\(\int (c+d x)^2 (a+b \sinh (e+f x)) \, dx\) [158]

Optimal result

Integrand size = 18, antiderivative size = 67 \[ \int (c+d x)^2 (a+b \sinh (e+f x)) \, dx=\frac {a (c+d x)^3}{3 d}+\frac {2 b d^2 \cosh (e+f x)}{f^3}+\frac {b (c+d x)^2 \cosh (e+f x)}{f}-\frac {2 b d (c+d x) \sinh (e+f x)}{f^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3398, 3377, 2718} \[ \int (c+d x)^2 (a+b \sinh (e+f x)) \, dx=\frac {a (c+d x)^3}{3 d}-\frac {2 b d (c+d x) \sinh (e+f x)}{f^2}+\frac {b (c+d x)^2 \cosh (e+f x)}{f}+\frac {2 b d^2 \cosh (e+f x)}{f^3} \]

[In]

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

[Out]

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

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)^2+b (c+d x)^2 \sinh (e+f x)\right ) \, dx \\ & = \frac {a (c+d x)^3}{3 d}+b \int (c+d x)^2 \sinh (e+f x) \, dx \\ & = \frac {a (c+d x)^3}{3 d}+\frac {b (c+d x)^2 \cosh (e+f x)}{f}-\frac {(2 b d) \int (c+d x) \cosh (e+f x) \, dx}{f} \\ & = \frac {a (c+d x)^3}{3 d}+\frac {b (c+d x)^2 \cosh (e+f x)}{f}-\frac {2 b d (c+d x) \sinh (e+f x)}{f^2}+\frac {\left (2 b d^2\right ) \int \sinh (e+f x) \, dx}{f^2} \\ & = \frac {a (c+d x)^3}{3 d}+\frac {2 b d^2 \cosh (e+f x)}{f^3}+\frac {b (c+d x)^2 \cosh (e+f x)}{f}-\frac {2 b d (c+d x) \sinh (e+f x)}{f^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.24 \[ \int (c+d x)^2 (a+b \sinh (e+f x)) \, dx=\frac {1}{3} a x \left (3 c^2+3 c d x+d^2 x^2\right )+\frac {b \left (c^2 f^2+2 c d f^2 x+d^2 \left (2+f^2 x^2\right )\right ) \cosh (e+f x)}{f^3}-\frac {2 b d (c+d x) \sinh (e+f x)}{f^2} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 1.28 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.24

method	result	size
parallelrisch	\(\frac {b \left (\left (d x +c \right )^{2} f^{2}+2 d^{2}\right ) \cosh \left (f x +e \right )-2 b d f \left (d x +c \right ) \sinh \left (f x +e \right )+\left (\frac {1}{3} d^{2} x^{2}+c d x +c^{2}\right ) x a \,f^{3}+b \,c^{2} f^{2}+2 b \,d^{2}}{f^{3}}\)	\(83\)
risch	\(\frac {a \,d^{2} x^{3}}{3}+a d c \,x^{2}+a x \,c^{2}+\frac {a \,c^{3}}{3 d}+\frac {b \left (d^{2} x^{2} f^{2}+2 c d \,f^{2} x +c^{2} f^{2}-2 d^{2} f x -2 c d f +2 d^{2}\right ) {\mathrm e}^{f x +e}}{2 f^{3}}+\frac {b \left (d^{2} x^{2} f^{2}+2 c d \,f^{2} x +c^{2} f^{2}+2 d^{2} f x +2 c d f +2 d^{2}\right ) {\mathrm e}^{-f x -e}}{2 f^{3}}\)	\(146\)
parts	\(\frac {a \left (d x +c \right )^{3}}{3 d}+\frac {b \left (\frac {d^{2} \left (\left (f x +e \right )^{2} \cosh \left (f x +e \right )-2 \left (f x +e \right ) \sinh \left (f x +e \right )+2 \cosh \left (f x +e \right )\right )}{f^{2}}-\frac {2 d^{2} e \left (\left (f x +e \right ) \cosh \left (f x +e \right )-\sinh \left (f x +e \right )\right )}{f^{2}}+\frac {2 d c \left (\left (f x +e \right ) \cosh \left (f x +e \right )-\sinh \left (f x +e \right )\right )}{f}+\frac {d^{2} e^{2} \cosh \left (f x +e \right )}{f^{2}}-\frac {2 d e c \cosh \left (f x +e \right )}{f}+c^{2} \cosh \left (f x +e \right )\right )}{f}\)	\(162\)
derivativedivides	\(\frac {\frac {d^{2} a \left (f x +e \right )^{3}}{3 f^{2}}+\frac {d^{2} b \left (\left (f x +e \right )^{2} \cosh \left (f x +e \right )-2 \left (f x +e \right ) \sinh \left (f x +e \right )+2 \cosh \left (f x +e \right )\right )}{f^{2}}-\frac {d^{2} e a \left (f x +e \right )^{2}}{f^{2}}-\frac {2 d^{2} e b \left (\left (f x +e \right ) \cosh \left (f x +e \right )-\sinh \left (f x +e \right )\right )}{f^{2}}+\frac {d c a \left (f x +e \right )^{2}}{f}+\frac {2 d c b \left (\left (f x +e \right ) \cosh \left (f x +e \right )-\sinh \left (f x +e \right )\right )}{f}+\frac {d^{2} e^{2} a \left (f x +e \right )}{f^{2}}+\frac {d^{2} e^{2} b \cosh \left (f x +e \right )}{f^{2}}-\frac {2 d e c a \left (f x +e \right )}{f}-\frac {2 d e c b \cosh \left (f x +e \right )}{f}+a \,c^{2} \left (f x +e \right )+b \,c^{2} \cosh \left (f x +e \right )}{f}\)	\(240\)
default	\(\frac {\frac {d^{2} a \left (f x +e \right )^{3}}{3 f^{2}}+\frac {d^{2} b \left (\left (f x +e \right )^{2} \cosh \left (f x +e \right )-2 \left (f x +e \right ) \sinh \left (f x +e \right )+2 \cosh \left (f x +e \right )\right )}{f^{2}}-\frac {d^{2} e a \left (f x +e \right )^{2}}{f^{2}}-\frac {2 d^{2} e b \left (\left (f x +e \right ) \cosh \left (f x +e \right )-\sinh \left (f x +e \right )\right )}{f^{2}}+\frac {d c a \left (f x +e \right )^{2}}{f}+\frac {2 d c b \left (\left (f x +e \right ) \cosh \left (f x +e \right )-\sinh \left (f x +e \right )\right )}{f}+\frac {d^{2} e^{2} a \left (f x +e \right )}{f^{2}}+\frac {d^{2} e^{2} b \cosh \left (f x +e \right )}{f^{2}}-\frac {2 d e c a \left (f x +e \right )}{f}-\frac {2 d e c b \cosh \left (f x +e \right )}{f}+a \,c^{2} \left (f x +e \right )+b \,c^{2} \cosh \left (f x +e \right )}{f}\)	\(240\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.52 \[ \int (c+d x)^2 (a+b \sinh (e+f x)) \, dx=\frac {a d^{2} f^{3} x^{3} + 3 \, a c d f^{3} x^{2} + 3 \, a c^{2} f^{3} x + 3 \, {\left (b d^{2} f^{2} x^{2} + 2 \, b c d f^{2} x + b c^{2} f^{2} + 2 \, b d^{2}\right )} \cosh \left (f x + e\right ) - 6 \, {\left (b d^{2} f x + b c d f\right )} \sinh \left (f x + e\right )}{3 \, f^{3}} \]

[In]

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

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 151 vs. \(2 (65) = 130\).

Time = 0.22 (sec) , antiderivative size = 151, normalized size of antiderivative = 2.25 \[ \int (c+d x)^2 (a+b \sinh (e+f x)) \, dx=\begin {cases} a c^{2} x + a c d x^{2} + \frac {a d^{2} x^{3}}{3} + \frac {b c^{2} \cosh {\left (e + f x \right )}}{f} + \frac {2 b c d x \cosh {\left (e + f x \right )}}{f} - \frac {2 b c d \sinh {\left (e + f x \right )}}{f^{2}} + \frac {b d^{2} x^{2} \cosh {\left (e + f x \right )}}{f} - \frac {2 b d^{2} x \sinh {\left (e + f x \right )}}{f^{2}} + \frac {2 b d^{2} \cosh {\left (e + f x \right )}}{f^{3}} & \text {for}\: f \neq 0 \\\left (a + b \sinh {\left (e \right )}\right ) \left (c^{2} x + c d x^{2} + \frac {d^{2} x^{3}}{3}\right ) & \text {otherwise} \end {cases} \]

[In]

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

[Out]

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 139 vs. \(2 (65) = 130\).

Time = 0.20 (sec) , antiderivative size = 139, normalized size of antiderivative = 2.07 \[ \int (c+d x)^2 (a+b \sinh (e+f x)) \, dx=\frac {1}{3} \, a d^{2} x^{3} + a c d x^{2} + a c^{2} x + b c d {\left (\frac {{\left (f x e^{e} - e^{e}\right )} e^{\left (f x\right )}}{f^{2}} + \frac {{\left (f x + 1\right )} e^{\left (-f x - e\right )}}{f^{2}}\right )} + \frac {1}{2} \, b d^{2} {\left (\frac {{\left (f^{2} x^{2} e^{e} - 2 \, f x e^{e} + 2 \, e^{e}\right )} e^{\left (f x\right )}}{f^{3}} + \frac {{\left (f^{2} x^{2} + 2 \, f x + 2\right )} e^{\left (-f x - e\right )}}{f^{3}}\right )} + \frac {b c^{2} \cosh \left (f x + e\right )}{f} \]

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 146 vs. \(2 (65) = 130\).

Time = 0.28 (sec) , antiderivative size = 146, normalized size of antiderivative = 2.18 \[ \int (c+d x)^2 (a+b \sinh (e+f x)) \, dx=\frac {1}{3} \, a d^{2} x^{3} + a c d x^{2} + a c^{2} x + \frac {{\left (b d^{2} f^{2} x^{2} + 2 \, b c d f^{2} x + b c^{2} f^{2} - 2 \, b d^{2} f x - 2 \, b c d f + 2 \, b d^{2}\right )} e^{\left (f x + e\right )}}{2 \, f^{3}} + \frac {{\left (b d^{2} f^{2} x^{2} + 2 \, b c d f^{2} x + b c^{2} f^{2} + 2 \, b d^{2} f x + 2 \, b c d f + 2 \, b d^{2}\right )} e^{\left (-f x - e\right )}}{2 \, f^{3}} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.64 \[ \int (c+d x)^2 (a+b \sinh (e+f x)) \, dx=\frac {a\,d^2\,x^3}{3}+\frac {\mathrm {cosh}\left (e+f\,x\right )\,\left (b\,c^2\,f^2+2\,b\,d^2\right )}{f^3}+a\,c^2\,x+a\,c\,d\,x^2-\frac {2\,b\,d^2\,x\,\mathrm {sinh}\left (e+f\,x\right )}{f^2}+\frac {b\,d^2\,x^2\,\mathrm {cosh}\left (e+f\,x\right )}{f}-\frac {2\,b\,c\,d\,\mathrm {sinh}\left (e+f\,x\right )}{f^2}+\frac {2\,b\,c\,d\,x\,\mathrm {cosh}\left (e+f\,x\right )}{f} \]

[In]

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

[Out]

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