\(\int \frac {x^2 \sinh ^2(c+d x)}{a+b \cosh (c+d x)} \, dx\) [229]

Optimal result

Integrand size = 24, antiderivative size = 370 \[ \int \frac {x^2 \sinh ^2(c+d x)}{a+b \cosh (c+d x)} \, dx=-\frac {a x^3}{3 b^2}-\frac {2 x \cosh (c+d x)}{b d^2}+\frac {\sqrt {a^2-b^2} x^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )}{b^2 d}-\frac {\sqrt {a^2-b^2} x^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )}{b^2 d}+\frac {2 \sqrt {a^2-b^2} x \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )}{b^2 d^2}-\frac {2 \sqrt {a^2-b^2} x \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )}{b^2 d^2}-\frac {2 \sqrt {a^2-b^2} \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )}{b^2 d^3}+\frac {2 \sqrt {a^2-b^2} \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )}{b^2 d^3}+\frac {2 \sinh (c+d x)}{b d^3}+\frac {x^2 \sinh (c+d x)}{b d} \] Output:

-1/3*a*x^3/b^2-2*x*cosh(d*x+c)/b/d^2+(a^2-b^2)^(1/2)*x^2*ln(1+b*exp(d*x+c) 
/(a-(a^2-b^2)^(1/2)))/b^2/d-(a^2-b^2)^(1/2)*x^2*ln(1+b*exp(d*x+c)/(a+(a^2- 
b^2)^(1/2)))/b^2/d+2*(a^2-b^2)^(1/2)*x*polylog(2,-b*exp(d*x+c)/(a-(a^2-b^2 
)^(1/2)))/b^2/d^2-2*(a^2-b^2)^(1/2)*x*polylog(2,-b*exp(d*x+c)/(a+(a^2-b^2) 
^(1/2)))/b^2/d^2-2*(a^2-b^2)^(1/2)*polylog(3,-b*exp(d*x+c)/(a-(a^2-b^2)^(1 
/2)))/b^2/d^3+2*(a^2-b^2)^(1/2)*polylog(3,-b*exp(d*x+c)/(a+(a^2-b^2)^(1/2) 
))/b^2/d^3+2*sinh(d*x+c)/b/d^3+x^2*sinh(d*x+c)/b/d
 

Mathematica [A] (verified)

Time = 0.49 (sec) , antiderivative size = 293, normalized size of antiderivative = 0.79 \[ \int \frac {x^2 \sinh ^2(c+d x)}{a+b \cosh (c+d x)} \, dx=\frac {-a d^3 x^3+3 \sqrt {a^2-b^2} \left (d^2 x^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )-d^2 x^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )+2 d x \operatorname {PolyLog}\left (2,\frac {b e^{c+d x}}{-a+\sqrt {a^2-b^2}}\right )-2 d x \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )-2 \operatorname {PolyLog}\left (3,\frac {b e^{c+d x}}{-a+\sqrt {a^2-b^2}}\right )+2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )\right )+3 b \cosh (d x) \left (-2 d x \cosh (c)+\left (2+d^2 x^2\right ) \sinh (c)\right )+3 b \left (\left (2+d^2 x^2\right ) \cosh (c)-2 d x \sinh (c)\right ) \sinh (d x)}{3 b^2 d^3} \] Input:

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

Output:

(-(a*d^3*x^3) + 3*Sqrt[a^2 - b^2]*(d^2*x^2*Log[1 + (b*E^(c + d*x))/(a - Sq 
rt[a^2 - b^2])] - d^2*x^2*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 - b^2])] + 
 2*d*x*PolyLog[2, (b*E^(c + d*x))/(-a + Sqrt[a^2 - b^2])] - 2*d*x*PolyLog[ 
2, -((b*E^(c + d*x))/(a + Sqrt[a^2 - b^2]))] - 2*PolyLog[3, (b*E^(c + d*x) 
)/(-a + Sqrt[a^2 - b^2])] + 2*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 - 
 b^2]))]) + 3*b*Cosh[d*x]*(-2*d*x*Cosh[c] + (2 + d^2*x^2)*Sinh[c]) + 3*b*( 
(2 + d^2*x^2)*Cosh[c] - 2*d*x*Sinh[c])*Sinh[d*x])/(3*b^2*d^3)
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 1.97 (sec) , antiderivative size = 356, normalized size of antiderivative = 0.96, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.708, Rules used = {6100, 15, 3042, 3777, 26, 3042, 26, 3777, 3042, 3117, 3801, 2694, 27, 2620, 3011, 2720, 7143}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 6100

\(\displaystyle \frac {\left (a^2-b^2\right ) \int \frac {x^2}{a+b \cosh (c+d x)}dx}{b^2}-\frac {a \int x^2dx}{b^2}+\frac {\int x^2 \cosh (c+d x)dx}{b}\)

\(\Big \downarrow \) 15

\(\displaystyle \frac {\left (a^2-b^2\right ) \int \frac {x^2}{a+b \cosh (c+d x)}dx}{b^2}+\frac {\int x^2 \cosh (c+d x)dx}{b}-\frac {a x^3}{3 b^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\left (a^2-b^2\right ) \int \frac {x^2}{a+b \sin \left (i c+i d x+\frac {\pi }{2}\right )}dx}{b^2}+\frac {\int x^2 \sin \left (i c+i d x+\frac {\pi }{2}\right )dx}{b}-\frac {a x^3}{3 b^2}\)

\(\Big \downarrow \) 3777

\(\displaystyle \frac {\left (a^2-b^2\right ) \int \frac {x^2}{a+b \sin \left (i c+i d x+\frac {\pi }{2}\right )}dx}{b^2}+\frac {\frac {x^2 \sinh (c+d x)}{d}-\frac {2 i \int -i x \sinh (c+d x)dx}{d}}{b}-\frac {a x^3}{3 b^2}\)

\(\Big \downarrow \) 26

\(\displaystyle \frac {\left (a^2-b^2\right ) \int \frac {x^2}{a+b \sin \left (i c+i d x+\frac {\pi }{2}\right )}dx}{b^2}+\frac {\frac {x^2 \sinh (c+d x)}{d}-\frac {2 \int x \sinh (c+d x)dx}{d}}{b}-\frac {a x^3}{3 b^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\left (a^2-b^2\right ) \int \frac {x^2}{a+b \sin \left (i c+i d x+\frac {\pi }{2}\right )}dx}{b^2}+\frac {\frac {x^2 \sinh (c+d x)}{d}-\frac {2 \int -i x \sin (i c+i d x)dx}{d}}{b}-\frac {a x^3}{3 b^2}\)

\(\Big \downarrow \) 26

\(\displaystyle \frac {\left (a^2-b^2\right ) \int \frac {x^2}{a+b \sin \left (i c+i d x+\frac {\pi }{2}\right )}dx}{b^2}+\frac {\frac {x^2 \sinh (c+d x)}{d}+\frac {2 i \int x \sin (i c+i d x)dx}{d}}{b}-\frac {a x^3}{3 b^2}\)

\(\Big \downarrow \) 3777

\(\displaystyle \frac {\left (a^2-b^2\right ) \int \frac {x^2}{a+b \sin \left (i c+i d x+\frac {\pi }{2}\right )}dx}{b^2}+\frac {\frac {x^2 \sinh (c+d x)}{d}+\frac {2 i \left (\frac {i x \cosh (c+d x)}{d}-\frac {i \int \cosh (c+d x)dx}{d}\right )}{d}}{b}-\frac {a x^3}{3 b^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\left (a^2-b^2\right ) \int \frac {x^2}{a+b \sin \left (i c+i d x+\frac {\pi }{2}\right )}dx}{b^2}+\frac {\frac {x^2 \sinh (c+d x)}{d}+\frac {2 i \left (\frac {i x \cosh (c+d x)}{d}-\frac {i \int \sin \left (i c+i d x+\frac {\pi }{2}\right )dx}{d}\right )}{d}}{b}-\frac {a x^3}{3 b^2}\)

\(\Big \downarrow \) 3117

\(\displaystyle \frac {\left (a^2-b^2\right ) \int \frac {x^2}{a+b \sin \left (i c+i d x+\frac {\pi }{2}\right )}dx}{b^2}-\frac {a x^3}{3 b^2}+\frac {\frac {x^2 \sinh (c+d x)}{d}+\frac {2 i \left (\frac {i x \cosh (c+d x)}{d}-\frac {i \sinh (c+d x)}{d^2}\right )}{d}}{b}\)

\(\Big \downarrow \) 3801

\(\displaystyle \frac {2 \left (a^2-b^2\right ) \int \frac {e^{c+d x} x^2}{2 e^{c+d x} a+b e^{2 (c+d x)}+b}dx}{b^2}-\frac {a x^3}{3 b^2}+\frac {\frac {x^2 \sinh (c+d x)}{d}+\frac {2 i \left (\frac {i x \cosh (c+d x)}{d}-\frac {i \sinh (c+d x)}{d^2}\right )}{d}}{b}\)

\(\Big \downarrow \) 2694

\(\displaystyle \frac {2 \left (a^2-b^2\right ) \left (\frac {b \int \frac {e^{c+d x} x^2}{2 \left (a+b e^{c+d x}-\sqrt {a^2-b^2}\right )}dx}{\sqrt {a^2-b^2}}-\frac {b \int \frac {e^{c+d x} x^2}{2 \left (a+b e^{c+d x}+\sqrt {a^2-b^2}\right )}dx}{\sqrt {a^2-b^2}}\right )}{b^2}-\frac {a x^3}{3 b^2}+\frac {\frac {x^2 \sinh (c+d x)}{d}+\frac {2 i \left (\frac {i x \cosh (c+d x)}{d}-\frac {i \sinh (c+d x)}{d^2}\right )}{d}}{b}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 \left (a^2-b^2\right ) \left (\frac {b \int \frac {e^{c+d x} x^2}{a+b e^{c+d x}-\sqrt {a^2-b^2}}dx}{2 \sqrt {a^2-b^2}}-\frac {b \int \frac {e^{c+d x} x^2}{a+b e^{c+d x}+\sqrt {a^2-b^2}}dx}{2 \sqrt {a^2-b^2}}\right )}{b^2}-\frac {a x^3}{3 b^2}+\frac {\frac {x^2 \sinh (c+d x)}{d}+\frac {2 i \left (\frac {i x \cosh (c+d x)}{d}-\frac {i \sinh (c+d x)}{d^2}\right )}{d}}{b}\)

\(\Big \downarrow \) 2620

\(\displaystyle \frac {2 \left (a^2-b^2\right ) \left (\frac {b \left (\frac {x^2 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}+1\right )}{b d}-\frac {2 \int x \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2-b^2}}+1\right )dx}{b d}\right )}{2 \sqrt {a^2-b^2}}-\frac {b \left (\frac {x^2 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2-b^2}+a}+1\right )}{b d}-\frac {2 \int x \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2-b^2}}+1\right )dx}{b d}\right )}{2 \sqrt {a^2-b^2}}\right )}{b^2}-\frac {a x^3}{3 b^2}+\frac {\frac {x^2 \sinh (c+d x)}{d}+\frac {2 i \left (\frac {i x \cosh (c+d x)}{d}-\frac {i \sinh (c+d x)}{d^2}\right )}{d}}{b}\)

\(\Big \downarrow \) 3011

\(\displaystyle \frac {2 \left (a^2-b^2\right ) \left (\frac {b \left (\frac {x^2 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}+1\right )}{b d}-\frac {2 \left (\frac {\int \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )dx}{d}-\frac {x \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )}{d}\right )}{b d}\right )}{2 \sqrt {a^2-b^2}}-\frac {b \left (\frac {x^2 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2-b^2}+a}+1\right )}{b d}-\frac {2 \left (\frac {\int \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )dx}{d}-\frac {x \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )}{d}\right )}{b d}\right )}{2 \sqrt {a^2-b^2}}\right )}{b^2}-\frac {a x^3}{3 b^2}+\frac {\frac {x^2 \sinh (c+d x)}{d}+\frac {2 i \left (\frac {i x \cosh (c+d x)}{d}-\frac {i \sinh (c+d x)}{d^2}\right )}{d}}{b}\)

\(\Big \downarrow \) 2720

\(\displaystyle \frac {2 \left (a^2-b^2\right ) \left (\frac {b \left (\frac {x^2 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}+1\right )}{b d}-\frac {2 \left (\frac {\int e^{-c-d x} \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )de^{c+d x}}{d^2}-\frac {x \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )}{d}\right )}{b d}\right )}{2 \sqrt {a^2-b^2}}-\frac {b \left (\frac {x^2 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2-b^2}+a}+1\right )}{b d}-\frac {2 \left (\frac {\int e^{-c-d x} \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )de^{c+d x}}{d^2}-\frac {x \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )}{d}\right )}{b d}\right )}{2 \sqrt {a^2-b^2}}\right )}{b^2}-\frac {a x^3}{3 b^2}+\frac {\frac {x^2 \sinh (c+d x)}{d}+\frac {2 i \left (\frac {i x \cosh (c+d x)}{d}-\frac {i \sinh (c+d x)}{d^2}\right )}{d}}{b}\)

\(\Big \downarrow \) 7143

\(\displaystyle \frac {2 \left (a^2-b^2\right ) \left (\frac {b \left (\frac {x^2 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}+1\right )}{b d}-\frac {2 \left (\frac {\operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )}{d^2}-\frac {x \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )}{d}\right )}{b d}\right )}{2 \sqrt {a^2-b^2}}-\frac {b \left (\frac {x^2 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2-b^2}+a}+1\right )}{b d}-\frac {2 \left (\frac {\operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )}{d^2}-\frac {x \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )}{d}\right )}{b d}\right )}{2 \sqrt {a^2-b^2}}\right )}{b^2}-\frac {a x^3}{3 b^2}+\frac {\frac {x^2 \sinh (c+d x)}{d}+\frac {2 i \left (\frac {i x \cosh (c+d x)}{d}-\frac {i \sinh (c+d x)}{d^2}\right )}{d}}{b}\)

Input:

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

Output:

-1/3*(a*x^3)/b^2 + (2*(a^2 - b^2)*((b*((x^2*Log[1 + (b*E^(c + d*x))/(a - S 
qrt[a^2 - b^2])])/(b*d) - (2*(-((x*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[ 
a^2 - b^2]))])/d) + PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 - b^2]))]/d 
^2))/(b*d)))/(2*Sqrt[a^2 - b^2]) - (b*((x^2*Log[1 + (b*E^(c + d*x))/(a + S 
qrt[a^2 - b^2])])/(b*d) - (2*(-((x*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[ 
a^2 - b^2]))])/d) + PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 - b^2]))]/d 
^2))/(b*d)))/(2*Sqrt[a^2 - b^2])))/b^2 + ((x^2*Sinh[c + d*x])/d + ((2*I)*( 
(I*x*Cosh[c + d*x])/d - (I*Sinh[c + d*x])/d^2))/d)/b
 

Defintions of rubi rules used

rule 15

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

rule 26

Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a])   I 
nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 2620

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] - Si 
mp[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 2694

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]}, Simp[2*(c/q)   Int 
[(f + g*x)^m*(F^u/(b - q + 2*c*F^u)), x], x] - Simp[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 2720

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] 
   Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct 
ionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ 
[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) 
*(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
 

rule 3011

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.) 
*(x_))^(m_.), x_Symbol] :> Simp[(-(f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + 
b*x)))^n]/(b*c*n*Log[F])), x] + Simp[g*(m/(b*c*n*Log[F]))   Int[(f + g*x)^( 
m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e 
, f, g, n}, x] && GtQ[m, 0]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3117

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; 
 FreeQ[{c, d}, x]
 

rule 3777

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[( 
-(c + d*x)^m)*(Cos[e + f*x]/f), x] + Simp[d*(m/f)   Int[(c + d*x)^(m - 1)*C 
os[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]
 

rule 3801

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

rule 6100

Int[(((e_.) + (f_.)*(x_))^(m_.)*Sinh[(c_.) + (d_.)*(x_)]^(n_))/(Cosh[(c_.) 
+ (d_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[-a/b^2   Int[(e + f*x)^m*Sin 
h[c + d*x]^(n - 2), x], x] + (Simp[1/b   Int[(e + f*x)^m*Sinh[c + d*x]^(n - 
 2)*Cosh[c + d*x], x], x] + Simp[(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 7143

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S 
ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d 
, e, n, p}, x] && EqQ[b*d, a*e]
 

Maple [F]

\[\int \frac {x^{2} \sinh \left (d x +c \right )^{2}}{a +b \cosh \left (d x +c \right )}d x\]

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 937 vs. \(2 (336) = 672\).

Time = 0.11 (sec) , antiderivative size = 937, normalized size of antiderivative = 2.53 \[ \int \frac {x^2 \sinh ^2(c+d x)}{a+b \cosh (c+d x)} \, dx=\text {Too large to display} \] Input:

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

Output:

-1/6*(2*a*d^3*x^3*cosh(d*x + c) + 3*b*d^2*x^2 + 6*b*d*x - 3*(b*d^2*x^2 - 2 
*b*d*x + 2*b)*cosh(d*x + c)^2 - 3*(b*d^2*x^2 - 2*b*d*x + 2*b)*sinh(d*x + c 
)^2 - 12*(b*d*x*cosh(d*x + c) + b*d*x*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) + 12*(b*d*x*cosh(d*x + c) + b*d*x 
*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) + 6*(b*c^2*cosh(d*x + c) + b*c^2*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) - 6*(b*c^2*cosh(d*x + c) + b*c^2*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 
) - 6*((b*d^2*x^2 - b*c^2)*cosh(d*x + c) + (b*d^2*x^2 - b*c^2)*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) + 6*((b*d^2*x^2 
 - b*c^2)*cosh(d*x + c) + (b*d^2*x^2 - b*c^2)*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*sin 
h(d*x + c))*sqrt((a^2 - b^2)/b^2) + b)/b) + 12*(b*cosh(d*x + c) + b*sinh(d 
*x + c))*sqrt((a^2 - b^2)/b^2)*polylog(3, -(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) - 12*( 
b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 - b^2)/b^2)*polylog(3, -(a...
 

Sympy [F]

\[ \int \frac {x^2 \sinh ^2(c+d x)}{a+b \cosh (c+d x)} \, dx=\int \frac {x^{2} \sinh ^{2}{\left (c + d x \right )}}{a + b \cosh {\left (c + d x \right )}}\, dx \] Input:

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

Output:

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

Maxima [F(-2)]

Exception generated. \[ \int \frac {x^2 \sinh ^2(c+d x)}{a+b \cosh (c+d x)} \, dx=\text {Exception raised: ValueError} \] Input:

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

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(a-b>0)', see `assume?` for more 
details)Is
 

Giac [F]

\[ \int \frac {x^2 \sinh ^2(c+d x)}{a+b \cosh (c+d x)} \, dx=\int { \frac {x^{2} \sinh \left (d x + c\right )^{2}}{b \cosh \left (d x + c\right ) + a} \,d x } \] Input:

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {x^2 \sinh ^2(c+d x)}{a+b \cosh (c+d x)} \, dx=\int \frac {x^2\,{\mathrm {sinh}\left (c+d\,x\right )}^2}{a+b\,\mathrm {cosh}\left (c+d\,x\right )} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {x^2 \sinh ^2(c+d x)}{a+b \cosh (c+d x)} \, dx=\frac {3 e^{2 d x +2 c} b^{2} d^{2} x^{2}-6 e^{2 d x +2 c} b^{2} d x +6 e^{2 d x +2 c} b^{2}-24 e^{d x +c} \left (\int \frac {x^{2}}{e^{2 d x +2 c} b +2 e^{d x +c} a +b}d x \right ) a^{3} d^{3}+24 e^{d x +c} \left (\int \frac {x^{2}}{e^{2 d x +2 c} b +2 e^{d x +c} a +b}d x \right ) a \,b^{2} d^{3}-2 e^{d x +c} a b \,d^{3} x^{3}-12 e^{d x} \left (\int \frac {x^{2}}{e^{3 d x +2 c} b +2 e^{2 d x +c} a +e^{d x} b}d x \right ) a^{2} b \,d^{3}+12 e^{d x} \left (\int \frac {x^{2}}{e^{3 d x +2 c} b +2 e^{2 d x +c} a +e^{d x} b}d x \right ) b^{3} d^{3}-12 a^{2} d^{2} x^{2}-24 a^{2} d x -24 a^{2}+9 b^{2} d^{2} x^{2}+18 b^{2} d x +18 b^{2}}{6 e^{d x +c} b^{3} d^{3}} \] Input:

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

Output:

(3*e**(2*c + 2*d*x)*b**2*d**2*x**2 - 6*e**(2*c + 2*d*x)*b**2*d*x + 6*e**(2 
*c + 2*d*x)*b**2 - 24*e**(c + d*x)*int(x**2/(e**(2*c + 2*d*x)*b + 2*e**(c 
+ d*x)*a + b),x)*a**3*d**3 + 24*e**(c + d*x)*int(x**2/(e**(2*c + 2*d*x)*b 
+ 2*e**(c + d*x)*a + b),x)*a*b**2*d**3 - 2*e**(c + d*x)*a*b*d**3*x**3 - 12 
*e**(d*x)*int(x**2/(e**(2*c + 3*d*x)*b + 2*e**(c + 2*d*x)*a + e**(d*x)*b), 
x)*a**2*b*d**3 + 12*e**(d*x)*int(x**2/(e**(2*c + 3*d*x)*b + 2*e**(c + 2*d* 
x)*a + e**(d*x)*b),x)*b**3*d**3 - 12*a**2*d**2*x**2 - 24*a**2*d*x - 24*a** 
2 + 9*b**2*d**2*x**2 + 18*b**2*d*x + 18*b**2)/(6*e**(c + d*x)*b**3*d**3)