\(\int (c+d x)^2 \cosh (a+b x) \, dx\) [3]

Optimal result

Integrand size = 14, antiderivative size = 49 \[ \int (c+d x)^2 \cosh (a+b x) \, dx=-\frac {2 d (c+d x) \cosh (a+b x)}{b^2}+\frac {2 d^2 \sinh (a+b x)}{b^3}+\frac {(c+d x)^2 \sinh (a+b x)}{b} \]

[Out]

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

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3377, 2717} \[ \int (c+d x)^2 \cosh (a+b x) \, dx=\frac {2 d^2 \sinh (a+b x)}{b^3}-\frac {2 d (c+d x) \cosh (a+b x)}{b^2}+\frac {(c+d x)^2 \sinh (a+b x)}{b} \]

[In]

Int[(c + d*x)^2*Cosh[a + b*x],x]

[Out]

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

Rule 2717

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[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]

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

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.90 \[ \int (c+d x)^2 \cosh (a+b x) \, dx=\frac {-2 b d (c+d x) \cosh (a+b x)+\left (2 d^2+b^2 (c+d x)^2\right ) \sinh (a+b x)}{b^3} \]

[In]

Integrate[(c + d*x)^2*Cosh[a + b*x],x]

[Out]

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

Maple [A] (verified)

Time = 0.18 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.57

method	result	size
parallelrisch	\(\frac {2 x \tanh \left (\frac {b x}{2}+\frac {a}{2}\right )^{2} b \,d^{2}+2 \left (-\left (d x +c \right )^{2} b^{2}-2 d^{2}\right ) \tanh \left (\frac {b x}{2}+\frac {a}{2}\right )+4 d \left (\frac {d x}{2}+c \right ) b}{b^{3} \left (\tanh \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}-1\right )}\)	\(77\)
parts	\(\frac {\sinh \left (b x +a \right ) x^{2} d^{2}}{b}+\frac {2 \sinh \left (b x +a \right ) c d x}{b}+\frac {\sinh \left (b x +a \right ) c^{2}}{b}-\frac {2 d \left (\frac {d \left (\left (b x +a \right ) \cosh \left (b x +a \right )-\sinh \left (b x +a \right )\right )}{b}-\frac {d a \cosh \left (b x +a \right )}{b}+c \cosh \left (b x +a \right )\right )}{b^{2}}\)	\(99\)
risch	\(\frac {\left (x^{2} d^{2} b^{2}+2 b^{2} c d x +b^{2} c^{2}-2 b \,d^{2} x -2 b c d +2 d^{2}\right ) {\mathrm e}^{b x +a}}{2 b^{3}}-\frac {\left (x^{2} d^{2} b^{2}+2 b^{2} c d x +b^{2} c^{2}+2 b \,d^{2} x +2 b c d +2 d^{2}\right ) {\mathrm e}^{-b x -a}}{2 b^{3}}\)	\(113\)
derivativedivides	\(\frac {\frac {d^{2} \left (\left (b x +a \right )^{2} \sinh \left (b x +a \right )-2 \left (b x +a \right ) \cosh \left (b x +a \right )+2 \sinh \left (b x +a \right )\right )}{b^{2}}-\frac {2 d^{2} a \left (\left (b x +a \right ) \sinh \left (b x +a \right )-\cosh \left (b x +a \right )\right )}{b^{2}}+\frac {2 d c \left (\left (b x +a \right ) \sinh \left (b x +a \right )-\cosh \left (b x +a \right )\right )}{b}+\frac {d^{2} a^{2} \sinh \left (b x +a \right )}{b^{2}}-\frac {2 d a c \sinh \left (b x +a \right )}{b}+c^{2} \sinh \left (b x +a \right )}{b}\)	\(147\)
default	\(\frac {\frac {d^{2} \left (\left (b x +a \right )^{2} \sinh \left (b x +a \right )-2 \left (b x +a \right ) \cosh \left (b x +a \right )+2 \sinh \left (b x +a \right )\right )}{b^{2}}-\frac {2 d^{2} a \left (\left (b x +a \right ) \sinh \left (b x +a \right )-\cosh \left (b x +a \right )\right )}{b^{2}}+\frac {2 d c \left (\left (b x +a \right ) \sinh \left (b x +a \right )-\cosh \left (b x +a \right )\right )}{b}+\frac {d^{2} a^{2} \sinh \left (b x +a \right )}{b^{2}}-\frac {2 d a c \sinh \left (b x +a \right )}{b}+c^{2} \sinh \left (b x +a \right )}{b}\)	\(147\)
meijerg	\(\frac {4 i d^{2} \cosh \left (a \right ) \sqrt {\pi }\, \left (\frac {i x b \cosh \left (b x \right )}{2 \sqrt {\pi }}-\frac {i \left (\frac {3 x^{2} b^{2}}{2}+3\right ) \sinh \left (b x \right )}{6 \sqrt {\pi }}\right )}{b^{3}}+\frac {4 d^{2} \sinh \left (a \right ) \sqrt {\pi }\, \left (-\frac {1}{2 \sqrt {\pi }}+\frac {\left (\frac {x^{2} b^{2}}{2}+1\right ) \cosh \left (b x \right )}{2 \sqrt {\pi }}-\frac {x b \sinh \left (b x \right )}{2 \sqrt {\pi }}\right )}{b^{3}}-\frac {4 d c \cosh \left (a \right ) \sqrt {\pi }\, \left (-\frac {1}{2 \sqrt {\pi }}+\frac {\cosh \left (b x \right )}{2 \sqrt {\pi }}-\frac {x b \sinh \left (b x \right )}{2 \sqrt {\pi }}\right )}{b^{2}}+\frac {2 d c \sinh \left (a \right ) \left (\cosh \left (b x \right ) x b -\sinh \left (b x \right )\right )}{b^{2}}+\frac {c^{2} \cosh \left (a \right ) \sinh \left (b x \right )}{b}-\frac {c^{2} \sinh \left (a \right ) \sqrt {\pi }\, \left (\frac {1}{\sqrt {\pi }}-\frac {\cosh \left (b x \right )}{\sqrt {\pi }}\right )}{b}\)	\(198\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.31 \[ \int (c+d x)^2 \cosh (a+b x) \, dx=-\frac {2 \, {\left (b d^{2} x + b c d\right )} \cosh \left (b x + a\right ) - {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2} + 2 \, d^{2}\right )} \sinh \left (b x + a\right )}{b^{3}} \]

[In]

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

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 112 vs. \(2 (48) = 96\).

Time = 0.21 (sec) , antiderivative size = 112, normalized size of antiderivative = 2.29 \[ \int (c+d x)^2 \cosh (a+b x) \, dx=\begin {cases} \frac {c^{2} \sinh {\left (a + b x \right )}}{b} + \frac {2 c d x \sinh {\left (a + b x \right )}}{b} + \frac {d^{2} x^{2} \sinh {\left (a + b x \right )}}{b} - \frac {2 c d \cosh {\left (a + b x \right )}}{b^{2}} - \frac {2 d^{2} x \cosh {\left (a + b x \right )}}{b^{2}} + \frac {2 d^{2} \sinh {\left (a + b x \right )}}{b^{3}} & \text {for}\: b \neq 0 \\\left (c^{2} x + c d x^{2} + \frac {d^{2} x^{3}}{3}\right ) \cosh {\left (a \right )} & \text {otherwise} \end {cases} \]

[In]

integrate((d*x+c)**2*cosh(b*x+a),x)

[Out]

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 135 vs. \(2 (49) = 98\).

Time = 0.19 (sec) , antiderivative size = 135, normalized size of antiderivative = 2.76 \[ \int (c+d x)^2 \cosh (a+b x) \, dx=\frac {c^{2} e^{\left (b x + a\right )}}{2 \, b} + \frac {{\left (b x e^{a} - e^{a}\right )} c d e^{\left (b x\right )}}{b^{2}} - \frac {c^{2} e^{\left (-b x - a\right )}}{2 \, b} - \frac {{\left (b x + 1\right )} c d e^{\left (-b x - a\right )}}{b^{2}} + \frac {{\left (b^{2} x^{2} e^{a} - 2 \, b x e^{a} + 2 \, e^{a}\right )} d^{2} e^{\left (b x\right )}}{2 \, b^{3}} - \frac {{\left (b^{2} x^{2} + 2 \, b x + 2\right )} d^{2} e^{\left (-b x - a\right )}}{2 \, b^{3}} \]

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 112 vs. \(2 (49) = 98\).

Time = 0.25 (sec) , antiderivative size = 112, normalized size of antiderivative = 2.29 \[ \int (c+d x)^2 \cosh (a+b x) \, dx=\frac {{\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2} - 2 \, b d^{2} x - 2 \, b c d + 2 \, d^{2}\right )} e^{\left (b x + a\right )}}{2 \, b^{3}} - \frac {{\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2} + 2 \, b d^{2} x + 2 \, b c d + 2 \, d^{2}\right )} e^{\left (-b x - a\right )}}{2 \, b^{3}} \]

[In]

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

[Out]

1/2*(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 - 1/2*(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

Mupad [B] (verification not implemented)

Time = 1.77 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.67 \[ \int (c+d x)^2 \cosh (a+b x) \, dx=\frac {\mathrm {sinh}\left (a+b\,x\right )\,\left (b^2\,c^2+2\,d^2\right )}{b^3}+\frac {d^2\,x^2\,\mathrm {sinh}\left (a+b\,x\right )}{b}-\frac {2\,c\,d\,\mathrm {cosh}\left (a+b\,x\right )}{b^2}-\frac {2\,d^2\,x\,\mathrm {cosh}\left (a+b\,x\right )}{b^2}+\frac {2\,c\,d\,x\,\mathrm {sinh}\left (a+b\,x\right )}{b} \]

[In]

int(cosh(a + b*x)*(c + d*x)^2,x)

[Out]

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