\(\int \frac {x^2 \sinh (c+d x)}{a+b \cosh (c+d x)} \, dx\) [223]

Optimal result

Integrand size = 22, antiderivative size = 245 \[ \int \frac {x^2 \sinh (c+d x)}{a+b \cosh (c+d x)} \, dx=-\frac {x^3}{3 b}+\frac {x^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )}{b d}+\frac {x^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )}{b d}+\frac {2 x \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )}{b d^2}+\frac {2 x \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )}{b d^2}-\frac {2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )}{b d^3}-\frac {2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )}{b d^3} \] Output:

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

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.00 \[ \int \frac {x^2 \sinh (c+d x)}{a+b \cosh (c+d x)} \, dx=-\frac {x^3}{3 b}+\frac {x^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )}{b d}+\frac {x^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )}{b d}+\frac {2 x \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )}{b d^2}+\frac {2 x \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )}{b d^2}-\frac {2 \operatorname {PolyLog}\left (3,\frac {b e^{c+d x}}{-a+\sqrt {a^2-b^2}}\right )}{b d^3}-\frac {2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )}{b d^3} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 1.00 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.02, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {6096, 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.

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

\(\Big \downarrow \) 6096

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 7143

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

Input:

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

Output:

-1/3*x^3/b + (x^2*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 - b^2])])/(b*d) + 
(x^2*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 - b^2])])/(b*d) - (2*(-((x*Poly 
Log[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*(-((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)
 

Defintions of rubi rules used

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 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 6096

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

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

Input:

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

Output:

int(x^2*sinh(d*x+c)/(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. 497 vs. \(2 (223) = 446\).

Time = 0.09 (sec) , antiderivative size = 497, normalized size of antiderivative = 2.03 \[ \int \frac {x^2 \sinh (c+d x)}{a+b \cosh (c+d x)} \, dx=-\frac {d^{3} x^{3} - 6 \, d x {\rm Li}_2\left (-\frac {a \cosh \left (d x + c\right ) + a \sinh \left (d x + c\right ) + {\left (b \cosh \left (d x + c\right ) + b \sinh \left (d x + c\right )\right )} \sqrt {\frac {a^{2} - b^{2}}{b^{2}}} + b}{b} + 1\right ) - 6 \, d x {\rm Li}_2\left (-\frac {a \cosh \left (d x + c\right ) + a \sinh \left (d x + c\right ) - {\left (b \cosh \left (d x + c\right ) + b \sinh \left (d x + c\right )\right )} \sqrt {\frac {a^{2} - b^{2}}{b^{2}}} + b}{b} + 1\right ) - 3 \, c^{2} \log \left (2 \, b \cosh \left (d x + c\right ) + 2 \, b \sinh \left (d x + c\right ) + 2 \, b \sqrt {\frac {a^{2} - b^{2}}{b^{2}}} + 2 \, a\right ) - 3 \, c^{2} \log \left (2 \, b \cosh \left (d x + c\right ) + 2 \, b \sinh \left (d x + c\right ) - 2 \, b \sqrt {\frac {a^{2} - b^{2}}{b^{2}}} + 2 \, a\right ) - 3 \, {\left (d^{2} x^{2} - c^{2}\right )} \log \left (\frac {a \cosh \left (d x + c\right ) + a \sinh \left (d x + c\right ) + {\left (b \cosh \left (d x + c\right ) + b \sinh \left (d x + c\right )\right )} \sqrt {\frac {a^{2} - b^{2}}{b^{2}}} + b}{b}\right ) - 3 \, {\left (d^{2} x^{2} - c^{2}\right )} \log \left (\frac {a \cosh \left (d x + c\right ) + a \sinh \left (d x + c\right ) - {\left (b \cosh \left (d x + c\right ) + b \sinh \left (d x + c\right )\right )} \sqrt {\frac {a^{2} - b^{2}}{b^{2}}} + b}{b}\right ) + 6 \, {\rm polylog}\left (3, -\frac {a \cosh \left (d x + c\right ) + a \sinh \left (d x + c\right ) + {\left (b \cosh \left (d x + c\right ) + b \sinh \left (d x + c\right )\right )} \sqrt {\frac {a^{2} - b^{2}}{b^{2}}}}{b}\right ) + 6 \, {\rm polylog}\left (3, -\frac {a \cosh \left (d x + c\right ) + a \sinh \left (d x + c\right ) - {\left (b \cosh \left (d x + c\right ) + b \sinh \left (d x + c\right )\right )} \sqrt {\frac {a^{2} - b^{2}}{b^{2}}}}{b}\right )}{3 \, b d^{3}} \] Input:

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

Output:

-1/3*(d^3*x^3 - 6*d*x*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*d*x*dilo 
g(-(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) - 3*c^2*log(2*b*cosh(d*x + c) + 2*b*si 
nh(d*x + c) + 2*b*sqrt((a^2 - b^2)/b^2) + 2*a) - 3*c^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) - 3*(d^2*x^2 - c 
^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) - 3*(d^2*x^2 - c^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*polylog(3, -(a*cosh(d*x + c) + a*sinh(d*x + c) + (b*co 
sh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 - b^2)/b^2))/b) + 6*polylog(3, -( 
a*cosh(d*x + c) + a*sinh(d*x + c) - (b*cosh(d*x + c) + b*sinh(d*x + c))*sq 
rt((a^2 - b^2)/b^2))/b))/(b*d^3)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

e**(2*c)*int((e**(2*d*x)*x**2)/(e**(2*c + 2*d*x)*b + 2*e**(c + d*x)*a + b) 
,x) - int(x**2/(e**(2*c + 2*d*x)*b + 2*e**(c + d*x)*a + b),x)