3.1.21 \(\int \frac {1}{(c+d x)^3 (a+a \coth (e+f x))} \, dx\) [21]

Optimal. Leaf size=211 \[ -\frac {f}{2 a d^2 (c+d x)}-\frac {f^2 \cosh \left (2 e-\frac {2 c f}{d}\right ) \text {Chi}\left (\frac {2 c f}{d}+2 f x\right )}{a d^3}-\frac {1}{2 d (c+d x)^2 (a+a \coth (e+f x))}+\frac {f}{d^2 (c+d x) (a+a \coth (e+f x))}+\frac {f^2 \text {Chi}\left (\frac {2 c f}{d}+2 f x\right ) \sinh \left (2 e-\frac {2 c f}{d}\right )}{a d^3}+\frac {f^2 \cosh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (\frac {2 c f}{d}+2 f x\right )}{a d^3}-\frac {f^2 \sinh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (\frac {2 c f}{d}+2 f x\right )}{a d^3} \]

[Out]

-1/2*f/a/d^2/(d*x+c)-f^2*Chi(2*c*f/d+2*f*x)*cosh(-2*e+2*c*f/d)/a/d^3-1/2/d/(d*x+c)^2/(a+a*coth(f*x+e))+f/d^2/(
d*x+c)/(a+a*coth(f*x+e))+f^2*cosh(-2*e+2*c*f/d)*Shi(2*c*f/d+2*f*x)/a/d^3-f^2*Chi(2*c*f/d+2*f*x)*sinh(-2*e+2*c*
f/d)/a/d^3+f^2*Shi(2*c*f/d+2*f*x)*sinh(-2*e+2*c*f/d)/a/d^3

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Rubi [A]
time = 0.22, antiderivative size = 211, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3806, 3805, 3384, 3379, 3382} \begin {gather*} \frac {f^2 \text {Chi}\left (2 x f+\frac {2 c f}{d}\right ) \sinh \left (2 e-\frac {2 c f}{d}\right )}{a d^3}-\frac {f^2 \text {Chi}\left (2 x f+\frac {2 c f}{d}\right ) \cosh \left (2 e-\frac {2 c f}{d}\right )}{a d^3}-\frac {f^2 \sinh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (2 x f+\frac {2 c f}{d}\right )}{a d^3}+\frac {f^2 \cosh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (2 x f+\frac {2 c f}{d}\right )}{a d^3}+\frac {f}{d^2 (c+d x) (a \coth (e+f x)+a)}-\frac {f}{2 a d^2 (c+d x)}-\frac {1}{2 d (c+d x)^2 (a \coth (e+f x)+a)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((c + d*x)^3*(a + a*Coth[e + f*x])),x]

[Out]

-1/2*f/(a*d^2*(c + d*x)) - (f^2*Cosh[2*e - (2*c*f)/d]*CoshIntegral[(2*c*f)/d + 2*f*x])/(a*d^3) - 1/(2*d*(c + d
*x)^2*(a + a*Coth[e + f*x])) + f/(d^2*(c + d*x)*(a + a*Coth[e + f*x])) + (f^2*CoshIntegral[(2*c*f)/d + 2*f*x]*
Sinh[2*e - (2*c*f)/d])/(a*d^3) + (f^2*Cosh[2*e - (2*c*f)/d]*SinhIntegral[(2*c*f)/d + 2*f*x])/(a*d^3) - (f^2*Si
nh[2*e - (2*c*f)/d]*SinhIntegral[(2*c*f)/d + 2*f*x])/(a*d^3)

Rule 3379

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[I*(SinhIntegral[c*f*(fz/
d) + f*fz*x]/d), x] /; FreeQ[{c, d, e, f, fz}, x] && EqQ[d*e - c*f*fz*I, 0]

Rule 3382

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CoshIntegral[c*f*(fz/d)
+ f*fz*x]/d, x] /; FreeQ[{c, d, e, f, fz}, x] && EqQ[d*(e - Pi/2) - c*f*fz*I, 0]

Rule 3384

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

Rule 3805

Int[1/(((c_.) + (d_.)*(x_))^2*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])), x_Symbol] :> -Simp[(d*(c + d*x)*(a + b*
Tan[e + f*x]))^(-1), x] + (-Dist[f/(a*d), Int[Sin[2*e + 2*f*x]/(c + d*x), x], x] + Dist[f/(b*d), Int[Cos[2*e +
2*f*x]/(c + d*x), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[a^2 + b^2, 0]

Rule 3806

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

Rubi steps

\begin {align*} \int \frac {1}{(c+d x)^3 (a+a \coth (e+f x))} \, dx &=-\frac {f}{2 a d^2 (c+d x)}-\frac {1}{2 d (c+d x)^2 (a+a \coth (e+f x))}-\frac {f \int \frac {1}{(c+d x)^2 (a+a \coth (e+f x))} \, dx}{d}\\ &=-\frac {f}{2 a d^2 (c+d x)}-\frac {1}{2 d (c+d x)^2 (a+a \coth (e+f x))}+\frac {f}{d^2 (c+d x) (a+a \coth (e+f x))}+\frac {\left (i f^2\right ) \int \frac {\sin \left (2 \left (i e+\frac {\pi }{2}\right )+2 i f x\right )}{c+d x} \, dx}{a d^2}+\frac {f^2 \int \frac {\cos \left (2 \left (i e+\frac {\pi }{2}\right )+2 i f x\right )}{c+d x} \, dx}{a d^2}\\ &=-\frac {f}{2 a d^2 (c+d x)}-\frac {1}{2 d (c+d x)^2 (a+a \coth (e+f x))}+\frac {f}{d^2 (c+d x) (a+a \coth (e+f x))}-\frac {\left (f^2 \cosh \left (2 e-\frac {2 c f}{d}\right )\right ) \int \frac {\cosh \left (\frac {2 c f}{d}+2 f x\right )}{c+d x} \, dx}{a d^2}+\frac {\left (f^2 \cosh \left (2 e-\frac {2 c f}{d}\right )\right ) \int \frac {\sinh \left (\frac {2 c f}{d}+2 f x\right )}{c+d x} \, dx}{a d^2}+\frac {\left (f^2 \sinh \left (2 e-\frac {2 c f}{d}\right )\right ) \int \frac {\cosh \left (\frac {2 c f}{d}+2 f x\right )}{c+d x} \, dx}{a d^2}-\frac {\left (f^2 \sinh \left (2 e-\frac {2 c f}{d}\right )\right ) \int \frac {\sinh \left (\frac {2 c f}{d}+2 f x\right )}{c+d x} \, dx}{a d^2}\\ &=-\frac {f}{2 a d^2 (c+d x)}-\frac {f^2 \cosh \left (2 e-\frac {2 c f}{d}\right ) \text {Chi}\left (\frac {2 c f}{d}+2 f x\right )}{a d^3}-\frac {1}{2 d (c+d x)^2 (a+a \coth (e+f x))}+\frac {f}{d^2 (c+d x) (a+a \coth (e+f x))}+\frac {f^2 \text {Chi}\left (\frac {2 c f}{d}+2 f x\right ) \sinh \left (2 e-\frac {2 c f}{d}\right )}{a d^3}+\frac {f^2 \cosh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (\frac {2 c f}{d}+2 f x\right )}{a d^3}-\frac {f^2 \sinh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (\frac {2 c f}{d}+2 f x\right )}{a d^3}\\ \end {align*}

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Mathematica [A]
time = 0.76, size = 265, normalized size = 1.26 \begin {gather*} -\frac {\text {csch}(e+f x) \left (\cosh \left (\frac {c f}{d}\right )+\sinh \left (\frac {c f}{d}\right )\right ) \left (d \left (d \cosh \left (e+f \left (-\frac {c}{d}+x\right )\right )+(-d+2 c f+2 d f x) \cosh \left (e+f \left (\frac {c}{d}+x\right )\right )+d \sinh \left (e+f \left (-\frac {c}{d}+x\right )\right )+d \sinh \left (e+f \left (\frac {c}{d}+x\right )\right )-2 c f \sinh \left (e+f \left (\frac {c}{d}+x\right )\right )-2 d f x \sinh \left (e+f \left (\frac {c}{d}+x\right )\right )\right )+4 f^2 (c+d x)^2 \text {Chi}\left (\frac {2 f (c+d x)}{d}\right ) \left (\cosh \left (e-\frac {f (c+d x)}{d}\right )-\sinh \left (e-\frac {f (c+d x)}{d}\right )\right )+4 f^2 (c+d x)^2 \left (-\cosh \left (e-\frac {f (c+d x)}{d}\right )+\sinh \left (e-\frac {f (c+d x)}{d}\right )\right ) \text {Shi}\left (\frac {2 f (c+d x)}{d}\right )\right )}{4 a d^3 (c+d x)^2 (1+\coth (e+f x))} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((c + d*x)^3*(a + a*Coth[e + f*x])),x]

[Out]

-1/4*(Csch[e + f*x]*(Cosh[(c*f)/d] + Sinh[(c*f)/d])*(d*(d*Cosh[e + f*(-(c/d) + x)] + (-d + 2*c*f + 2*d*f*x)*Co
sh[e + f*(c/d + x)] + d*Sinh[e + f*(-(c/d) + x)] + d*Sinh[e + f*(c/d + x)] - 2*c*f*Sinh[e + f*(c/d + x)] - 2*d
*f*x*Sinh[e + f*(c/d + x)]) + 4*f^2*(c + d*x)^2*CoshIntegral[(2*f*(c + d*x))/d]*(Cosh[e - (f*(c + d*x))/d] - S
inh[e - (f*(c + d*x))/d]) + 4*f^2*(c + d*x)^2*(-Cosh[e - (f*(c + d*x))/d] + Sinh[e - (f*(c + d*x))/d])*SinhInt
egral[(2*f*(c + d*x))/d]))/(a*d^3*(c + d*x)^2*(1 + Coth[e + f*x]))

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Maple [A]
time = 10.43, size = 210, normalized size = 1.00

method	result	size
risch	\(-\frac {1}{4 a d \left (d x +c \right )^{2}}-\frac {f^{3} {\mathrm e}^{-2 f x -2 e} x}{2 a d \left (d^{2} x^{2} f^{2}+2 c d \,f^{2} x +c^{2} f^{2}\right )}-\frac {f^{3} {\mathrm e}^{-2 f x -2 e} c}{2 a \,d^{2} \left (d^{2} x^{2} f^{2}+2 c d \,f^{2} x +c^{2} f^{2}\right )}+\frac {f^{2} {\mathrm e}^{-2 f x -2 e}}{4 a d \left (d^{2} x^{2} f^{2}+2 c d \,f^{2} x +c^{2} f^{2}\right )}+\frac {f^{2} {\mathrm e}^{\frac {2 c f -2 d e}{d}} \expIntegral \left (1, 2 f x +2 e +\frac {2 c f -2 d e}{d}\right )}{a \,d^{3}}\)	\(210\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(d*x+c)^3/(a+a*coth(f*x+e)),x,method=_RETURNVERBOSE)

[Out]

-1/4/a/d/(d*x+c)^2-1/2*f^3/a*exp(-2*f*x-2*e)/d/(d^2*f^2*x^2+2*c*d*f^2*x+c^2*f^2)*x-1/2*f^3/a*exp(-2*f*x-2*e)/d
^2/(d^2*f^2*x^2+2*c*d*f^2*x+c^2*f^2)*c+1/4*f^2/a*exp(-2*f*x-2*e)/d/(d^2*f^2*x^2+2*c*d*f^2*x+c^2*f^2)+f^2/a/d^3
*exp(2*(c*f-d*e)/d)*Ei(1,2*f*x+2*e+2*(c*f-d*e)/d)

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Maxima [A]
time = 0.65, size = 69, normalized size = 0.33 \begin {gather*} -\frac {1}{4 \, {\left (a d^{3} x^{2} + 2 \, a c d^{2} x + a c^{2} d\right )}} + \frac {e^{\left (\frac {2 \, c f}{d} - 2 \, e\right )} E_{3}\left (\frac {2 \, {\left (d x + c\right )} f}{d}\right )}{2 \, {\left (d x + c\right )}^{2} a d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x+c)^3/(a+a*coth(f*x+e)),x, algorithm="maxima")

[Out]

-1/4/(a*d^3*x^2 + 2*a*c*d^2*x + a*c^2*d) + 1/2*e^(2*c*f/d - 2*e)*exp_integral_e(3, 2*(d*x + c)*f/d)/((d*x + c)
^2*a*d)

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Fricas [A]
time = 0.35, size = 380, normalized size = 1.80 \begin {gather*} -\frac {2 \, {\left (d^{2} f^{2} x^{2} + 2 \, c d f^{2} x + c^{2} f^{2}\right )} {\rm Ei}\left (-\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) \cosh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) \cosh \left (-\frac {2 \, {\left (c f - d \cosh \left (1\right ) - d \sinh \left (1\right )\right )}}{d}\right ) + {\left (d^{2} f x + c d f\right )} \cosh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) - {\left (d^{2} f x + c d f - 2 \, {\left (d^{2} f^{2} x^{2} + 2 \, c d f^{2} x + c^{2} f^{2}\right )} {\rm Ei}\left (-\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) \cosh \left (-\frac {2 \, {\left (c f - d \cosh \left (1\right ) - d \sinh \left (1\right )\right )}}{d}\right ) - d^{2}\right )} \sinh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) - 2 \, {\left ({\left (d^{2} f^{2} x^{2} + 2 \, c d f^{2} x + c^{2} f^{2}\right )} {\rm Ei}\left (-\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) \cosh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) + {\left (d^{2} f^{2} x^{2} + 2 \, c d f^{2} x + c^{2} f^{2}\right )} {\rm Ei}\left (-\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) \sinh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right )\right )} \sinh \left (-\frac {2 \, {\left (c f - d \cosh \left (1\right ) - d \sinh \left (1\right )\right )}}{d}\right )}{2 \, {\left ({\left (a d^{5} x^{2} + 2 \, a c d^{4} x + a c^{2} d^{3}\right )} \cosh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) + {\left (a d^{5} x^{2} + 2 \, a c d^{4} x + a c^{2} d^{3}\right )} \sinh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right )\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x+c)^3/(a+a*coth(f*x+e)),x, algorithm="fricas")

[Out]

-1/2*(2*(d^2*f^2*x^2 + 2*c*d*f^2*x + c^2*f^2)*Ei(-2*(d*f*x + c*f)/d)*cosh(f*x + cosh(1) + sinh(1))*cosh(-2*(c*
f - d*cosh(1) - d*sinh(1))/d) + (d^2*f*x + c*d*f)*cosh(f*x + cosh(1) + sinh(1)) - (d^2*f*x + c*d*f - 2*(d^2*f^
2*x^2 + 2*c*d*f^2*x + c^2*f^2)*Ei(-2*(d*f*x + c*f)/d)*cosh(-2*(c*f - d*cosh(1) - d*sinh(1))/d) - d^2)*sinh(f*x
+ cosh(1) + sinh(1)) - 2*((d^2*f^2*x^2 + 2*c*d*f^2*x + c^2*f^2)*Ei(-2*(d*f*x + c*f)/d)*cosh(f*x + cosh(1) + s
inh(1)) + (d^2*f^2*x^2 + 2*c*d*f^2*x + c^2*f^2)*Ei(-2*(d*f*x + c*f)/d)*sinh(f*x + cosh(1) + sinh(1)))*sinh(-2*
(c*f - d*cosh(1) - d*sinh(1))/d))/((a*d^5*x^2 + 2*a*c*d^4*x + a*c^2*d^3)*cosh(f*x + cosh(1) + sinh(1)) + (a*d^
5*x^2 + 2*a*c*d^4*x + a*c^2*d^3)*sinh(f*x + cosh(1) + sinh(1)))

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {1}{c^{3} \coth {\left (e + f x \right )} + c^{3} + 3 c^{2} d x \coth {\left (e + f x \right )} + 3 c^{2} d x + 3 c d^{2} x^{2} \coth {\left (e + f x \right )} + 3 c d^{2} x^{2} + d^{3} x^{3} \coth {\left (e + f x \right )} + d^{3} x^{3}}\, dx}{a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x+c)**3/(a+a*coth(f*x+e)),x)

[Out]

Integral(1/(c**3*coth(e + f*x) + c**3 + 3*c**2*d*x*coth(e + f*x) + 3*c**2*d*x + 3*c*d**2*x**2*coth(e + f*x) +
3*c*d**2*x**2 + d**3*x**3*coth(e + f*x) + d**3*x**3), x)/a

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Giac [A]
time = 0.45, size = 175, normalized size = 0.83 \begin {gather*} -\frac {4 \, d^{2} f^{2} x^{2} {\rm Ei}\left (-\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) e^{\left (\frac {2 \, c f}{d}\right )} + 8 \, c d f^{2} x {\rm Ei}\left (-\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) e^{\left (\frac {2 \, c f}{d}\right )} + 4 \, c^{2} f^{2} {\rm Ei}\left (-\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) e^{\left (\frac {2 \, c f}{d}\right )} + 2 \, d^{2} f x e^{\left (-2 \, f x\right )} + 2 \, c d f e^{\left (-2 \, f x\right )} - d^{2} e^{\left (-2 \, f x\right )} + d^{2} e^{\left (2 \, e\right )}}{4 \, {\left (a d^{5} x^{2} e^{\left (2 \, e\right )} + 2 \, a c d^{4} x e^{\left (2 \, e\right )} + a c^{2} d^{3} e^{\left (2 \, e\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x+c)^3/(a+a*coth(f*x+e)),x, algorithm="giac")

[Out]

-1/4*(4*d^2*f^2*x^2*Ei(-2*(d*f*x + c*f)/d)*e^(2*c*f/d) + 8*c*d*f^2*x*Ei(-2*(d*f*x + c*f)/d)*e^(2*c*f/d) + 4*c^
2*f^2*Ei(-2*(d*f*x + c*f)/d)*e^(2*c*f/d) + 2*d^2*f*x*e^(-2*f*x) + 2*c*d*f*e^(-2*f*x) - d^2*e^(-2*f*x) + d^2*e^
(2*e))/(a*d^5*x^2*e^(2*e) + 2*a*c*d^4*x*e^(2*e) + a*c^2*d^3*e^(2*e))

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {1}{\left (a+a\,\mathrm {coth}\left (e+f\,x\right )\right )\,{\left (c+d\,x\right )}^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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