3.1.59 \(\int \frac {c+d x}{(a+b \coth (e+f x))^2} \, dx\) [59]

Optimal. Leaf size=196 \[ -\frac {(c+d x)^2}{2 \left (a^2-b^2\right ) d}+\frac {(b d-2 a c f-2 a d f x)^2}{4 a (a-b) (a+b)^2 d f^2}+\frac {b (c+d x)}{\left (a^2-b^2\right ) f (a+b \coth (e+f x))}+\frac {b (b d-2 a c f-2 a d f x) \log \left (1-\frac {(a-b) e^{-2 (e+f x)}}{a+b}\right )}{\left (a^2-b^2\right )^2 f^2}+\frac {a b d \text {PolyLog}\left (2,\frac {(a-b) e^{-2 (e+f x)}}{a+b}\right )}{\left (a^2-b^2\right )^2 f^2} \]

[Out]

-1/2*(d*x+c)^2/(a^2-b^2)/d+1/4*(-2*a*d*f*x-2*a*c*f+b*d)^2/a/(a-b)/(a+b)^2/d/f^2+b*(d*x+c)/(a^2-b^2)/f/(a+b*cot
h(f*x+e))+b*(-2*a*d*f*x-2*a*c*f+b*d)*ln(1+(-a+b)/(a+b)/exp(2*f*x+2*e))/(a^2-b^2)^2/f^2+a*b*d*polylog(2,(a-b)/(
a+b)/exp(2*f*x+2*e))/(a^2-b^2)^2/f^2

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Rubi [A]
time = 0.21, antiderivative size = 196, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {3814, 3812, 2221, 2317, 2438} \begin {gather*} \frac {b (-2 a c f-2 a d f x+b d) \log \left (1-\frac {(a-b) e^{-2 (e+f x)}}{a+b}\right )}{f^2 \left (a^2-b^2\right )^2}+\frac {b (c+d x)}{f \left (a^2-b^2\right ) (a+b \coth (e+f x))}-\frac {(c+d x)^2}{2 d \left (a^2-b^2\right )}+\frac {a b d \text {Li}_2\left (\frac {(a-b) e^{-2 (e+f x)}}{a+b}\right )}{f^2 \left (a^2-b^2\right )^2}+\frac {(-2 a c f-2 a d f x+b d)^2}{4 a d f^2 (a-b) (a+b)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*(c + d*x)^2/((a^2 - b^2)*d) + (b*d - 2*a*c*f - 2*a*d*f*x)^2/(4*a*(a - b)*(a + b)^2*d*f^2) + (b*(c + d*x))
/((a^2 - b^2)*f*(a + b*Coth[e + f*x])) + (b*(b*d - 2*a*c*f - 2*a*d*f*x)*Log[1 - (a - b)/((a + b)*E^(2*(e + f*x
)))])/((a^2 - b^2)^2*f^2) + (a*b*d*PolyLog[2, (a - b)/((a + b)*E^(2*(e + f*x)))])/((a^2 - b^2)^2*f^2)

Rule 2221

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

Rule 2317

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 3812

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*tan[(e_.) + Pi*(k_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(c + d*x)^
(m + 1)/(d*(m + 1)*(a + I*b)), x] + Dist[2*I*b, Int[(c + d*x)^m*E^(2*I*k*Pi)*(E^Simp[2*I*(e + f*x), x]/((a + I
*b)^2 + (a^2 + b^2)*E^(2*I*k*Pi)*E^Simp[2*I*(e + f*x), x])), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && Integer
Q[4*k] && NeQ[a^2 + b^2, 0] && IGtQ[m, 0]

Rule 3814

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

Rubi steps

\begin {align*} \int \frac {c+d x}{(a+b \coth (e+f x))^2} \, dx &=-\frac {(c+d x)^2}{2 \left (a^2-b^2\right ) d}+\frac {b (c+d x)}{\left (a^2-b^2\right ) f (a+b \coth (e+f x))}-\frac {i \int \frac {-i b d+2 i a c f+2 i a d f x}{a+b \coth (e+f x)} \, dx}{\left (a^2-b^2\right ) f}\\ &=-\frac {(c+d x)^2}{2 \left (a^2-b^2\right ) d}+\frac {(b d-2 a c f-2 a d f x)^2}{4 a (a-b) (a+b)^2 d f^2}+\frac {b (c+d x)}{\left (a^2-b^2\right ) f (a+b \coth (e+f x))}+\frac {(2 i b) \int \frac {e^{-2 (e+f x)} (-i b d+2 i a c f+2 i a d f x)}{(a+b)^2+\left (-a^2+b^2\right ) e^{-2 (e+f x)}} \, dx}{\left (a^2-b^2\right ) f}\\ &=-\frac {(c+d x)^2}{2 \left (a^2-b^2\right ) d}+\frac {(b d-2 a c f-2 a d f x)^2}{4 a (a-b) (a+b)^2 d f^2}+\frac {b (c+d x)}{\left (a^2-b^2\right ) f (a+b \coth (e+f x))}+\frac {b (b d-2 a c f-2 a d f x) \log \left (1-\frac {(a-b) e^{-2 (e+f x)}}{a+b}\right )}{\left (a^2-b^2\right )^2 f^2}+\frac {(2 a b d) \int \log \left (1+\frac {\left (-a^2+b^2\right ) e^{-2 (e+f x)}}{(a+b)^2}\right ) \, dx}{\left (a^2-b^2\right )^2 f}\\ &=-\frac {(c+d x)^2}{2 \left (a^2-b^2\right ) d}+\frac {(b d-2 a c f-2 a d f x)^2}{4 a (a-b) (a+b)^2 d f^2}+\frac {b (c+d x)}{\left (a^2-b^2\right ) f (a+b \coth (e+f x))}+\frac {b (b d-2 a c f-2 a d f x) \log \left (1-\frac {(a-b) e^{-2 (e+f x)}}{a+b}\right )}{\left (a^2-b^2\right )^2 f^2}-\frac {(a b d) \text {Subst}\left (\int \frac {\log \left (1+\frac {\left (-a^2+b^2\right ) x}{(a+b)^2}\right )}{x} \, dx,x,e^{-2 (e+f x)}\right )}{\left (a^2-b^2\right )^2 f^2}\\ &=-\frac {(c+d x)^2}{2 \left (a^2-b^2\right ) d}+\frac {(b d-2 a c f-2 a d f x)^2}{4 a (a-b) (a+b)^2 d f^2}+\frac {b (c+d x)}{\left (a^2-b^2\right ) f (a+b \coth (e+f x))}+\frac {b (b d-2 a c f-2 a d f x) \log \left (1-\frac {(a-b) e^{-2 (e+f x)}}{a+b}\right )}{\left (a^2-b^2\right )^2 f^2}+\frac {a b d \text {Li}_2\left (\frac {(a-b) e^{-2 (e+f x)}}{a+b}\right )}{\left (a^2-b^2\right )^2 f^2}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 3.87, size = 477, normalized size = 2.43 \begin {gather*} \frac {\text {csch}^2(e+f x) (b \cosh (e+f x)+a \sinh (e+f x)) \left (2 b \left (-a^2+b^2\right ) f (c+d x) \sinh (e+f x)-\left (a^2-b^2\right ) (e+f x) (-2 c f+d (e-f x)) (b \cosh (e+f x)+a \sinh (e+f x))+2 b d (a (e+f x)-b \log (b \cosh (e+f x)+a \sinh (e+f x))) (b \cosh (e+f x)+a \sinh (e+f x))+4 a d e (a (e+f x)-b \log (b \cosh (e+f x)+a \sinh (e+f x))) (b \cosh (e+f x)+a \sinh (e+f x))-4 a c f (a (e+f x)-b \log (b \cosh (e+f x)+a \sinh (e+f x))) (b \cosh (e+f x)+a \sinh (e+f x))-2 a d \left (a \sqrt {1-\frac {b^2}{a^2}} e^{-\tanh ^{-1}\left (\frac {b}{a}\right )} (e+f x)^2-i b (e+f x) \left (\pi -2 i \tanh ^{-1}\left (\frac {b}{a}\right )\right )+i b \pi \log \left (1+e^{2 (e+f x)}\right )-2 b \left (e+f x+\tanh ^{-1}\left (\frac {b}{a}\right )\right ) \log \left (1-e^{-2 \left (e+f x+\tanh ^{-1}\left (\frac {b}{a}\right )\right )}\right )-i b \pi \log (\cosh (e+f x))+2 b \tanh ^{-1}\left (\frac {b}{a}\right ) \log \left (i \sinh \left (e+f x+\tanh ^{-1}\left (\frac {b}{a}\right )\right )\right )+b \text {PolyLog}\left (2,e^{-2 \left (e+f x+\tanh ^{-1}\left (\frac {b}{a}\right )\right )}\right )\right ) (b \cosh (e+f x)+a \sinh (e+f x))\right )}{2 (-a+b) (a+b) \left (a^2-b^2\right ) f^2 (a+b \coth (e+f x))^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Csch[e + f*x]^2*(b*Cosh[e + f*x] + a*Sinh[e + f*x])*(2*b*(-a^2 + b^2)*f*(c + d*x)*Sinh[e + f*x] - (a^2 - b^2)
*(e + f*x)*(-2*c*f + d*(e - f*x))*(b*Cosh[e + f*x] + a*Sinh[e + f*x]) + 2*b*d*(a*(e + f*x) - b*Log[b*Cosh[e +
f*x] + a*Sinh[e + f*x]])*(b*Cosh[e + f*x] + a*Sinh[e + f*x]) + 4*a*d*e*(a*(e + f*x) - b*Log[b*Cosh[e + f*x] +
a*Sinh[e + f*x]])*(b*Cosh[e + f*x] + a*Sinh[e + f*x]) - 4*a*c*f*(a*(e + f*x) - b*Log[b*Cosh[e + f*x] + a*Sinh[
e + f*x]])*(b*Cosh[e + f*x] + a*Sinh[e + f*x]) - 2*a*d*((a*Sqrt[1 - b^2/a^2]*(e + f*x)^2)/E^ArcTanh[b/a] - I*b
*(e + f*x)*(Pi - (2*I)*ArcTanh[b/a]) + I*b*Pi*Log[1 + E^(2*(e + f*x))] - 2*b*(e + f*x + ArcTanh[b/a])*Log[1 -
E^(-2*(e + f*x + ArcTanh[b/a]))] - I*b*Pi*Log[Cosh[e + f*x]] + 2*b*ArcTanh[b/a]*Log[I*Sinh[e + f*x + ArcTanh[b
/a]]] + b*PolyLog[2, E^(-2*(e + f*x + ArcTanh[b/a]))])*(b*Cosh[e + f*x] + a*Sinh[e + f*x])))/(2*(-a + b)*(a +
b)*(a^2 - b^2)*f^2*(a + b*Coth[e + f*x])^2)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(619\) vs. \(2(195)=390\).
time = 5.24, size = 620, normalized size = 3.16

method	result	size
risch	\(\frac {d \,x^{2}}{2 a^{2}+4 a b +2 b^{2}}+\frac {c x}{a^{2}+2 a b +b^{2}}-\frac {2 \left (d x +c \right ) b^{2}}{\left (a -b \right ) f \left (a^{2}+2 a b +b^{2}\right ) \left (a \,{\mathrm e}^{2 f x +2 e}+b \,{\mathrm e}^{2 f x +2 e}-a +b \right )}+\frac {b^{2} d \ln \left (a \,{\mathrm e}^{2 f x +2 e}+b \,{\mathrm e}^{2 f x +2 e}-a +b \right )}{\left (a^{2}+2 a b +b^{2}\right ) f^{2} \left (a -b \right )^{2}}-\frac {2 b^{2} d \ln \left ({\mathrm e}^{f x +e}\right )}{\left (a^{2}+2 a b +b^{2}\right ) f^{2} \left (a -b \right )^{2}}-\frac {2 b a c \ln \left (a \,{\mathrm e}^{2 f x +2 e}+b \,{\mathrm e}^{2 f x +2 e}-a +b \right )}{\left (a^{2}+2 a b +b^{2}\right ) f \left (a -b \right )^{2}}+\frac {4 b a c \ln \left ({\mathrm e}^{f x +e}\right )}{\left (a^{2}+2 a b +b^{2}\right ) f \left (a -b \right )^{2}}+\frac {2 b a d e \ln \left (a \,{\mathrm e}^{2 f x +2 e}+b \,{\mathrm e}^{2 f x +2 e}-a +b \right )}{\left (a^{2}+2 a b +b^{2}\right ) f^{2} \left (a -b \right )^{2}}-\frac {4 b a d e \ln \left ({\mathrm e}^{f x +e}\right )}{\left (a^{2}+2 a b +b^{2}\right ) f^{2} \left (a -b \right )^{2}}-\frac {2 b a d \ln \left (1-\frac {\left (a +b \right ) {\mathrm e}^{2 f x +2 e}}{a -b}\right ) x}{\left (a^{2}+2 a b +b^{2}\right ) f \left (a -b \right )^{2}}-\frac {2 b a d \ln \left (1-\frac {\left (a +b \right ) {\mathrm e}^{2 f x +2 e}}{a -b}\right ) e}{\left (a^{2}+2 a b +b^{2}\right ) f^{2} \left (a -b \right )^{2}}+\frac {2 b a d \,x^{2}}{\left (a^{2}+2 a b +b^{2}\right ) \left (a -b \right )^{2}}+\frac {4 b a d e x}{\left (a^{2}+2 a b +b^{2}\right ) f \left (a -b \right )^{2}}+\frac {2 b a d \,e^{2}}{\left (a^{2}+2 a b +b^{2}\right ) f^{2} \left (a -b \right )^{2}}-\frac {b a d \polylog \left (2, \frac {\left (a +b \right ) {\mathrm e}^{2 f x +2 e}}{a -b}\right )}{\left (a^{2}+2 a b +b^{2}\right ) f^{2} \left (a -b \right )^{2}}\)	\(620\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2/(a^2+2*a*b+b^2)*d*x^2+1/(a^2+2*a*b+b^2)*c*x-2/(a-b)/f/(a^2+2*a*b+b^2)*(d*x+c)*b^2/(a*exp(2*f*x+2*e)+b*exp(
2*f*x+2*e)-a+b)+1/(a^2+2*a*b+b^2)/f^2*b^2/(a-b)^2*d*ln(a*exp(2*f*x+2*e)+b*exp(2*f*x+2*e)-a+b)-2/(a^2+2*a*b+b^2
)/f^2*b^2/(a-b)^2*d*ln(exp(f*x+e))-2/(a^2+2*a*b+b^2)/f*b/(a-b)^2*a*c*ln(a*exp(2*f*x+2*e)+b*exp(2*f*x+2*e)-a+b)
+4/(a^2+2*a*b+b^2)/f*b/(a-b)^2*a*c*ln(exp(f*x+e))+2/(a^2+2*a*b+b^2)/f^2*b/(a-b)^2*a*d*e*ln(a*exp(2*f*x+2*e)+b*
exp(2*f*x+2*e)-a+b)-4/(a^2+2*a*b+b^2)/f^2*b/(a-b)^2*a*d*e*ln(exp(f*x+e))-2/(a^2+2*a*b+b^2)/f*b/(a-b)^2*a*d*ln(
1-(a+b)*exp(2*f*x+2*e)/(a-b))*x-2/(a^2+2*a*b+b^2)/f^2*b/(a-b)^2*a*d*ln(1-(a+b)*exp(2*f*x+2*e)/(a-b))*e+2/(a^2+
2*a*b+b^2)*b/(a-b)^2*a*d*x^2+4/(a^2+2*a*b+b^2)/f*b/(a-b)^2*a*d*e*x+2/(a^2+2*a*b+b^2)/f^2*b/(a-b)^2*a*d*e^2-1/(
a^2+2*a*b+b^2)/f^2*b/(a-b)^2*a*d*polylog(2,(a+b)*exp(2*f*x+2*e)/(a-b))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(8*a*b*f*integrate(x/(a^4*f*e^(2*f*x + 2*e) + 2*a^3*b*f*e^(2*f*x + 2*e) - 2*a*b^3*f*e^(2*f*x + 2*e) - b^4
*f*e^(2*f*x + 2*e) - a^4*f + 2*a^2*b^2*f - b^4*f), x) + 2*b^2*(2*(f*x + e)/((a^4 - 2*a^2*b^2 + b^4)*f^2) - log
((a + b)*e^(2*f*x + 2*e) - a + b)/((a^4 - 2*a^2*b^2 + b^4)*f^2)) + ((a^2*f - b^2*f)*x^2*e^(2*f*x + 2*e) - 4*b^
2*x - (a^2*f - 2*a*b*f + b^2*f)*x^2)/(a^4*f - 2*a^2*b^2*f + b^4*f - (a^4*f + 2*a^3*b*f - 2*a*b^3*f - b^4*f)*e^
(2*f*x + 2*e)))*d - c*(2*a*b*log(-(a - b)*e^(-2*f*x - 2*e) + a + b)/((a^4 - 2*a^2*b^2 + b^4)*f) + 2*b^2/((a^4
- 2*a^2*b^2 + b^4 - (a^4 - 2*a^3*b + 2*a*b^3 - b^4)*e^(-2*f*x - 2*e))*f) - (f*x + e)/((a^2 + 2*a*b + b^2)*f))

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 2489 vs. \(2 (197) = 394\).
time = 0.41, size = 2489, normalized size = 12.70 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*((a^3 + a^2*b - a*b^2 - b^3)*d*f^2*x^2 + 2*(a^3 + a^2*b - a*b^2 - b^3)*c*f^2*x - 4*(a^2*b - a*b^2)*d*cosh
(1)^2 - 4*(a^2*b - a*b^2)*d*sinh(1)^2 + 4*(a*b^2 - b^3)*c*f - ((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*f^2*x^2 - 4*(
a^2*b + a*b^2)*d*cosh(1)^2 - 4*(a^2*b + a*b^2)*d*sinh(1)^2 + 2*((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*c*f^2 - 2*(a*b
^2 + b^3)*d*f)*x + 4*(2*(a^2*b + a*b^2)*c*f - (a*b^2 + b^3)*d)*cosh(1) + 4*(2*(a^2*b + a*b^2)*c*f - 2*(a^2*b +
a*b^2)*d*cosh(1) - (a*b^2 + b^3)*d)*sinh(1))*cosh(f*x + cosh(1) + sinh(1))^2 - 2*((a^3 + 3*a^2*b + 3*a*b^2 +
b^3)*d*f^2*x^2 - 4*(a^2*b + a*b^2)*d*cosh(1)^2 - 4*(a^2*b + a*b^2)*d*sinh(1)^2 + 2*((a^3 + 3*a^2*b + 3*a*b^2 +
b^3)*c*f^2 - 2*(a*b^2 + b^3)*d*f)*x + 4*(2*(a^2*b + a*b^2)*c*f - (a*b^2 + b^3)*d)*cosh(1) + 4*(2*(a^2*b + a*b
^2)*c*f - 2*(a^2*b + a*b^2)*d*cosh(1) - (a*b^2 + b^3)*d)*sinh(1))*cosh(f*x + cosh(1) + sinh(1))*sinh(f*x + cos
h(1) + sinh(1)) - ((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*f^2*x^2 - 4*(a^2*b + a*b^2)*d*cosh(1)^2 - 4*(a^2*b + a*b^
2)*d*sinh(1)^2 + 2*((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*c*f^2 - 2*(a*b^2 + b^3)*d*f)*x + 4*(2*(a^2*b + a*b^2)*c*f
- (a*b^2 + b^3)*d)*cosh(1) + 4*(2*(a^2*b + a*b^2)*c*f - 2*(a^2*b + a*b^2)*d*cosh(1) - (a*b^2 + b^3)*d)*sinh(1)
)*sinh(f*x + cosh(1) + sinh(1))^2 + 4*(2*(a^2*b - a*b^2)*c*f - (a*b^2 - b^3)*d)*cosh(1) + 4*((a^2*b + a*b^2)*d
*cosh(f*x + cosh(1) + sinh(1))^2 + 2*(a^2*b + a*b^2)*d*cosh(f*x + cosh(1) + sinh(1))*sinh(f*x + cosh(1) + sinh
(1)) + (a^2*b + a*b^2)*d*sinh(f*x + cosh(1) + sinh(1))^2 - (a^2*b - a*b^2)*d)*dilog(sqrt((a + b)/(a - b))*(cos
h(f*x + cosh(1) + sinh(1)) + sinh(f*x + cosh(1) + sinh(1)))) + 4*((a^2*b + a*b^2)*d*cosh(f*x + cosh(1) + sinh(
1))^2 + 2*(a^2*b + a*b^2)*d*cosh(f*x + cosh(1) + sinh(1))*sinh(f*x + cosh(1) + sinh(1)) + (a^2*b + a*b^2)*d*si
nh(f*x + cosh(1) + sinh(1))^2 - (a^2*b - a*b^2)*d)*dilog(-sqrt((a + b)/(a - b))*(cosh(f*x + cosh(1) + sinh(1))
+ sinh(f*x + cosh(1) + sinh(1)))) - 2*(2*(a^2*b - a*b^2)*c*f - 2*(a^2*b - a*b^2)*d*cosh(1) - (2*(a^2*b + a*b^
2)*c*f - 2*(a^2*b + a*b^2)*d*cosh(1) - 2*(a^2*b + a*b^2)*d*sinh(1) - (a*b^2 + b^3)*d)*cosh(f*x + cosh(1) + sin
h(1))^2 - 2*(a^2*b - a*b^2)*d*sinh(1) - 2*(2*(a^2*b + a*b^2)*c*f - 2*(a^2*b + a*b^2)*d*cosh(1) - 2*(a^2*b + a*
b^2)*d*sinh(1) - (a*b^2 + b^3)*d)*cosh(f*x + cosh(1) + sinh(1))*sinh(f*x + cosh(1) + sinh(1)) - (2*(a^2*b + a*
b^2)*c*f - 2*(a^2*b + a*b^2)*d*cosh(1) - 2*(a^2*b + a*b^2)*d*sinh(1) - (a*b^2 + b^3)*d)*sinh(f*x + cosh(1) + s
inh(1))^2 - (a*b^2 - b^3)*d)*log(2*(a + b)*cosh(f*x + cosh(1) + sinh(1)) + 2*(a + b)*sinh(f*x + cosh(1) + sinh
(1)) + 2*(a - b)*sqrt((a + b)/(a - b))) - 2*(2*(a^2*b - a*b^2)*c*f - 2*(a^2*b - a*b^2)*d*cosh(1) - (2*(a^2*b +
a*b^2)*c*f - 2*(a^2*b + a*b^2)*d*cosh(1) - 2*(a^2*b + a*b^2)*d*sinh(1) - (a*b^2 + b^3)*d)*cosh(f*x + cosh(1)
+ sinh(1))^2 - 2*(a^2*b - a*b^2)*d*sinh(1) - 2*(2*(a^2*b + a*b^2)*c*f - 2*(a^2*b + a*b^2)*d*cosh(1) - 2*(a^2*b
+ a*b^2)*d*sinh(1) - (a*b^2 + b^3)*d)*cosh(f*x + cosh(1) + sinh(1))*sinh(f*x + cosh(1) + sinh(1)) - (2*(a^2*b
+ a*b^2)*c*f - 2*(a^2*b + a*b^2)*d*cosh(1) - 2*(a^2*b + a*b^2)*d*sinh(1) - (a*b^2 + b^3)*d)*sinh(f*x + cosh(1
) + sinh(1))^2 - (a*b^2 - b^3)*d)*log(2*(a + b)*cosh(f*x + cosh(1) + sinh(1)) + 2*(a + b)*sinh(f*x + cosh(1) +
sinh(1)) - 2*(a - b)*sqrt((a + b)/(a - b))) - 4*((a^2*b - a*b^2)*d*f*x + (a^2*b - a*b^2)*d*cosh(1) - ((a^2*b
+ a*b^2)*d*f*x + (a^2*b + a*b^2)*d*cosh(1) + (a^2*b + a*b^2)*d*sinh(1))*cosh(f*x + cosh(1) + sinh(1))^2 + (a^2
*b - a*b^2)*d*sinh(1) - 2*((a^2*b + a*b^2)*d*f*x + (a^2*b + a*b^2)*d*cosh(1) + (a^2*b + a*b^2)*d*sinh(1))*cosh
(f*x + cosh(1) + sinh(1))*sinh(f*x + cosh(1) + sinh(1)) - ((a^2*b + a*b^2)*d*f*x + (a^2*b + a*b^2)*d*cosh(1) +
(a^2*b + a*b^2)*d*sinh(1))*sinh(f*x + cosh(1) + sinh(1))^2)*log(sqrt((a + b)/(a - b))*(cosh(f*x + cosh(1) + s
inh(1)) + sinh(f*x + cosh(1) + sinh(1))) + 1) - 4*((a^2*b - a*b^2)*d*f*x + (a^2*b - a*b^2)*d*cosh(1) - ((a^2*b
+ a*b^2)*d*f*x + (a^2*b + a*b^2)*d*cosh(1) + (a^2*b + a*b^2)*d*sinh(1))*cosh(f*x + cosh(1) + sinh(1))^2 + (a^
2*b - a*b^2)*d*sinh(1) - 2*((a^2*b + a*b^2)*d*f*x + (a^2*b + a*b^2)*d*cosh(1) + (a^2*b + a*b^2)*d*sinh(1))*cos
h(f*x + cosh(1) + sinh(1))*sinh(f*x + cosh(1) + sinh(1)) - ((a^2*b + a*b^2)*d*f*x + (a^2*b + a*b^2)*d*cosh(1)
+ (a^2*b + a*b^2)*d*sinh(1))*sinh(f*x + cosh(1) + sinh(1))^2)*log(-sqrt((a + b)/(a - b))*(cosh(f*x + cosh(1) +
sinh(1)) + sinh(f*x + cosh(1) + sinh(1))) + 1) + 4*(2*(a^2*b - a*b^2)*c*f - 2*(a^2*b - a*b^2)*d*cosh(1) - (a*
b^2 - b^3)*d)*sinh(1))/((a^5 + a^4*b - 2*a^3*b^2 - 2*a^2*b^3 + a*b^4 + b^5)*f^2*cosh(f*x + cosh(1) + sinh(1))^
2 + 2*(a^5 + a^4*b - 2*a^3*b^2 - 2*a^2*b^3 + a*b^4 + b^5)*f^2*cosh(f*x + cosh(1) + sinh(1))*sinh(f*x + cosh(1)
+ sinh(1)) + (a^5 + a^4*b - 2*a^3*b^2 - 2*a^2*b^3 + a*b^4 + b^5)*f^2*sinh(f*x + cosh(1) + sinh(1))^2 - (a^5 -
a^4*b - 2*a^3*b^2 + 2*a^2*b^3 + a*b^4 - b^5)*f^2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {c+d\,x}{{\left (a+b\,\mathrm {coth}\left (e+f\,x\right )\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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