\(\int (c+d x)^4 \sinh ^2(a+b x) \, dx\) [8]

Optimal result

Integrand size = 16, antiderivative size = 162 \[ \int (c+d x)^4 \sinh ^2(a+b x) \, dx=-\frac {3 d^4 x}{4 b^4}-\frac {d (c+d x)^3}{2 b^2}-\frac {(c+d x)^5}{10 d}+\frac {3 d^4 \cosh (a+b x) \sinh (a+b x)}{4 b^5}+\frac {3 d^2 (c+d x)^2 \cosh (a+b x) \sinh (a+b x)}{2 b^3}+\frac {(c+d x)^4 \cosh (a+b x) \sinh (a+b x)}{2 b}-\frac {3 d^3 (c+d x) \sinh ^2(a+b x)}{2 b^4}-\frac {d (c+d x)^3 \sinh ^2(a+b x)}{b^2} \] Output:

-3/4*d^4*x/b^4-1/2*d*(d*x+c)^3/b^2-1/10*(d*x+c)^5/d+3/4*d^4*cosh(b*x+a)*si 
nh(b*x+a)/b^5+3/2*d^2*(d*x+c)^2*cosh(b*x+a)*sinh(b*x+a)/b^3+1/2*(d*x+c)^4* 
cosh(b*x+a)*sinh(b*x+a)/b-3/2*d^3*(d*x+c)*sinh(b*x+a)^2/b^4-d*(d*x+c)^3*si 
nh(b*x+a)^2/b^2
 

Mathematica [A] (verified)

Time = 0.35 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.81 \[ \int (c+d x)^4 \sinh ^2(a+b x) \, dx=\frac {-8 b^5 x \left (5 c^4+10 c^3 d x+10 c^2 d^2 x^2+5 c d^3 x^3+d^4 x^4\right )-20 b d (c+d x) \left (3 d^2+2 b^2 (c+d x)^2\right ) \cosh (2 (a+b x))+10 \left (3 d^4+6 b^2 d^2 (c+d x)^2+2 b^4 (c+d x)^4\right ) \sinh (2 (a+b x))}{80 b^5} \] Input:

Integrate[(c + d*x)^4*Sinh[a + b*x]^2,x]
 

Output:

(-8*b^5*x*(5*c^4 + 10*c^3*d*x + 10*c^2*d^2*x^2 + 5*c*d^3*x^3 + d^4*x^4) - 
20*b*d*(c + d*x)*(3*d^2 + 2*b^2*(c + d*x)^2)*Cosh[2*(a + b*x)] + 10*(3*d^4 
 + 6*b^2*d^2*(c + d*x)^2 + 2*b^4*(c + d*x)^4)*Sinh[2*(a + b*x)])/(80*b^5)
 

Rubi [A] (verified)

Time = 0.54 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.03, number of steps used = 14, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.875, Rules used = {3042, 25, 3792, 17, 25, 3042, 25, 3792, 17, 25, 3042, 25, 3115, 24}

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.

\(\displaystyle \int (c+d x)^4 \sinh ^2(a+b x) \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int -(c+d x)^4 \sin (i a+i b x)^2dx\)

\(\Big \downarrow \) 25

\(\displaystyle -\int (c+d x)^4 \sin (i a+i b x)^2dx\)

\(\Big \downarrow \) 3792

\(\displaystyle -\frac {3 d^2 \int -(c+d x)^2 \sinh ^2(a+b x)dx}{b^2}-\frac {1}{2} \int (c+d x)^4dx-\frac {d (c+d x)^3 \sinh ^2(a+b x)}{b^2}+\frac {(c+d x)^4 \sinh (a+b x) \cosh (a+b x)}{2 b}\)

\(\Big \downarrow \) 17

\(\displaystyle -\frac {3 d^2 \int -(c+d x)^2 \sinh ^2(a+b x)dx}{b^2}-\frac {d (c+d x)^3 \sinh ^2(a+b x)}{b^2}+\frac {(c+d x)^4 \sinh (a+b x) \cosh (a+b x)}{2 b}-\frac {(c+d x)^5}{10 d}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {3 d^2 \int (c+d x)^2 \sinh ^2(a+b x)dx}{b^2}-\frac {d (c+d x)^3 \sinh ^2(a+b x)}{b^2}+\frac {(c+d x)^4 \sinh (a+b x) \cosh (a+b x)}{2 b}-\frac {(c+d x)^5}{10 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {3 d^2 \int -(c+d x)^2 \sin (i a+i b x)^2dx}{b^2}-\frac {d (c+d x)^3 \sinh ^2(a+b x)}{b^2}+\frac {(c+d x)^4 \sinh (a+b x) \cosh (a+b x)}{2 b}-\frac {(c+d x)^5}{10 d}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {3 d^2 \int (c+d x)^2 \sin (i a+i b x)^2dx}{b^2}-\frac {d (c+d x)^3 \sinh ^2(a+b x)}{b^2}+\frac {(c+d x)^4 \sinh (a+b x) \cosh (a+b x)}{2 b}-\frac {(c+d x)^5}{10 d}\)

\(\Big \downarrow \) 3792

\(\displaystyle -\frac {3 d^2 \left (\frac {d^2 \int -\sinh ^2(a+b x)dx}{2 b^2}+\frac {1}{2} \int (c+d x)^2dx+\frac {d (c+d x) \sinh ^2(a+b x)}{2 b^2}-\frac {(c+d x)^2 \sinh (a+b x) \cosh (a+b x)}{2 b}\right )}{b^2}-\frac {d (c+d x)^3 \sinh ^2(a+b x)}{b^2}+\frac {(c+d x)^4 \sinh (a+b x) \cosh (a+b x)}{2 b}-\frac {(c+d x)^5}{10 d}\)

\(\Big \downarrow \) 17

\(\displaystyle -\frac {3 d^2 \left (\frac {d^2 \int -\sinh ^2(a+b x)dx}{2 b^2}+\frac {d (c+d x) \sinh ^2(a+b x)}{2 b^2}-\frac {(c+d x)^2 \sinh (a+b x) \cosh (a+b x)}{2 b}+\frac {(c+d x)^3}{6 d}\right )}{b^2}-\frac {d (c+d x)^3 \sinh ^2(a+b x)}{b^2}+\frac {(c+d x)^4 \sinh (a+b x) \cosh (a+b x)}{2 b}-\frac {(c+d x)^5}{10 d}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {3 d^2 \left (-\frac {d^2 \int \sinh ^2(a+b x)dx}{2 b^2}+\frac {d (c+d x) \sinh ^2(a+b x)}{2 b^2}-\frac {(c+d x)^2 \sinh (a+b x) \cosh (a+b x)}{2 b}+\frac {(c+d x)^3}{6 d}\right )}{b^2}-\frac {d (c+d x)^3 \sinh ^2(a+b x)}{b^2}+\frac {(c+d x)^4 \sinh (a+b x) \cosh (a+b x)}{2 b}-\frac {(c+d x)^5}{10 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {3 d^2 \left (-\frac {d^2 \int -\sin (i a+i b x)^2dx}{2 b^2}+\frac {d (c+d x) \sinh ^2(a+b x)}{2 b^2}-\frac {(c+d x)^2 \sinh (a+b x) \cosh (a+b x)}{2 b}+\frac {(c+d x)^3}{6 d}\right )}{b^2}-\frac {d (c+d x)^3 \sinh ^2(a+b x)}{b^2}+\frac {(c+d x)^4 \sinh (a+b x) \cosh (a+b x)}{2 b}-\frac {(c+d x)^5}{10 d}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {3 d^2 \left (\frac {d^2 \int \sin (i a+i b x)^2dx}{2 b^2}+\frac {d (c+d x) \sinh ^2(a+b x)}{2 b^2}-\frac {(c+d x)^2 \sinh (a+b x) \cosh (a+b x)}{2 b}+\frac {(c+d x)^3}{6 d}\right )}{b^2}-\frac {d (c+d x)^3 \sinh ^2(a+b x)}{b^2}+\frac {(c+d x)^4 \sinh (a+b x) \cosh (a+b x)}{2 b}-\frac {(c+d x)^5}{10 d}\)

\(\Big \downarrow \) 3115

\(\displaystyle -\frac {3 d^2 \left (\frac {d^2 \left (\frac {\int 1dx}{2}-\frac {\sinh (a+b x) \cosh (a+b x)}{2 b}\right )}{2 b^2}+\frac {d (c+d x) \sinh ^2(a+b x)}{2 b^2}-\frac {(c+d x)^2 \sinh (a+b x) \cosh (a+b x)}{2 b}+\frac {(c+d x)^3}{6 d}\right )}{b^2}-\frac {d (c+d x)^3 \sinh ^2(a+b x)}{b^2}+\frac {(c+d x)^4 \sinh (a+b x) \cosh (a+b x)}{2 b}-\frac {(c+d x)^5}{10 d}\)

\(\Big \downarrow \) 24

\(\displaystyle -\frac {3 d^2 \left (\frac {d (c+d x) \sinh ^2(a+b x)}{2 b^2}+\frac {d^2 \left (\frac {x}{2}-\frac {\sinh (a+b x) \cosh (a+b x)}{2 b}\right )}{2 b^2}-\frac {(c+d x)^2 \sinh (a+b x) \cosh (a+b x)}{2 b}+\frac {(c+d x)^3}{6 d}\right )}{b^2}-\frac {d (c+d x)^3 \sinh ^2(a+b x)}{b^2}+\frac {(c+d x)^4 \sinh (a+b x) \cosh (a+b x)}{2 b}-\frac {(c+d x)^5}{10 d}\)

Input:

Int[(c + d*x)^4*Sinh[a + b*x]^2,x]
 

Output:

-1/10*(c + d*x)^5/d + ((c + d*x)^4*Cosh[a + b*x]*Sinh[a + b*x])/(2*b) - (d 
*(c + d*x)^3*Sinh[a + b*x]^2)/b^2 - (3*d^2*((c + d*x)^3/(6*d) - ((c + d*x) 
^2*Cosh[a + b*x]*Sinh[a + b*x])/(2*b) + (d*(c + d*x)*Sinh[a + b*x]^2)/(2*b 
^2) + (d^2*(x/2 - (Cosh[a + b*x]*Sinh[a + b*x])/(2*b)))/(2*b^2)))/b^2
 

Defintions of rubi rules used

rule 17

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

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 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 3792

Int[((c_.) + (d_.)*(x_))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbo 
l] :> Simp[d*m*(c + d*x)^(m - 1)*((b*Sin[e + f*x])^n/(f^2*n^2)), x] + (-Sim 
p[b*(c + d*x)^m*Cos[e + f*x]*((b*Sin[e + f*x])^(n - 1)/(f*n)), x] + Simp[b^ 
2*((n - 1)/n)   Int[(c + d*x)^m*(b*Sin[e + f*x])^(n - 2), x], x] - Simp[d^2 
*m*((m - 1)/(f^2*n^2))   Int[(c + d*x)^(m - 2)*(b*Sin[e + f*x])^n, x], x]) 
/; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1] && GtQ[m, 1]
 

Maple [A] (verified)

Time = 0.45 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.90

method	result	size
parallelrisch	\(\frac {\left (2 \left (d x +c \right )^{4} b^{4}+6 d^{2} \left (d x +c \right )^{2} b^{2}+3 d^{4}\right ) \sinh \left (2 b x +2 a \right )-4 b \left (d \left (\left (d x +c \right )^{2} b^{2}+\frac {3 d^{2}}{2}\right ) \left (d x +c \right ) \cosh \left (2 b x +2 a \right )+x \left (\frac {1}{5} d^{4} x^{4}+c \,d^{3} x^{3}+2 c^{2} d^{2} x^{2}+2 c^{3} d x +c^{4}\right ) b^{4}-b^{2} c^{3} d -\frac {3 d^{3} c}{2}\right )}{8 b^{5}}\)	\(145\)
risch	\(-\frac {d^{4} x^{5}}{10}-\frac {d^{3} c \,x^{4}}{2}-d^{2} c^{2} x^{3}-d \,c^{3} x^{2}-\frac {c^{4} x}{2}-\frac {c^{5}}{10 d}+\frac {\left (2 d^{4} x^{4} b^{4}+8 b^{4} c \,d^{3} x^{3}+12 b^{4} c^{2} d^{2} x^{2}-4 b^{3} d^{4} x^{3}+8 b^{4} c^{3} d x -12 b^{3} c \,d^{3} x^{2}+2 b^{4} c^{4}-12 b^{3} c^{2} d^{2} x +6 b^{2} d^{4} x^{2}-4 b^{3} c^{3} d +12 b^{2} c \,d^{3} x +6 b^{2} c^{2} d^{2}-6 b \,d^{4} x -6 b c \,d^{3}+3 d^{4}\right ) {\mathrm e}^{2 b x +2 a}}{16 b^{5}}-\frac {\left (2 d^{4} x^{4} b^{4}+8 b^{4} c \,d^{3} x^{3}+12 b^{4} c^{2} d^{2} x^{2}+4 b^{3} d^{4} x^{3}+8 b^{4} c^{3} d x +12 b^{3} c \,d^{3} x^{2}+2 b^{4} c^{4}+12 b^{3} c^{2} d^{2} x +6 b^{2} d^{4} x^{2}+4 b^{3} c^{3} d +12 b^{2} c \,d^{3} x +6 b^{2} c^{2} d^{2}+6 b \,d^{4} x +6 b c \,d^{3}+3 d^{4}\right ) {\mathrm e}^{-2 b x -2 a}}{16 b^{5}}\)	\(383\)
orering	\(\frac {\left (2 b^{6} d^{6} x^{7}+14 b^{6} c \,d^{5} x^{6}+42 b^{6} c^{2} d^{4} x^{5}+70 b^{6} c^{3} d^{3} x^{4}+70 b^{6} c^{4} d^{2} x^{3}-20 b^{4} d^{6} x^{5}+40 b^{6} c^{5} d \,x^{2}-100 b^{4} c \,d^{5} x^{4}+10 b^{6} c^{6} x -200 b^{4} c^{2} d^{4} x^{3}-210 b^{4} c^{3} d^{3} x^{2}-120 b^{4} c^{4} d^{2} x -80 b^{2} d^{6} x^{3}-20 b^{4} c^{5} d -255 b^{2} c \,d^{5} x^{2}-270 b^{2} c^{2} d^{4} x -45 b^{2} c^{3} d^{3}-90 d^{6} x -15 c \,d^{5}\right ) \sinh \left (b x +a \right )^{2}}{10 b^{6} \left (d x +c \right )^{2}}+\frac {\left (26 b^{4} d^{5} x^{5}+130 b^{4} c \,d^{4} x^{4}+260 b^{4} c^{2} d^{3} x^{3}+260 b^{4} c^{3} d^{2} x^{2}+130 b^{4} c^{4} d x +110 b^{2} d^{5} x^{3}+10 b^{4} c^{5}+330 b^{2} c \,d^{4} x^{2}+330 b^{2} c^{2} d^{3} x +30 b^{2} c^{3} d^{2}+135 d^{5} x +15 c \,d^{4}\right ) \left (4 \left (d x +c \right )^{3} \sinh \left (b x +a \right )^{2} d +2 \left (d x +c \right )^{4} \sinh \left (b x +a \right ) b \cosh \left (b x +a \right )\right )}{40 b^{6} \left (d x +c \right )^{5}}-\frac {x \left (2 d^{4} x^{4} b^{4}+10 b^{4} c \,d^{3} x^{3}+20 b^{4} c^{2} d^{2} x^{2}+20 b^{4} c^{3} d x +10 b^{4} c^{4}+10 b^{2} d^{4} x^{2}+30 b^{2} c \,d^{3} x +30 b^{2} c^{2} d^{2}+15 d^{4}\right ) \left (12 \left (d x +c \right )^{2} \sinh \left (b x +a \right )^{2} d^{2}+16 \left (d x +c \right )^{3} \sinh \left (b x +a \right ) d b \cosh \left (b x +a \right )+2 \left (d x +c \right )^{4} b^{2} \cosh \left (b x +a \right )^{2}+2 \left (d x +c \right )^{4} \sinh \left (b x +a \right )^{2} b^{2}\right )}{40 b^{6} \left (d x +c \right )^{4}}\)	\(610\)
derivativedivides	\(\text {Expression too large to display}\)	\(910\)
default	\(\text {Expression too large to display}\)	\(910\)

Input:

int((d*x+c)^4*sinh(b*x+a)^2,x,method=_RETURNVERBOSE)
 

Output:

1/8*((2*(d*x+c)^4*b^4+6*d^2*(d*x+c)^2*b^2+3*d^4)*sinh(2*b*x+2*a)-4*b*(d*(( 
d*x+c)^2*b^2+3/2*d^2)*(d*x+c)*cosh(2*b*x+2*a)+x*(1/5*d^4*x^4+c*d^3*x^3+2*c 
^2*d^2*x^2+2*c^3*d*x+c^4)*b^4-b^2*c^3*d-3/2*d^3*c))/b^5
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 312 vs. \(2 (148) = 296\).

Time = 0.09 (sec) , antiderivative size = 312, normalized size of antiderivative = 1.93 \[ \int (c+d x)^4 \sinh ^2(a+b x) \, dx=-\frac {2 \, b^{5} d^{4} x^{5} + 10 \, b^{5} c d^{3} x^{4} + 20 \, b^{5} c^{2} d^{2} x^{3} + 20 \, b^{5} c^{3} d x^{2} + 10 \, b^{5} c^{4} x + 5 \, {\left (2 \, b^{3} d^{4} x^{3} + 6 \, b^{3} c d^{3} x^{2} + 2 \, b^{3} c^{3} d + 3 \, b c d^{3} + 3 \, {\left (2 \, b^{3} c^{2} d^{2} + b d^{4}\right )} x\right )} \cosh \left (b x + a\right )^{2} - 5 \, {\left (2 \, b^{4} d^{4} x^{4} + 8 \, b^{4} c d^{3} x^{3} + 2 \, b^{4} c^{4} + 6 \, b^{2} c^{2} d^{2} + 3 \, d^{4} + 6 \, {\left (2 \, b^{4} c^{2} d^{2} + b^{2} d^{4}\right )} x^{2} + 4 \, {\left (2 \, b^{4} c^{3} d + 3 \, b^{2} c d^{3}\right )} x\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + 5 \, {\left (2 \, b^{3} d^{4} x^{3} + 6 \, b^{3} c d^{3} x^{2} + 2 \, b^{3} c^{3} d + 3 \, b c d^{3} + 3 \, {\left (2 \, b^{3} c^{2} d^{2} + b d^{4}\right )} x\right )} \sinh \left (b x + a\right )^{2}}{20 \, b^{5}} \] Input:

integrate((d*x+c)^4*sinh(b*x+a)^2,x, algorithm="fricas")
 

Output:

-1/20*(2*b^5*d^4*x^5 + 10*b^5*c*d^3*x^4 + 20*b^5*c^2*d^2*x^3 + 20*b^5*c^3* 
d*x^2 + 10*b^5*c^4*x + 5*(2*b^3*d^4*x^3 + 6*b^3*c*d^3*x^2 + 2*b^3*c^3*d + 
3*b*c*d^3 + 3*(2*b^3*c^2*d^2 + b*d^4)*x)*cosh(b*x + a)^2 - 5*(2*b^4*d^4*x^ 
4 + 8*b^4*c*d^3*x^3 + 2*b^4*c^4 + 6*b^2*c^2*d^2 + 3*d^4 + 6*(2*b^4*c^2*d^2 
 + b^2*d^4)*x^2 + 4*(2*b^4*c^3*d + 3*b^2*c*d^3)*x)*cosh(b*x + a)*sinh(b*x 
+ a) + 5*(2*b^3*d^4*x^3 + 6*b^3*c*d^3*x^2 + 2*b^3*c^3*d + 3*b*c*d^3 + 3*(2 
*b^3*c^2*d^2 + b*d^4)*x)*sinh(b*x + a)^2)/b^5
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 660 vs. \(2 (156) = 312\).

Time = 0.49 (sec) , antiderivative size = 660, normalized size of antiderivative = 4.07 \[ \int (c+d x)^4 \sinh ^2(a+b x) \, dx =\text {Too large to display} \] Input:

integrate((d*x+c)**4*sinh(b*x+a)**2,x)
 

Output:

Piecewise((c**4*x*sinh(a + b*x)**2/2 - c**4*x*cosh(a + b*x)**2/2 + c**3*d* 
x**2*sinh(a + b*x)**2 - c**3*d*x**2*cosh(a + b*x)**2 + c**2*d**2*x**3*sinh 
(a + b*x)**2 - c**2*d**2*x**3*cosh(a + b*x)**2 + c*d**3*x**4*sinh(a + b*x) 
**2/2 - c*d**3*x**4*cosh(a + b*x)**2/2 + d**4*x**5*sinh(a + b*x)**2/10 - d 
**4*x**5*cosh(a + b*x)**2/10 + c**4*sinh(a + b*x)*cosh(a + b*x)/(2*b) + 2* 
c**3*d*x*sinh(a + b*x)*cosh(a + b*x)/b + 3*c**2*d**2*x**2*sinh(a + b*x)*co 
sh(a + b*x)/b + 2*c*d**3*x**3*sinh(a + b*x)*cosh(a + b*x)/b + d**4*x**4*si 
nh(a + b*x)*cosh(a + b*x)/(2*b) - c**3*d*sinh(a + b*x)**2/b**2 - 3*c**2*d* 
*2*x*sinh(a + b*x)**2/(2*b**2) - 3*c**2*d**2*x*cosh(a + b*x)**2/(2*b**2) - 
 3*c*d**3*x**2*sinh(a + b*x)**2/(2*b**2) - 3*c*d**3*x**2*cosh(a + b*x)**2/ 
(2*b**2) - d**4*x**3*sinh(a + b*x)**2/(2*b**2) - d**4*x**3*cosh(a + b*x)** 
2/(2*b**2) + 3*c**2*d**2*sinh(a + b*x)*cosh(a + b*x)/(2*b**3) + 3*c*d**3*x 
*sinh(a + b*x)*cosh(a + b*x)/b**3 + 3*d**4*x**2*sinh(a + b*x)*cosh(a + b*x 
)/(2*b**3) - 3*c*d**3*sinh(a + b*x)**2/(2*b**4) - 3*d**4*x*sinh(a + b*x)** 
2/(4*b**4) - 3*d**4*x*cosh(a + b*x)**2/(4*b**4) + 3*d**4*sinh(a + b*x)*cos 
h(a + b*x)/(4*b**5), Ne(b, 0)), ((c**4*x + 2*c**3*d*x**2 + 2*c**2*d**2*x** 
3 + c*d**3*x**4 + d**4*x**5/5)*sinh(a)**2, True))
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 382 vs. \(2 (148) = 296\).

Time = 0.06 (sec) , antiderivative size = 382, normalized size of antiderivative = 2.36 \[ \int (c+d x)^4 \sinh ^2(a+b x) \, dx=-\frac {1}{4} \, {\left (4 \, x^{2} - \frac {{\left (2 \, b x e^{\left (2 \, a\right )} - e^{\left (2 \, a\right )}\right )} e^{\left (2 \, b x\right )}}{b^{2}} + \frac {{\left (2 \, b x + 1\right )} e^{\left (-2 \, b x - 2 \, a\right )}}{b^{2}}\right )} c^{3} d - \frac {1}{8} \, {\left (8 \, x^{3} - \frac {3 \, {\left (2 \, b^{2} x^{2} e^{\left (2 \, a\right )} - 2 \, b x e^{\left (2 \, a\right )} + e^{\left (2 \, a\right )}\right )} e^{\left (2 \, b x\right )}}{b^{3}} + \frac {3 \, {\left (2 \, b^{2} x^{2} + 2 \, b x + 1\right )} e^{\left (-2 \, b x - 2 \, a\right )}}{b^{3}}\right )} c^{2} d^{2} - \frac {1}{8} \, {\left (4 \, x^{4} - \frac {{\left (4 \, b^{3} x^{3} e^{\left (2 \, a\right )} - 6 \, b^{2} x^{2} e^{\left (2 \, a\right )} + 6 \, b x e^{\left (2 \, a\right )} - 3 \, e^{\left (2 \, a\right )}\right )} e^{\left (2 \, b x\right )}}{b^{4}} + \frac {{\left (4 \, b^{3} x^{3} + 6 \, b^{2} x^{2} + 6 \, b x + 3\right )} e^{\left (-2 \, b x - 2 \, a\right )}}{b^{4}}\right )} c d^{3} - \frac {1}{80} \, {\left (8 \, x^{5} - \frac {5 \, {\left (2 \, b^{4} x^{4} e^{\left (2 \, a\right )} - 4 \, b^{3} x^{3} e^{\left (2 \, a\right )} + 6 \, b^{2} x^{2} e^{\left (2 \, a\right )} - 6 \, b x e^{\left (2 \, a\right )} + 3 \, e^{\left (2 \, a\right )}\right )} e^{\left (2 \, b x\right )}}{b^{5}} + \frac {5 \, {\left (2 \, b^{4} x^{4} + 4 \, b^{3} x^{3} + 6 \, b^{2} x^{2} + 6 \, b x + 3\right )} e^{\left (-2 \, b x - 2 \, a\right )}}{b^{5}}\right )} d^{4} - \frac {1}{8} \, c^{4} {\left (4 \, x - \frac {e^{\left (2 \, b x + 2 \, a\right )}}{b} + \frac {e^{\left (-2 \, b x - 2 \, a\right )}}{b}\right )} \] Input:

integrate((d*x+c)^4*sinh(b*x+a)^2,x, algorithm="maxima")
 

Output:

-1/4*(4*x^2 - (2*b*x*e^(2*a) - e^(2*a))*e^(2*b*x)/b^2 + (2*b*x + 1)*e^(-2* 
b*x - 2*a)/b^2)*c^3*d - 1/8*(8*x^3 - 3*(2*b^2*x^2*e^(2*a) - 2*b*x*e^(2*a) 
+ e^(2*a))*e^(2*b*x)/b^3 + 3*(2*b^2*x^2 + 2*b*x + 1)*e^(-2*b*x - 2*a)/b^3) 
*c^2*d^2 - 1/8*(4*x^4 - (4*b^3*x^3*e^(2*a) - 6*b^2*x^2*e^(2*a) + 6*b*x*e^( 
2*a) - 3*e^(2*a))*e^(2*b*x)/b^4 + (4*b^3*x^3 + 6*b^2*x^2 + 6*b*x + 3)*e^(- 
2*b*x - 2*a)/b^4)*c*d^3 - 1/80*(8*x^5 - 5*(2*b^4*x^4*e^(2*a) - 4*b^3*x^3*e 
^(2*a) + 6*b^2*x^2*e^(2*a) - 6*b*x*e^(2*a) + 3*e^(2*a))*e^(2*b*x)/b^5 + 5* 
(2*b^4*x^4 + 4*b^3*x^3 + 6*b^2*x^2 + 6*b*x + 3)*e^(-2*b*x - 2*a)/b^5)*d^4 
- 1/8*c^4*(4*x - e^(2*b*x + 2*a)/b + e^(-2*b*x - 2*a)/b)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 374 vs. \(2 (148) = 296\).

Time = 0.13 (sec) , antiderivative size = 374, normalized size of antiderivative = 2.31 \[ \int (c+d x)^4 \sinh ^2(a+b x) \, dx=-\frac {1}{10} \, d^{4} x^{5} - \frac {1}{2} \, c d^{3} x^{4} - c^{2} d^{2} x^{3} - c^{3} d x^{2} - \frac {1}{2} \, c^{4} x + \frac {{\left (2 \, b^{4} d^{4} x^{4} + 8 \, b^{4} c d^{3} x^{3} + 12 \, b^{4} c^{2} d^{2} x^{2} - 4 \, b^{3} d^{4} x^{3} + 8 \, b^{4} c^{3} d x - 12 \, b^{3} c d^{3} x^{2} + 2 \, b^{4} c^{4} - 12 \, b^{3} c^{2} d^{2} x + 6 \, b^{2} d^{4} x^{2} - 4 \, b^{3} c^{3} d + 12 \, b^{2} c d^{3} x + 6 \, b^{2} c^{2} d^{2} - 6 \, b d^{4} x - 6 \, b c d^{3} + 3 \, d^{4}\right )} e^{\left (2 \, b x + 2 \, a\right )}}{16 \, b^{5}} - \frac {{\left (2 \, b^{4} d^{4} x^{4} + 8 \, b^{4} c d^{3} x^{3} + 12 \, b^{4} c^{2} d^{2} x^{2} + 4 \, b^{3} d^{4} x^{3} + 8 \, b^{4} c^{3} d x + 12 \, b^{3} c d^{3} x^{2} + 2 \, b^{4} c^{4} + 12 \, b^{3} c^{2} d^{2} x + 6 \, b^{2} d^{4} x^{2} + 4 \, b^{3} c^{3} d + 12 \, b^{2} c d^{3} x + 6 \, b^{2} c^{2} d^{2} + 6 \, b d^{4} x + 6 \, b c d^{3} + 3 \, d^{4}\right )} e^{\left (-2 \, b x - 2 \, a\right )}}{16 \, b^{5}} \] Input:

integrate((d*x+c)^4*sinh(b*x+a)^2,x, algorithm="giac")
 

Output:

-1/10*d^4*x^5 - 1/2*c*d^3*x^4 - c^2*d^2*x^3 - c^3*d*x^2 - 1/2*c^4*x + 1/16 
*(2*b^4*d^4*x^4 + 8*b^4*c*d^3*x^3 + 12*b^4*c^2*d^2*x^2 - 4*b^3*d^4*x^3 + 8 
*b^4*c^3*d*x - 12*b^3*c*d^3*x^2 + 2*b^4*c^4 - 12*b^3*c^2*d^2*x + 6*b^2*d^4 
*x^2 - 4*b^3*c^3*d + 12*b^2*c*d^3*x + 6*b^2*c^2*d^2 - 6*b*d^4*x - 6*b*c*d^ 
3 + 3*d^4)*e^(2*b*x + 2*a)/b^5 - 1/16*(2*b^4*d^4*x^4 + 8*b^4*c*d^3*x^3 + 1 
2*b^4*c^2*d^2*x^2 + 4*b^3*d^4*x^3 + 8*b^4*c^3*d*x + 12*b^3*c*d^3*x^2 + 2*b 
^4*c^4 + 12*b^3*c^2*d^2*x + 6*b^2*d^4*x^2 + 4*b^3*c^3*d + 12*b^2*c*d^3*x + 
 6*b^2*c^2*d^2 + 6*b*d^4*x + 6*b*c*d^3 + 3*d^4)*e^(-2*b*x - 2*a)/b^5
 

Mupad [B] (verification not implemented)

Time = 2.03 (sec) , antiderivative size = 334, normalized size of antiderivative = 2.06 \[ \int (c+d x)^4 \sinh ^2(a+b x) \, dx=\frac {c^4\,\mathrm {sinh}\left (2\,a+2\,b\,x\right )}{4\,b}-\frac {d^4\,x^5}{10}-c^3\,d\,x^2-\frac {c\,d^3\,x^4}{2}-\frac {c^4\,x}{2}+\frac {3\,d^4\,\mathrm {sinh}\left (2\,a+2\,b\,x\right )}{8\,b^5}-c^2\,d^2\,x^3-\frac {c^3\,d\,\mathrm {cosh}\left (2\,a+2\,b\,x\right )}{2\,b^2}-\frac {3\,c\,d^3\,\mathrm {cosh}\left (2\,a+2\,b\,x\right )}{4\,b^4}-\frac {3\,d^4\,x\,\mathrm {cosh}\left (2\,a+2\,b\,x\right )}{4\,b^4}+\frac {3\,c^2\,d^2\,\mathrm {sinh}\left (2\,a+2\,b\,x\right )}{4\,b^3}-\frac {d^4\,x^3\,\mathrm {cosh}\left (2\,a+2\,b\,x\right )}{2\,b^2}+\frac {d^4\,x^4\,\mathrm {sinh}\left (2\,a+2\,b\,x\right )}{4\,b}+\frac {3\,d^4\,x^2\,\mathrm {sinh}\left (2\,a+2\,b\,x\right )}{4\,b^3}+\frac {3\,c^2\,d^2\,x^2\,\mathrm {sinh}\left (2\,a+2\,b\,x\right )}{2\,b}+\frac {c^3\,d\,x\,\mathrm {sinh}\left (2\,a+2\,b\,x\right )}{b}+\frac {3\,c\,d^3\,x\,\mathrm {sinh}\left (2\,a+2\,b\,x\right )}{2\,b^3}-\frac {3\,c^2\,d^2\,x\,\mathrm {cosh}\left (2\,a+2\,b\,x\right )}{2\,b^2}-\frac {3\,c\,d^3\,x^2\,\mathrm {cosh}\left (2\,a+2\,b\,x\right )}{2\,b^2}+\frac {c\,d^3\,x^3\,\mathrm {sinh}\left (2\,a+2\,b\,x\right )}{b} \] Input:

int(sinh(a + b*x)^2*(c + d*x)^4,x)
 

Output:

(c^4*sinh(2*a + 2*b*x))/(4*b) - (d^4*x^5)/10 - c^3*d*x^2 - (c*d^3*x^4)/2 - 
 (c^4*x)/2 + (3*d^4*sinh(2*a + 2*b*x))/(8*b^5) - c^2*d^2*x^3 - (c^3*d*cosh 
(2*a + 2*b*x))/(2*b^2) - (3*c*d^3*cosh(2*a + 2*b*x))/(4*b^4) - (3*d^4*x*co 
sh(2*a + 2*b*x))/(4*b^4) + (3*c^2*d^2*sinh(2*a + 2*b*x))/(4*b^3) - (d^4*x^ 
3*cosh(2*a + 2*b*x))/(2*b^2) + (d^4*x^4*sinh(2*a + 2*b*x))/(4*b) + (3*d^4* 
x^2*sinh(2*a + 2*b*x))/(4*b^3) + (3*c^2*d^2*x^2*sinh(2*a + 2*b*x))/(2*b) + 
 (c^3*d*x*sinh(2*a + 2*b*x))/b + (3*c*d^3*x*sinh(2*a + 2*b*x))/(2*b^3) - ( 
3*c^2*d^2*x*cosh(2*a + 2*b*x))/(2*b^2) - (3*c*d^3*x^2*cosh(2*a + 2*b*x))/( 
2*b^2) + (c*d^3*x^3*sinh(2*a + 2*b*x))/b
 

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 576, normalized size of antiderivative = 3.56 \[ \int (c+d x)^4 \sinh ^2(a+b x) \, dx=\frac {15 e^{4 b x +4 a} d^{4}-10 b^{4} c^{4}-15 d^{4}+10 e^{4 b x +4 a} b^{4} d^{4} x^{4}-20 e^{4 b x +4 a} b^{3} c^{3} d -20 e^{4 b x +4 a} b^{3} d^{4} x^{3}+30 e^{4 b x +4 a} b^{2} c^{2} d^{2}+30 e^{4 b x +4 a} b^{2} d^{4} x^{2}-30 e^{4 b x +4 a} b c \,d^{3}-30 e^{4 b x +4 a} b \,d^{4} x -40 e^{2 b x +2 a} b^{5} c^{4} x -8 e^{2 b x +2 a} b^{5} d^{4} x^{5}-40 b^{4} c^{3} d x -60 b^{4} c^{2} d^{2} x^{2}-40 b^{4} c \,d^{3} x^{3}-60 b^{3} c^{2} d^{2} x -60 b^{3} c \,d^{3} x^{2}-60 b^{2} c \,d^{3} x +10 e^{4 b x +4 a} b^{4} c^{4}-10 b^{4} d^{4} x^{4}-20 b^{3} c^{3} d -20 b^{3} d^{4} x^{3}-30 b^{2} c^{2} d^{2}-30 b^{2} d^{4} x^{2}-30 b c \,d^{3}-30 b \,d^{4} x +40 e^{4 b x +4 a} b^{4} c^{3} d x +60 e^{4 b x +4 a} b^{4} c^{2} d^{2} x^{2}+40 e^{4 b x +4 a} b^{4} c \,d^{3} x^{3}-60 e^{4 b x +4 a} b^{3} c^{2} d^{2} x -60 e^{4 b x +4 a} b^{3} c \,d^{3} x^{2}+60 e^{4 b x +4 a} b^{2} c \,d^{3} x -80 e^{2 b x +2 a} b^{5} c^{3} d \,x^{2}-80 e^{2 b x +2 a} b^{5} c^{2} d^{2} x^{3}-40 e^{2 b x +2 a} b^{5} c \,d^{3} x^{4}}{80 e^{2 b x +2 a} b^{5}} \] Input:

int((d*x+c)^4*sinh(b*x+a)^2,x)
 

Output:

(10*e**(4*a + 4*b*x)*b**4*c**4 + 40*e**(4*a + 4*b*x)*b**4*c**3*d*x + 60*e* 
*(4*a + 4*b*x)*b**4*c**2*d**2*x**2 + 40*e**(4*a + 4*b*x)*b**4*c*d**3*x**3 
+ 10*e**(4*a + 4*b*x)*b**4*d**4*x**4 - 20*e**(4*a + 4*b*x)*b**3*c**3*d - 6 
0*e**(4*a + 4*b*x)*b**3*c**2*d**2*x - 60*e**(4*a + 4*b*x)*b**3*c*d**3*x**2 
 - 20*e**(4*a + 4*b*x)*b**3*d**4*x**3 + 30*e**(4*a + 4*b*x)*b**2*c**2*d**2 
 + 60*e**(4*a + 4*b*x)*b**2*c*d**3*x + 30*e**(4*a + 4*b*x)*b**2*d**4*x**2 
- 30*e**(4*a + 4*b*x)*b*c*d**3 - 30*e**(4*a + 4*b*x)*b*d**4*x + 15*e**(4*a 
 + 4*b*x)*d**4 - 40*e**(2*a + 2*b*x)*b**5*c**4*x - 80*e**(2*a + 2*b*x)*b** 
5*c**3*d*x**2 - 80*e**(2*a + 2*b*x)*b**5*c**2*d**2*x**3 - 40*e**(2*a + 2*b 
*x)*b**5*c*d**3*x**4 - 8*e**(2*a + 2*b*x)*b**5*d**4*x**5 - 10*b**4*c**4 - 
40*b**4*c**3*d*x - 60*b**4*c**2*d**2*x**2 - 40*b**4*c*d**3*x**3 - 10*b**4* 
d**4*x**4 - 20*b**3*c**3*d - 60*b**3*c**2*d**2*x - 60*b**3*c*d**3*x**2 - 2 
0*b**3*d**4*x**3 - 30*b**2*c**2*d**2 - 60*b**2*c*d**3*x - 30*b**2*d**4*x** 
2 - 30*b*c*d**3 - 30*b*d**4*x - 15*d**4)/(80*e**(2*a + 2*b*x)*b**5)