\(\int (c+d x)^3 \sinh ^2(a+b x) \, dx\) [9]

Optimal result

Integrand size = 16, antiderivative size = 124 \[ \int (c+d x)^3 \sinh ^2(a+b x) \, dx=-\frac {3 d (c+d x)^2}{8 b^2}-\frac {(c+d x)^4}{8 d}+\frac {3 d^2 (c+d x) \cosh (a+b x) \sinh (a+b x)}{4 b^3}+\frac {(c+d x)^3 \cosh (a+b x) \sinh (a+b x)}{2 b}-\frac {3 d^3 \sinh ^2(a+b x)}{8 b^4}-\frac {3 d (c+d x)^2 \sinh ^2(a+b x)}{4 b^2} \] Output:

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

Mathematica [A] (verified)

Time = 0.25 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.84 \[ \int (c+d x)^3 \sinh ^2(a+b x) \, dx=\frac {-2 b^4 x \left (4 c^3+6 c^2 d x+4 c d^2 x^2+d^3 x^3\right )-3 d \left (d^2+2 b^2 (c+d x)^2\right ) \cosh (2 (a+b x))+2 b (c+d x) \left (3 d^2+2 b^2 (c+d x)^2\right ) \sinh (2 (a+b x))}{16 b^4} \] Input:

Integrate[(c + d*x)^3*Sinh[a + b*x]^2,x]
 

Output:

(-2*b^4*x*(4*c^3 + 6*c^2*d*x + 4*c*d^2*x^2 + d^3*x^3) - 3*d*(d^2 + 2*b^2*( 
c + d*x)^2)*Cosh[2*(a + b*x)] + 2*b*(c + d*x)*(3*d^2 + 2*b^2*(c + d*x)^2)* 
Sinh[2*(a + b*x)])/(16*b^4)
 

Rubi [A] (verified)

Time = 0.37 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.04, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.562, Rules used = {3042, 25, 3792, 17, 25, 3042, 25, 3791, 17}

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.

\(\displaystyle \int (c+d x)^3 \sinh ^2(a+b x) \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int -(c+d x)^3 \sin (i a+i b x)^2dx\)

\(\Big \downarrow \) 25

\(\displaystyle -\int (c+d x)^3 \sin (i a+i b x)^2dx\)

\(\Big \downarrow \) 3792

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

\(\Big \downarrow \) 17

\(\displaystyle -\frac {3 d^2 \int -\left ((c+d x) \sinh ^2(a+b x)\right )dx}{2 b^2}-\frac {3 d (c+d x)^2 \sinh ^2(a+b x)}{4 b^2}+\frac {(c+d x)^3 \sinh (a+b x) \cosh (a+b x)}{2 b}-\frac {(c+d x)^4}{8 d}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {3 d^2 \int (c+d x) \sinh ^2(a+b x)dx}{2 b^2}-\frac {3 d (c+d x)^2 \sinh ^2(a+b x)}{4 b^2}+\frac {(c+d x)^3 \sinh (a+b x) \cosh (a+b x)}{2 b}-\frac {(c+d x)^4}{8 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {3 d^2 \int -\left ((c+d x) \sin (i a+i b x)^2\right )dx}{2 b^2}-\frac {3 d (c+d x)^2 \sinh ^2(a+b x)}{4 b^2}+\frac {(c+d x)^3 \sinh (a+b x) \cosh (a+b x)}{2 b}-\frac {(c+d x)^4}{8 d}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {3 d^2 \int (c+d x) \sin (i a+i b x)^2dx}{2 b^2}-\frac {3 d (c+d x)^2 \sinh ^2(a+b x)}{4 b^2}+\frac {(c+d x)^3 \sinh (a+b x) \cosh (a+b x)}{2 b}-\frac {(c+d x)^4}{8 d}\)

\(\Big \downarrow \) 3791

\(\displaystyle -\frac {3 d^2 \left (\frac {1}{2} \int (c+d x)dx+\frac {d \sinh ^2(a+b x)}{4 b^2}-\frac {(c+d x) \sinh (a+b x) \cosh (a+b x)}{2 b}\right )}{2 b^2}-\frac {3 d (c+d x)^2 \sinh ^2(a+b x)}{4 b^2}+\frac {(c+d x)^3 \sinh (a+b x) \cosh (a+b x)}{2 b}-\frac {(c+d x)^4}{8 d}\)

\(\Big \downarrow \) 17

\(\displaystyle -\frac {3 d^2 \left (\frac {d \sinh ^2(a+b x)}{4 b^2}-\frac {(c+d x) \sinh (a+b x) \cosh (a+b x)}{2 b}+\frac {(c+d x)^2}{4 d}\right )}{2 b^2}-\frac {3 d (c+d x)^2 \sinh ^2(a+b x)}{4 b^2}+\frac {(c+d x)^3 \sinh (a+b x) \cosh (a+b x)}{2 b}-\frac {(c+d x)^4}{8 d}\)

Input:

Int[(c + d*x)^3*Sinh[a + b*x]^2,x]
 

Output:

-1/8*(c + d*x)^4/d + ((c + d*x)^3*Cosh[a + b*x]*Sinh[a + b*x])/(2*b) - (3* 
d*(c + d*x)^2*Sinh[a + b*x]^2)/(4*b^2) - (3*d^2*((c + d*x)^2/(4*d) - ((c + 
 d*x)*Cosh[a + b*x]*Sinh[a + b*x])/(2*b) + (d*Sinh[a + b*x]^2)/(4*b^2)))/( 
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 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 3791

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

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 = 0.36 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.98

method	result	size
parallelrisch	\(\frac {4 \left (\left (d x +c \right )^{2} b^{2}+\frac {3 d^{2}}{2}\right ) b \left (d x +c \right ) \sinh \left (2 b x +2 a \right )-6 \left (\left (d x +c \right )^{2} b^{2}+\frac {d^{2}}{2}\right ) d \cosh \left (2 b x +2 a \right )+\left (-2 d^{3} x^{4}-8 d^{2} c \,x^{3}-12 d \,c^{2} x^{2}-8 c^{3} x \right ) b^{4}+6 b^{2} c^{2} d +3 d^{3}}{16 b^{4}}\)	\(121\)
risch	\(-\frac {d^{3} x^{4}}{8}-\frac {d^{2} c \,x^{3}}{2}-\frac {3 d \,c^{2} x^{2}}{4}-\frac {c^{3} x}{2}-\frac {c^{4}}{8 d}+\frac {\left (4 d^{3} x^{3} b^{3}+12 b^{3} c \,d^{2} x^{2}+12 b^{3} c^{2} d x -6 b^{2} d^{3} x^{2}+4 b^{3} c^{3}-12 b^{2} c \,d^{2} x -6 b^{2} c^{2} d +6 b \,d^{3} x +6 b c \,d^{2}-3 d^{3}\right ) {\mathrm e}^{2 b x +2 a}}{32 b^{4}}-\frac {\left (4 d^{3} x^{3} b^{3}+12 b^{3} c \,d^{2} x^{2}+12 b^{3} c^{2} d x +6 b^{2} d^{3} x^{2}+4 b^{3} c^{3}+12 b^{2} c \,d^{2} x +6 b^{2} c^{2} d +6 b \,d^{3} x +6 b c \,d^{2}+3 d^{3}\right ) {\mathrm e}^{-2 b x -2 a}}{32 b^{4}}\)	\(252\)
orering	\(\frac {\left (b^{4} d^{5} x^{6}+6 b^{4} c \,d^{4} x^{5}+15 b^{4} c^{2} d^{3} x^{4}+20 b^{4} c^{3} d^{2} x^{3}+14 b^{4} c^{4} d \,x^{2}-6 b^{2} d^{5} x^{4}+4 b^{4} c^{5} x -24 b^{2} c \,d^{4} x^{3}-39 b^{2} c^{2} d^{3} x^{2}-30 b^{2} c^{3} d^{2} x -6 b^{2} c^{4} d -15 d^{5} x^{2}-30 d^{4} c x -6 d^{3} c^{2}\right ) \sinh \left (b x +a \right )^{2}}{4 b^{4} \left (d x +c \right )^{2}}+\frac {\left (5 b^{2} d^{4} x^{4}+20 b^{2} c \,d^{3} x^{3}+30 b^{2} c^{2} d^{2} x^{2}+20 b^{2} c^{3} d x +2 b^{2} c^{4}+12 d^{4} x^{2}+24 d^{3} c x +3 d^{2} c^{2}\right ) \left (3 \left (d x +c \right )^{2} \sinh \left (b x +a \right )^{2} d +2 \left (d x +c \right )^{3} \sinh \left (b x +a \right ) b \cosh \left (b x +a \right )\right )}{8 \left (d x +c \right )^{4} b^{4}}-\frac {x \left (b^{2} d^{3} x^{3}+4 b^{2} c \,d^{2} x^{2}+6 b^{2} c^{2} d x +4 b^{2} c^{3}+3 d^{3} x +6 d^{2} c \right ) \left (6 \left (d x +c \right ) \sinh \left (b x +a \right )^{2} d^{2}+12 \left (d x +c \right )^{2} \sinh \left (b x +a \right ) d b \cosh \left (b x +a \right )+2 \left (d x +c \right )^{3} b^{2} \cosh \left (b x +a \right )^{2}+2 \left (d x +c \right )^{3} \sinh \left (b x +a \right )^{2} b^{2}\right )}{16 b^{4} \left (d x +c \right )^{3}}\)	\(455\)
derivativedivides	\(\frac {\frac {d^{3} \left (\frac {\left (b x +a \right )^{3} \cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{2}-\frac {\left (b x +a \right )^{4}}{8}-\frac {3 \left (b x +a \right )^{2} \cosh \left (b x +a \right )^{2}}{4}+\frac {3 \left (b x +a \right ) \cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{4}+\frac {3 \left (b x +a \right )^{2}}{8}-\frac {3 \cosh \left (b x +a \right )^{2}}{8}\right )}{b^{3}}-\frac {3 d^{3} a \left (\frac {\left (b x +a \right )^{2} \cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{2}-\frac {\left (b x +a \right )^{3}}{6}-\frac {\left (b x +a \right ) \cosh \left (b x +a \right )^{2}}{2}+\frac {\cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{4}+\frac {b x}{4}+\frac {a}{4}\right )}{b^{3}}+\frac {3 d^{2} c \left (\frac {\left (b x +a \right )^{2} \cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{2}-\frac {\left (b x +a \right )^{3}}{6}-\frac {\left (b x +a \right ) \cosh \left (b x +a \right )^{2}}{2}+\frac {\cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{4}+\frac {b x}{4}+\frac {a}{4}\right )}{b^{2}}+\frac {3 d^{3} a^{2} \left (\frac {\left (b x +a \right ) \cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{2}-\frac {\left (b x +a \right )^{2}}{4}-\frac {\cosh \left (b x +a \right )^{2}}{4}\right )}{b^{3}}-\frac {6 d^{2} a c \left (\frac {\left (b x +a \right ) \cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{2}-\frac {\left (b x +a \right )^{2}}{4}-\frac {\cosh \left (b x +a \right )^{2}}{4}\right )}{b^{2}}+\frac {3 d \,c^{2} \left (\frac {\left (b x +a \right ) \cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{2}-\frac {\left (b x +a \right )^{2}}{4}-\frac {\cosh \left (b x +a \right )^{2}}{4}\right )}{b}-\frac {d^{3} a^{3} \left (\frac {\cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{2}-\frac {b x}{2}-\frac {a}{2}\right )}{b^{3}}+\frac {3 d^{2} a^{2} c \left (\frac {\cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{2}-\frac {b x}{2}-\frac {a}{2}\right )}{b^{2}}-\frac {3 d a \,c^{2} \left (\frac {\cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{2}-\frac {b x}{2}-\frac {a}{2}\right )}{b}+c^{3} \left (\frac {\cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{2}-\frac {b x}{2}-\frac {a}{2}\right )}{b}\)	\(523\)
default	\(\frac {\frac {d^{3} \left (\frac {\left (b x +a \right )^{3} \cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{2}-\frac {\left (b x +a \right )^{4}}{8}-\frac {3 \left (b x +a \right )^{2} \cosh \left (b x +a \right )^{2}}{4}+\frac {3 \left (b x +a \right ) \cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{4}+\frac {3 \left (b x +a \right )^{2}}{8}-\frac {3 \cosh \left (b x +a \right )^{2}}{8}\right )}{b^{3}}-\frac {3 d^{3} a \left (\frac {\left (b x +a \right )^{2} \cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{2}-\frac {\left (b x +a \right )^{3}}{6}-\frac {\left (b x +a \right ) \cosh \left (b x +a \right )^{2}}{2}+\frac {\cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{4}+\frac {b x}{4}+\frac {a}{4}\right )}{b^{3}}+\frac {3 d^{2} c \left (\frac {\left (b x +a \right )^{2} \cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{2}-\frac {\left (b x +a \right )^{3}}{6}-\frac {\left (b x +a \right ) \cosh \left (b x +a \right )^{2}}{2}+\frac {\cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{4}+\frac {b x}{4}+\frac {a}{4}\right )}{b^{2}}+\frac {3 d^{3} a^{2} \left (\frac {\left (b x +a \right ) \cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{2}-\frac {\left (b x +a \right )^{2}}{4}-\frac {\cosh \left (b x +a \right )^{2}}{4}\right )}{b^{3}}-\frac {6 d^{2} a c \left (\frac {\left (b x +a \right ) \cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{2}-\frac {\left (b x +a \right )^{2}}{4}-\frac {\cosh \left (b x +a \right )^{2}}{4}\right )}{b^{2}}+\frac {3 d \,c^{2} \left (\frac {\left (b x +a \right ) \cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{2}-\frac {\left (b x +a \right )^{2}}{4}-\frac {\cosh \left (b x +a \right )^{2}}{4}\right )}{b}-\frac {d^{3} a^{3} \left (\frac {\cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{2}-\frac {b x}{2}-\frac {a}{2}\right )}{b^{3}}+\frac {3 d^{2} a^{2} c \left (\frac {\cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{2}-\frac {b x}{2}-\frac {a}{2}\right )}{b^{2}}-\frac {3 d a \,c^{2} \left (\frac {\cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{2}-\frac {b x}{2}-\frac {a}{2}\right )}{b}+c^{3} \left (\frac {\cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{2}-\frac {b x}{2}-\frac {a}{2}\right )}{b}\)	\(523\)

Input:

int((d*x+c)^3*sinh(b*x+a)^2,x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.69 \[ \int (c+d x)^3 \sinh ^2(a+b x) \, dx=-\frac {2 \, b^{4} d^{3} x^{4} + 8 \, b^{4} c d^{2} x^{3} + 12 \, b^{4} c^{2} d x^{2} + 8 \, b^{4} c^{3} x + 3 \, {\left (2 \, b^{2} d^{3} x^{2} + 4 \, b^{2} c d^{2} x + 2 \, b^{2} c^{2} d + d^{3}\right )} \cosh \left (b x + a\right )^{2} - 4 \, {\left (2 \, b^{3} d^{3} x^{3} + 6 \, b^{3} c d^{2} x^{2} + 2 \, b^{3} c^{3} + 3 \, b c d^{2} + 3 \, {\left (2 \, b^{3} c^{2} d + b d^{3}\right )} x\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + 3 \, {\left (2 \, b^{2} d^{3} x^{2} + 4 \, b^{2} c d^{2} x + 2 \, b^{2} c^{2} d + d^{3}\right )} \sinh \left (b x + a\right )^{2}}{16 \, b^{4}} \] Input:

integrate((d*x+c)^3*sinh(b*x+a)^2,x, algorithm="fricas")
 

Output:

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 456 vs. \(2 (119) = 238\).

Time = 0.37 (sec) , antiderivative size = 456, normalized size of antiderivative = 3.68 \[ \int (c+d x)^3 \sinh ^2(a+b x) \, dx=\begin {cases} \frac {c^{3} x \sinh ^{2}{\left (a + b x \right )}}{2} - \frac {c^{3} x \cosh ^{2}{\left (a + b x \right )}}{2} + \frac {3 c^{2} d x^{2} \sinh ^{2}{\left (a + b x \right )}}{4} - \frac {3 c^{2} d x^{2} \cosh ^{2}{\left (a + b x \right )}}{4} + \frac {c d^{2} x^{3} \sinh ^{2}{\left (a + b x \right )}}{2} - \frac {c d^{2} x^{3} \cosh ^{2}{\left (a + b x \right )}}{2} + \frac {d^{3} x^{4} \sinh ^{2}{\left (a + b x \right )}}{8} - \frac {d^{3} x^{4} \cosh ^{2}{\left (a + b x \right )}}{8} + \frac {c^{3} \sinh {\left (a + b x \right )} \cosh {\left (a + b x \right )}}{2 b} + \frac {3 c^{2} d x \sinh {\left (a + b x \right )} \cosh {\left (a + b x \right )}}{2 b} + \frac {3 c d^{2} x^{2} \sinh {\left (a + b x \right )} \cosh {\left (a + b x \right )}}{2 b} + \frac {d^{3} x^{3} \sinh {\left (a + b x \right )} \cosh {\left (a + b x \right )}}{2 b} - \frac {3 c^{2} d \sinh ^{2}{\left (a + b x \right )}}{4 b^{2}} - \frac {3 c d^{2} x \sinh ^{2}{\left (a + b x \right )}}{4 b^{2}} - \frac {3 c d^{2} x \cosh ^{2}{\left (a + b x \right )}}{4 b^{2}} - \frac {3 d^{3} x^{2} \sinh ^{2}{\left (a + b x \right )}}{8 b^{2}} - \frac {3 d^{3} x^{2} \cosh ^{2}{\left (a + b x \right )}}{8 b^{2}} + \frac {3 c d^{2} \sinh {\left (a + b x \right )} \cosh {\left (a + b x \right )}}{4 b^{3}} + \frac {3 d^{3} x \sinh {\left (a + b x \right )} \cosh {\left (a + b x \right )}}{4 b^{3}} - \frac {3 d^{3} \sinh ^{2}{\left (a + b x \right )}}{8 b^{4}} & \text {for}\: b \neq 0 \\\left (c^{3} x + \frac {3 c^{2} d x^{2}}{2} + c d^{2} x^{3} + \frac {d^{3} x^{4}}{4}\right ) \sinh ^{2}{\left (a \right )} & \text {otherwise} \end {cases} \] Input:

integrate((d*x+c)**3*sinh(b*x+a)**2,x)
 

Output:

Piecewise((c**3*x*sinh(a + b*x)**2/2 - c**3*x*cosh(a + b*x)**2/2 + 3*c**2* 
d*x**2*sinh(a + b*x)**2/4 - 3*c**2*d*x**2*cosh(a + b*x)**2/4 + c*d**2*x**3 
*sinh(a + b*x)**2/2 - c*d**2*x**3*cosh(a + b*x)**2/2 + d**3*x**4*sinh(a + 
b*x)**2/8 - d**3*x**4*cosh(a + b*x)**2/8 + c**3*sinh(a + b*x)*cosh(a + b*x 
)/(2*b) + 3*c**2*d*x*sinh(a + b*x)*cosh(a + b*x)/(2*b) + 3*c*d**2*x**2*sin 
h(a + b*x)*cosh(a + b*x)/(2*b) + d**3*x**3*sinh(a + b*x)*cosh(a + b*x)/(2* 
b) - 3*c**2*d*sinh(a + b*x)**2/(4*b**2) - 3*c*d**2*x*sinh(a + b*x)**2/(4*b 
**2) - 3*c*d**2*x*cosh(a + b*x)**2/(4*b**2) - 3*d**3*x**2*sinh(a + b*x)**2 
/(8*b**2) - 3*d**3*x**2*cosh(a + b*x)**2/(8*b**2) + 3*c*d**2*sinh(a + b*x) 
*cosh(a + b*x)/(4*b**3) + 3*d**3*x*sinh(a + b*x)*cosh(a + b*x)/(4*b**3) - 
3*d**3*sinh(a + b*x)**2/(8*b**4), Ne(b, 0)), ((c**3*x + 3*c**2*d*x**2/2 + 
c*d**2*x**3 + d**3*x**4/4)*sinh(a)**2, True))
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 263 vs. \(2 (112) = 224\).

Time = 0.05 (sec) , antiderivative size = 263, normalized size of antiderivative = 2.12 \[ \int (c+d x)^3 \sinh ^2(a+b x) \, dx=-\frac {3}{16} \, {\left (4 \, x^{2} - \frac {{\left (2 \, b x e^{\left (2 \, a\right )} - e^{\left (2 \, a\right )}\right )} e^{\left (2 \, b x\right )}}{b^{2}} + \frac {{\left (2 \, b x + 1\right )} e^{\left (-2 \, b x - 2 \, a\right )}}{b^{2}}\right )} c^{2} d - \frac {1}{16} \, {\left (8 \, x^{3} - \frac {3 \, {\left (2 \, b^{2} x^{2} e^{\left (2 \, a\right )} - 2 \, b x e^{\left (2 \, a\right )} + e^{\left (2 \, a\right )}\right )} e^{\left (2 \, b x\right )}}{b^{3}} + \frac {3 \, {\left (2 \, b^{2} x^{2} + 2 \, b x + 1\right )} e^{\left (-2 \, b x - 2 \, a\right )}}{b^{3}}\right )} c d^{2} - \frac {1}{32} \, {\left (4 \, x^{4} - \frac {{\left (4 \, b^{3} x^{3} e^{\left (2 \, a\right )} - 6 \, b^{2} x^{2} e^{\left (2 \, a\right )} + 6 \, b x e^{\left (2 \, a\right )} - 3 \, e^{\left (2 \, a\right )}\right )} e^{\left (2 \, b x\right )}}{b^{4}} + \frac {{\left (4 \, b^{3} x^{3} + 6 \, b^{2} x^{2} + 6 \, b x + 3\right )} e^{\left (-2 \, b x - 2 \, a\right )}}{b^{4}}\right )} d^{3} - \frac {1}{8} \, c^{3} {\left (4 \, x - \frac {e^{\left (2 \, b x + 2 \, a\right )}}{b} + \frac {e^{\left (-2 \, b x - 2 \, a\right )}}{b}\right )} \] Input:

integrate((d*x+c)^3*sinh(b*x+a)^2,x, algorithm="maxima")
 

Output:

-3/16*(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^2*d - 1/16*(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*d^2 - 1/32*(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)*d^3 - 1/8*c^3*(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. 243 vs. \(2 (112) = 224\).

Time = 0.14 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.96 \[ \int (c+d x)^3 \sinh ^2(a+b x) \, dx=-\frac {1}{8} \, d^{3} x^{4} - \frac {1}{2} \, c d^{2} x^{3} - \frac {3}{4} \, c^{2} d x^{2} - \frac {1}{2} \, c^{3} x + \frac {{\left (4 \, b^{3} d^{3} x^{3} + 12 \, b^{3} c d^{2} x^{2} + 12 \, b^{3} c^{2} d x - 6 \, b^{2} d^{3} x^{2} + 4 \, b^{3} c^{3} - 12 \, b^{2} c d^{2} x - 6 \, b^{2} c^{2} d + 6 \, b d^{3} x + 6 \, b c d^{2} - 3 \, d^{3}\right )} e^{\left (2 \, b x + 2 \, a\right )}}{32 \, b^{4}} - \frac {{\left (4 \, b^{3} d^{3} x^{3} + 12 \, b^{3} c d^{2} x^{2} + 12 \, b^{3} c^{2} d x + 6 \, b^{2} d^{3} x^{2} + 4 \, b^{3} c^{3} + 12 \, b^{2} c d^{2} x + 6 \, b^{2} c^{2} d + 6 \, b d^{3} x + 6 \, b c d^{2} + 3 \, d^{3}\right )} e^{\left (-2 \, b x - 2 \, a\right )}}{32 \, b^{4}} \] Input:

integrate((d*x+c)^3*sinh(b*x+a)^2,x, algorithm="giac")
 

Output:

-1/8*d^3*x^4 - 1/2*c*d^2*x^3 - 3/4*c^2*d*x^2 - 1/2*c^3*x + 1/32*(4*b^3*d^3 
*x^3 + 12*b^3*c*d^2*x^2 + 12*b^3*c^2*d*x - 6*b^2*d^3*x^2 + 4*b^3*c^3 - 12* 
b^2*c*d^2*x - 6*b^2*c^2*d + 6*b*d^3*x + 6*b*c*d^2 - 3*d^3)*e^(2*b*x + 2*a) 
/b^4 - 1/32*(4*b^3*d^3*x^3 + 12*b^3*c*d^2*x^2 + 12*b^3*c^2*d*x + 6*b^2*d^3 
*x^2 + 4*b^3*c^3 + 12*b^2*c*d^2*x + 6*b^2*c^2*d + 6*b*d^3*x + 6*b*c*d^2 + 
3*d^3)*e^(-2*b*x - 2*a)/b^4
 

Mupad [B] (verification not implemented)

Time = 1.77 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.85 \[ \int (c+d x)^3 \sinh ^2(a+b x) \, dx=-\frac {\frac {3\,d^3\,\mathrm {cosh}\left (2\,a+2\,b\,x\right )}{2}+4\,b^4\,c^3\,x-2\,b^3\,c^3\,\mathrm {sinh}\left (2\,a+2\,b\,x\right )+b^4\,d^3\,x^4+3\,b^2\,c^2\,d\,\mathrm {cosh}\left (2\,a+2\,b\,x\right )+6\,b^4\,c^2\,d\,x^2+4\,b^4\,c\,d^2\,x^3+3\,b^2\,d^3\,x^2\,\mathrm {cosh}\left (2\,a+2\,b\,x\right )-2\,b^3\,d^3\,x^3\,\mathrm {sinh}\left (2\,a+2\,b\,x\right )-3\,b\,c\,d^2\,\mathrm {sinh}\left (2\,a+2\,b\,x\right )-3\,b\,d^3\,x\,\mathrm {sinh}\left (2\,a+2\,b\,x\right )+6\,b^2\,c\,d^2\,x\,\mathrm {cosh}\left (2\,a+2\,b\,x\right )-6\,b^3\,c^2\,d\,x\,\mathrm {sinh}\left (2\,a+2\,b\,x\right )-6\,b^3\,c\,d^2\,x^2\,\mathrm {sinh}\left (2\,a+2\,b\,x\right )}{8\,b^4} \] Input:

int(sinh(a + b*x)^2*(c + d*x)^3,x)
 

Output:

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

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 382, normalized size of antiderivative = 3.08 \[ \int (c+d x)^3 \sinh ^2(a+b x) \, dx=\frac {4 e^{4 b x +4 a} b^{3} c^{3}+12 e^{4 b x +4 a} b^{3} c^{2} d x +12 e^{4 b x +4 a} b^{3} c \,d^{2} x^{2}+4 e^{4 b x +4 a} b^{3} d^{3} x^{3}-6 e^{4 b x +4 a} b^{2} c^{2} d -12 e^{4 b x +4 a} b^{2} c \,d^{2} x -6 e^{4 b x +4 a} b^{2} d^{3} x^{2}+6 e^{4 b x +4 a} b c \,d^{2}+6 e^{4 b x +4 a} b \,d^{3} x -3 e^{4 b x +4 a} d^{3}-16 e^{2 b x +2 a} b^{4} c^{3} x -24 e^{2 b x +2 a} b^{4} c^{2} d \,x^{2}-16 e^{2 b x +2 a} b^{4} c \,d^{2} x^{3}-4 e^{2 b x +2 a} b^{4} d^{3} x^{4}-4 b^{3} c^{3}-12 b^{3} c^{2} d x -12 b^{3} c \,d^{2} x^{2}-4 b^{3} d^{3} x^{3}-6 b^{2} c^{2} d -12 b^{2} c \,d^{2} x -6 b^{2} d^{3} x^{2}-6 b c \,d^{2}-6 b \,d^{3} x -3 d^{3}}{32 e^{2 b x +2 a} b^{4}} \] Input:

int((d*x+c)^3*sinh(b*x+a)^2,x)
 

Output:

(4*e**(4*a + 4*b*x)*b**3*c**3 + 12*e**(4*a + 4*b*x)*b**3*c**2*d*x + 12*e** 
(4*a + 4*b*x)*b**3*c*d**2*x**2 + 4*e**(4*a + 4*b*x)*b**3*d**3*x**3 - 6*e** 
(4*a + 4*b*x)*b**2*c**2*d - 12*e**(4*a + 4*b*x)*b**2*c*d**2*x - 6*e**(4*a 
+ 4*b*x)*b**2*d**3*x**2 + 6*e**(4*a + 4*b*x)*b*c*d**2 + 6*e**(4*a + 4*b*x) 
*b*d**3*x - 3*e**(4*a + 4*b*x)*d**3 - 16*e**(2*a + 2*b*x)*b**4*c**3*x - 24 
*e**(2*a + 2*b*x)*b**4*c**2*d*x**2 - 16*e**(2*a + 2*b*x)*b**4*c*d**2*x**3 
- 4*e**(2*a + 2*b*x)*b**4*d**3*x**4 - 4*b**3*c**3 - 12*b**3*c**2*d*x - 12* 
b**3*c*d**2*x**2 - 4*b**3*d**3*x**3 - 6*b**2*c**2*d - 12*b**2*c*d**2*x - 6 
*b**2*d**3*x**2 - 6*b*c*d**2 - 6*b*d**3*x - 3*d**3)/(32*e**(2*a + 2*b*x)*b 
**4)