Integrand size = 18, antiderivative size = 89 \[ \int (c+d x)^3 (a+b \sinh (e+f x)) \, dx=\frac {a (c+d x)^4}{4 d}+\frac {6 b d^2 (c+d x) \cosh (e+f x)}{f^3}+\frac {b (c+d x)^3 \cosh (e+f x)}{f}-\frac {6 b d^3 \sinh (e+f x)}{f^4}-\frac {3 b d (c+d x)^2 \sinh (e+f x)}{f^2} \] Output:
1/4*a*(d*x+c)^4/d+6*b*d^2*(d*x+c)*cosh(f*x+e)/f^3+b*(d*x+c)^3*cosh(f*x+e)/ f-6*b*d^3*sinh(f*x+e)/f^4-3*b*d*(d*x+c)^2*sinh(f*x+e)/f^2
Time = 0.27 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.38 \[ \int (c+d x)^3 (a+b \sinh (e+f x)) \, dx=\frac {1}{4} a x \left (4 c^3+6 c^2 d x+4 c d^2 x^2+d^3 x^3\right )+\frac {b (c+d x) \left (c^2 f^2+2 c d f^2 x+d^2 \left (6+f^2 x^2\right )\right ) \cosh (e+f x)}{f^3}-\frac {3 b d \left (c^2 f^2+2 c d f^2 x+d^2 \left (2+f^2 x^2\right )\right ) \sinh (e+f x)}{f^4} \] Input:
Integrate[(c + d*x)^3*(a + b*Sinh[e + f*x]),x]
Output:
(a*x*(4*c^3 + 6*c^2*d*x + 4*c*d^2*x^2 + d^3*x^3))/4 + (b*(c + d*x)*(c^2*f^ 2 + 2*c*d*f^2*x + d^2*(6 + f^2*x^2))*Cosh[e + f*x])/f^3 - (3*b*d*(c^2*f^2 + 2*c*d*f^2*x + d^2*(2 + f^2*x^2))*Sinh[e + f*x])/f^4
Time = 0.36 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3042, 3798, 2009}
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.
| \(\displaystyle \int (c+d x)^3 (a+b \sinh (e+f x)) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (c+d x)^3 (a-i b \sin (i e+i f x))dx\) |
| \(\Big \downarrow \) 3798 |
| \(\displaystyle \int \left (a (c+d x)^3+b (c+d x)^3 \sinh (e+f x)\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {a (c+d x)^4}{4 d}+\frac {6 b d^2 (c+d x) \cosh (e+f x)}{f^3}-\frac {3 b d (c+d x)^2 \sinh (e+f x)}{f^2}+\frac {b (c+d x)^3 \cosh (e+f x)}{f}-\frac {6 b d^3 \sinh (e+f x)}{f^4}\) |
Input:
Int[(c + d*x)^3*(a + b*Sinh[e + f*x]),x]
Output:
(a*(c + d*x)^4)/(4*d) + (6*b*d^2*(c + d*x)*Cosh[e + f*x])/f^3 + (b*(c + d* x)^3*Cosh[e + f*x])/f - (6*b*d^3*Sinh[e + f*x])/f^4 - (3*b*d*(c + d*x)^2*S inh[e + f*x])/f^2
Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(n_.) , x_Symbol] :> Int[ExpandIntegrand[(c + d*x)^m, (a + b*Sin[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[n, 0] && (EqQ[n, 1] || IGtQ[ m, 0] || NeQ[a^2 - b^2, 0])
Time = 0.32 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.24
| method | result | size |
| parallelrisch | \(\frac {b \left (d x +c \right ) f \left (\left (d x +c \right )^{2} f^{2}+6 d^{2}\right ) \cosh \left (f x +e \right )-3 b d \left (\left (d x +c \right )^{2} f^{2}+2 d^{2}\right ) \sinh \left (f x +e \right )+\left (a \left (\frac {1}{2} x^{2} d^{2}+c d x +c^{2}\right ) \left (\frac {d x}{2}+c \right ) x \,f^{3}+b \,c^{3} f^{2}+6 b c \,d^{2}\right ) f}{f^{4}}\) | \(110\) |
| risch | \(\frac {a \,d^{3} x^{4}}{4}+a \,d^{2} c \,x^{3}+\frac {3 a d \,c^{2} x^{2}}{2}+a \,c^{3} x +\frac {a \,c^{4}}{4 d}+\frac {b \left (d^{3} x^{3} f^{3}+3 c \,d^{2} f^{3} x^{2}+3 c^{2} d \,f^{3} x -3 d^{3} f^{2} x^{2}+c^{3} f^{3}-6 c \,d^{2} f^{2} x -3 c^{2} d \,f^{2}+6 d^{3} f x +6 c \,d^{2} f -6 d^{3}\right ) {\mathrm e}^{f x +e}}{2 f^{4}}+\frac {b \left (d^{3} x^{3} f^{3}+3 c \,d^{2} f^{3} x^{2}+3 c^{2} d \,f^{3} x +3 d^{3} f^{2} x^{2}+c^{3} f^{3}+6 c \,d^{2} f^{2} x +3 c^{2} d \,f^{2}+6 d^{3} f x +6 c \,d^{2} f +6 d^{3}\right ) {\mathrm e}^{-f x -e}}{2 f^{4}}\) | \(250\) |
| parts | \(\frac {a \left (d x +c \right )^{4}}{4 d}+\frac {b \left (\frac {d^{3} \left (\left (f x +e \right )^{3} \cosh \left (f x +e \right )-3 \left (f x +e \right )^{2} \sinh \left (f x +e \right )+6 \left (f x +e \right ) \cosh \left (f x +e \right )-6 \sinh \left (f x +e \right )\right )}{f^{3}}-\frac {3 d^{3} e \left (\left (f x +e \right )^{2} \cosh \left (f x +e \right )-2 \left (f x +e \right ) \sinh \left (f x +e \right )+2 \cosh \left (f x +e \right )\right )}{f^{3}}+\frac {3 d^{2} c \left (\left (f x +e \right )^{2} \cosh \left (f x +e \right )-2 \left (f x +e \right ) \sinh \left (f x +e \right )+2 \cosh \left (f x +e \right )\right )}{f^{2}}+\frac {3 d^{3} e^{2} \left (\left (f x +e \right ) \cosh \left (f x +e \right )-\sinh \left (f x +e \right )\right )}{f^{3}}-\frac {6 d^{2} e c \left (\left (f x +e \right ) \cosh \left (f x +e \right )-\sinh \left (f x +e \right )\right )}{f^{2}}+\frac {3 d \,c^{2} \left (\left (f x +e \right ) \cosh \left (f x +e \right )-\sinh \left (f x +e \right )\right )}{f}-\frac {d^{3} e^{3} \cosh \left (f x +e \right )}{f^{3}}+\frac {3 d^{2} e^{2} c \cosh \left (f x +e \right )}{f^{2}}-\frac {3 d e \,c^{2} \cosh \left (f x +e \right )}{f}+c^{3} \cosh \left (f x +e \right )\right )}{f}\) | \(323\) |
| orering | \(\frac {\left (d^{5} f^{4} x^{6}+6 c \,d^{4} f^{4} x^{5}+15 c^{2} d^{3} f^{4} x^{4}+20 c^{3} d^{2} f^{4} x^{3}+14 c^{4} d \,f^{4} x^{2}-24 d^{5} f^{2} x^{4}+4 c^{5} f^{4} x -96 c \,d^{4} f^{2} x^{3}-156 c^{2} d^{3} f^{2} x^{2}-120 c^{3} d^{2} f^{2} x -24 c^{4} d \,f^{2}-240 d^{5} x^{2}-480 c \,d^{4} x -96 c^{2} d^{3}\right ) \left (a +b \sinh \left (f x +e \right )\right )}{4 f^{4} \left (d x +c \right )^{2}}+\frac {\left (5 d^{4} f^{2} x^{4}+20 c \,d^{3} f^{2} x^{3}+30 c^{2} d^{2} f^{2} x^{2}+20 c^{3} d \,f^{2} x +2 c^{4} f^{2}+48 d^{4} x^{2}+96 c \,d^{3} x +12 d^{2} c^{2}\right ) \left (3 \left (d x +c \right )^{2} \left (a +b \sinh \left (f x +e \right )\right ) d +\left (d x +c \right )^{3} b f \cosh \left (f x +e \right )\right )}{2 f^{4} \left (d x +c \right )^{4}}-\frac {x \left (d^{3} f^{2} x^{3}+4 c \,d^{2} f^{2} x^{2}+6 c^{2} d \,f^{2} x +4 c^{3} f^{2}+12 d^{3} x +24 d^{2} c \right ) \left (6 \left (d x +c \right ) \left (a +b \sinh \left (f x +e \right )\right ) d^{2}+6 \left (d x +c \right )^{2} b f \cosh \left (f x +e \right ) d +\left (d x +c \right )^{3} b \,f^{2} \sinh \left (f x +e \right )\right )}{4 f^{4} \left (d x +c \right )^{3}}\) | \(428\) |
| derivativedivides | \(\frac {\frac {d^{3} a \left (f x +e \right )^{4}}{4 f^{3}}+\frac {d^{3} b \left (\left (f x +e \right )^{3} \cosh \left (f x +e \right )-3 \left (f x +e \right )^{2} \sinh \left (f x +e \right )+6 \left (f x +e \right ) \cosh \left (f x +e \right )-6 \sinh \left (f x +e \right )\right )}{f^{3}}-\frac {d^{3} e a \left (f x +e \right )^{3}}{f^{3}}-\frac {3 d^{3} e b \left (\left (f x +e \right )^{2} \cosh \left (f x +e \right )-2 \left (f x +e \right ) \sinh \left (f x +e \right )+2 \cosh \left (f x +e \right )\right )}{f^{3}}+\frac {d^{2} c a \left (f x +e \right )^{3}}{f^{2}}+\frac {3 d^{2} c b \left (\left (f x +e \right )^{2} \cosh \left (f x +e \right )-2 \left (f x +e \right ) \sinh \left (f x +e \right )+2 \cosh \left (f x +e \right )\right )}{f^{2}}+\frac {3 d^{3} e^{2} a \left (f x +e \right )^{2}}{2 f^{3}}+\frac {3 d^{3} e^{2} b \left (\left (f x +e \right ) \cosh \left (f x +e \right )-\sinh \left (f x +e \right )\right )}{f^{3}}-\frac {3 d^{2} e c a \left (f x +e \right )^{2}}{f^{2}}-\frac {6 d^{2} e c b \left (\left (f x +e \right ) \cosh \left (f x +e \right )-\sinh \left (f x +e \right )\right )}{f^{2}}+\frac {3 d \,c^{2} a \left (f x +e \right )^{2}}{2 f}+\frac {3 d \,c^{2} b \left (\left (f x +e \right ) \cosh \left (f x +e \right )-\sinh \left (f x +e \right )\right )}{f}-\frac {d^{3} e^{3} a \left (f x +e \right )}{f^{3}}-\frac {d^{3} e^{3} b \cosh \left (f x +e \right )}{f^{3}}+\frac {3 d^{2} e^{2} c a \left (f x +e \right )}{f^{2}}+\frac {3 d^{2} e^{2} c b \cosh \left (f x +e \right )}{f^{2}}-\frac {3 d e \,c^{2} a \left (f x +e \right )}{f}-\frac {3 d e \,c^{2} b \cosh \left (f x +e \right )}{f}+c^{3} a \left (f x +e \right )+b \,c^{3} \cosh \left (f x +e \right )}{f}\) | \(482\) |
| default | \(\frac {\frac {d^{3} a \left (f x +e \right )^{4}}{4 f^{3}}+\frac {d^{3} b \left (\left (f x +e \right )^{3} \cosh \left (f x +e \right )-3 \left (f x +e \right )^{2} \sinh \left (f x +e \right )+6 \left (f x +e \right ) \cosh \left (f x +e \right )-6 \sinh \left (f x +e \right )\right )}{f^{3}}-\frac {d^{3} e a \left (f x +e \right )^{3}}{f^{3}}-\frac {3 d^{3} e b \left (\left (f x +e \right )^{2} \cosh \left (f x +e \right )-2 \left (f x +e \right ) \sinh \left (f x +e \right )+2 \cosh \left (f x +e \right )\right )}{f^{3}}+\frac {d^{2} c a \left (f x +e \right )^{3}}{f^{2}}+\frac {3 d^{2} c b \left (\left (f x +e \right )^{2} \cosh \left (f x +e \right )-2 \left (f x +e \right ) \sinh \left (f x +e \right )+2 \cosh \left (f x +e \right )\right )}{f^{2}}+\frac {3 d^{3} e^{2} a \left (f x +e \right )^{2}}{2 f^{3}}+\frac {3 d^{3} e^{2} b \left (\left (f x +e \right ) \cosh \left (f x +e \right )-\sinh \left (f x +e \right )\right )}{f^{3}}-\frac {3 d^{2} e c a \left (f x +e \right )^{2}}{f^{2}}-\frac {6 d^{2} e c b \left (\left (f x +e \right ) \cosh \left (f x +e \right )-\sinh \left (f x +e \right )\right )}{f^{2}}+\frac {3 d \,c^{2} a \left (f x +e \right )^{2}}{2 f}+\frac {3 d \,c^{2} b \left (\left (f x +e \right ) \cosh \left (f x +e \right )-\sinh \left (f x +e \right )\right )}{f}-\frac {d^{3} e^{3} a \left (f x +e \right )}{f^{3}}-\frac {d^{3} e^{3} b \cosh \left (f x +e \right )}{f^{3}}+\frac {3 d^{2} e^{2} c a \left (f x +e \right )}{f^{2}}+\frac {3 d^{2} e^{2} c b \cosh \left (f x +e \right )}{f^{2}}-\frac {3 d e \,c^{2} a \left (f x +e \right )}{f}-\frac {3 d e \,c^{2} b \cosh \left (f x +e \right )}{f}+c^{3} a \left (f x +e \right )+b \,c^{3} \cosh \left (f x +e \right )}{f}\) | \(482\) |
Input:
int((d*x+c)^3*(a+b*sinh(f*x+e)),x,method=_RETURNVERBOSE)
Output:
(b*(d*x+c)*f*((d*x+c)^2*f^2+6*d^2)*cosh(f*x+e)-3*b*d*((d*x+c)^2*f^2+2*d^2) *sinh(f*x+e)+(a*(1/2*x^2*d^2+c*d*x+c^2)*(1/2*d*x+c)*x*f^3+b*c^3*f^2+6*b*c* d^2)*f)/f^4
Time = 0.09 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.89 \[ \int (c+d x)^3 (a+b \sinh (e+f x)) \, dx=\frac {a d^{3} f^{4} x^{4} + 4 \, a c d^{2} f^{4} x^{3} + 6 \, a c^{2} d f^{4} x^{2} + 4 \, a c^{3} f^{4} x + 4 \, {\left (b d^{3} f^{3} x^{3} + 3 \, b c d^{2} f^{3} x^{2} + b c^{3} f^{3} + 6 \, b c d^{2} f + 3 \, {\left (b c^{2} d f^{3} + 2 \, b d^{3} f\right )} x\right )} \cosh \left (f x + e\right ) - 12 \, {\left (b d^{3} f^{2} x^{2} + 2 \, b c d^{2} f^{2} x + b c^{2} d f^{2} + 2 \, b d^{3}\right )} \sinh \left (f x + e\right )}{4 \, f^{4}} \] Input:
integrate((d*x+c)^3*(a+b*sinh(f*x+e)),x, algorithm="fricas")
Output:
1/4*(a*d^3*f^4*x^4 + 4*a*c*d^2*f^4*x^3 + 6*a*c^2*d*f^4*x^2 + 4*a*c^3*f^4*x + 4*(b*d^3*f^3*x^3 + 3*b*c*d^2*f^3*x^2 + b*c^3*f^3 + 6*b*c*d^2*f + 3*(b*c ^2*d*f^3 + 2*b*d^3*f)*x)*cosh(f*x + e) - 12*(b*d^3*f^2*x^2 + 2*b*c*d^2*f^2 *x + b*c^2*d*f^2 + 2*b*d^3)*sinh(f*x + e))/f^4
Leaf count of result is larger than twice the leaf count of optimal. 264 vs. \(2 (88) = 176\).
Time = 0.28 (sec) , antiderivative size = 264, normalized size of antiderivative = 2.97 \[ \int (c+d x)^3 (a+b \sinh (e+f x)) \, dx=\begin {cases} a c^{3} x + \frac {3 a c^{2} d x^{2}}{2} + a c d^{2} x^{3} + \frac {a d^{3} x^{4}}{4} + \frac {b c^{3} \cosh {\left (e + f x \right )}}{f} + \frac {3 b c^{2} d x \cosh {\left (e + f x \right )}}{f} - \frac {3 b c^{2} d \sinh {\left (e + f x \right )}}{f^{2}} + \frac {3 b c d^{2} x^{2} \cosh {\left (e + f x \right )}}{f} - \frac {6 b c d^{2} x \sinh {\left (e + f x \right )}}{f^{2}} + \frac {6 b c d^{2} \cosh {\left (e + f x \right )}}{f^{3}} + \frac {b d^{3} x^{3} \cosh {\left (e + f x \right )}}{f} - \frac {3 b d^{3} x^{2} \sinh {\left (e + f x \right )}}{f^{2}} + \frac {6 b d^{3} x \cosh {\left (e + f x \right )}}{f^{3}} - \frac {6 b d^{3} \sinh {\left (e + f x \right )}}{f^{4}} & \text {for}\: f \neq 0 \\\left (a + b \sinh {\left (e \right )}\right ) \left (c^{3} x + \frac {3 c^{2} d x^{2}}{2} + c d^{2} x^{3} + \frac {d^{3} x^{4}}{4}\right ) & \text {otherwise} \end {cases} \] Input:
integrate((d*x+c)**3*(a+b*sinh(f*x+e)),x)
Output:
Piecewise((a*c**3*x + 3*a*c**2*d*x**2/2 + a*c*d**2*x**3 + a*d**3*x**4/4 + b*c**3*cosh(e + f*x)/f + 3*b*c**2*d*x*cosh(e + f*x)/f - 3*b*c**2*d*sinh(e + f*x)/f**2 + 3*b*c*d**2*x**2*cosh(e + f*x)/f - 6*b*c*d**2*x*sinh(e + f*x) /f**2 + 6*b*c*d**2*cosh(e + f*x)/f**3 + b*d**3*x**3*cosh(e + f*x)/f - 3*b* d**3*x**2*sinh(e + f*x)/f**2 + 6*b*d**3*x*cosh(e + f*x)/f**3 - 6*b*d**3*si nh(e + f*x)/f**4, Ne(f, 0)), ((a + b*sinh(e))*(c**3*x + 3*c**2*d*x**2/2 + c*d**2*x**3 + d**3*x**4/4), True))
Leaf count of result is larger than twice the leaf count of optimal. 234 vs. \(2 (87) = 174\).
Time = 0.05 (sec) , antiderivative size = 234, normalized size of antiderivative = 2.63 \[ \int (c+d x)^3 (a+b \sinh (e+f x)) \, dx=\frac {1}{4} \, a d^{3} x^{4} + a c d^{2} x^{3} + \frac {3}{2} \, a c^{2} d x^{2} + a c^{3} x + \frac {3}{2} \, b c^{2} d {\left (\frac {{\left (f x e^{e} - e^{e}\right )} e^{\left (f x\right )}}{f^{2}} + \frac {{\left (f x + 1\right )} e^{\left (-f x - e\right )}}{f^{2}}\right )} + \frac {3}{2} \, b c d^{2} {\left (\frac {{\left (f^{2} x^{2} e^{e} - 2 \, f x e^{e} + 2 \, e^{e}\right )} e^{\left (f x\right )}}{f^{3}} + \frac {{\left (f^{2} x^{2} + 2 \, f x + 2\right )} e^{\left (-f x - e\right )}}{f^{3}}\right )} + \frac {1}{2} \, b d^{3} {\left (\frac {{\left (f^{3} x^{3} e^{e} - 3 \, f^{2} x^{2} e^{e} + 6 \, f x e^{e} - 6 \, e^{e}\right )} e^{\left (f x\right )}}{f^{4}} + \frac {{\left (f^{3} x^{3} + 3 \, f^{2} x^{2} + 6 \, f x + 6\right )} e^{\left (-f x - e\right )}}{f^{4}}\right )} + \frac {b c^{3} \cosh \left (f x + e\right )}{f} \] Input:
integrate((d*x+c)^3*(a+b*sinh(f*x+e)),x, algorithm="maxima")
Output:
1/4*a*d^3*x^4 + a*c*d^2*x^3 + 3/2*a*c^2*d*x^2 + a*c^3*x + 3/2*b*c^2*d*((f* x*e^e - e^e)*e^(f*x)/f^2 + (f*x + 1)*e^(-f*x - e)/f^2) + 3/2*b*c*d^2*((f^2 *x^2*e^e - 2*f*x*e^e + 2*e^e)*e^(f*x)/f^3 + (f^2*x^2 + 2*f*x + 2)*e^(-f*x - e)/f^3) + 1/2*b*d^3*((f^3*x^3*e^e - 3*f^2*x^2*e^e + 6*f*x*e^e - 6*e^e)*e ^(f*x)/f^4 + (f^3*x^3 + 3*f^2*x^2 + 6*f*x + 6)*e^(-f*x - e)/f^4) + b*c^3*c osh(f*x + e)/f
Leaf count of result is larger than twice the leaf count of optimal. 258 vs. \(2 (87) = 174\).
Time = 0.12 (sec) , antiderivative size = 258, normalized size of antiderivative = 2.90 \[ \int (c+d x)^3 (a+b \sinh (e+f x)) \, dx=\frac {1}{4} \, a d^{3} x^{4} + a c d^{2} x^{3} + \frac {3}{2} \, a c^{2} d x^{2} + a c^{3} x + \frac {{\left (b d^{3} f^{3} x^{3} + 3 \, b c d^{2} f^{3} x^{2} + 3 \, b c^{2} d f^{3} x - 3 \, b d^{3} f^{2} x^{2} + b c^{3} f^{3} - 6 \, b c d^{2} f^{2} x - 3 \, b c^{2} d f^{2} + 6 \, b d^{3} f x + 6 \, b c d^{2} f - 6 \, b d^{3}\right )} e^{\left (f x + e\right )}}{2 \, f^{4}} + \frac {{\left (b d^{3} f^{3} x^{3} + 3 \, b c d^{2} f^{3} x^{2} + 3 \, b c^{2} d f^{3} x + 3 \, b d^{3} f^{2} x^{2} + b c^{3} f^{3} + 6 \, b c d^{2} f^{2} x + 3 \, b c^{2} d f^{2} + 6 \, b d^{3} f x + 6 \, b c d^{2} f + 6 \, b d^{3}\right )} e^{\left (-f x - e\right )}}{2 \, f^{4}} \] Input:
integrate((d*x+c)^3*(a+b*sinh(f*x+e)),x, algorithm="giac")
Output:
1/4*a*d^3*x^4 + a*c*d^2*x^3 + 3/2*a*c^2*d*x^2 + a*c^3*x + 1/2*(b*d^3*f^3*x ^3 + 3*b*c*d^2*f^3*x^2 + 3*b*c^2*d*f^3*x - 3*b*d^3*f^2*x^2 + b*c^3*f^3 - 6 *b*c*d^2*f^2*x - 3*b*c^2*d*f^2 + 6*b*d^3*f*x + 6*b*c*d^2*f - 6*b*d^3)*e^(f *x + e)/f^4 + 1/2*(b*d^3*f^3*x^3 + 3*b*c*d^2*f^3*x^2 + 3*b*c^2*d*f^3*x + 3 *b*d^3*f^2*x^2 + b*c^3*f^3 + 6*b*c*d^2*f^2*x + 3*b*c^2*d*f^2 + 6*b*d^3*f*x + 6*b*c*d^2*f + 6*b*d^3)*e^(-f*x - e)/f^4
Time = 1.08 (sec) , antiderivative size = 187, normalized size of antiderivative = 2.10 \[ \int (c+d x)^3 (a+b \sinh (e+f x)) \, dx=\frac {\mathrm {cosh}\left (e+f\,x\right )\,\left (b\,c^3\,f^2+6\,b\,c\,d^2\right )}{f^3}-\frac {3\,\mathrm {sinh}\left (e+f\,x\right )\,\left (b\,c^2\,d\,f^2+2\,b\,d^3\right )}{f^4}+\frac {a\,d^3\,x^4}{4}+a\,c^3\,x+\frac {3\,x\,\mathrm {cosh}\left (e+f\,x\right )\,\left (b\,c^2\,d\,f^2+2\,b\,d^3\right )}{f^3}+\frac {3\,a\,c^2\,d\,x^2}{2}+a\,c\,d^2\,x^3+\frac {b\,d^3\,x^3\,\mathrm {cosh}\left (e+f\,x\right )}{f}-\frac {3\,b\,d^3\,x^2\,\mathrm {sinh}\left (e+f\,x\right )}{f^2}-\frac {6\,b\,c\,d^2\,x\,\mathrm {sinh}\left (e+f\,x\right )}{f^2}+\frac {3\,b\,c\,d^2\,x^2\,\mathrm {cosh}\left (e+f\,x\right )}{f} \] Input:
int((a + b*sinh(e + f*x))*(c + d*x)^3,x)
Output:
(cosh(e + f*x)*(b*c^3*f^2 + 6*b*c*d^2))/f^3 - (3*sinh(e + f*x)*(2*b*d^3 + b*c^2*d*f^2))/f^4 + (a*d^3*x^4)/4 + a*c^3*x + (3*x*cosh(e + f*x)*(2*b*d^3 + b*c^2*d*f^2))/f^3 + (3*a*c^2*d*x^2)/2 + a*c*d^2*x^3 + (b*d^3*x^3*cosh(e + f*x))/f - (3*b*d^3*x^2*sinh(e + f*x))/f^2 - (6*b*c*d^2*x*sinh(e + f*x))/ f^2 + (3*b*c*d^2*x^2*cosh(e + f*x))/f
Time = 0.24 (sec) , antiderivative size = 213, normalized size of antiderivative = 2.39 \[ \int (c+d x)^3 (a+b \sinh (e+f x)) \, dx=\frac {4 \cosh \left (f x +e \right ) b \,c^{3} f^{3}+12 \cosh \left (f x +e \right ) b \,c^{2} d \,f^{3} x +12 \cosh \left (f x +e \right ) b c \,d^{2} f^{3} x^{2}+24 \cosh \left (f x +e \right ) b c \,d^{2} f +4 \cosh \left (f x +e \right ) b \,d^{3} f^{3} x^{3}+24 \cosh \left (f x +e \right ) b \,d^{3} f x -12 \sinh \left (f x +e \right ) b \,c^{2} d \,f^{2}-24 \sinh \left (f x +e \right ) b c \,d^{2} f^{2} x -12 \sinh \left (f x +e \right ) b \,d^{3} f^{2} x^{2}-24 \sinh \left (f x +e \right ) b \,d^{3}+4 a \,c^{3} f^{4} x +6 a \,c^{2} d \,f^{4} x^{2}+4 a c \,d^{2} f^{4} x^{3}+a \,d^{3} f^{4} x^{4}}{4 f^{4}} \] Input:
int((d*x+c)^3*(a+b*sinh(f*x+e)),x)
Output:
(4*cosh(e + f*x)*b*c**3*f**3 + 12*cosh(e + f*x)*b*c**2*d*f**3*x + 12*cosh( e + f*x)*b*c*d**2*f**3*x**2 + 24*cosh(e + f*x)*b*c*d**2*f + 4*cosh(e + f*x )*b*d**3*f**3*x**3 + 24*cosh(e + f*x)*b*d**3*f*x - 12*sinh(e + f*x)*b*c**2 *d*f**2 - 24*sinh(e + f*x)*b*c*d**2*f**2*x - 12*sinh(e + f*x)*b*d**3*f**2* x**2 - 24*sinh(e + f*x)*b*d**3 + 4*a*c**3*f**4*x + 6*a*c**2*d*f**4*x**2 + 4*a*c*d**2*f**4*x**3 + a*d**3*f**4*x**4)/(4*f**4)