3.230 \(\int \frac{x \sinh ^2(c+d x)}{a+b \cosh (c+d x)} \, dx\)

Optimal. Leaf size=244 \[ \frac{\sqrt{a^2-b^2} \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a-\sqrt{a^2-b^2}}\right )}{b^2 d^2}-\frac{\sqrt{a^2-b^2} \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{\sqrt{a^2-b^2}+a}\right )}{b^2 d^2}+\frac{x \sqrt{a^2-b^2} \log \left (\frac{b e^{c+d x}}{a-\sqrt{a^2-b^2}}+1\right )}{b^2 d}-\frac{x \sqrt{a^2-b^2} \log \left (\frac{b e^{c+d x}}{\sqrt{a^2-b^2}+a}+1\right )}{b^2 d}-\frac{a x^2}{2 b^2}-\frac{\cosh (c+d x)}{b d^2}+\frac{x \sinh (c+d x)}{b d} \]

[Out]

-(a*x^2)/(2*b^2) - Cosh[c + d*x]/(b*d^2) + (Sqrt[a^2 - b^2]*x*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 - b^2])])/
(b^2*d) - (Sqrt[a^2 - b^2]*x*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 - b^2])])/(b^2*d) + (Sqrt[a^2 - b^2]*PolyLo
g[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 - b^2]))])/(b^2*d^2) - (Sqrt[a^2 - b^2]*PolyLog[2, -((b*E^(c + d*x))/(a +
Sqrt[a^2 - b^2]))])/(b^2*d^2) + (x*Sinh[c + d*x])/(b*d)

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Rubi [A]  time = 0.417587, antiderivative size = 244, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.409, Rules used = {5566, 30, 3296, 2638, 3320, 2264, 2190, 2279, 2391} \[ \frac{\sqrt{a^2-b^2} \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a-\sqrt{a^2-b^2}}\right )}{b^2 d^2}-\frac{\sqrt{a^2-b^2} \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{\sqrt{a^2-b^2}+a}\right )}{b^2 d^2}+\frac{x \sqrt{a^2-b^2} \log \left (\frac{b e^{c+d x}}{a-\sqrt{a^2-b^2}}+1\right )}{b^2 d}-\frac{x \sqrt{a^2-b^2} \log \left (\frac{b e^{c+d x}}{\sqrt{a^2-b^2}+a}+1\right )}{b^2 d}-\frac{a x^2}{2 b^2}-\frac{\cosh (c+d x)}{b d^2}+\frac{x \sinh (c+d x)}{b d} \]

Antiderivative was successfully verified.

[In]

Int[(x*Sinh[c + d*x]^2)/(a + b*Cosh[c + d*x]),x]

[Out]

-(a*x^2)/(2*b^2) - Cosh[c + d*x]/(b*d^2) + (Sqrt[a^2 - b^2]*x*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 - b^2])])/
(b^2*d) - (Sqrt[a^2 - b^2]*x*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 - b^2])])/(b^2*d) + (Sqrt[a^2 - b^2]*PolyLo
g[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 - b^2]))])/(b^2*d^2) - (Sqrt[a^2 - b^2]*PolyLog[2, -((b*E^(c + d*x))/(a +
Sqrt[a^2 - b^2]))])/(b^2*d^2) + (x*Sinh[c + d*x])/(b*d)

Rule 5566

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

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

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 2638

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

Rule 3320

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

Rule 2264

Int[((F_)^(u_)*((f_.) + (g_.)*(x_))^(m_.))/((a_.) + (b_.)*(F_)^(u_) + (c_.)*(F_)^(v_)), x_Symbol] :> With[{q =
Rt[b^2 - 4*a*c, 2]}, Dist[(2*c)/q, Int[((f + g*x)^m*F^u)/(b - q + 2*c*F^u), x], x] - Dist[(2*c)/q, Int[((f +
g*x)^m*F^u)/(b + q + 2*c*F^u), x], x]] /; FreeQ[{F, a, b, c, f, g}, x] && EqQ[v, 2*u] && LinearQ[u, x] && NeQ[
b^2 - 4*a*c, 0] && IGtQ[m, 0]

Rule 2190

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*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), 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 2279

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 2391

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]

Rubi steps

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

Mathematica [A]  time = 0.949367, size = 187, normalized size = 0.77 \[ \frac{2 \sqrt{a^2-b^2} \left (\text{PolyLog}\left (2,\frac{b e^{c+d x}}{\sqrt{a^2-b^2}-a}\right )-\text{PolyLog}\left (2,-\frac{b e^{c+d x}}{\sqrt{a^2-b^2}+a}\right )+d x \left (\log \left (\frac{b e^{c+d x}}{a-\sqrt{a^2-b^2}}+1\right )-\log \left (\frac{b e^{c+d x}}{\sqrt{a^2-b^2}+a}+1\right )\right )\right )+a (c-d x) (c+d x)+2 b d x \sinh (c+d x)-2 b \cosh (c+d x)}{2 b^2 d^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(x*Sinh[c + d*x]^2)/(a + b*Cosh[c + d*x]),x]

[Out]

(a*(c - d*x)*(c + d*x) - 2*b*Cosh[c + d*x] + 2*Sqrt[a^2 - b^2]*(d*x*(Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 - b
^2])] - Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 - b^2])]) + PolyLog[2, (b*E^(c + d*x))/(-a + Sqrt[a^2 - b^2])] -
PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 - b^2]))]) + 2*b*d*x*Sinh[c + d*x])/(2*b^2*d^2)

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Maple [B]  time = 0.056, size = 862, normalized size = 3.5 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*sinh(d*x+c)^2/(a+b*cosh(d*x+c)),x)

[Out]

-1/2*a*x^2/b^2+1/2*(d*x-1)/b/d^2*exp(d*x+c)-1/2*(d*x+1)/b/d^2*exp(-d*x-c)+1/b^2/d/(a^2-b^2)^(1/2)*ln((-b*exp(d
*x+c)+(a^2-b^2)^(1/2)-a)/(-a+(a^2-b^2)^(1/2)))*x*a^2-1/d/(a^2-b^2)^(1/2)*ln((-b*exp(d*x+c)+(a^2-b^2)^(1/2)-a)/
(-a+(a^2-b^2)^(1/2)))*x-1/b^2/d/(a^2-b^2)^(1/2)*ln((b*exp(d*x+c)+(a^2-b^2)^(1/2)+a)/(a+(a^2-b^2)^(1/2)))*x*a^2
+1/d/(a^2-b^2)^(1/2)*ln((b*exp(d*x+c)+(a^2-b^2)^(1/2)+a)/(a+(a^2-b^2)^(1/2)))*x+1/b^2/d^2/(a^2-b^2)^(1/2)*ln((
-b*exp(d*x+c)+(a^2-b^2)^(1/2)-a)/(-a+(a^2-b^2)^(1/2)))*c*a^2-1/d^2/(a^2-b^2)^(1/2)*ln((-b*exp(d*x+c)+(a^2-b^2)
^(1/2)-a)/(-a+(a^2-b^2)^(1/2)))*c-1/b^2/d^2/(a^2-b^2)^(1/2)*ln((b*exp(d*x+c)+(a^2-b^2)^(1/2)+a)/(a+(a^2-b^2)^(
1/2)))*c*a^2+1/d^2/(a^2-b^2)^(1/2)*ln((b*exp(d*x+c)+(a^2-b^2)^(1/2)+a)/(a+(a^2-b^2)^(1/2)))*c+1/b^2/d^2/(a^2-b
^2)^(1/2)*dilog((-b*exp(d*x+c)+(a^2-b^2)^(1/2)-a)/(-a+(a^2-b^2)^(1/2)))*a^2-1/d^2/(a^2-b^2)^(1/2)*dilog((-b*ex
p(d*x+c)+(a^2-b^2)^(1/2)-a)/(-a+(a^2-b^2)^(1/2)))-1/b^2/d^2/(a^2-b^2)^(1/2)*dilog((b*exp(d*x+c)+(a^2-b^2)^(1/2
)+a)/(a+(a^2-b^2)^(1/2)))*a^2+1/d^2/(a^2-b^2)^(1/2)*dilog((b*exp(d*x+c)+(a^2-b^2)^(1/2)+a)/(a+(a^2-b^2)^(1/2))
)-2/b^2/d^2*c/(-a^2+b^2)^(1/2)*arctan(1/2*(2*b*exp(d*x+c)+2*a)/(-a^2+b^2)^(1/2))*a^2+2/d^2*c/(-a^2+b^2)^(1/2)*
arctan(1/2*(2*b*exp(d*x+c)+2*a)/(-a^2+b^2)^(1/2))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*sinh(d*x+c)^2/(a+b*cosh(d*x+c)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.11274, size = 1666, normalized size = 6.83 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*sinh(d*x+c)^2/(a+b*cosh(d*x+c)),x, algorithm="fricas")

[Out]

-1/2*(a*d^2*x^2*cosh(d*x + c) + b*d*x - (b*d*x - b)*cosh(d*x + c)^2 - (b*d*x - b)*sinh(d*x + c)^2 - 2*(b*cosh(
d*x + c) + b*sinh(d*x + c))*sqrt((a^2 - b^2)/b^2)*dilog(-(a*cosh(d*x + c) + a*sinh(d*x + c) + (b*cosh(d*x + c)
+ b*sinh(d*x + c))*sqrt((a^2 - b^2)/b^2) + b)/b + 1) + 2*(b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 - b^2)
/b^2)*dilog(-(a*cosh(d*x + c) + a*sinh(d*x + c) - (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 - b^2)/b^2) +
b)/b + 1) - 2*(b*c*cosh(d*x + c) + b*c*sinh(d*x + c))*sqrt((a^2 - b^2)/b^2)*log(2*b*cosh(d*x + c) + 2*b*sinh(d
*x + c) + 2*b*sqrt((a^2 - b^2)/b^2) + 2*a) + 2*(b*c*cosh(d*x + c) + b*c*sinh(d*x + c))*sqrt((a^2 - b^2)/b^2)*l
og(2*b*cosh(d*x + c) + 2*b*sinh(d*x + c) - 2*b*sqrt((a^2 - b^2)/b^2) + 2*a) - 2*((b*d*x + b*c)*cosh(d*x + c) +
(b*d*x + b*c)*sinh(d*x + c))*sqrt((a^2 - b^2)/b^2)*log((a*cosh(d*x + c) + a*sinh(d*x + c) + (b*cosh(d*x + c)
+ b*sinh(d*x + c))*sqrt((a^2 - b^2)/b^2) + b)/b) + 2*((b*d*x + b*c)*cosh(d*x + c) + (b*d*x + b*c)*sinh(d*x + c
))*sqrt((a^2 - b^2)/b^2)*log((a*cosh(d*x + c) + a*sinh(d*x + c) - (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^
2 - b^2)/b^2) + b)/b) + (a*d^2*x^2 - 2*(b*d*x - b)*cosh(d*x + c))*sinh(d*x + c) + b)/(b^2*d^2*cosh(d*x + c) +
b^2*d^2*sinh(d*x + c))

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*sinh(d*x+c)**2/(a+b*cosh(d*x+c)),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x \sinh \left (d x + c\right )^{2}}{b \cosh \left (d x + c\right ) + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*sinh(d*x+c)^2/(a+b*cosh(d*x+c)),x, algorithm="giac")

[Out]

integrate(x*sinh(d*x + c)^2/(b*cosh(d*x + c) + a), x)