x + co
sh(1) + sinh(1))^2 + 2*(32*d^3*f^4*x^4 + 32*c^3*f^3 + 24*c^2*d*f^2 + 12*c*d^2*f + 32*(4*c*d^2*f^4 + d^3*f^3)*x
^3 + 3*d^3 + 24*(8*c^2*d*f^4 + 4*c*d^2*f^3 + d^3*f^2)*x^2 + 4*(32*c^3*f^4 + 24*c^2*d*f^3 + 12*c*d^2*f^2 + 3*d^
3*f)*x)*cosh(f*x + cosh(1) + sinh(1))*sinh(f*x + cosh(1) + sinh(1)) + (32*d^3*f^4*x^4 - 32*c^3*f^3 - 24*c^2*d*
f^2 - 12*c*d^2*f + 32*(4*c*d^2*f^4 - d^3*f^3)*x^3 - 3*d^3 + 24*(8*c^2*d*f^4 - 4*c*d^2*f^3 - d^3*f^2)*x^2 + 4*(
32*c^3*f^4 - 24*c^2*d*f^3 - 12*c*d^2*f^2 - 3*d^3*f)*x)*sinh(f*x + cosh(1) + sinh(1))^2 + 192*(2*c^2*d*f^3 + 2*
c*d^2*f^2 + d^3*f)*x)/(a^2*f^4*cosh(f*x + cosh(1) + sinh(1))^2 + 2*a^2*f^4*cosh(f*x + cosh(1) + sinh(1))*sinh(
f*x + cosh(1) + sinh(1)) + a^2*f^4*sinh(f*x + cosh(1) + sinh(1))^2)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 2193 vs. \(2 (236) = 472\).
time = 1.18, size = 2193, normalized size = 9.53 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((32*c**3*f**4*x*tanh(e + f*x)**2/(128*a**2*f**4*tanh(e + f*x)**2 + 256*a**2*f**4*tanh(e + f*x) + 128
*a**2*f**4) + 64*c**3*f**4*x*tanh(e + f*x)/(128*a**2*f**4*tanh(e + f*x)**2 + 256*a**2*f**4*tanh(e + f*x) + 128
*a**2*f**4) + 32*c**3*f**4*x/(128*a**2*f**4*tanh(e + f*x)**2 + 256*a**2*f**4*tanh(e + f*x) + 128*a**2*f**4) +
96*c**3*f**3*tanh(e + f*x)/(128*a**2*f**4*tanh(e + f*x)**2 + 256*a**2*f**4*tanh(e + f*x) + 128*a**2*f**4) + 64
*c**3*f**3/(128*a**2*f**4*tanh(e + f*x)**2 + 256*a**2*f**4*tanh(e + f*x) + 128*a**2*f**4) + 48*c**2*d*f**4*x**
2*tanh(e + f*x)**2/(128*a**2*f**4*tanh(e + f*x)**2 + 256*a**2*f**4*tanh(e + f*x) + 128*a**2*f**4) + 96*c**2*d*
f**4*x**2*tanh(e + f*x)/(128*a**2*f**4*tanh(e + f*x)**2 + 256*a**2*f**4*tanh(e + f*x) + 128*a**2*f**4) + 48*c*
*2*d*f**4*x**2/(128*a**2*f**4*tanh(e + f*x)**2 + 256*a**2*f**4*tanh(e + f*x) + 128*a**2*f**4) - 120*c**2*d*f**
3*x*tanh(e + f*x)**2/(128*a**2*f**4*tanh(e + f*x)**2 + 256*a**2*f**4*tanh(e + f*x) + 128*a**2*f**4) + 48*c**2*
d*f**3*x*tanh(e + f*x)/(128*a**2*f**4*tanh(e + f*x)**2 + 256*a**2*f**4*tanh(e + f*x) + 128*a**2*f**4) + 72*c**
2*d*f**3*x/(128*a**2*f**4*tanh(e + f*x)**2 + 256*a**2*f**4*tanh(e + f*x) + 128*a**2*f**4) + 120*c**2*d*f**2*ta
nh(e + f*x)/(128*a**2*f**4*tanh(e + f*x)**2 + 256*a**2*f**4*tanh(e + f*x) + 128*a**2*f**4) + 96*c**2*d*f**2/(1
28*a**2*f**4*tanh(e + f*x)**2 + 256*a**2*f**4*tanh(e + f*x) + 128*a**2*f**4) + 32*c*d**2*f**4*x**3*tanh(e + f*
x)**2/(128*a**2*f**4*tanh(e + f*x)**2 + 256*a**2*f**4*tanh(e + f*x) + 128*a**2*f**4) + 64*c*d**2*f**4*x**3*tan
h(e + f*x)/(128*a**2*f**4*tanh(e + f*x)**2 + 256*a**2*f**4*tanh(e + f*x) + 128*a**2*f**4) + 32*c*d**2*f**4*x**
3/(128*a**2*f**4*tanh(e + f*x)**2 + 256*a**2*f**4*tanh(e + f*x) + 128*a**2*f**4) - 120*c*d**2*f**3*x**2*tanh(e
+ f*x)**2/(128*a**2*f**4*tanh(e + f*x)**2 + 256*a**2*f**4*tanh(e + f*x) + 128*a**2*f**4) + 48*c*d**2*f**3*x**
2*tanh(e + f*x)/(128*a**2*f**4*tanh(e + f*x)**2 + 256*a**2*f**4*tanh(e + f*x) + 128*a**2*f**4) + 72*c*d**2*f**
3*x**2/(128*a**2*f**4*tanh(e + f*x)**2 + 256*a**2*f**4*tanh(e + f*x) + 128*a**2*f**4) - 108*c*d**2*f**2*x*tanh
(e + f*x)**2/(128*a**2*f**4*tanh(e + f*x)**2 + 256*a**2*f**4*tanh(e + f*x) + 128*a**2*f**4) + 24*c*d**2*f**2*x
*tanh(e + f*x)/(128*a**2*f**4*tanh(e + f*x)**2 + 256*a**2*f**4*tanh(e + f*x) + 128*a**2*f**4) + 84*c*d**2*f**2
*x/(128*a**2*f**4*tanh(e + f*x)**2 + 256*a**2*f**4*tanh(e + f*x) + 128*a**2*f**4) + 108*c*d**2*f*tanh(e + f*x)
/(128*a**2*f**4*tanh(e + f*x)**2 + 256*a**2*f**4*tanh(e + f*x) + 128*a**2*f**4) + 96*c*d**2*f/(128*a**2*f**4*t
anh(e + f*x)**2 + 256*a**2*f**4*tanh(e + f*x) + 128*a**2*f**4) + 8*d**3*f**4*x**4*tanh(e + f*x)**2/(128*a**2*f
**4*tanh(e + f*x)**2 + 256*a**2*f**4*tanh(e + f*x) + 128*a**2*f**4) + 16*d**3*f**4*x**4*tanh(e + f*x)/(128*a**
2*f**4*tanh(e + f*x)**2 + 256*a**2*f**4*tanh(e + f*x) + 128*a**2*f**4) + 8*d**3*f**4*x**4/(128*a**2*f**4*tanh(
e + f*x)**2 + 256*a**2*f**4*tanh(e + f*x) + 128*a**2*f**4) - 40*d**3*f**3*x**3*tanh(e + f*x)**2/(128*a**2*f**4
*tanh(e + f*x)**2 + 256*a**2*f**4*tanh(e + f*x) + 128*a**2*f**4) + 16*d**3*f**3*x**3*tanh(e + f*x)/(128*a**2*f
**4*tanh(e + f*x)**2 + 256*a**2*f**4*tanh(e + f*x) + 128*a**2*f**4) + 24*d**3*f**3*x**3/(128*a**2*f**4*tanh(e
+ f*x)**2 + 256*a**2*f**4*tanh(e + f*x) + 128*a**2*f**4) - 54*d**3*f**2*x**2*tanh(e + f*x)**2/(128*a**2*f**4*t
anh(e + f*x)**2 + 256*a**2*f**4*tanh(e + f*x) + 128*a**2*f**4) + 12*d**3*f**2*x**2*tanh(e + f*x)/(128*a**2*f**
4*tanh(e + f*x)**2 + 256*a**2*f**4*tanh(e + f*x) + 128*a**2*f**4) + 42*d**3*f**2*x**2/(128*a**2*f**4*tanh(e +
f*x)**2 + 256*a**2*f**4*tanh(e + f*x) + 128*a**2*f**4) - 51*d**3*f*x*tanh(e + f*x)**2/(128*a**2*f**4*tanh(e +
f*x)**2 + 256*a**2*f**4*tanh(e + f*x) + 128*a**2*f**4) + 6*d**3*f*x*tanh(e + f*x)/(128*a**2*f**4*tanh(e + f*x)
**2 + 256*a**2*f**4*tanh(e + f*x) + 128*a**2*f**4) + 45*d**3*f*x/(128*a**2*f**4*tanh(e + f*x)**2 + 256*a**2*f*
*4*tanh(e + f*x) + 128*a**2*f**4) + 51*d**3*tanh(e + f*x)/(128*a**2*f**4*tanh(e + f*x)**2 + 256*a**2*f**4*tanh
(e + f*x) + 128*a**2*f**4) + 48*d**3/(128*a**2*f**4*tanh(e + f*x)**2 + 256*a**2*f**4*tanh(e + f*x) + 128*a**2*
f**4), Ne(f, 0)), ((c**3*x + 3*c**2*d*x**2/2 + c*d**2*x**3 + d**3*x**4/4)/(a*coth(e) + a)**2, True))

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Giac [A]
time = 0.42, size = 368, normalized size = 1.60 \begin {gather*} \frac {{\left (32 \, d^{3} f^{4} x^{4} e^{\left (4 \, f x + 4 \, e\right )} + 128 \, c d^{2} f^{4} x^{3} e^{\left (4 \, f x + 4 \, e\right )} + 192 \, c^{2} d f^{4} x^{2} e^{\left (4 \, f x + 4 \, e\right )} + 128 \, d^{3} f^{3} x^{3} e^{\left (2 \, f x + 2 \, e\right )} - 32 \, d^{3} f^{3} x^{3} + 128 \, c^{3} f^{4} x e^{\left (4 \, f x + 4 \, e\right )} + 384 \, c d^{2} f^{3} x^{2} e^{\left (2 \, f x + 2 \, e\right )} - 96 \, c d^{2} f^{3} x^{2} + 384 \, c^{2} d f^{3} x e^{\left (2 \, f x + 2 \, e\right )} + 192 \, d^{3} f^{2} x^{2} e^{\left (2 \, f x + 2 \, e\right )} - 96 \, c^{2} d f^{3} x - 24 \, d^{3} f^{2} x^{2} + 128 \, c^{3} f^{3} e^{\left (2 \, f x + 2 \, e\right )} + 384 \, c d^{2} f^{2} x e^{\left (2 \, f x + 2 \, e\right )} - 32 \, c^{3} f^{3} - 48 \, c d^{2} f^{2} x + 192 \, c^{2} d f^{2} e^{\left (2 \, f x + 2 \, e\right )} + 192 \, d^{3} f x e^{\left (2 \, f x + 2 \, e\right )} - 24 \, c^{2} d f^{2} - 12 \, d^{3} f x + 192 \, c d^{2} f e^{\left (2 \, f x + 2 \, e\right )} - 12 \, c d^{2} f + 96 \, d^{3} e^{\left (2 \, f x + 2 \, e\right )} - 3 \, d^{3}\right )} e^{\left (-4 \, f x - 4 \, e\right )}}{512 \, a^{2} f^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/512*(32*d^3*f^4*x^4*e^(4*f*x + 4*e) + 128*c*d^2*f^4*x^3*e^(4*f*x + 4*e) + 192*c^2*d*f^4*x^2*e^(4*f*x + 4*e)
+ 128*d^3*f^3*x^3*e^(2*f*x + 2*e) - 32*d^3*f^3*x^3 + 128*c^3*f^4*x*e^(4*f*x + 4*e) + 384*c*d^2*f^3*x^2*e^(2*f*
x + 2*e) - 96*c*d^2*f^3*x^2 + 384*c^2*d*f^3*x*e^(2*f*x + 2*e) + 192*d^3*f^2*x^2*e^(2*f*x + 2*e) - 96*c^2*d*f^3
*x - 24*d^3*f^2*x^2 + 128*c^3*f^3*e^(2*f*x + 2*e) + 384*c*d^2*f^2*x*e^(2*f*x + 2*e) - 32*c^3*f^3 - 48*c*d^2*f^
2*x + 192*c^2*d*f^2*e^(2*f*x + 2*e) + 192*d^3*f*x*e^(2*f*x + 2*e) - 24*c^2*d*f^2 - 12*d^3*f*x + 192*c*d^2*f*e^
(2*f*x + 2*e) - 12*c*d^2*f + 96*d^3*e^(2*f*x + 2*e) - 3*d^3)*e^(-4*f*x - 4*e)/(a^2*f^4)

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Mupad [B]
time = 1.33, size = 266, normalized size = 1.16 \begin {gather*} {\mathrm {e}}^{-2\,e-2\,f\,x}\,\left (\frac {4\,c^3\,f^3+6\,c^2\,d\,f^2+6\,c\,d^2\,f+3\,d^3}{16\,a^2\,f^4}+\frac {d^3\,x^3}{4\,a^2\,f}+\frac {3\,d\,x\,\left (2\,c^2\,f^2+2\,c\,d\,f+d^2\right )}{8\,a^2\,f^3}+\frac {3\,d^2\,x^2\,\left (d+2\,c\,f\right )}{8\,a^2\,f^2}\right )-{\mathrm {e}}^{-4\,e-4\,f\,x}\,\left (\frac {32\,c^3\,f^3+24\,c^2\,d\,f^2+12\,c\,d^2\,f+3\,d^3}{512\,a^2\,f^4}+\frac {d^3\,x^3}{16\,a^2\,f}+\frac {3\,d\,x\,\left (8\,c^2\,f^2+4\,c\,d\,f+d^2\right )}{128\,a^2\,f^3}+\frac {3\,d^2\,x^2\,\left (d+4\,c\,f\right )}{64\,a^2\,f^2}\right )+\frac {c^3\,x}{4\,a^2}+\frac {d^3\,x^4}{16\,a^2}+\frac {3\,c^2\,d\,x^2}{8\,a^2}+\frac {c\,d^2\,x^3}{4\,a^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

exp(- 2*e - 2*f*x)*((3*d^3 + 4*c^3*f^3 + 6*c^2*d*f^2 + 6*c*d^2*f)/(16*a^2*f^4) + (d^3*x^3)/(4*a^2*f) + (3*d*x*
(d^2 + 2*c^2*f^2 + 2*c*d*f))/(8*a^2*f^3) + (3*d^2*x^2*(d + 2*c*f))/(8*a^2*f^2)) - exp(- 4*e - 4*f*x)*((3*d^3 +
32*c^3*f^3 + 24*c^2*d*f^2 + 12*c*d^2*f)/(512*a^2*f^4) + (d^3*x^3)/(16*a^2*f) + (3*d*x*(d^2 + 8*c^2*f^2 + 4*c*
d*f))/(128*a^2*f^3) + (3*d^2*x^2*(d + 4*c*f))/(64*a^2*f^2)) + (c^3*x)/(4*a^2) + (d^3*x^4)/(16*a^2) + (3*c^2*d*
x^2)/(8*a^2) + (c*d^2*x^3)/(4*a^2)

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