Integrand size = 16, antiderivative size = 123 \[ \int (c+d x)^2 \sinh ^3(a+b x) \, dx=-\frac {14 d^2 \cosh (a+b x)}{9 b^3}-\frac {2 (c+d x)^2 \cosh (a+b x)}{3 b}+\frac {2 d^2 \cosh ^3(a+b x)}{27 b^3}+\frac {4 d (c+d x) \sinh (a+b x)}{3 b^2}+\frac {(c+d x)^2 \cosh (a+b x) \sinh ^2(a+b x)}{3 b}-\frac {2 d (c+d x) \sinh ^3(a+b x)}{9 b^2} \] Output:
-14/9*d^2*cosh(b*x+a)/b^3-2/3*(d*x+c)^2*cosh(b*x+a)/b+2/27*d^2*cosh(b*x+a) ^3/b^3+4/3*d*(d*x+c)*sinh(b*x+a)/b^2+1/3*(d*x+c)^2*cosh(b*x+a)*sinh(b*x+a) ^2/b-2/9*d*(d*x+c)*sinh(b*x+a)^3/b^2
Time = 0.25 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.70 \[ \int (c+d x)^2 \sinh ^3(a+b x) \, dx=\frac {-81 \left (2 d^2+b^2 (c+d x)^2\right ) \cosh (a+b x)+\left (2 d^2+9 b^2 (c+d x)^2\right ) \cosh (3 (a+b x))-6 b d (c+d x) (-27 \sinh (a+b x)+\sinh (3 (a+b x)))}{108 b^3} \] Input:
Integrate[(c + d*x)^2*Sinh[a + b*x]^3,x]
Output:
(-81*(2*d^2 + b^2*(c + d*x)^2)*Cosh[a + b*x] + (2*d^2 + 9*b^2*(c + d*x)^2) *Cosh[3*(a + b*x)] - 6*b*d*(c + d*x)*(-27*Sinh[a + b*x] + Sinh[3*(a + b*x) ]))/(108*b^3)
Result contains complex when optimal does not.
Time = 0.66 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.22, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.938, Rules used = {3042, 26, 3792, 26, 3042, 26, 3113, 2009, 3777, 3042, 3777, 26, 3042, 26, 3118}
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.
| \(\displaystyle \int (c+d x)^2 \sinh ^3(a+b x) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int i (c+d x)^2 \sin (i a+i b x)^3dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \int (c+d x)^2 \sin (i a+i b x)^3dx\) |
| \(\Big \downarrow \) 3792 |
| \(\displaystyle i \left (\frac {2 d^2 \int -i \sinh ^3(a+b x)dx}{9 b^2}+\frac {2}{3} \int i (c+d x)^2 \sinh (a+b x)dx+\frac {2 i d (c+d x) \sinh ^3(a+b x)}{9 b^2}-\frac {i (c+d x)^2 \sinh ^2(a+b x) \cosh (a+b x)}{3 b}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \left (-\frac {2 i d^2 \int \sinh ^3(a+b x)dx}{9 b^2}+\frac {2}{3} i \int (c+d x)^2 \sinh (a+b x)dx+\frac {2 i d (c+d x) \sinh ^3(a+b x)}{9 b^2}-\frac {i (c+d x)^2 \sinh ^2(a+b x) \cosh (a+b x)}{3 b}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle i \left (-\frac {2 i d^2 \int i \sin (i a+i b x)^3dx}{9 b^2}+\frac {2}{3} i \int -i (c+d x)^2 \sin (i a+i b x)dx+\frac {2 i d (c+d x) \sinh ^3(a+b x)}{9 b^2}-\frac {i (c+d x)^2 \sinh ^2(a+b x) \cosh (a+b x)}{3 b}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \left (\frac {2 d^2 \int \sin (i a+i b x)^3dx}{9 b^2}+\frac {2}{3} \int (c+d x)^2 \sin (i a+i b x)dx+\frac {2 i d (c+d x) \sinh ^3(a+b x)}{9 b^2}-\frac {i (c+d x)^2 \sinh ^2(a+b x) \cosh (a+b x)}{3 b}\right )\) |
| \(\Big \downarrow \) 3113 |
| \(\displaystyle i \left (\frac {2 i d^2 \int \left (1-\cosh ^2(a+b x)\right )d\cosh (a+b x)}{9 b^3}+\frac {2}{3} \int (c+d x)^2 \sin (i a+i b x)dx+\frac {2 i d (c+d x) \sinh ^3(a+b x)}{9 b^2}-\frac {i (c+d x)^2 \sinh ^2(a+b x) \cosh (a+b x)}{3 b}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle i \left (\frac {2}{3} \int (c+d x)^2 \sin (i a+i b x)dx+\frac {2 i d^2 \left (\cosh (a+b x)-\frac {1}{3} \cosh ^3(a+b x)\right )}{9 b^3}+\frac {2 i d (c+d x) \sinh ^3(a+b x)}{9 b^2}-\frac {i (c+d x)^2 \sinh ^2(a+b x) \cosh (a+b x)}{3 b}\right )\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle i \left (\frac {2}{3} \left (\frac {i (c+d x)^2 \cosh (a+b x)}{b}-\frac {2 i d \int (c+d x) \cosh (a+b x)dx}{b}\right )+\frac {2 i d^2 \left (\cosh (a+b x)-\frac {1}{3} \cosh ^3(a+b x)\right )}{9 b^3}+\frac {2 i d (c+d x) \sinh ^3(a+b x)}{9 b^2}-\frac {i (c+d x)^2 \sinh ^2(a+b x) \cosh (a+b x)}{3 b}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle i \left (\frac {2}{3} \left (\frac {i (c+d x)^2 \cosh (a+b x)}{b}-\frac {2 i d \int (c+d x) \sin \left (i a+i b x+\frac {\pi }{2}\right )dx}{b}\right )+\frac {2 i d^2 \left (\cosh (a+b x)-\frac {1}{3} \cosh ^3(a+b x)\right )}{9 b^3}+\frac {2 i d (c+d x) \sinh ^3(a+b x)}{9 b^2}-\frac {i (c+d x)^2 \sinh ^2(a+b x) \cosh (a+b x)}{3 b}\right )\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle i \left (\frac {2}{3} \left (\frac {i (c+d x)^2 \cosh (a+b x)}{b}-\frac {2 i d \left (\frac {(c+d x) \sinh (a+b x)}{b}-\frac {i d \int -i \sinh (a+b x)dx}{b}\right )}{b}\right )+\frac {2 i d^2 \left (\cosh (a+b x)-\frac {1}{3} \cosh ^3(a+b x)\right )}{9 b^3}+\frac {2 i d (c+d x) \sinh ^3(a+b x)}{9 b^2}-\frac {i (c+d x)^2 \sinh ^2(a+b x) \cosh (a+b x)}{3 b}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \left (\frac {2}{3} \left (\frac {i (c+d x)^2 \cosh (a+b x)}{b}-\frac {2 i d \left (\frac {(c+d x) \sinh (a+b x)}{b}-\frac {d \int \sinh (a+b x)dx}{b}\right )}{b}\right )+\frac {2 i d^2 \left (\cosh (a+b x)-\frac {1}{3} \cosh ^3(a+b x)\right )}{9 b^3}+\frac {2 i d (c+d x) \sinh ^3(a+b x)}{9 b^2}-\frac {i (c+d x)^2 \sinh ^2(a+b x) \cosh (a+b x)}{3 b}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle i \left (\frac {2}{3} \left (\frac {i (c+d x)^2 \cosh (a+b x)}{b}-\frac {2 i d \left (\frac {(c+d x) \sinh (a+b x)}{b}-\frac {d \int -i \sin (i a+i b x)dx}{b}\right )}{b}\right )+\frac {2 i d^2 \left (\cosh (a+b x)-\frac {1}{3} \cosh ^3(a+b x)\right )}{9 b^3}+\frac {2 i d (c+d x) \sinh ^3(a+b x)}{9 b^2}-\frac {i (c+d x)^2 \sinh ^2(a+b x) \cosh (a+b x)}{3 b}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \left (\frac {2}{3} \left (\frac {i (c+d x)^2 \cosh (a+b x)}{b}-\frac {2 i d \left (\frac {(c+d x) \sinh (a+b x)}{b}+\frac {i d \int \sin (i a+i b x)dx}{b}\right )}{b}\right )+\frac {2 i d^2 \left (\cosh (a+b x)-\frac {1}{3} \cosh ^3(a+b x)\right )}{9 b^3}+\frac {2 i d (c+d x) \sinh ^3(a+b x)}{9 b^2}-\frac {i (c+d x)^2 \sinh ^2(a+b x) \cosh (a+b x)}{3 b}\right )\) |
| \(\Big \downarrow \) 3118 |
| \(\displaystyle i \left (\frac {2 i d^2 \left (\cosh (a+b x)-\frac {1}{3} \cosh ^3(a+b x)\right )}{9 b^3}+\frac {2 i d (c+d x) \sinh ^3(a+b x)}{9 b^2}+\frac {2}{3} \left (\frac {i (c+d x)^2 \cosh (a+b x)}{b}-\frac {2 i d \left (\frac {(c+d x) \sinh (a+b x)}{b}-\frac {d \cosh (a+b x)}{b^2}\right )}{b}\right )-\frac {i (c+d x)^2 \sinh ^2(a+b x) \cosh (a+b x)}{3 b}\right )\) |
Input:
Int[(c + d*x)^2*Sinh[a + b*x]^3,x]
Output:
I*((((2*I)/9)*d^2*(Cosh[a + b*x] - Cosh[a + b*x]^3/3))/b^3 - ((I/3)*(c + d *x)^2*Cosh[a + b*x]*Sinh[a + b*x]^2)/b + (((2*I)/9)*d*(c + d*x)*Sinh[a + b *x]^3)/b^2 + (2*((I*(c + d*x)^2*Cosh[a + b*x])/b - ((2*I)*d*(-((d*Cosh[a + b*x])/b^2) + ((c + d*x)*Sinh[a + b*x])/b))/b))/3)
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Simp[-d^(-1) Subst[Int[Exp and[(1 - x^2)^((n - 1)/2), x], x], x, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]
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]
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]
Time = 0.59 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.86
| method | result | size |
| parallelrisch | \(\frac {\left (9 \left (d x +c \right )^{2} b^{2}+2 d^{2}\right ) \cosh \left (3 b x +3 a \right )-6 b d \left (d x +c \right ) \sinh \left (3 b x +3 a \right )+\left (-81 \left (d x +c \right )^{2} b^{2}-162 d^{2}\right ) \cosh \left (b x +a \right )+162 b d \left (d x +c \right ) \sinh \left (b x +a \right )-72 b^{2} c^{2}-160 d^{2}}{108 b^{3}}\) | \(106\) |
| risch | \(\frac {\left (9 d^{2} x^{2} b^{2}+18 b^{2} c d x +9 b^{2} c^{2}-6 b \,d^{2} x -6 b c d +2 d^{2}\right ) {\mathrm e}^{3 b x +3 a}}{216 b^{3}}-\frac {3 \left (d^{2} x^{2} b^{2}+2 b^{2} c d x +b^{2} c^{2}-2 b \,d^{2} x -2 b c d +2 d^{2}\right ) {\mathrm e}^{b x +a}}{8 b^{3}}-\frac {3 \left (d^{2} x^{2} b^{2}+2 b^{2} c d x +b^{2} c^{2}+2 b \,d^{2} x +2 b c d +2 d^{2}\right ) {\mathrm e}^{-b x -a}}{8 b^{3}}+\frac {\left (9 d^{2} x^{2} b^{2}+18 b^{2} c d x +9 b^{2} c^{2}+6 b \,d^{2} x +6 b c d +2 d^{2}\right ) {\mathrm e}^{-3 b x -3 a}}{216 b^{3}}\) | \(231\) |
| derivativedivides | \(\frac {\frac {d^{2} \left (-\frac {2 \left (b x +a \right )^{2} \cosh \left (b x +a \right )}{3}+\frac {\left (b x +a \right )^{2} \cosh \left (b x +a \right ) \sinh \left (b x +a \right )^{2}}{3}+\frac {4 \left (b x +a \right ) \sinh \left (b x +a \right )}{3}-\frac {40 \cosh \left (b x +a \right )}{27}-\frac {2 \left (b x +a \right ) \sinh \left (b x +a \right )^{3}}{9}+\frac {2 \cosh \left (b x +a \right ) \sinh \left (b x +a \right )^{2}}{27}\right )}{b^{2}}-\frac {2 d^{2} a \left (-\frac {2 \left (b x +a \right ) \cosh \left (b x +a \right )}{3}+\frac {\left (b x +a \right ) \cosh \left (b x +a \right ) \sinh \left (b x +a \right )^{2}}{3}+\frac {2 \sinh \left (b x +a \right )}{3}-\frac {\sinh \left (b x +a \right )^{3}}{9}\right )}{b^{2}}+\frac {2 d c \left (-\frac {2 \left (b x +a \right ) \cosh \left (b x +a \right )}{3}+\frac {\left (b x +a \right ) \cosh \left (b x +a \right ) \sinh \left (b x +a \right )^{2}}{3}+\frac {2 \sinh \left (b x +a \right )}{3}-\frac {\sinh \left (b x +a \right )^{3}}{9}\right )}{b}+\frac {d^{2} a^{2} \left (-\frac {2}{3}+\frac {\sinh \left (b x +a \right )^{2}}{3}\right ) \cosh \left (b x +a \right )}{b^{2}}-\frac {2 d a c \left (-\frac {2}{3}+\frac {\sinh \left (b x +a \right )^{2}}{3}\right ) \cosh \left (b x +a \right )}{b}+c^{2} \left (-\frac {2}{3}+\frac {\sinh \left (b x +a \right )^{2}}{3}\right ) \cosh \left (b x +a \right )}{b}\) | \(302\) |
| default | \(\frac {\frac {d^{2} \left (-\frac {2 \left (b x +a \right )^{2} \cosh \left (b x +a \right )}{3}+\frac {\left (b x +a \right )^{2} \cosh \left (b x +a \right ) \sinh \left (b x +a \right )^{2}}{3}+\frac {4 \left (b x +a \right ) \sinh \left (b x +a \right )}{3}-\frac {40 \cosh \left (b x +a \right )}{27}-\frac {2 \left (b x +a \right ) \sinh \left (b x +a \right )^{3}}{9}+\frac {2 \cosh \left (b x +a \right ) \sinh \left (b x +a \right )^{2}}{27}\right )}{b^{2}}-\frac {2 d^{2} a \left (-\frac {2 \left (b x +a \right ) \cosh \left (b x +a \right )}{3}+\frac {\left (b x +a \right ) \cosh \left (b x +a \right ) \sinh \left (b x +a \right )^{2}}{3}+\frac {2 \sinh \left (b x +a \right )}{3}-\frac {\sinh \left (b x +a \right )^{3}}{9}\right )}{b^{2}}+\frac {2 d c \left (-\frac {2 \left (b x +a \right ) \cosh \left (b x +a \right )}{3}+\frac {\left (b x +a \right ) \cosh \left (b x +a \right ) \sinh \left (b x +a \right )^{2}}{3}+\frac {2 \sinh \left (b x +a \right )}{3}-\frac {\sinh \left (b x +a \right )^{3}}{9}\right )}{b}+\frac {d^{2} a^{2} \left (-\frac {2}{3}+\frac {\sinh \left (b x +a \right )^{2}}{3}\right ) \cosh \left (b x +a \right )}{b^{2}}-\frac {2 d a c \left (-\frac {2}{3}+\frac {\sinh \left (b x +a \right )^{2}}{3}\right ) \cosh \left (b x +a \right )}{b}+c^{2} \left (-\frac {2}{3}+\frac {\sinh \left (b x +a \right )^{2}}{3}\right ) \cosh \left (b x +a \right )}{b}\) | \(302\) |
| orering | \(-\frac {40 d \left (9 d^{4} x^{4} b^{4}+36 b^{4} c \,d^{3} x^{3}+54 b^{4} c^{2} d^{2} x^{2}+36 b^{4} c^{3} d x +9 b^{4} c^{4}+b^{2} d^{4} x^{2}+2 b^{2} c \,d^{3} x +b^{2} c^{2} d^{2}-12 d^{4}\right ) \sinh \left (b x +a \right )^{3}}{81 b^{6} \left (d x +c \right )^{3}}+\frac {2 \left (45 d^{4} x^{4} b^{4}+180 b^{4} c \,d^{3} x^{3}+270 b^{4} c^{2} d^{2} x^{2}+180 b^{4} c^{3} d x +45 b^{4} c^{4}-26 b^{2} d^{4} x^{2}-52 b^{2} c \,d^{3} x -26 b^{2} c^{2} d^{2}-180 d^{4}\right ) \left (2 \left (d x +c \right ) \sinh \left (b x +a \right )^{3} d +3 \left (d x +c \right )^{2} \sinh \left (b x +a \right )^{2} b \cosh \left (b x +a \right )\right )}{81 b^{6} \left (d x +c \right )^{4}}+\frac {8 d \left (3 d^{2} x^{2} b^{2}+6 b^{2} c d x +3 b^{2} c^{2}+5 d^{2}\right ) \left (2 d^{2} \sinh \left (b x +a \right )^{3}+12 \left (d x +c \right ) \sinh \left (b x +a \right )^{2} d b \cosh \left (b x +a \right )+6 \left (d x +c \right )^{2} \sinh \left (b x +a \right ) b^{2} \cosh \left (b x +a \right )^{2}+3 \left (d x +c \right )^{2} \sinh \left (b x +a \right )^{3} b^{2}\right )}{27 b^{6} \left (d x +c \right )^{3}}-\frac {\left (9 d^{2} x^{2} b^{2}+18 b^{2} c d x +9 b^{2} c^{2}+20 d^{2}\right ) \left (18 d^{2} \sinh \left (b x +a \right )^{2} b \cosh \left (b x +a \right )+36 \left (d x +c \right ) \sinh \left (b x +a \right ) d \,b^{2} \cosh \left (b x +a \right )^{2}+18 \left (d x +c \right ) \sinh \left (b x +a \right )^{3} d \,b^{2}+6 \left (d x +c \right )^{2} b^{3} \cosh \left (b x +a \right )^{3}+21 \left (d x +c \right )^{2} \sinh \left (b x +a \right )^{2} b^{3} \cosh \left (b x +a \right )\right )}{81 b^{6} \left (d x +c \right )^{2}}\) | \(545\) |
Input:
int((d*x+c)^2*sinh(b*x+a)^3,x,method=_RETURNVERBOSE)
Output:
1/108*((9*(d*x+c)^2*b^2+2*d^2)*cosh(3*b*x+3*a)-6*b*d*(d*x+c)*sinh(3*b*x+3* a)+(-81*(d*x+c)^2*b^2-162*d^2)*cosh(b*x+a)+162*b*d*(d*x+c)*sinh(b*x+a)-72* b^2*c^2-160*d^2)/b^3
Time = 0.10 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.62 \[ \int (c+d x)^2 \sinh ^3(a+b x) \, dx=\frac {{\left (9 \, b^{2} d^{2} x^{2} + 18 \, b^{2} c d x + 9 \, b^{2} c^{2} + 2 \, d^{2}\right )} \cosh \left (b x + a\right )^{3} + 3 \, {\left (9 \, b^{2} d^{2} x^{2} + 18 \, b^{2} c d x + 9 \, b^{2} c^{2} + 2 \, d^{2}\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} - 6 \, {\left (b d^{2} x + b c d\right )} \sinh \left (b x + a\right )^{3} - 81 \, {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2} + 2 \, d^{2}\right )} \cosh \left (b x + a\right ) + 18 \, {\left (9 \, b d^{2} x + 9 \, b c d - {\left (b d^{2} x + b c d\right )} \cosh \left (b x + a\right )^{2}\right )} \sinh \left (b x + a\right )}{108 \, b^{3}} \] Input:
integrate((d*x+c)^2*sinh(b*x+a)^3,x, algorithm="fricas")
Output:
1/108*((9*b^2*d^2*x^2 + 18*b^2*c*d*x + 9*b^2*c^2 + 2*d^2)*cosh(b*x + a)^3 + 3*(9*b^2*d^2*x^2 + 18*b^2*c*d*x + 9*b^2*c^2 + 2*d^2)*cosh(b*x + a)*sinh( b*x + a)^2 - 6*(b*d^2*x + b*c*d)*sinh(b*x + a)^3 - 81*(b^2*d^2*x^2 + 2*b^2 *c*d*x + b^2*c^2 + 2*d^2)*cosh(b*x + a) + 18*(9*b*d^2*x + 9*b*c*d - (b*d^2 *x + b*c*d)*cosh(b*x + a)^2)*sinh(b*x + a))/b^3
Leaf count of result is larger than twice the leaf count of optimal. 284 vs. \(2 (121) = 242\).
Time = 0.34 (sec) , antiderivative size = 284, normalized size of antiderivative = 2.31 \[ \int (c+d x)^2 \sinh ^3(a+b x) \, dx=\begin {cases} \frac {c^{2} \sinh ^{2}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{b} - \frac {2 c^{2} \cosh ^{3}{\left (a + b x \right )}}{3 b} + \frac {2 c d x \sinh ^{2}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{b} - \frac {4 c d x \cosh ^{3}{\left (a + b x \right )}}{3 b} + \frac {d^{2} x^{2} \sinh ^{2}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{b} - \frac {2 d^{2} x^{2} \cosh ^{3}{\left (a + b x \right )}}{3 b} - \frac {14 c d \sinh ^{3}{\left (a + b x \right )}}{9 b^{2}} + \frac {4 c d \sinh {\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{3 b^{2}} - \frac {14 d^{2} x \sinh ^{3}{\left (a + b x \right )}}{9 b^{2}} + \frac {4 d^{2} x \sinh {\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{3 b^{2}} + \frac {14 d^{2} \sinh ^{2}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{9 b^{3}} - \frac {40 d^{2} \cosh ^{3}{\left (a + b x \right )}}{27 b^{3}} & \text {for}\: b \neq 0 \\\left (c^{2} x + c d x^{2} + \frac {d^{2} x^{3}}{3}\right ) \sinh ^{3}{\left (a \right )} & \text {otherwise} \end {cases} \] Input:
integrate((d*x+c)**2*sinh(b*x+a)**3,x)
Output:
Piecewise((c**2*sinh(a + b*x)**2*cosh(a + b*x)/b - 2*c**2*cosh(a + b*x)**3 /(3*b) + 2*c*d*x*sinh(a + b*x)**2*cosh(a + b*x)/b - 4*c*d*x*cosh(a + b*x)* *3/(3*b) + d**2*x**2*sinh(a + b*x)**2*cosh(a + b*x)/b - 2*d**2*x**2*cosh(a + b*x)**3/(3*b) - 14*c*d*sinh(a + b*x)**3/(9*b**2) + 4*c*d*sinh(a + b*x)* cosh(a + b*x)**2/(3*b**2) - 14*d**2*x*sinh(a + b*x)**3/(9*b**2) + 4*d**2*x *sinh(a + b*x)*cosh(a + b*x)**2/(3*b**2) + 14*d**2*sinh(a + b*x)**2*cosh(a + b*x)/(9*b**3) - 40*d**2*cosh(a + b*x)**3/(27*b**3), Ne(b, 0)), ((c**2*x + c*d*x**2 + d**2*x**3/3)*sinh(a)**3, True))
Leaf count of result is larger than twice the leaf count of optimal. 269 vs. \(2 (111) = 222\).
Time = 0.05 (sec) , antiderivative size = 269, normalized size of antiderivative = 2.19 \[ \int (c+d x)^2 \sinh ^3(a+b x) \, dx=\frac {1}{36} \, c d {\left (\frac {{\left (3 \, b x e^{\left (3 \, a\right )} - e^{\left (3 \, a\right )}\right )} e^{\left (3 \, b x\right )}}{b^{2}} - \frac {27 \, {\left (b x e^{a} - e^{a}\right )} e^{\left (b x\right )}}{b^{2}} - \frac {27 \, {\left (b x + 1\right )} e^{\left (-b x - a\right )}}{b^{2}} + \frac {{\left (3 \, b x + 1\right )} e^{\left (-3 \, b x - 3 \, a\right )}}{b^{2}}\right )} + \frac {1}{24} \, c^{2} {\left (\frac {e^{\left (3 \, b x + 3 \, a\right )}}{b} - \frac {9 \, e^{\left (b x + a\right )}}{b} - \frac {9 \, e^{\left (-b x - a\right )}}{b} + \frac {e^{\left (-3 \, b x - 3 \, a\right )}}{b}\right )} + \frac {1}{216} \, d^{2} {\left (\frac {{\left (9 \, b^{2} x^{2} e^{\left (3 \, a\right )} - 6 \, b x e^{\left (3 \, a\right )} + 2 \, e^{\left (3 \, a\right )}\right )} e^{\left (3 \, b x\right )}}{b^{3}} - \frac {81 \, {\left (b^{2} x^{2} e^{a} - 2 \, b x e^{a} + 2 \, e^{a}\right )} e^{\left (b x\right )}}{b^{3}} - \frac {81 \, {\left (b^{2} x^{2} + 2 \, b x + 2\right )} e^{\left (-b x - a\right )}}{b^{3}} + \frac {{\left (9 \, b^{2} x^{2} + 6 \, b x + 2\right )} e^{\left (-3 \, b x - 3 \, a\right )}}{b^{3}}\right )} \] Input:
integrate((d*x+c)^2*sinh(b*x+a)^3,x, algorithm="maxima")
Output:
1/36*c*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^2*(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) + 1/216*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)
Leaf count of result is larger than twice the leaf count of optimal. 230 vs. \(2 (111) = 222\).
Time = 0.12 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.87 \[ \int (c+d x)^2 \sinh ^3(a+b x) \, dx=\frac {{\left (9 \, b^{2} d^{2} x^{2} + 18 \, b^{2} c d x + 9 \, b^{2} c^{2} - 6 \, b d^{2} x - 6 \, b c d + 2 \, d^{2}\right )} e^{\left (3 \, b x + 3 \, a\right )}}{216 \, b^{3}} - \frac {3 \, {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2} - 2 \, b d^{2} x - 2 \, b c d + 2 \, d^{2}\right )} e^{\left (b x + a\right )}}{8 \, b^{3}} - \frac {3 \, {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2} + 2 \, b d^{2} x + 2 \, b c d + 2 \, d^{2}\right )} e^{\left (-b x - a\right )}}{8 \, b^{3}} + \frac {{\left (9 \, b^{2} d^{2} x^{2} + 18 \, b^{2} c d x + 9 \, b^{2} c^{2} + 6 \, b d^{2} x + 6 \, b c d + 2 \, d^{2}\right )} e^{\left (-3 \, b x - 3 \, a\right )}}{216 \, b^{3}} \] Input:
integrate((d*x+c)^2*sinh(b*x+a)^3,x, algorithm="giac")
Output:
1/216*(9*b^2*d^2*x^2 + 18*b^2*c*d*x + 9*b^2*c^2 - 6*b*d^2*x - 6*b*c*d + 2* d^2)*e^(3*b*x + 3*a)/b^3 - 3/8*(b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2 - 2*b* d^2*x - 2*b*c*d + 2*d^2)*e^(b*x + a)/b^3 - 3/8*(b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2 + 2*b*d^2*x + 2*b*c*d + 2*d^2)*e^(-b*x - a)/b^3 + 1/216*(9*b^2*d ^2*x^2 + 18*b^2*c*d*x + 9*b^2*c^2 + 6*b*d^2*x + 6*b*c*d + 2*d^2)*e^(-3*b*x - 3*a)/b^3
Time = 1.67 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.50 \[ \int (c+d x)^2 \sinh ^3(a+b x) \, dx=-\frac {\frac {3\,d^2\,\mathrm {cosh}\left (a+b\,x\right )}{2}-\frac {d^2\,\mathrm {cosh}\left (3\,a+3\,b\,x\right )}{54}+\frac {3\,b^2\,c^2\,\mathrm {cosh}\left (a+b\,x\right )}{4}-\frac {b^2\,c^2\,\mathrm {cosh}\left (3\,a+3\,b\,x\right )}{12}+\frac {3\,b^2\,d^2\,x^2\,\mathrm {cosh}\left (a+b\,x\right )}{4}+\frac {b\,c\,d\,\mathrm {sinh}\left (3\,a+3\,b\,x\right )}{18}-\frac {3\,b\,d^2\,x\,\mathrm {sinh}\left (a+b\,x\right )}{2}-\frac {b^2\,d^2\,x^2\,\mathrm {cosh}\left (3\,a+3\,b\,x\right )}{12}+\frac {b\,d^2\,x\,\mathrm {sinh}\left (3\,a+3\,b\,x\right )}{18}-\frac {3\,b\,c\,d\,\mathrm {sinh}\left (a+b\,x\right )}{2}-\frac {b^2\,c\,d\,x\,\mathrm {cosh}\left (3\,a+3\,b\,x\right )}{6}+\frac {3\,b^2\,c\,d\,x\,\mathrm {cosh}\left (a+b\,x\right )}{2}}{b^3} \] Input:
int(sinh(a + b*x)^3*(c + d*x)^2,x)
Output:
-((3*d^2*cosh(a + b*x))/2 - (d^2*cosh(3*a + 3*b*x))/54 + (3*b^2*c^2*cosh(a + b*x))/4 - (b^2*c^2*cosh(3*a + 3*b*x))/12 + (3*b^2*d^2*x^2*cosh(a + b*x) )/4 + (b*c*d*sinh(3*a + 3*b*x))/18 - (3*b*d^2*x*sinh(a + b*x))/2 - (b^2*d^ 2*x^2*cosh(3*a + 3*b*x))/12 + (b*d^2*x*sinh(3*a + 3*b*x))/18 - (3*b*c*d*si nh(a + b*x))/2 - (b^2*c*d*x*cosh(3*a + 3*b*x))/6 + (3*b^2*c*d*x*cosh(a + b *x))/2)/b^3
Time = 0.23 (sec) , antiderivative size = 374, normalized size of antiderivative = 3.04 \[ \int (c+d x)^2 \sinh ^3(a+b x) \, dx=\frac {9 e^{6 b x +6 a} b^{2} c^{2}+18 e^{6 b x +6 a} b^{2} c d x +9 e^{6 b x +6 a} b^{2} d^{2} x^{2}-6 e^{6 b x +6 a} b c d -6 e^{6 b x +6 a} b \,d^{2} x +2 e^{6 b x +6 a} d^{2}-81 e^{4 b x +4 a} b^{2} c^{2}-162 e^{4 b x +4 a} b^{2} c d x -81 e^{4 b x +4 a} b^{2} d^{2} x^{2}+162 e^{4 b x +4 a} b c d +162 e^{4 b x +4 a} b \,d^{2} x -162 e^{4 b x +4 a} d^{2}-81 e^{2 b x +2 a} b^{2} c^{2}-162 e^{2 b x +2 a} b^{2} c d x -81 e^{2 b x +2 a} b^{2} d^{2} x^{2}-162 e^{2 b x +2 a} b c d -162 e^{2 b x +2 a} b \,d^{2} x -162 e^{2 b x +2 a} d^{2}+9 b^{2} c^{2}+18 b^{2} c d x +9 b^{2} d^{2} x^{2}+6 b c d +6 b \,d^{2} x +2 d^{2}}{216 e^{3 b x +3 a} b^{3}} \] Input:
int((d*x+c)^2*sinh(b*x+a)^3,x)
Output:
(9*e**(6*a + 6*b*x)*b**2*c**2 + 18*e**(6*a + 6*b*x)*b**2*c*d*x + 9*e**(6*a + 6*b*x)*b**2*d**2*x**2 - 6*e**(6*a + 6*b*x)*b*c*d - 6*e**(6*a + 6*b*x)*b *d**2*x + 2*e**(6*a + 6*b*x)*d**2 - 81*e**(4*a + 4*b*x)*b**2*c**2 - 162*e* *(4*a + 4*b*x)*b**2*c*d*x - 81*e**(4*a + 4*b*x)*b**2*d**2*x**2 + 162*e**(4 *a + 4*b*x)*b*c*d + 162*e**(4*a + 4*b*x)*b*d**2*x - 162*e**(4*a + 4*b*x)*d **2 - 81*e**(2*a + 2*b*x)*b**2*c**2 - 162*e**(2*a + 2*b*x)*b**2*c*d*x - 81 *e**(2*a + 2*b*x)*b**2*d**2*x**2 - 162*e**(2*a + 2*b*x)*b*c*d - 162*e**(2* a + 2*b*x)*b*d**2*x - 162*e**(2*a + 2*b*x)*d**2 + 9*b**2*c**2 + 18*b**2*c* d*x + 9*b**2*d**2*x**2 + 6*b*c*d + 6*b*d**2*x + 2*d**2)/(216*e**(3*a + 3*b *x)*b**3)