\(\int (c+d x)^4 \sinh ^3(a+b x) \, dx\) [16]

Optimal result

Integrand size = 16, antiderivative size = 225 \[ \int (c+d x)^4 \sinh ^3(a+b x) \, dx=-\frac {488 d^4 \cosh (a+b x)}{27 b^5}-\frac {80 d^2 (c+d x)^2 \cosh (a+b x)}{9 b^3}-\frac {2 (c+d x)^4 \cosh (a+b x)}{3 b}+\frac {8 d^4 \cosh ^3(a+b x)}{81 b^5}+\frac {160 d^3 (c+d x) \sinh (a+b x)}{9 b^4}+\frac {8 d (c+d x)^3 \sinh (a+b x)}{3 b^2}+\frac {4 d^2 (c+d x)^2 \cosh (a+b x) \sinh ^2(a+b x)}{9 b^3}+\frac {(c+d x)^4 \cosh (a+b x) \sinh ^2(a+b x)}{3 b}-\frac {8 d^3 (c+d x) \sinh ^3(a+b x)}{27 b^4}-\frac {4 d (c+d x)^3 \sinh ^3(a+b x)}{9 b^2} \]

[Out]

-488/27*d^4*cosh(b*x+a)/b^5-80/9*d^2*(d*x+c)^2*cosh(b*x+a)/b^3-2/3*(d*x+c)^4*cosh(b*x+a)/b+8/81*d^4*cosh(b*x+a
)^3/b^5+160/9*d^3*(d*x+c)*sinh(b*x+a)/b^4+8/3*d*(d*x+c)^3*sinh(b*x+a)/b^2+4/9*d^2*(d*x+c)^2*cosh(b*x+a)*sinh(b
*x+a)^2/b^3+1/3*(d*x+c)^4*cosh(b*x+a)*sinh(b*x+a)^2/b-8/27*d^3*(d*x+c)*sinh(b*x+a)^3/b^4-4/9*d*(d*x+c)^3*sinh(
b*x+a)^3/b^2

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3392, 3377, 2718, 2713} \[ \int (c+d x)^4 \sinh ^3(a+b x) \, dx=\frac {8 d^4 \cosh ^3(a+b x)}{81 b^5}-\frac {488 d^4 \cosh (a+b x)}{27 b^5}-\frac {8 d^3 (c+d x) \sinh ^3(a+b x)}{27 b^4}+\frac {160 d^3 (c+d x) \sinh (a+b x)}{9 b^4}-\frac {80 d^2 (c+d x)^2 \cosh (a+b x)}{9 b^3}+\frac {4 d^2 (c+d x)^2 \sinh ^2(a+b x) \cosh (a+b x)}{9 b^3}-\frac {4 d (c+d x)^3 \sinh ^3(a+b x)}{9 b^2}+\frac {8 d (c+d x)^3 \sinh (a+b x)}{3 b^2}-\frac {2 (c+d x)^4 \cosh (a+b x)}{3 b}+\frac {(c+d x)^4 \sinh ^2(a+b x) \cosh (a+b x)}{3 b} \]

[In]

Int[(c + d*x)^4*Sinh[a + b*x]^3,x]

[Out]

(-488*d^4*Cosh[a + b*x])/(27*b^5) - (80*d^2*(c + d*x)^2*Cosh[a + b*x])/(9*b^3) - (2*(c + d*x)^4*Cosh[a + b*x])
/(3*b) + (8*d^4*Cosh[a + b*x]^3)/(81*b^5) + (160*d^3*(c + d*x)*Sinh[a + b*x])/(9*b^4) + (8*d*(c + d*x)^3*Sinh[
a + b*x])/(3*b^2) + (4*d^2*(c + d*x)^2*Cosh[a + b*x]*Sinh[a + b*x]^2)/(9*b^3) + ((c + d*x)^4*Cosh[a + b*x]*Sin
h[a + b*x]^2)/(3*b) - (8*d^3*(c + d*x)*Sinh[a + b*x]^3)/(27*b^4) - (4*d*(c + d*x)^3*Sinh[a + b*x]^3)/(9*b^2)

Rule 2713

Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Dist[-d^(-1), Subst[Int[Expand[(1 - x^2)^((n - 1)/2), x], x], x
, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]

Rule 2718

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

Rule 3377

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

Rule 3392

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

Rubi steps \begin{align*} \text {integral}& = \frac {(c+d x)^4 \cosh (a+b x) \sinh ^2(a+b x)}{3 b}-\frac {4 d (c+d x)^3 \sinh ^3(a+b x)}{9 b^2}-\frac {2}{3} \int (c+d x)^4 \sinh (a+b x) \, dx+\frac {\left (4 d^2\right ) \int (c+d x)^2 \sinh ^3(a+b x) \, dx}{3 b^2} \\ & = -\frac {2 (c+d x)^4 \cosh (a+b x)}{3 b}+\frac {4 d^2 (c+d x)^2 \cosh (a+b x) \sinh ^2(a+b x)}{9 b^3}+\frac {(c+d x)^4 \cosh (a+b x) \sinh ^2(a+b x)}{3 b}-\frac {8 d^3 (c+d x) \sinh ^3(a+b x)}{27 b^4}-\frac {4 d (c+d x)^3 \sinh ^3(a+b x)}{9 b^2}+\frac {(8 d) \int (c+d x)^3 \cosh (a+b x) \, dx}{3 b}-\frac {\left (8 d^2\right ) \int (c+d x)^2 \sinh (a+b x) \, dx}{9 b^2}+\frac {\left (8 d^4\right ) \int \sinh ^3(a+b x) \, dx}{27 b^4} \\ & = -\frac {8 d^2 (c+d x)^2 \cosh (a+b x)}{9 b^3}-\frac {2 (c+d x)^4 \cosh (a+b x)}{3 b}+\frac {8 d (c+d x)^3 \sinh (a+b x)}{3 b^2}+\frac {4 d^2 (c+d x)^2 \cosh (a+b x) \sinh ^2(a+b x)}{9 b^3}+\frac {(c+d x)^4 \cosh (a+b x) \sinh ^2(a+b x)}{3 b}-\frac {8 d^3 (c+d x) \sinh ^3(a+b x)}{27 b^4}-\frac {4 d (c+d x)^3 \sinh ^3(a+b x)}{9 b^2}-\frac {\left (8 d^2\right ) \int (c+d x)^2 \sinh (a+b x) \, dx}{b^2}+\frac {\left (16 d^3\right ) \int (c+d x) \cosh (a+b x) \, dx}{9 b^3}-\frac {\left (8 d^4\right ) \text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cosh (a+b x)\right )}{27 b^5} \\ & = -\frac {8 d^4 \cosh (a+b x)}{27 b^5}-\frac {80 d^2 (c+d x)^2 \cosh (a+b x)}{9 b^3}-\frac {2 (c+d x)^4 \cosh (a+b x)}{3 b}+\frac {8 d^4 \cosh ^3(a+b x)}{81 b^5}+\frac {16 d^3 (c+d x) \sinh (a+b x)}{9 b^4}+\frac {8 d (c+d x)^3 \sinh (a+b x)}{3 b^2}+\frac {4 d^2 (c+d x)^2 \cosh (a+b x) \sinh ^2(a+b x)}{9 b^3}+\frac {(c+d x)^4 \cosh (a+b x) \sinh ^2(a+b x)}{3 b}-\frac {8 d^3 (c+d x) \sinh ^3(a+b x)}{27 b^4}-\frac {4 d (c+d x)^3 \sinh ^3(a+b x)}{9 b^2}+\frac {\left (16 d^3\right ) \int (c+d x) \cosh (a+b x) \, dx}{b^3}-\frac {\left (16 d^4\right ) \int \sinh (a+b x) \, dx}{9 b^4} \\ & = -\frac {56 d^4 \cosh (a+b x)}{27 b^5}-\frac {80 d^2 (c+d x)^2 \cosh (a+b x)}{9 b^3}-\frac {2 (c+d x)^4 \cosh (a+b x)}{3 b}+\frac {8 d^4 \cosh ^3(a+b x)}{81 b^5}+\frac {160 d^3 (c+d x) \sinh (a+b x)}{9 b^4}+\frac {8 d (c+d x)^3 \sinh (a+b x)}{3 b^2}+\frac {4 d^2 (c+d x)^2 \cosh (a+b x) \sinh ^2(a+b x)}{9 b^3}+\frac {(c+d x)^4 \cosh (a+b x) \sinh ^2(a+b x)}{3 b}-\frac {8 d^3 (c+d x) \sinh ^3(a+b x)}{27 b^4}-\frac {4 d (c+d x)^3 \sinh ^3(a+b x)}{9 b^2}-\frac {\left (16 d^4\right ) \int \sinh (a+b x) \, dx}{b^4} \\ & = -\frac {488 d^4 \cosh (a+b x)}{27 b^5}-\frac {80 d^2 (c+d x)^2 \cosh (a+b x)}{9 b^3}-\frac {2 (c+d x)^4 \cosh (a+b x)}{3 b}+\frac {8 d^4 \cosh ^3(a+b x)}{81 b^5}+\frac {160 d^3 (c+d x) \sinh (a+b x)}{9 b^4}+\frac {8 d (c+d x)^3 \sinh (a+b x)}{3 b^2}+\frac {4 d^2 (c+d x)^2 \cosh (a+b x) \sinh ^2(a+b x)}{9 b^3}+\frac {(c+d x)^4 \cosh (a+b x) \sinh ^2(a+b x)}{3 b}-\frac {8 d^3 (c+d x) \sinh ^3(a+b x)}{27 b^4}-\frac {4 d (c+d x)^3 \sinh ^3(a+b x)}{9 b^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.62 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.67 \[ \int (c+d x)^4 \sinh ^3(a+b x) \, dx=\frac {-243 \left (24 d^4+12 b^2 d^2 (c+d x)^2+b^4 (c+d x)^4\right ) \cosh (a+b x)+\left (8 d^4+36 b^2 d^2 (c+d x)^2+27 b^4 (c+d x)^4\right ) \cosh (3 (a+b x))-24 b d (c+d x) \left (-242 d^2-39 b^2 (c+d x)^2+\left (2 d^2+3 b^2 (c+d x)^2\right ) \cosh (2 (a+b x))\right ) \sinh (a+b x)}{324 b^5} \]

[In]

Integrate[(c + d*x)^4*Sinh[a + b*x]^3,x]

[Out]

(-243*(24*d^4 + 12*b^2*d^2*(c + d*x)^2 + b^4*(c + d*x)^4)*Cosh[a + b*x] + (8*d^4 + 36*b^2*d^2*(c + d*x)^2 + 27
*b^4*(c + d*x)^4)*Cosh[3*(a + b*x)] - 24*b*d*(c + d*x)*(-242*d^2 - 39*b^2*(c + d*x)^2 + (2*d^2 + 3*b^2*(c + d*
x)^2)*Cosh[2*(a + b*x)])*Sinh[a + b*x])/(324*b^5)

Maple [A] (verified)

Time = 1.90 (sec) , antiderivative size = 181, normalized size of antiderivative = 0.80

method	result	size
parallelrisch	\(\frac {\left (27 \left (d x +c \right )^{4} b^{4}+36 d^{2} \left (d x +c \right )^{2} b^{2}+8 d^{4}\right ) \cosh \left (3 b x +3 a \right )-36 d \left (\left (d x +c \right )^{2} b^{2}+\frac {2 d^{2}}{3}\right ) b \left (d x +c \right ) \sinh \left (3 b x +3 a \right )+\left (-243 \left (d x +c \right )^{4} b^{4}-2916 d^{2} \left (d x +c \right )^{2} b^{2}-5832 d^{4}\right ) \cosh \left (b x +a \right )+972 d \left (\left (d x +c \right )^{2} b^{2}+6 d^{2}\right ) b \left (d x +c \right ) \sinh \left (b x +a \right )-216 b^{4} c^{4}-2880 b^{2} c^{2} d^{2}-5824 d^{4}}{324 b^{5}}\)	\(181\)
risch	\(\frac {\left (27 d^{4} x^{4} b^{4}+108 b^{4} c \,d^{3} x^{3}+162 b^{4} c^{2} d^{2} x^{2}-36 b^{3} d^{4} x^{3}+108 b^{4} c^{3} d x -108 b^{3} c \,d^{3} x^{2}+27 b^{4} c^{4}-108 b^{3} c^{2} d^{2} x +36 b^{2} d^{4} x^{2}-36 b^{3} c^{3} d +72 b^{2} c \,d^{3} x +36 b^{2} c^{2} d^{2}-24 b \,d^{4} x -24 b c \,d^{3}+8 d^{4}\right ) {\mathrm e}^{3 b x +3 a}}{648 b^{5}}-\frac {3 \left (d^{4} x^{4} b^{4}+4 b^{4} c \,d^{3} x^{3}+6 b^{4} c^{2} d^{2} x^{2}-4 b^{3} d^{4} x^{3}+4 b^{4} c^{3} d x -12 b^{3} c \,d^{3} x^{2}+b^{4} c^{4}-12 b^{3} c^{2} d^{2} x +12 b^{2} d^{4} x^{2}-4 b^{3} c^{3} d +24 b^{2} c \,d^{3} x +12 b^{2} c^{2} d^{2}-24 b \,d^{4} x -24 b c \,d^{3}+24 d^{4}\right ) {\mathrm e}^{b x +a}}{8 b^{5}}-\frac {3 \left (d^{4} x^{4} b^{4}+4 b^{4} c \,d^{3} x^{3}+6 b^{4} c^{2} d^{2} x^{2}+4 b^{3} d^{4} x^{3}+4 b^{4} c^{3} d x +12 b^{3} c \,d^{3} x^{2}+b^{4} c^{4}+12 b^{3} c^{2} d^{2} x +12 b^{2} d^{4} x^{2}+4 b^{3} c^{3} d +24 b^{2} c \,d^{3} x +12 b^{2} c^{2} d^{2}+24 b \,d^{4} x +24 b c \,d^{3}+24 d^{4}\right ) {\mathrm e}^{-b x -a}}{8 b^{5}}+\frac {\left (27 d^{4} x^{4} b^{4}+108 b^{4} c \,d^{3} x^{3}+162 b^{4} c^{2} d^{2} x^{2}+36 b^{3} d^{4} x^{3}+108 b^{4} c^{3} d x +108 b^{3} c \,d^{3} x^{2}+27 b^{4} c^{4}+108 b^{3} c^{2} d^{2} x +36 b^{2} d^{4} x^{2}+36 b^{3} c^{3} d +72 b^{2} c \,d^{3} x +36 b^{2} c^{2} d^{2}+24 b \,d^{4} x +24 b c \,d^{3}+8 d^{4}\right ) {\mathrm e}^{-3 b x -3 a}}{648 b^{5}}\)	\(655\)
derivativedivides	\(\text {Expression too large to display}\)	\(1139\)
default	\(\text {Expression too large to display}\)	\(1139\)

[In]

int((d*x+c)^4*sinh(b*x+a)^3,x,method=_RETURNVERBOSE)

[Out]

1/324*((27*(d*x+c)^4*b^4+36*d^2*(d*x+c)^2*b^2+8*d^4)*cosh(3*b*x+3*a)-36*d*((d*x+c)^2*b^2+2/3*d^2)*b*(d*x+c)*si
nh(3*b*x+3*a)+(-243*(d*x+c)^4*b^4-2916*d^2*(d*x+c)^2*b^2-5832*d^4)*cosh(b*x+a)+972*d*((d*x+c)^2*b^2+6*d^2)*b*(
d*x+c)*sinh(b*x+a)-216*b^4*c^4-2880*b^2*c^2*d^2-5824*d^4)/b^5

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 528 vs. \(2 (205) = 410\).

Time = 0.25 (sec) , antiderivative size = 528, normalized size of antiderivative = 2.35 \[ \int (c+d x)^4 \sinh ^3(a+b x) \, dx=\frac {{\left (27 \, b^{4} d^{4} x^{4} + 108 \, b^{4} c d^{3} x^{3} + 27 \, b^{4} c^{4} + 36 \, b^{2} c^{2} d^{2} + 8 \, d^{4} + 18 \, {\left (9 \, b^{4} c^{2} d^{2} + 2 \, b^{2} d^{4}\right )} x^{2} + 36 \, {\left (3 \, b^{4} c^{3} d + 2 \, b^{2} c d^{3}\right )} x\right )} \cosh \left (b x + a\right )^{3} + 3 \, {\left (27 \, b^{4} d^{4} x^{4} + 108 \, b^{4} c d^{3} x^{3} + 27 \, b^{4} c^{4} + 36 \, b^{2} c^{2} d^{2} + 8 \, d^{4} + 18 \, {\left (9 \, b^{4} c^{2} d^{2} + 2 \, b^{2} d^{4}\right )} x^{2} + 36 \, {\left (3 \, b^{4} c^{3} d + 2 \, b^{2} c d^{3}\right )} x\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} - 12 \, {\left (3 \, b^{3} d^{4} x^{3} + 9 \, b^{3} c d^{3} x^{2} + 3 \, b^{3} c^{3} d + 2 \, b c d^{3} + {\left (9 \, b^{3} c^{2} d^{2} + 2 \, b d^{4}\right )} x\right )} \sinh \left (b x + a\right )^{3} - 243 \, {\left (b^{4} d^{4} x^{4} + 4 \, b^{4} c d^{3} x^{3} + b^{4} c^{4} + 12 \, b^{2} c^{2} d^{2} + 24 \, d^{4} + 6 \, {\left (b^{4} c^{2} d^{2} + 2 \, b^{2} d^{4}\right )} x^{2} + 4 \, {\left (b^{4} c^{3} d + 6 \, b^{2} c d^{3}\right )} x\right )} \cosh \left (b x + a\right ) + 36 \, {\left (27 \, b^{3} d^{4} x^{3} + 81 \, b^{3} c d^{3} x^{2} + 27 \, b^{3} c^{3} d + 162 \, b c d^{3} - {\left (3 \, b^{3} d^{4} x^{3} + 9 \, b^{3} c d^{3} x^{2} + 3 \, b^{3} c^{3} d + 2 \, b c d^{3} + {\left (9 \, b^{3} c^{2} d^{2} + 2 \, b d^{4}\right )} x\right )} \cosh \left (b x + a\right )^{2} + 81 \, {\left (b^{3} c^{2} d^{2} + 2 \, b d^{4}\right )} x\right )} \sinh \left (b x + a\right )}{324 \, b^{5}} \]

[In]

integrate((d*x+c)^4*sinh(b*x+a)^3,x, algorithm="fricas")

[Out]

1/324*((27*b^4*d^4*x^4 + 108*b^4*c*d^3*x^3 + 27*b^4*c^4 + 36*b^2*c^2*d^2 + 8*d^4 + 18*(9*b^4*c^2*d^2 + 2*b^2*d
^4)*x^2 + 36*(3*b^4*c^3*d + 2*b^2*c*d^3)*x)*cosh(b*x + a)^3 + 3*(27*b^4*d^4*x^4 + 108*b^4*c*d^3*x^3 + 27*b^4*c
^4 + 36*b^2*c^2*d^2 + 8*d^4 + 18*(9*b^4*c^2*d^2 + 2*b^2*d^4)*x^2 + 36*(3*b^4*c^3*d + 2*b^2*c*d^3)*x)*cosh(b*x
+ a)*sinh(b*x + a)^2 - 12*(3*b^3*d^4*x^3 + 9*b^3*c*d^3*x^2 + 3*b^3*c^3*d + 2*b*c*d^3 + (9*b^3*c^2*d^2 + 2*b*d^
4)*x)*sinh(b*x + a)^3 - 243*(b^4*d^4*x^4 + 4*b^4*c*d^3*x^3 + b^4*c^4 + 12*b^2*c^2*d^2 + 24*d^4 + 6*(b^4*c^2*d^
2 + 2*b^2*d^4)*x^2 + 4*(b^4*c^3*d + 6*b^2*c*d^3)*x)*cosh(b*x + a) + 36*(27*b^3*d^4*x^3 + 81*b^3*c*d^3*x^2 + 27
*b^3*c^3*d + 162*b*c*d^3 - (3*b^3*d^4*x^3 + 9*b^3*c*d^3*x^2 + 3*b^3*c^3*d + 2*b*c*d^3 + (9*b^3*c^2*d^2 + 2*b*d
^4)*x)*cosh(b*x + a)^2 + 81*(b^3*c^2*d^2 + 2*b*d^4)*x)*sinh(b*x + a))/b^5

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 772 vs. \(2 (226) = 452\).

Time = 0.67 (sec) , antiderivative size = 772, normalized size of antiderivative = 3.43 \[ \int (c+d x)^4 \sinh ^3(a+b x) \, dx=\begin {cases} \frac {c^{4} \sinh ^{2}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{b} - \frac {2 c^{4} \cosh ^{3}{\left (a + b x \right )}}{3 b} + \frac {4 c^{3} d x \sinh ^{2}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{b} - \frac {8 c^{3} d x \cosh ^{3}{\left (a + b x \right )}}{3 b} + \frac {6 c^{2} d^{2} x^{2} \sinh ^{2}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{b} - \frac {4 c^{2} d^{2} x^{2} \cosh ^{3}{\left (a + b x \right )}}{b} + \frac {4 c d^{3} x^{3} \sinh ^{2}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{b} - \frac {8 c d^{3} x^{3} \cosh ^{3}{\left (a + b x \right )}}{3 b} + \frac {d^{4} x^{4} \sinh ^{2}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{b} - \frac {2 d^{4} x^{4} \cosh ^{3}{\left (a + b x \right )}}{3 b} - \frac {28 c^{3} d \sinh ^{3}{\left (a + b x \right )}}{9 b^{2}} + \frac {8 c^{3} d \sinh {\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{3 b^{2}} - \frac {28 c^{2} d^{2} x \sinh ^{3}{\left (a + b x \right )}}{3 b^{2}} + \frac {8 c^{2} d^{2} x \sinh {\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{b^{2}} - \frac {28 c d^{3} x^{2} \sinh ^{3}{\left (a + b x \right )}}{3 b^{2}} + \frac {8 c d^{3} x^{2} \sinh {\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{b^{2}} - \frac {28 d^{4} x^{3} \sinh ^{3}{\left (a + b x \right )}}{9 b^{2}} + \frac {8 d^{4} x^{3} \sinh {\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{3 b^{2}} + \frac {28 c^{2} d^{2} \sinh ^{2}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{3 b^{3}} - \frac {80 c^{2} d^{2} \cosh ^{3}{\left (a + b x \right )}}{9 b^{3}} + \frac {56 c d^{3} x \sinh ^{2}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{3 b^{3}} - \frac {160 c d^{3} x \cosh ^{3}{\left (a + b x \right )}}{9 b^{3}} + \frac {28 d^{4} x^{2} \sinh ^{2}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{3 b^{3}} - \frac {80 d^{4} x^{2} \cosh ^{3}{\left (a + b x \right )}}{9 b^{3}} - \frac {488 c d^{3} \sinh ^{3}{\left (a + b x \right )}}{27 b^{4}} + \frac {160 c d^{3} \sinh {\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{9 b^{4}} - \frac {488 d^{4} x \sinh ^{3}{\left (a + b x \right )}}{27 b^{4}} + \frac {160 d^{4} x \sinh {\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{9 b^{4}} + \frac {488 d^{4} \sinh ^{2}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{27 b^{5}} - \frac {1456 d^{4} \cosh ^{3}{\left (a + b x \right )}}{81 b^{5}} & \text {for}\: b \neq 0 \\\left (c^{4} x + 2 c^{3} d x^{2} + 2 c^{2} d^{2} x^{3} + c d^{3} x^{4} + \frac {d^{4} x^{5}}{5}\right ) \sinh ^{3}{\left (a \right )} & \text {otherwise} \end {cases} \]

[In]

integrate((d*x+c)**4*sinh(b*x+a)**3,x)

[Out]

Piecewise((c**4*sinh(a + b*x)**2*cosh(a + b*x)/b - 2*c**4*cosh(a + b*x)**3/(3*b) + 4*c**3*d*x*sinh(a + b*x)**2
*cosh(a + b*x)/b - 8*c**3*d*x*cosh(a + b*x)**3/(3*b) + 6*c**2*d**2*x**2*sinh(a + b*x)**2*cosh(a + b*x)/b - 4*c
**2*d**2*x**2*cosh(a + b*x)**3/b + 4*c*d**3*x**3*sinh(a + b*x)**2*cosh(a + b*x)/b - 8*c*d**3*x**3*cosh(a + b*x
)**3/(3*b) + d**4*x**4*sinh(a + b*x)**2*cosh(a + b*x)/b - 2*d**4*x**4*cosh(a + b*x)**3/(3*b) - 28*c**3*d*sinh(
a + b*x)**3/(9*b**2) + 8*c**3*d*sinh(a + b*x)*cosh(a + b*x)**2/(3*b**2) - 28*c**2*d**2*x*sinh(a + b*x)**3/(3*b
**2) + 8*c**2*d**2*x*sinh(a + b*x)*cosh(a + b*x)**2/b**2 - 28*c*d**3*x**2*sinh(a + b*x)**3/(3*b**2) + 8*c*d**3
*x**2*sinh(a + b*x)*cosh(a + b*x)**2/b**2 - 28*d**4*x**3*sinh(a + b*x)**3/(9*b**2) + 8*d**4*x**3*sinh(a + b*x)
*cosh(a + b*x)**2/(3*b**2) + 28*c**2*d**2*sinh(a + b*x)**2*cosh(a + b*x)/(3*b**3) - 80*c**2*d**2*cosh(a + b*x)
**3/(9*b**3) + 56*c*d**3*x*sinh(a + b*x)**2*cosh(a + b*x)/(3*b**3) - 160*c*d**3*x*cosh(a + b*x)**3/(9*b**3) +
28*d**4*x**2*sinh(a + b*x)**2*cosh(a + b*x)/(3*b**3) - 80*d**4*x**2*cosh(a + b*x)**3/(9*b**3) - 488*c*d**3*sin
h(a + b*x)**3/(27*b**4) + 160*c*d**3*sinh(a + b*x)*cosh(a + b*x)**2/(9*b**4) - 488*d**4*x*sinh(a + b*x)**3/(27
*b**4) + 160*d**4*x*sinh(a + b*x)*cosh(a + b*x)**2/(9*b**4) + 488*d**4*sinh(a + b*x)**2*cosh(a + b*x)/(27*b**5
) - 1456*d**4*cosh(a + b*x)**3/(81*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)**3, True))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 639 vs. \(2 (205) = 410\).

Time = 0.22 (sec) , antiderivative size = 639, normalized size of antiderivative = 2.84 \[ \int (c+d x)^4 \sinh ^3(a+b x) \, dx=\frac {1}{18} \, c^{3} d {\left (\frac {{\left (3 \, b x e^{\left (3 \, a\right )} - e^{\left (3 \, a\right )}\right )} e^{\left (3 \, b x\right )}}{b^{2}} - \frac {27 \, {\left (b x e^{a} - e^{a}\right )} e^{\left (b x\right )}}{b^{2}} - \frac {27 \, {\left (b x + 1\right )} e^{\left (-b x - a\right )}}{b^{2}} + \frac {{\left (3 \, b x + 1\right )} e^{\left (-3 \, b x - 3 \, a\right )}}{b^{2}}\right )} + \frac {1}{24} \, c^{4} {\left (\frac {e^{\left (3 \, b x + 3 \, a\right )}}{b} - \frac {9 \, e^{\left (b x + a\right )}}{b} - \frac {9 \, e^{\left (-b x - a\right )}}{b} + \frac {e^{\left (-3 \, b x - 3 \, a\right )}}{b}\right )} + \frac {1}{36} \, c^{2} d^{2} {\left (\frac {{\left (9 \, b^{2} x^{2} e^{\left (3 \, a\right )} - 6 \, b x e^{\left (3 \, a\right )} + 2 \, e^{\left (3 \, a\right )}\right )} e^{\left (3 \, b x\right )}}{b^{3}} - \frac {81 \, {\left (b^{2} x^{2} e^{a} - 2 \, b x e^{a} + 2 \, e^{a}\right )} e^{\left (b x\right )}}{b^{3}} - \frac {81 \, {\left (b^{2} x^{2} + 2 \, b x + 2\right )} e^{\left (-b x - a\right )}}{b^{3}} + \frac {{\left (9 \, b^{2} x^{2} + 6 \, b x + 2\right )} e^{\left (-3 \, b x - 3 \, a\right )}}{b^{3}}\right )} + \frac {1}{54} \, c d^{3} {\left (\frac {{\left (9 \, b^{3} x^{3} e^{\left (3 \, a\right )} - 9 \, b^{2} x^{2} e^{\left (3 \, a\right )} + 6 \, b x e^{\left (3 \, a\right )} - 2 \, e^{\left (3 \, a\right )}\right )} e^{\left (3 \, b x\right )}}{b^{4}} - \frac {81 \, {\left (b^{3} x^{3} e^{a} - 3 \, b^{2} x^{2} e^{a} + 6 \, b x e^{a} - 6 \, e^{a}\right )} e^{\left (b x\right )}}{b^{4}} - \frac {81 \, {\left (b^{3} x^{3} + 3 \, b^{2} x^{2} + 6 \, b x + 6\right )} e^{\left (-b x - a\right )}}{b^{4}} + \frac {{\left (9 \, b^{3} x^{3} + 9 \, b^{2} x^{2} + 6 \, b x + 2\right )} e^{\left (-3 \, b x - 3 \, a\right )}}{b^{4}}\right )} + \frac {1}{648} \, d^{4} {\left (\frac {{\left (27 \, b^{4} x^{4} e^{\left (3 \, a\right )} - 36 \, b^{3} x^{3} e^{\left (3 \, a\right )} + 36 \, b^{2} x^{2} e^{\left (3 \, a\right )} - 24 \, b x e^{\left (3 \, a\right )} + 8 \, e^{\left (3 \, a\right )}\right )} e^{\left (3 \, b x\right )}}{b^{5}} - \frac {243 \, {\left (b^{4} x^{4} e^{a} - 4 \, b^{3} x^{3} e^{a} + 12 \, b^{2} x^{2} e^{a} - 24 \, b x e^{a} + 24 \, e^{a}\right )} e^{\left (b x\right )}}{b^{5}} - \frac {243 \, {\left (b^{4} x^{4} + 4 \, b^{3} x^{3} + 12 \, b^{2} x^{2} + 24 \, b x + 24\right )} e^{\left (-b x - a\right )}}{b^{5}} + \frac {{\left (27 \, b^{4} x^{4} + 36 \, b^{3} x^{3} + 36 \, b^{2} x^{2} + 24 \, b x + 8\right )} e^{\left (-3 \, b x - 3 \, a\right )}}{b^{5}}\right )} \]

[In]

integrate((d*x+c)^4*sinh(b*x+a)^3,x, algorithm="maxima")

[Out]

1/18*c^3*d*((3*b*x*e^(3*a) - e^(3*a))*e^(3*b*x)/b^2 - 27*(b*x*e^a - e^a)*e^(b*x)/b^2 - 27*(b*x + 1)*e^(-b*x -
a)/b^2 + (3*b*x + 1)*e^(-3*b*x - 3*a)/b^2) + 1/24*c^4*(e^(3*b*x + 3*a)/b - 9*e^(b*x + a)/b - 9*e^(-b*x - a)/b
+ e^(-3*b*x - 3*a)/b) + 1/36*c^2*d^2*((9*b^2*x^2*e^(3*a) - 6*b*x*e^(3*a) + 2*e^(3*a))*e^(3*b*x)/b^3 - 81*(b^2*
x^2*e^a - 2*b*x*e^a + 2*e^a)*e^(b*x)/b^3 - 81*(b^2*x^2 + 2*b*x + 2)*e^(-b*x - a)/b^3 + (9*b^2*x^2 + 6*b*x + 2)
*e^(-3*b*x - 3*a)/b^3) + 1/54*c*d^3*((9*b^3*x^3*e^(3*a) - 9*b^2*x^2*e^(3*a) + 6*b*x*e^(3*a) - 2*e^(3*a))*e^(3*
b*x)/b^4 - 81*(b^3*x^3*e^a - 3*b^2*x^2*e^a + 6*b*x*e^a - 6*e^a)*e^(b*x)/b^4 - 81*(b^3*x^3 + 3*b^2*x^2 + 6*b*x
+ 6)*e^(-b*x - a)/b^4 + (9*b^3*x^3 + 9*b^2*x^2 + 6*b*x + 2)*e^(-3*b*x - 3*a)/b^4) + 1/648*d^4*((27*b^4*x^4*e^(
3*a) - 36*b^3*x^3*e^(3*a) + 36*b^2*x^2*e^(3*a) - 24*b*x*e^(3*a) + 8*e^(3*a))*e^(3*b*x)/b^5 - 243*(b^4*x^4*e^a
- 4*b^3*x^3*e^a + 12*b^2*x^2*e^a - 24*b*x*e^a + 24*e^a)*e^(b*x)/b^5 - 243*(b^4*x^4 + 4*b^3*x^3 + 12*b^2*x^2 +
24*b*x + 24)*e^(-b*x - a)/b^5 + (27*b^4*x^4 + 36*b^3*x^3 + 36*b^2*x^2 + 24*b*x + 8)*e^(-3*b*x - 3*a)/b^5)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 654 vs. \(2 (205) = 410\).

Time = 0.27 (sec) , antiderivative size = 654, normalized size of antiderivative = 2.91 \[ \int (c+d x)^4 \sinh ^3(a+b x) \, dx=\frac {{\left (27 \, b^{4} d^{4} x^{4} + 108 \, b^{4} c d^{3} x^{3} + 162 \, b^{4} c^{2} d^{2} x^{2} - 36 \, b^{3} d^{4} x^{3} + 108 \, b^{4} c^{3} d x - 108 \, b^{3} c d^{3} x^{2} + 27 \, b^{4} c^{4} - 108 \, b^{3} c^{2} d^{2} x + 36 \, b^{2} d^{4} x^{2} - 36 \, b^{3} c^{3} d + 72 \, b^{2} c d^{3} x + 36 \, b^{2} c^{2} d^{2} - 24 \, b d^{4} x - 24 \, b c d^{3} + 8 \, d^{4}\right )} e^{\left (3 \, b x + 3 \, a\right )}}{648 \, b^{5}} - \frac {3 \, {\left (b^{4} d^{4} x^{4} + 4 \, b^{4} c d^{3} x^{3} + 6 \, b^{4} c^{2} d^{2} x^{2} - 4 \, b^{3} d^{4} x^{3} + 4 \, b^{4} c^{3} d x - 12 \, b^{3} c d^{3} x^{2} + b^{4} c^{4} - 12 \, b^{3} c^{2} d^{2} x + 12 \, b^{2} d^{4} x^{2} - 4 \, b^{3} c^{3} d + 24 \, b^{2} c d^{3} x + 12 \, b^{2} c^{2} d^{2} - 24 \, b d^{4} x - 24 \, b c d^{3} + 24 \, d^{4}\right )} e^{\left (b x + a\right )}}{8 \, b^{5}} - \frac {3 \, {\left (b^{4} d^{4} x^{4} + 4 \, b^{4} c d^{3} x^{3} + 6 \, b^{4} c^{2} d^{2} x^{2} + 4 \, b^{3} d^{4} x^{3} + 4 \, b^{4} c^{3} d x + 12 \, b^{3} c d^{3} x^{2} + b^{4} c^{4} + 12 \, b^{3} c^{2} d^{2} x + 12 \, b^{2} d^{4} x^{2} + 4 \, b^{3} c^{3} d + 24 \, b^{2} c d^{3} x + 12 \, b^{2} c^{2} d^{2} + 24 \, b d^{4} x + 24 \, b c d^{3} + 24 \, d^{4}\right )} e^{\left (-b x - a\right )}}{8 \, b^{5}} + \frac {{\left (27 \, b^{4} d^{4} x^{4} + 108 \, b^{4} c d^{3} x^{3} + 162 \, b^{4} c^{2} d^{2} x^{2} + 36 \, b^{3} d^{4} x^{3} + 108 \, b^{4} c^{3} d x + 108 \, b^{3} c d^{3} x^{2} + 27 \, b^{4} c^{4} + 108 \, b^{3} c^{2} d^{2} x + 36 \, b^{2} d^{4} x^{2} + 36 \, b^{3} c^{3} d + 72 \, b^{2} c d^{3} x + 36 \, b^{2} c^{2} d^{2} + 24 \, b d^{4} x + 24 \, b c d^{3} + 8 \, d^{4}\right )} e^{\left (-3 \, b x - 3 \, a\right )}}{648 \, b^{5}} \]

[In]

integrate((d*x+c)^4*sinh(b*x+a)^3,x, algorithm="giac")

[Out]

1/648*(27*b^4*d^4*x^4 + 108*b^4*c*d^3*x^3 + 162*b^4*c^2*d^2*x^2 - 36*b^3*d^4*x^3 + 108*b^4*c^3*d*x - 108*b^3*c
*d^3*x^2 + 27*b^4*c^4 - 108*b^3*c^2*d^2*x + 36*b^2*d^4*x^2 - 36*b^3*c^3*d + 72*b^2*c*d^3*x + 36*b^2*c^2*d^2 -
24*b*d^4*x - 24*b*c*d^3 + 8*d^4)*e^(3*b*x + 3*a)/b^5 - 3/8*(b^4*d^4*x^4 + 4*b^4*c*d^3*x^3 + 6*b^4*c^2*d^2*x^2
- 4*b^3*d^4*x^3 + 4*b^4*c^3*d*x - 12*b^3*c*d^3*x^2 + b^4*c^4 - 12*b^3*c^2*d^2*x + 12*b^2*d^4*x^2 - 4*b^3*c^3*d
+ 24*b^2*c*d^3*x + 12*b^2*c^2*d^2 - 24*b*d^4*x - 24*b*c*d^3 + 24*d^4)*e^(b*x + a)/b^5 - 3/8*(b^4*d^4*x^4 + 4*
b^4*c*d^3*x^3 + 6*b^4*c^2*d^2*x^2 + 4*b^3*d^4*x^3 + 4*b^4*c^3*d*x + 12*b^3*c*d^3*x^2 + b^4*c^4 + 12*b^3*c^2*d^
2*x + 12*b^2*d^4*x^2 + 4*b^3*c^3*d + 24*b^2*c*d^3*x + 12*b^2*c^2*d^2 + 24*b*d^4*x + 24*b*c*d^3 + 24*d^4)*e^(-b
*x - a)/b^5 + 1/648*(27*b^4*d^4*x^4 + 108*b^4*c*d^3*x^3 + 162*b^4*c^2*d^2*x^2 + 36*b^3*d^4*x^3 + 108*b^4*c^3*d
*x + 108*b^3*c*d^3*x^2 + 27*b^4*c^4 + 108*b^3*c^2*d^2*x + 36*b^2*d^4*x^2 + 36*b^3*c^3*d + 72*b^2*c*d^3*x + 36*
b^2*c^2*d^2 + 24*b*d^4*x + 24*b*c*d^3 + 8*d^4)*e^(-3*b*x - 3*a)/b^5

Mupad [B] (verification not implemented)

Time = 1.29 (sec) , antiderivative size = 532, normalized size of antiderivative = 2.36 \[ \int (c+d x)^4 \sinh ^3(a+b x) \, dx=\frac {\mathrm {cosh}\left (a+b\,x\right )\,{\mathrm {sinh}\left (a+b\,x\right )}^2\,\left (27\,b^4\,c^4+252\,b^2\,c^2\,d^2+488\,d^4\right )}{27\,b^5}-\frac {2\,{\mathrm {cosh}\left (a+b\,x\right )}^3\,\left (27\,b^4\,c^4+360\,b^2\,c^2\,d^2+728\,d^4\right )}{81\,b^5}-\frac {4\,{\mathrm {sinh}\left (a+b\,x\right )}^3\,\left (21\,b^2\,c^3\,d+122\,c\,d^3\right )}{27\,b^4}+\frac {8\,{\mathrm {cosh}\left (a+b\,x\right )}^2\,\mathrm {sinh}\left (a+b\,x\right )\,\left (3\,b^2\,c^3\,d+20\,c\,d^3\right )}{9\,b^4}-\frac {2\,d^4\,x^4\,{\mathrm {cosh}\left (a+b\,x\right )}^3}{3\,b}-\frac {8\,x\,{\mathrm {cosh}\left (a+b\,x\right )}^3\,\left (3\,b^2\,c^3\,d+20\,c\,d^3\right )}{9\,b^3}-\frac {28\,d^4\,x^3\,{\mathrm {sinh}\left (a+b\,x\right )}^3}{9\,b^2}-\frac {4\,x\,{\mathrm {sinh}\left (a+b\,x\right )}^3\,\left (63\,b^2\,c^2\,d^2+122\,d^4\right )}{27\,b^4}-\frac {4\,x^2\,{\mathrm {cosh}\left (a+b\,x\right )}^3\,\left (9\,b^2\,c^2\,d^2+20\,d^4\right )}{9\,b^3}+\frac {2\,x^2\,\mathrm {cosh}\left (a+b\,x\right )\,{\mathrm {sinh}\left (a+b\,x\right )}^2\,\left (9\,b^2\,c^2\,d^2+14\,d^4\right )}{3\,b^3}-\frac {8\,c\,d^3\,x^3\,{\mathrm {cosh}\left (a+b\,x\right )}^3}{3\,b}+\frac {d^4\,x^4\,\mathrm {cosh}\left (a+b\,x\right )\,{\mathrm {sinh}\left (a+b\,x\right )}^2}{b}+\frac {8\,d^4\,x^3\,{\mathrm {cosh}\left (a+b\,x\right )}^2\,\mathrm {sinh}\left (a+b\,x\right )}{3\,b^2}-\frac {28\,c\,d^3\,x^2\,{\mathrm {sinh}\left (a+b\,x\right )}^3}{3\,b^2}+\frac {8\,x\,{\mathrm {cosh}\left (a+b\,x\right )}^2\,\mathrm {sinh}\left (a+b\,x\right )\,\left (9\,b^2\,c^2\,d^2+20\,d^4\right )}{9\,b^4}+\frac {4\,x\,\mathrm {cosh}\left (a+b\,x\right )\,{\mathrm {sinh}\left (a+b\,x\right )}^2\,\left (3\,b^2\,c^3\,d+14\,c\,d^3\right )}{3\,b^3}+\frac {4\,c\,d^3\,x^3\,\mathrm {cosh}\left (a+b\,x\right )\,{\mathrm {sinh}\left (a+b\,x\right )}^2}{b}+\frac {8\,c\,d^3\,x^2\,{\mathrm {cosh}\left (a+b\,x\right )}^2\,\mathrm {sinh}\left (a+b\,x\right )}{b^2} \]

[In]

int(sinh(a + b*x)^3*(c + d*x)^4,x)

[Out]

(cosh(a + b*x)*sinh(a + b*x)^2*(488*d^4 + 27*b^4*c^4 + 252*b^2*c^2*d^2))/(27*b^5) - (2*cosh(a + b*x)^3*(728*d^
4 + 27*b^4*c^4 + 360*b^2*c^2*d^2))/(81*b^5) - (4*sinh(a + b*x)^3*(122*c*d^3 + 21*b^2*c^3*d))/(27*b^4) + (8*cos
h(a + b*x)^2*sinh(a + b*x)*(20*c*d^3 + 3*b^2*c^3*d))/(9*b^4) - (2*d^4*x^4*cosh(a + b*x)^3)/(3*b) - (8*x*cosh(a
+ b*x)^3*(20*c*d^3 + 3*b^2*c^3*d))/(9*b^3) - (28*d^4*x^3*sinh(a + b*x)^3)/(9*b^2) - (4*x*sinh(a + b*x)^3*(122
*d^4 + 63*b^2*c^2*d^2))/(27*b^4) - (4*x^2*cosh(a + b*x)^3*(20*d^4 + 9*b^2*c^2*d^2))/(9*b^3) + (2*x^2*cosh(a +
b*x)*sinh(a + b*x)^2*(14*d^4 + 9*b^2*c^2*d^2))/(3*b^3) - (8*c*d^3*x^3*cosh(a + b*x)^3)/(3*b) + (d^4*x^4*cosh(a
+ b*x)*sinh(a + b*x)^2)/b + (8*d^4*x^3*cosh(a + b*x)^2*sinh(a + b*x))/(3*b^2) - (28*c*d^3*x^2*sinh(a + b*x)^3
)/(3*b^2) + (8*x*cosh(a + b*x)^2*sinh(a + b*x)*(20*d^4 + 9*b^2*c^2*d^2))/(9*b^4) + (4*x*cosh(a + b*x)*sinh(a +
b*x)^2*(14*c*d^3 + 3*b^2*c^3*d))/(3*b^3) + (4*c*d^3*x^3*cosh(a + b*x)*sinh(a + b*x)^2)/b + (8*c*d^3*x^2*cosh(
a + b*x)^2*sinh(a + b*x))/b^2