\(\int (c+d x)^3 (a+a \cosh (e+f x)) \, dx\) [99]

Optimal result

Integrand size = 18, antiderivative size = 89 \[ \int (c+d x)^3 (a+a \cosh (e+f x)) \, dx=\frac {a (c+d x)^4}{4 d}-\frac {6 a d^3 \cosh (e+f x)}{f^4}-\frac {3 a d (c+d x)^2 \cosh (e+f x)}{f^2}+\frac {6 a d^2 (c+d x) \sinh (e+f x)}{f^3}+\frac {a (c+d x)^3 \sinh (e+f x)}{f} \]

[Out]

1/4*a*(d*x+c)^4/d-6*a*d^3*cosh(f*x+e)/f^4-3*a*d*(d*x+c)^2*cosh(f*x+e)/f^2+6*a*d^2*(d*x+c)*sinh(f*x+e)/f^3+a*(d
*x+c)^3*sinh(f*x+e)/f

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3398, 3377, 2718} \[ \int (c+d x)^3 (a+a \cosh (e+f x)) \, dx=\frac {6 a d^2 (c+d x) \sinh (e+f x)}{f^3}-\frac {3 a d (c+d x)^2 \cosh (e+f x)}{f^2}+\frac {a (c+d x)^3 \sinh (e+f x)}{f}+\frac {a (c+d x)^4}{4 d}-\frac {6 a d^3 \cosh (e+f x)}{f^4} \]

[In]

Int[(c + d*x)^3*(a + a*Cosh[e + f*x]),x]

[Out]

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

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 3398

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])

Rubi steps \begin{align*} \text {integral}& = \int \left (a (c+d x)^3+a (c+d x)^3 \cosh (e+f x)\right ) \, dx \\ & = \frac {a (c+d x)^4}{4 d}+a \int (c+d x)^3 \cosh (e+f x) \, dx \\ & = \frac {a (c+d x)^4}{4 d}+\frac {a (c+d x)^3 \sinh (e+f x)}{f}-\frac {(3 a d) \int (c+d x)^2 \sinh (e+f x) \, dx}{f} \\ & = \frac {a (c+d x)^4}{4 d}-\frac {3 a d (c+d x)^2 \cosh (e+f x)}{f^2}+\frac {a (c+d x)^3 \sinh (e+f x)}{f}+\frac {\left (6 a d^2\right ) \int (c+d x) \cosh (e+f x) \, dx}{f^2} \\ & = \frac {a (c+d x)^4}{4 d}-\frac {3 a d (c+d x)^2 \cosh (e+f x)}{f^2}+\frac {6 a d^2 (c+d x) \sinh (e+f x)}{f^3}+\frac {a (c+d x)^3 \sinh (e+f x)}{f}-\frac {\left (6 a d^3\right ) \int \sinh (e+f x) \, dx}{f^3} \\ & = \frac {a (c+d x)^4}{4 d}-\frac {6 a d^3 \cosh (e+f x)}{f^4}-\frac {3 a d (c+d x)^2 \cosh (e+f x)}{f^2}+\frac {6 a d^2 (c+d x) \sinh (e+f x)}{f^3}+\frac {a (c+d x)^3 \sinh (e+f x)}{f} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.34 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.37 \[ \int (c+d x)^3 (a+a \cosh (e+f x)) \, dx=a \left (\frac {1}{4} x \left (4 c^3+6 c^2 d x+4 c d^2 x^2+d^3 x^3\right )-\frac {3 d \left (c^2 f^2+2 c d f^2 x+d^2 \left (2+f^2 x^2\right )\right ) \cosh (e+f x)}{f^4}+\frac {(c+d x) \left (c^2 f^2+2 c d f^2 x+d^2 \left (6+f^2 x^2\right )\right ) \sinh (e+f x)}{f^3}\right ) \]

[In]

Integrate[(c + d*x)^3*(a + a*Cosh[e + f*x]),x]

[Out]

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

Maple [A] (verified)

Time = 0.23 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.17

method	result	size
parallelrisch	\(\frac {a \left (\left (d x +c \right ) \left (\left (d x +c \right )^{2} f^{2}+6 d^{2}\right ) f \sinh \left (f x +e \right )-3 d \left (\left (d x +c \right )^{2} f^{2}+2 d^{2}\right ) \cosh \left (f x +e \right )+\left (\frac {1}{2} x^{2} d^{2}+c d x +c^{2}\right ) x \left (\frac {d x}{2}+c \right ) f^{4}-3 c^{2} d \,f^{2}-6 d^{3}\right )}{f^{4}}\)	\(104\)
risch	\(\frac {a \,d^{3} x^{4}}{4}+a \,d^{2} c \,x^{3}+\frac {3 a d \,c^{2} x^{2}}{2}+a \,c^{3} x +\frac {a \,c^{4}}{4 d}+\frac {a \left (d^{3} x^{3} f^{3}+3 c \,d^{2} f^{3} x^{2}+3 c^{2} d \,f^{3} x -3 d^{3} f^{2} x^{2}+c^{3} f^{3}-6 c \,d^{2} f^{2} x -3 c^{2} d \,f^{2}+6 d^{3} f x +6 c \,d^{2} f -6 d^{3}\right ) {\mathrm e}^{f x +e}}{2 f^{4}}-\frac {a \left (d^{3} x^{3} f^{3}+3 c \,d^{2} f^{3} x^{2}+3 c^{2} d \,f^{3} x +3 d^{3} f^{2} x^{2}+c^{3} f^{3}+6 c \,d^{2} f^{2} x +3 c^{2} d \,f^{2}+6 d^{3} f x +6 c \,d^{2} f +6 d^{3}\right ) {\mathrm e}^{-f x -e}}{2 f^{4}}\)	\(250\)
parts	\(\frac {a \left (d x +c \right )^{4}}{4 d}+\frac {a \left (\frac {d^{3} \left (\left (f x +e \right )^{3} \sinh \left (f x +e \right )-3 \left (f x +e \right )^{2} \cosh \left (f x +e \right )+6 \left (f x +e \right ) \sinh \left (f x +e \right )-6 \cosh \left (f x +e \right )\right )}{f^{3}}-\frac {3 d^{3} e \left (\left (f x +e \right )^{2} \sinh \left (f x +e \right )-2 \left (f x +e \right ) \cosh \left (f x +e \right )+2 \sinh \left (f x +e \right )\right )}{f^{3}}+\frac {3 d^{2} c \left (\left (f x +e \right )^{2} \sinh \left (f x +e \right )-2 \left (f x +e \right ) \cosh \left (f x +e \right )+2 \sinh \left (f x +e \right )\right )}{f^{2}}+\frac {3 d^{3} e^{2} \left (\left (f x +e \right ) \sinh \left (f x +e \right )-\cosh \left (f x +e \right )\right )}{f^{3}}-\frac {6 d^{2} e c \left (\left (f x +e \right ) \sinh \left (f x +e \right )-\cosh \left (f x +e \right )\right )}{f^{2}}+\frac {3 d \,c^{2} \left (\left (f x +e \right ) \sinh \left (f x +e \right )-\cosh \left (f x +e \right )\right )}{f}-\frac {d^{3} e^{3} \sinh \left (f x +e \right )}{f^{3}}+\frac {3 d^{2} e^{2} c \sinh \left (f x +e \right )}{f^{2}}-\frac {3 d e \,c^{2} \sinh \left (f x +e \right )}{f}+c^{3} \sinh \left (f x +e \right )\right )}{f}\)	\(323\)
derivativedivides	\(\frac {\frac {d^{3} a \left (f x +e \right )^{4}}{4 f^{3}}+\frac {d^{3} a \left (\left (f x +e \right )^{3} \sinh \left (f x +e \right )-3 \left (f x +e \right )^{2} \cosh \left (f x +e \right )+6 \left (f x +e \right ) \sinh \left (f x +e \right )-6 \cosh \left (f x +e \right )\right )}{f^{3}}-\frac {d^{3} e a \left (f x +e \right )^{3}}{f^{3}}-\frac {3 d^{3} e a \left (\left (f x +e \right )^{2} \sinh \left (f x +e \right )-2 \left (f x +e \right ) \cosh \left (f x +e \right )+2 \sinh \left (f x +e \right )\right )}{f^{3}}+\frac {d^{2} c a \left (f x +e \right )^{3}}{f^{2}}+\frac {3 d^{2} c a \left (\left (f x +e \right )^{2} \sinh \left (f x +e \right )-2 \left (f x +e \right ) \cosh \left (f x +e \right )+2 \sinh \left (f x +e \right )\right )}{f^{2}}+\frac {3 d^{3} e^{2} a \left (f x +e \right )^{2}}{2 f^{3}}+\frac {3 d^{3} e^{2} a \left (\left (f x +e \right ) \sinh \left (f x +e \right )-\cosh \left (f x +e \right )\right )}{f^{3}}-\frac {3 d^{2} e c a \left (f x +e \right )^{2}}{f^{2}}-\frac {6 d^{2} e c a \left (\left (f x +e \right ) \sinh \left (f x +e \right )-\cosh \left (f x +e \right )\right )}{f^{2}}+\frac {3 d \,c^{2} a \left (f x +e \right )^{2}}{2 f}+\frac {3 d \,c^{2} a \left (\left (f x +e \right ) \sinh \left (f x +e \right )-\cosh \left (f x +e \right )\right )}{f}-\frac {d^{3} e^{3} a \left (f x +e \right )}{f^{3}}-\frac {d^{3} e^{3} a \sinh \left (f x +e \right )}{f^{3}}+\frac {3 d^{2} e^{2} c a \left (f x +e \right )}{f^{2}}+\frac {3 d^{2} e^{2} c a \sinh \left (f x +e \right )}{f^{2}}-\frac {3 d e \,c^{2} a \left (f x +e \right )}{f}-\frac {3 d e \,c^{2} a \sinh \left (f x +e \right )}{f}+c^{3} a \left (f x +e \right )+c^{3} a \sinh \left (f x +e \right )}{f}\)	\(482\)
default	\(\frac {\frac {d^{3} a \left (f x +e \right )^{4}}{4 f^{3}}+\frac {d^{3} a \left (\left (f x +e \right )^{3} \sinh \left (f x +e \right )-3 \left (f x +e \right )^{2} \cosh \left (f x +e \right )+6 \left (f x +e \right ) \sinh \left (f x +e \right )-6 \cosh \left (f x +e \right )\right )}{f^{3}}-\frac {d^{3} e a \left (f x +e \right )^{3}}{f^{3}}-\frac {3 d^{3} e a \left (\left (f x +e \right )^{2} \sinh \left (f x +e \right )-2 \left (f x +e \right ) \cosh \left (f x +e \right )+2 \sinh \left (f x +e \right )\right )}{f^{3}}+\frac {d^{2} c a \left (f x +e \right )^{3}}{f^{2}}+\frac {3 d^{2} c a \left (\left (f x +e \right )^{2} \sinh \left (f x +e \right )-2 \left (f x +e \right ) \cosh \left (f x +e \right )+2 \sinh \left (f x +e \right )\right )}{f^{2}}+\frac {3 d^{3} e^{2} a \left (f x +e \right )^{2}}{2 f^{3}}+\frac {3 d^{3} e^{2} a \left (\left (f x +e \right ) \sinh \left (f x +e \right )-\cosh \left (f x +e \right )\right )}{f^{3}}-\frac {3 d^{2} e c a \left (f x +e \right )^{2}}{f^{2}}-\frac {6 d^{2} e c a \left (\left (f x +e \right ) \sinh \left (f x +e \right )-\cosh \left (f x +e \right )\right )}{f^{2}}+\frac {3 d \,c^{2} a \left (f x +e \right )^{2}}{2 f}+\frac {3 d \,c^{2} a \left (\left (f x +e \right ) \sinh \left (f x +e \right )-\cosh \left (f x +e \right )\right )}{f}-\frac {d^{3} e^{3} a \left (f x +e \right )}{f^{3}}-\frac {d^{3} e^{3} a \sinh \left (f x +e \right )}{f^{3}}+\frac {3 d^{2} e^{2} c a \left (f x +e \right )}{f^{2}}+\frac {3 d^{2} e^{2} c a \sinh \left (f x +e \right )}{f^{2}}-\frac {3 d e \,c^{2} a \left (f x +e \right )}{f}-\frac {3 d e \,c^{2} a \sinh \left (f x +e \right )}{f}+c^{3} a \left (f x +e \right )+c^{3} a \sinh \left (f x +e \right )}{f}\)	\(482\)

[In]

int((d*x+c)^3*(a+a*cosh(f*x+e)),x,method=_RETURNVERBOSE)

[Out]

a*((d*x+c)*((d*x+c)^2*f^2+6*d^2)*f*sinh(f*x+e)-3*d*((d*x+c)^2*f^2+2*d^2)*cosh(f*x+e)+(1/2*x^2*d^2+c*d*x+c^2)*x
*(1/2*d*x+c)*f^4-3*c^2*d*f^2-6*d^3)/f^4

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.89 \[ \int (c+d x)^3 (a+a \cosh (e+f x)) \, dx=\frac {a d^{3} f^{4} x^{4} + 4 \, a c d^{2} f^{4} x^{3} + 6 \, a c^{2} d f^{4} x^{2} + 4 \, a c^{3} f^{4} x - 12 \, {\left (a d^{3} f^{2} x^{2} + 2 \, a c d^{2} f^{2} x + a c^{2} d f^{2} + 2 \, a d^{3}\right )} \cosh \left (f x + e\right ) + 4 \, {\left (a d^{3} f^{3} x^{3} + 3 \, a c d^{2} f^{3} x^{2} + a c^{3} f^{3} + 6 \, a c d^{2} f + 3 \, {\left (a c^{2} d f^{3} + 2 \, a d^{3} f\right )} x\right )} \sinh \left (f x + e\right )}{4 \, f^{4}} \]

[In]

integrate((d*x+c)^3*(a+a*cosh(f*x+e)),x, algorithm="fricas")

[Out]

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 - 12*(a*d^3*f^2*x^2 + 2*a*c*d^2*f^2
*x + a*c^2*d*f^2 + 2*a*d^3)*cosh(f*x + e) + 4*(a*d^3*f^3*x^3 + 3*a*c*d^2*f^3*x^2 + a*c^3*f^3 + 6*a*c*d^2*f + 3
*(a*c^2*d*f^3 + 2*a*d^3*f)*x)*sinh(f*x + e))/f^4

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 264 vs. \(2 (88) = 176\).

Time = 0.30 (sec) , antiderivative size = 264, normalized size of antiderivative = 2.97 \[ \int (c+d x)^3 (a+a \cosh (e+f x)) \, dx=\begin {cases} a c^{3} x + \frac {a c^{3} \sinh {\left (e + f x \right )}}{f} + \frac {3 a c^{2} d x^{2}}{2} + \frac {3 a c^{2} d x \sinh {\left (e + f x \right )}}{f} - \frac {3 a c^{2} d \cosh {\left (e + f x \right )}}{f^{2}} + a c d^{2} x^{3} + \frac {3 a c d^{2} x^{2} \sinh {\left (e + f x \right )}}{f} - \frac {6 a c d^{2} x \cosh {\left (e + f x \right )}}{f^{2}} + \frac {6 a c d^{2} \sinh {\left (e + f x \right )}}{f^{3}} + \frac {a d^{3} x^{4}}{4} + \frac {a d^{3} x^{3} \sinh {\left (e + f x \right )}}{f} - \frac {3 a d^{3} x^{2} \cosh {\left (e + f x \right )}}{f^{2}} + \frac {6 a d^{3} x \sinh {\left (e + f x \right )}}{f^{3}} - \frac {6 a d^{3} \cosh {\left (e + f x \right )}}{f^{4}} & \text {for}\: f \neq 0 \\\left (a \cosh {\left (e \right )} + a\right ) \left (c^{3} x + \frac {3 c^{2} d x^{2}}{2} + c d^{2} x^{3} + \frac {d^{3} x^{4}}{4}\right ) & \text {otherwise} \end {cases} \]

[In]

integrate((d*x+c)**3*(a+a*cosh(f*x+e)),x)

[Out]

Piecewise((a*c**3*x + a*c**3*sinh(e + f*x)/f + 3*a*c**2*d*x**2/2 + 3*a*c**2*d*x*sinh(e + f*x)/f - 3*a*c**2*d*c
osh(e + f*x)/f**2 + a*c*d**2*x**3 + 3*a*c*d**2*x**2*sinh(e + f*x)/f - 6*a*c*d**2*x*cosh(e + f*x)/f**2 + 6*a*c*
d**2*sinh(e + f*x)/f**3 + a*d**3*x**4/4 + a*d**3*x**3*sinh(e + f*x)/f - 3*a*d**3*x**2*cosh(e + f*x)/f**2 + 6*a
*d**3*x*sinh(e + f*x)/f**3 - 6*a*d**3*cosh(e + f*x)/f**4, Ne(f, 0)), ((a*cosh(e) + a)*(c**3*x + 3*c**2*d*x**2/
2 + c*d**2*x**3 + d**3*x**4/4), True))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 237 vs. \(2 (87) = 174\).

Time = 0.20 (sec) , antiderivative size = 237, normalized size of antiderivative = 2.66 \[ \int (c+d x)^3 (a+a \cosh (e+f x)) \, dx=\frac {1}{4} \, a d^{3} x^{4} + a c d^{2} x^{3} + \frac {3}{2} \, a c^{2} d x^{2} + a c^{3} x + \frac {3}{2} \, a c^{2} d {\left (\frac {{\left (f x e^{e} - e^{e}\right )} e^{\left (f x\right )}}{f^{2}} - \frac {{\left (f x + 1\right )} e^{\left (-f x - e\right )}}{f^{2}}\right )} + \frac {3}{2} \, a c d^{2} {\left (\frac {{\left (f^{2} x^{2} e^{e} - 2 \, f x e^{e} + 2 \, e^{e}\right )} e^{\left (f x\right )}}{f^{3}} - \frac {{\left (f^{2} x^{2} + 2 \, f x + 2\right )} e^{\left (-f x - e\right )}}{f^{3}}\right )} + \frac {1}{2} \, a d^{3} {\left (\frac {{\left (f^{3} x^{3} e^{e} - 3 \, f^{2} x^{2} e^{e} + 6 \, f x e^{e} - 6 \, e^{e}\right )} e^{\left (f x\right )}}{f^{4}} - \frac {{\left (f^{3} x^{3} + 3 \, f^{2} x^{2} + 6 \, f x + 6\right )} e^{\left (-f x - e\right )}}{f^{4}}\right )} + \frac {a c^{3} \sinh \left (f x + e\right )}{f} \]

[In]

integrate((d*x+c)^3*(a+a*cosh(f*x+e)),x, algorithm="maxima")

[Out]

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*a*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*a*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*a*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) + a*c^3*sinh(f*x + e)/f

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 258 vs. \(2 (87) = 174\).

Time = 0.28 (sec) , antiderivative size = 258, normalized size of antiderivative = 2.90 \[ \int (c+d x)^3 (a+a \cosh (e+f x)) \, dx=\frac {1}{4} \, a d^{3} x^{4} + a c d^{2} x^{3} + \frac {3}{2} \, a c^{2} d x^{2} + a c^{3} x + \frac {{\left (a d^{3} f^{3} x^{3} + 3 \, a c d^{2} f^{3} x^{2} + 3 \, a c^{2} d f^{3} x - 3 \, a d^{3} f^{2} x^{2} + a c^{3} f^{3} - 6 \, a c d^{2} f^{2} x - 3 \, a c^{2} d f^{2} + 6 \, a d^{3} f x + 6 \, a c d^{2} f - 6 \, a d^{3}\right )} e^{\left (f x + e\right )}}{2 \, f^{4}} - \frac {{\left (a d^{3} f^{3} x^{3} + 3 \, a c d^{2} f^{3} x^{2} + 3 \, a c^{2} d f^{3} x + 3 \, a d^{3} f^{2} x^{2} + a c^{3} f^{3} + 6 \, a c d^{2} f^{2} x + 3 \, a c^{2} d f^{2} + 6 \, a d^{3} f x + 6 \, a c d^{2} f + 6 \, a d^{3}\right )} e^{\left (-f x - e\right )}}{2 \, f^{4}} \]

[In]

integrate((d*x+c)^3*(a+a*cosh(f*x+e)),x, algorithm="giac")

[Out]

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*(a*d^3*f^3*x^3 + 3*a*c*d^2*f^3*x^2 + 3*a*c^2*d*f
^3*x - 3*a*d^3*f^2*x^2 + a*c^3*f^3 - 6*a*c*d^2*f^2*x - 3*a*c^2*d*f^2 + 6*a*d^3*f*x + 6*a*c*d^2*f - 6*a*d^3)*e^
(f*x + e)/f^4 - 1/2*(a*d^3*f^3*x^3 + 3*a*c*d^2*f^3*x^2 + 3*a*c^2*d*f^3*x + 3*a*d^3*f^2*x^2 + a*c^3*f^3 + 6*a*c
*d^2*f^2*x + 3*a*c^2*d*f^2 + 6*a*d^3*f*x + 6*a*c*d^2*f + 6*a*d^3)*e^(-f*x - e)/f^4

Mupad [B] (verification not implemented)

Time = 1.84 (sec) , antiderivative size = 187, normalized size of antiderivative = 2.10 \[ \int (c+d x)^3 (a+a \cosh (e+f x)) \, dx=\frac {\mathrm {sinh}\left (e+f\,x\right )\,\left (a\,c^3\,f^2+6\,a\,c\,d^2\right )}{f^3}-\frac {3\,\mathrm {cosh}\left (e+f\,x\right )\,\left (a\,c^2\,d\,f^2+2\,a\,d^3\right )}{f^4}+\frac {a\,d^3\,x^4}{4}+a\,c^3\,x+\frac {3\,x\,\mathrm {sinh}\left (e+f\,x\right )\,\left (a\,c^2\,d\,f^2+2\,a\,d^3\right )}{f^3}+\frac {3\,a\,c^2\,d\,x^2}{2}+a\,c\,d^2\,x^3-\frac {3\,a\,d^3\,x^2\,\mathrm {cosh}\left (e+f\,x\right )}{f^2}+\frac {a\,d^3\,x^3\,\mathrm {sinh}\left (e+f\,x\right )}{f}-\frac {6\,a\,c\,d^2\,x\,\mathrm {cosh}\left (e+f\,x\right )}{f^2}+\frac {3\,a\,c\,d^2\,x^2\,\mathrm {sinh}\left (e+f\,x\right )}{f} \]

[In]

int((a + a*cosh(e + f*x))*(c + d*x)^3,x)

[Out]

(sinh(e + f*x)*(a*c^3*f^2 + 6*a*c*d^2))/f^3 - (3*cosh(e + f*x)*(2*a*d^3 + a*c^2*d*f^2))/f^4 + (a*d^3*x^4)/4 +
a*c^3*x + (3*x*sinh(e + f*x)*(2*a*d^3 + a*c^2*d*f^2))/f^3 + (3*a*c^2*d*x^2)/2 + a*c*d^2*x^3 - (3*a*d^3*x^2*cos
h(e + f*x))/f^2 + (a*d^3*x^3*sinh(e + f*x))/f - (6*a*c*d^2*x*cosh(e + f*x))/f^2 + (3*a*c*d^2*x^2*sinh(e + f*x)
)/f