\(\int \frac {x \sinh ^3(c+d x)}{a+b \cosh (c+d x)} \, dx\) [236]

Optimal result

Integrand size = 22, antiderivative size = 288 \[ \int \frac {x \sinh ^3(c+d x)}{a+b \cosh (c+d x)} \, dx=\frac {x}{4 b d}-\frac {\left (a^2-b^2\right ) x^2}{2 b^3}-\frac {a x \cosh (c+d x)}{b^2 d}+\frac {\left (a^2-b^2\right ) x \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )}{b^3 d}+\frac {\left (a^2-b^2\right ) x \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )}{b^3 d}+\frac {\left (a^2-b^2\right ) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )}{b^3 d^2}+\frac {\left (a^2-b^2\right ) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )}{b^3 d^2}+\frac {a \sinh (c+d x)}{b^2 d^2}-\frac {\cosh (c+d x) \sinh (c+d x)}{4 b d^2}+\frac {x \sinh ^2(c+d x)}{2 b d} \] Output:

1/4*x/b/d-1/2*(a^2-b^2)*x^2/b^3-a*x*cosh(d*x+c)/b^2/d+(a^2-b^2)*x*ln(1+b*e 
xp(d*x+c)/(a-(a^2-b^2)^(1/2)))/b^3/d+(a^2-b^2)*x*ln(1+b*exp(d*x+c)/(a+(a^2 
-b^2)^(1/2)))/b^3/d+(a^2-b^2)*polylog(2,-b*exp(d*x+c)/(a-(a^2-b^2)^(1/2))) 
/b^3/d^2+(a^2-b^2)*polylog(2,-b*exp(d*x+c)/(a+(a^2-b^2)^(1/2)))/b^3/d^2+a* 
sinh(d*x+c)/b^2/d^2-1/4*cosh(d*x+c)*sinh(d*x+c)/b/d^2+1/2*x*sinh(d*x+c)^2/ 
b/d
 

Mathematica [A] (verified)

Time = 1.52 (sec) , antiderivative size = 374, normalized size of antiderivative = 1.30 \[ \int \frac {x \sinh ^3(c+d x)}{a+b \cosh (c+d x)} \, dx=\frac {-8 a b d x \cosh (c+d x)+2 b^2 d x \cosh (2 (c+d x))+4 \left (a^2-b^2\right ) \left (2 c (c+d x)-(c+d x)^2+\frac {4 a \sqrt {-\left (a^2-b^2\right )^2} c \arctan \left (\frac {a+b e^{c+d x}}{\sqrt {-a^2+b^2}}\right )}{\left (a^2-b^2\right )^{3/2}}+\frac {4 a \sqrt {-\left (a^2-b^2\right )^2} c \text {arctanh}\left (\frac {a+b e^{c+d x}}{\sqrt {a^2-b^2}}\right )}{\left (-a^2+b^2\right )^{3/2}}+2 (c+d x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )+2 (c+d x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )-2 c \log \left (b+2 a e^{c+d x}+b e^{2 (c+d x)}\right )+2 \operatorname {PolyLog}\left (2,\frac {b e^{c+d x}}{-a+\sqrt {a^2-b^2}}\right )+2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )\right )+8 a b \sinh (c+d x)-b^2 \sinh (2 (c+d x))}{8 b^3 d^2} \] Input:

Integrate[(x*Sinh[c + d*x]^3)/(a + b*Cosh[c + d*x]),x]
 

Output:

(-8*a*b*d*x*Cosh[c + d*x] + 2*b^2*d*x*Cosh[2*(c + d*x)] + 4*(a^2 - b^2)*(2 
*c*(c + d*x) - (c + d*x)^2 + (4*a*Sqrt[-(a^2 - b^2)^2]*c*ArcTan[(a + b*E^( 
c + d*x))/Sqrt[-a^2 + b^2]])/(a^2 - b^2)^(3/2) + (4*a*Sqrt[-(a^2 - b^2)^2] 
*c*ArcTanh[(a + b*E^(c + d*x))/Sqrt[a^2 - b^2]])/(-a^2 + b^2)^(3/2) + 2*(c 
 + d*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 - b^2])] + 2*(c + d*x)*Log[1 
 + (b*E^(c + d*x))/(a + Sqrt[a^2 - b^2])] - 2*c*Log[b + 2*a*E^(c + d*x) + 
b*E^(2*(c + d*x))] + 2*PolyLog[2, (b*E^(c + d*x))/(-a + Sqrt[a^2 - b^2])] 
+ 2*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 - b^2]))]) + 8*a*b*Sinh[c + 
 d*x] - b^2*Sinh[2*(c + d*x)])/(8*b^3*d^2)
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 1.38 (sec) , antiderivative size = 264, normalized size of antiderivative = 0.92, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.682, Rules used = {6100, 3042, 26, 3777, 3042, 3117, 5895, 3042, 25, 3115, 24, 6096, 2620, 2715, 2838}

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.

\(\displaystyle \int \frac {x \sinh ^3(c+d x)}{a+b \cosh (c+d x)} \, dx\)

\(\Big \downarrow \) 6100

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3777

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3117

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

\(\Big \downarrow \) 5895

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3115

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

\(\Big \downarrow \) 24

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

\(\Big \downarrow \) 6096

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

Input:

Int[(x*Sinh[c + d*x]^3)/(a + b*Cosh[c + d*x]),x]
 

Output:

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

Defintions of rubi rules used

rule 24

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]
 

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 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 2715

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] 
:> Simp[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 2838

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 3042

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

rule 3115

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* 
x]*((b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[b^2*((n - 1)/n)   Int[(b*Sin 
[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[ 
2*n]
 

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 5895

Int[Cosh[(a_.) + (b_.)*(x_)^(n_.)]*(x_)^(m_.)*Sinh[(a_.) + (b_.)*(x_)^(n_.) 
]^(p_.), x_Symbol] :> Simp[x^(m - n + 1)*(Sinh[a + b*x^n]^(p + 1)/(b*n*(p + 
 1))), x] - Simp[(m - n + 1)/(b*n*(p + 1))   Int[x^(m - n)*Sinh[a + b*x^n]^ 
(p + 1), x], x] /; FreeQ[{a, b, p}, x] && LtQ[0, n, m + 1] && NeQ[p, -1]
 

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 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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(859\) vs. \(2(268)=536\).

Time = 12.74 (sec) , antiderivative size = 860, normalized size of antiderivative = 2.99

method	result	size
risch	\(\frac {\left (2 d x -1\right ) {\mathrm e}^{2 d x +2 c}}{16 b \,d^{2}}+\frac {c^{2}}{d^{2} b}-\frac {\operatorname {dilog}\left (\frac {-{\mathrm e}^{d x +c} b +\sqrt {a^{2}-b^{2}}-a}{-a +\sqrt {a^{2}-b^{2}}}\right )}{d^{2} b}+\frac {\left (2 d x +1\right ) {\mathrm e}^{-2 d x -2 c}}{16 b \,d^{2}}-\frac {\ln \left (\frac {-{\mathrm e}^{d x +c} b +\sqrt {a^{2}-b^{2}}-a}{-a +\sqrt {a^{2}-b^{2}}}\right ) c}{d^{2} b}-\frac {\ln \left (\frac {{\mathrm e}^{d x +c} b +\sqrt {a^{2}-b^{2}}+a}{a +\sqrt {a^{2}-b^{2}}}\right ) x}{d b}-\frac {\ln \left (\frac {{\mathrm e}^{d x +c} b +\sqrt {a^{2}-b^{2}}+a}{a +\sqrt {a^{2}-b^{2}}}\right ) c}{d^{2} b}+\frac {c \ln \left ({\mathrm e}^{2 d x +2 c} b +2 \,{\mathrm e}^{d x +c} a +b \right )}{d^{2} b}-\frac {2 c \ln \left ({\mathrm e}^{d x +c}\right )}{d^{2} b}+\frac {2 c x}{d b}-\frac {\ln \left (\frac {-{\mathrm e}^{d x +c} b +\sqrt {a^{2}-b^{2}}-a}{-a +\sqrt {a^{2}-b^{2}}}\right ) x}{d b}-\frac {\operatorname {dilog}\left (\frac {{\mathrm e}^{d x +c} b +\sqrt {a^{2}-b^{2}}+a}{a +\sqrt {a^{2}-b^{2}}}\right )}{d^{2} b}-\frac {a \left (d x +1\right ) {\mathrm e}^{-d x -c}}{2 b^{2} d^{2}}-\frac {2 a^{2} c x}{d \,b^{3}}+\frac {\ln \left (\frac {-{\mathrm e}^{d x +c} b +\sqrt {a^{2}-b^{2}}-a}{-a +\sqrt {a^{2}-b^{2}}}\right ) a^{2} c}{d^{2} b^{3}}+\frac {\ln \left (\frac {{\mathrm e}^{d x +c} b +\sqrt {a^{2}-b^{2}}+a}{a +\sqrt {a^{2}-b^{2}}}\right ) a^{2} c}{d^{2} b^{3}}-\frac {c \,a^{2} \ln \left ({\mathrm e}^{2 d x +2 c} b +2 \,{\mathrm e}^{d x +c} a +b \right )}{d^{2} b^{3}}+\frac {2 c \,a^{2} \ln \left ({\mathrm e}^{d x +c}\right )}{d^{2} b^{3}}-\frac {a \left (d x -1\right ) {\mathrm e}^{d x +c}}{2 b^{2} d^{2}}+\frac {\ln \left (\frac {-{\mathrm e}^{d x +c} b +\sqrt {a^{2}-b^{2}}-a}{-a +\sqrt {a^{2}-b^{2}}}\right ) a^{2} x}{d \,b^{3}}+\frac {\ln \left (\frac {{\mathrm e}^{d x +c} b +\sqrt {a^{2}-b^{2}}+a}{a +\sqrt {a^{2}-b^{2}}}\right ) a^{2} x}{d \,b^{3}}-\frac {a^{2} c^{2}}{d^{2} b^{3}}+\frac {a^{2} \operatorname {dilog}\left (\frac {{\mathrm e}^{d x +c} b +\sqrt {a^{2}-b^{2}}+a}{a +\sqrt {a^{2}-b^{2}}}\right )}{d^{2} b^{3}}+\frac {a^{2} \operatorname {dilog}\left (\frac {-{\mathrm e}^{d x +c} b +\sqrt {a^{2}-b^{2}}-a}{-a +\sqrt {a^{2}-b^{2}}}\right )}{d^{2} b^{3}}-\frac {x^{2} a^{2}}{2 b^{3}}+\frac {x^{2}}{2 b}\)	\(860\)

Input:

int(x*sinh(d*x+c)^3/(a+b*cosh(d*x+c)),x,method=_RETURNVERBOSE)
 

Output:

1/16*(2*d*x-1)/b/d^2*exp(2*d*x+2*c)+1/d^2/b*c^2-1/d^2/b*dilog((-exp(d*x+c) 
*b+(a^2-b^2)^(1/2)-a)/(-a+(a^2-b^2)^(1/2)))+1/16*(2*d*x+1)/b/d^2*exp(-2*d* 
x-2*c)-1/d^2/b*ln((-exp(d*x+c)*b+(a^2-b^2)^(1/2)-a)/(-a+(a^2-b^2)^(1/2)))* 
c-1/d/b*ln((exp(d*x+c)*b+(a^2-b^2)^(1/2)+a)/(a+(a^2-b^2)^(1/2)))*x-1/d^2/b 
*ln((exp(d*x+c)*b+(a^2-b^2)^(1/2)+a)/(a+(a^2-b^2)^(1/2)))*c+1/d^2/b*c*ln(e 
xp(2*d*x+2*c)*b+2*exp(d*x+c)*a+b)-2/d^2/b*c*ln(exp(d*x+c))+2/d/b*c*x-1/d/b 
*ln((-exp(d*x+c)*b+(a^2-b^2)^(1/2)-a)/(-a+(a^2-b^2)^(1/2)))*x-1/d^2/b*dilo 
g((exp(d*x+c)*b+(a^2-b^2)^(1/2)+a)/(a+(a^2-b^2)^(1/2)))-1/2*a*(d*x+1)/b^2/ 
d^2*exp(-d*x-c)-2/d/b^3*a^2*c*x+1/d^2/b^3*ln((-exp(d*x+c)*b+(a^2-b^2)^(1/2 
)-a)/(-a+(a^2-b^2)^(1/2)))*a^2*c+1/d^2/b^3*ln((exp(d*x+c)*b+(a^2-b^2)^(1/2 
)+a)/(a+(a^2-b^2)^(1/2)))*a^2*c-1/d^2/b^3*c*a^2*ln(exp(2*d*x+2*c)*b+2*exp( 
d*x+c)*a+b)+2/d^2/b^3*c*a^2*ln(exp(d*x+c))-1/2*a*(d*x-1)/b^2/d^2*exp(d*x+c 
)+1/d/b^3*ln((-exp(d*x+c)*b+(a^2-b^2)^(1/2)-a)/(-a+(a^2-b^2)^(1/2)))*a^2*x 
+1/d/b^3*ln((exp(d*x+c)*b+(a^2-b^2)^(1/2)+a)/(a+(a^2-b^2)^(1/2)))*a^2*x-1/ 
d^2/b^3*a^2*c^2+1/d^2/b^3*a^2*dilog((exp(d*x+c)*b+(a^2-b^2)^(1/2)+a)/(a+(a 
^2-b^2)^(1/2)))+1/d^2/b^3*a^2*dilog((-exp(d*x+c)*b+(a^2-b^2)^(1/2)-a)/(-a+ 
(a^2-b^2)^(1/2)))-1/2*x^2/b^3*a^2+1/2*x^2/b
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1196 vs. \(2 (266) = 532\).

Time = 0.10 (sec) , antiderivative size = 1196, normalized size of antiderivative = 4.15 \[ \int \frac {x \sinh ^3(c+d x)}{a+b \cosh (c+d x)} \, dx=\text {Too large to display} \] Input:

integrate(x*sinh(d*x+c)^3/(a+b*cosh(d*x+c)),x, algorithm="fricas")
 

Output:

1/16*((2*b^2*d*x - b^2)*cosh(d*x + c)^4 + (2*b^2*d*x - b^2)*sinh(d*x + c)^ 
4 + 2*b^2*d*x - 8*(a*b*d*x - a*b)*cosh(d*x + c)^3 - 4*(2*a*b*d*x - 2*a*b - 
 (2*b^2*d*x - b^2)*cosh(d*x + c))*sinh(d*x + c)^3 - 8*((a^2 - b^2)*d^2*x^2 
 - 2*(a^2 - b^2)*c^2)*cosh(d*x + c)^2 - 2*(4*(a^2 - b^2)*d^2*x^2 - 8*(a^2 
- b^2)*c^2 - 3*(2*b^2*d*x - b^2)*cosh(d*x + c)^2 + 12*(a*b*d*x - a*b)*cosh 
(d*x + c))*sinh(d*x + c)^2 + b^2 - 8*(a*b*d*x + a*b)*cosh(d*x + c) + 16*(( 
a^2 - b^2)*cosh(d*x + c)^2 + 2*(a^2 - b^2)*cosh(d*x + c)*sinh(d*x + c) + ( 
a^2 - b^2)*sinh(d*x + c)^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) + 16*( 
(a^2 - b^2)*cosh(d*x + c)^2 + 2*(a^2 - b^2)*cosh(d*x + c)*sinh(d*x + c) + 
(a^2 - b^2)*sinh(d*x + c)^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) - 16* 
((a^2 - b^2)*c*cosh(d*x + c)^2 + 2*(a^2 - b^2)*c*cosh(d*x + c)*sinh(d*x + 
c) + (a^2 - b^2)*c*sinh(d*x + 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) - 16*((a^2 - b^2)*c*cosh(d*x + c)^2 
 + 2*(a^2 - b^2)*c*cosh(d*x + c)*sinh(d*x + c) + (a^2 - b^2)*c*sinh(d*x + 
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) + 16*(((a^2 - b^2)*d*x + (a^2 - b^2)*c)*cosh(d*x + c)^2 + 2*((a^2 
 - b^2)*d*x + (a^2 - b^2)*c)*cosh(d*x + c)*sinh(d*x + c) + ((a^2 - b^2)*d* 
x + (a^2 - b^2)*c)*sinh(d*x + c)^2)*log((a*cosh(d*x + c) + a*sinh(d*x +...
                                                                                    
                                                                                    
 

Sympy [F]

\[ \int \frac {x \sinh ^3(c+d x)}{a+b \cosh (c+d x)} \, dx=\int \frac {x \sinh ^{3}{\left (c + d x \right )}}{a + b \cosh {\left (c + d x \right )}}\, dx \] Input:

integrate(x*sinh(d*x+c)**3/(a+b*cosh(d*x+c)),x)
 

Output:

Integral(x*sinh(c + d*x)**3/(a + b*cosh(c + d*x)), x)
 

Maxima [F]

\[ \int \frac {x \sinh ^3(c+d x)}{a+b \cosh (c+d x)} \, dx=\int { \frac {x \sinh \left (d x + c\right )^{3}}{b \cosh \left (d x + c\right ) + a} \,d x } \] Input:

integrate(x*sinh(d*x+c)^3/(a+b*cosh(d*x+c)),x, algorithm="maxima")
 

Output:

1/16*(8*(a^2*d^2*e^(2*c) - b^2*d^2*e^(2*c))*x^2 + (2*b^2*d*x*e^(4*c) - b^2 
*e^(4*c))*e^(2*d*x) - 8*(a*b*d*x*e^(3*c) - a*b*e^(3*c))*e^(d*x) - 8*(a*b*d 
*x*e^c + a*b*e^c)*e^(-d*x) + (2*b^2*d*x + b^2)*e^(-2*d*x))*e^(-2*c)/(b^3*d 
^2) - 1/8*integrate(16*((a^3*e^c - a*b^2*e^c)*x*e^(d*x) + (a^2*b - b^3)*x) 
/(b^4*e^(2*d*x + 2*c) + 2*a*b^3*e^(d*x + c) + b^4), x)
 

Giac [F]

\[ \int \frac {x \sinh ^3(c+d x)}{a+b \cosh (c+d x)} \, dx=\int { \frac {x \sinh \left (d x + c\right )^{3}}{b \cosh \left (d x + c\right ) + a} \,d x } \] Input:

integrate(x*sinh(d*x+c)^3/(a+b*cosh(d*x+c)),x, algorithm="giac")
 

Output:

integrate(x*sinh(d*x + c)^3/(b*cosh(d*x + c) + a), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x \sinh ^3(c+d x)}{a+b \cosh (c+d x)} \, dx=\int \frac {x\,{\mathrm {sinh}\left (c+d\,x\right )}^3}{a+b\,\mathrm {cosh}\left (c+d\,x\right )} \,d x \] Input:

int((x*sinh(c + d*x)^3)/(a + b*cosh(c + d*x)),x)
 

Output:

int((x*sinh(c + d*x)^3)/(a + b*cosh(c + d*x)), x)
 

Reduce [F]

\[ \int \frac {x \sinh ^3(c+d x)}{a+b \cosh (c+d x)} \, dx=\frac {2 e^{4 d x +4 c} b^{4} d x -e^{4 d x +4 c} b^{4}-8 e^{3 d x +3 c} a \,b^{3} d x +8 e^{3 d x +3 c} a \,b^{3}-64 e^{2 d x +2 c} \left (\int \frac {x}{e^{4 d x +4 c} b +2 e^{3 d x +3 c} a +e^{2 d x +2 c} b}d x \right ) a^{4} b \,d^{2}+96 e^{2 d x +2 c} \left (\int \frac {x}{e^{4 d x +4 c} b +2 e^{3 d x +3 c} a +e^{2 d x +2 c} b}d x \right ) a^{2} b^{3} d^{2}-32 e^{2 d x +2 c} \left (\int \frac {x}{e^{4 d x +4 c} b +2 e^{3 d x +3 c} a +e^{2 d x +2 c} b}d x \right ) b^{5} d^{2}+8 e^{2 d x +2 c} a^{2} b^{2} d^{2} x^{2}-8 e^{2 d x +2 c} b^{4} d^{2} x^{2}-128 e^{2 d x +c} \left (\int \frac {x}{e^{3 d x +2 c} b +2 e^{2 d x +c} a +e^{d x} b}d x \right ) a^{5} d^{2}+224 e^{2 d x +c} \left (\int \frac {x}{e^{3 d x +2 c} b +2 e^{2 d x +c} a +e^{d x} b}d x \right ) a^{3} b^{2} d^{2}-96 e^{2 d x +c} \left (\int \frac {x}{e^{3 d x +2 c} b +2 e^{2 d x +c} a +e^{d x} b}d x \right ) a \,b^{4} d^{2}+32 e^{d x +c} a^{3} b d x +32 e^{d x +c} a^{3} b -40 e^{d x +c} a \,b^{3} d x -40 e^{d x +c} a \,b^{3}-32 a^{4} d x -16 a^{4}+48 a^{2} b^{2} d x +24 a^{2} b^{2}-14 b^{4} d x -7 b^{4}}{16 e^{2 d x +2 c} b^{5} d^{2}} \] Input:

int(x*sinh(d*x+c)^3/(a+b*cosh(d*x+c)),x)
 

Output:

(2*e**(4*c + 4*d*x)*b**4*d*x - e**(4*c + 4*d*x)*b**4 - 8*e**(3*c + 3*d*x)* 
a*b**3*d*x + 8*e**(3*c + 3*d*x)*a*b**3 - 64*e**(2*c + 2*d*x)*int(x/(e**(4* 
c + 4*d*x)*b + 2*e**(3*c + 3*d*x)*a + e**(2*c + 2*d*x)*b),x)*a**4*b*d**2 + 
 96*e**(2*c + 2*d*x)*int(x/(e**(4*c + 4*d*x)*b + 2*e**(3*c + 3*d*x)*a + e* 
*(2*c + 2*d*x)*b),x)*a**2*b**3*d**2 - 32*e**(2*c + 2*d*x)*int(x/(e**(4*c + 
 4*d*x)*b + 2*e**(3*c + 3*d*x)*a + e**(2*c + 2*d*x)*b),x)*b**5*d**2 + 8*e* 
*(2*c + 2*d*x)*a**2*b**2*d**2*x**2 - 8*e**(2*c + 2*d*x)*b**4*d**2*x**2 - 1 
28*e**(c + 2*d*x)*int(x/(e**(2*c + 3*d*x)*b + 2*e**(c + 2*d*x)*a + e**(d*x 
)*b),x)*a**5*d**2 + 224*e**(c + 2*d*x)*int(x/(e**(2*c + 3*d*x)*b + 2*e**(c 
 + 2*d*x)*a + e**(d*x)*b),x)*a**3*b**2*d**2 - 96*e**(c + 2*d*x)*int(x/(e** 
(2*c + 3*d*x)*b + 2*e**(c + 2*d*x)*a + e**(d*x)*b),x)*a*b**4*d**2 + 32*e** 
(c + d*x)*a**3*b*d*x + 32*e**(c + d*x)*a**3*b - 40*e**(c + d*x)*a*b**3*d*x 
 - 40*e**(c + d*x)*a*b**3 - 32*a**4*d*x - 16*a**4 + 48*a**2*b**2*d*x + 24* 
a**2*b**2 - 14*b**4*d*x - 7*b**4)/(16*e**(2*c + 2*d*x)*b**5*d**2)