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

Optimal result

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

[Out]

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

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, 2718} \[ \int (c+d x)^2 \sinh (a+b x) \, dx=\frac {2 d^2 \cosh (a+b x)}{b^3}-\frac {2 d (c+d x) \sinh (a+b x)}{b^2}+\frac {(c+d x)^2 \cosh (a+b x)}{b} \]

[In]

Int[(c + d*x)^2*Sinh[a + b*x],x]

[Out]

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

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

Mathematica [A] (verified)

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

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.99 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.86

method	result	size
parallelrisch	\(\frac {-2 x d \,b^{2} \left (\frac {d x}{2}+c \right ) \tanh \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}+4 b d \left (d x +c \right ) \tanh \left (\frac {b x}{2}+\frac {a}{2}\right )+\left (-d^{2} x^{2}-2 c d x -2 c^{2}\right ) b^{2}-4 d^{2}}{b^{3} \left (\tanh \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}-1\right )}\)	\(91\)
parts	\(\frac {\cosh \left (b x +a \right ) d^{2} x^{2}}{b}+\frac {2 \cosh \left (b x +a \right ) c d x}{b}+\frac {\cosh \left (b x +a \right ) c^{2}}{b}-\frac {2 d \left (\frac {d \left (\left (b x +a \right ) \sinh \left (b x +a \right )-\cosh \left (b x +a \right )\right )}{b}-\frac {d a \sinh \left (b x +a \right )}{b}+c \sinh \left (b x +a \right )\right )}{b^{2}}\)	\(99\)
risch	\(\frac {\left (b^{2} d^{2} x^{2}+2 b^{2} c d x +b^{2} c^{2}-2 b \,d^{2} x -2 b c d +2 d^{2}\right ) {\mathrm e}^{b x +a}}{2 b^{3}}+\frac {\left (b^{2} d^{2} x^{2}+2 b^{2} c d x +b^{2} c^{2}+2 b \,d^{2} x +2 b c d +2 d^{2}\right ) {\mathrm e}^{-b x -a}}{2 b^{3}}\)	\(113\)
derivativedivides	\(\frac {\frac {d^{2} \left (\left (b x +a \right )^{2} \cosh \left (b x +a \right )-2 \left (b x +a \right ) \sinh \left (b x +a \right )+2 \cosh \left (b x +a \right )\right )}{b^{2}}-\frac {2 d^{2} a \left (\left (b x +a \right ) \cosh \left (b x +a \right )-\sinh \left (b x +a \right )\right )}{b^{2}}+\frac {2 d c \left (\left (b x +a \right ) \cosh \left (b x +a \right )-\sinh \left (b x +a \right )\right )}{b}+\frac {d^{2} a^{2} \cosh \left (b x +a \right )}{b^{2}}-\frac {2 d a c \cosh \left (b x +a \right )}{b}+c^{2} \cosh \left (b x +a \right )}{b}\)	\(147\)
default	\(\frac {\frac {d^{2} \left (\left (b x +a \right )^{2} \cosh \left (b x +a \right )-2 \left (b x +a \right ) \sinh \left (b x +a \right )+2 \cosh \left (b x +a \right )\right )}{b^{2}}-\frac {2 d^{2} a \left (\left (b x +a \right ) \cosh \left (b x +a \right )-\sinh \left (b x +a \right )\right )}{b^{2}}+\frac {2 d c \left (\left (b x +a \right ) \cosh \left (b x +a \right )-\sinh \left (b x +a \right )\right )}{b}+\frac {d^{2} a^{2} \cosh \left (b x +a \right )}{b^{2}}-\frac {2 d a c \cosh \left (b x +a \right )}{b}+c^{2} \cosh \left (b x +a \right )}{b}\)	\(147\)
meijerg	\(\frac {4 i d^{2} \sinh \left (a \right ) \sqrt {\pi }\, \left (\frac {i b x \cosh \left (b x \right )}{2 \sqrt {\pi }}-\frac {i \left (\frac {3 b^{2} x^{2}}{2}+3\right ) \sinh \left (b x \right )}{6 \sqrt {\pi }}\right )}{b^{3}}+\frac {4 d^{2} \cosh \left (a \right ) \sqrt {\pi }\, \left (-\frac {1}{2 \sqrt {\pi }}+\frac {\left (\frac {b^{2} x^{2}}{2}+1\right ) \cosh \left (b x \right )}{2 \sqrt {\pi }}-\frac {x b \sinh \left (b x \right )}{2 \sqrt {\pi }}\right )}{b^{3}}-\frac {4 d c \sinh \left (a \right ) \sqrt {\pi }\, \left (-\frac {1}{2 \sqrt {\pi }}+\frac {\cosh \left (b x \right )}{2 \sqrt {\pi }}-\frac {x b \sinh \left (b x \right )}{2 \sqrt {\pi }}\right )}{b^{2}}-\frac {2 d c \cosh \left (a \right ) \left (-\cosh \left (b x \right ) b x +\sinh \left (b x \right )\right )}{b^{2}}+\frac {c^{2} \sinh \left (a \right ) \sinh \left (b x \right )}{b}-\frac {c^{2} \cosh \left (a \right ) \sqrt {\pi }\, \left (\frac {1}{\sqrt {\pi }}-\frac {\cosh \left (b x \right )}{\sqrt {\pi }}\right )}{b}\)	\(197\)

[In]

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

[Out]

(-2*x*d*b^2*(1/2*d*x+c)*tanh(1/2*b*x+1/2*a)^2+4*b*d*(d*x+c)*tanh(1/2*b*x+1/2*a)+(-d^2*x^2-2*c*d*x-2*c^2)*b^2-4
*d^2)/b^3/(tanh(1/2*b*x+1/2*a)^2-1)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.27 \[ \int (c+d x)^2 \sinh (a+b x) \, dx=\frac {{\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2} + 2 \, d^{2}\right )} \cosh \left (b x + a\right ) - 2 \, {\left (b d^{2} x + b c d\right )} \sinh \left (b x + a\right )}{b^{3}} \]

[In]

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

[Out]

((b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2 + 2*d^2)*cosh(b*x + a) - 2*(b*d^2*x + b*c*d)*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 \[ \int (c+d x)^2 \sinh (a+b x) \, dx=\begin {cases} \frac {c^{2} \cosh {\left (a + b x \right )}}{b} + \frac {2 c d x \cosh {\left (a + b x \right )}}{b} + \frac {d^{2} x^{2} \cosh {\left (a + b x \right )}}{b} - \frac {2 c d \sinh {\left (a + b x \right )}}{b^{2}} - \frac {2 d^{2} x \sinh {\left (a + b x \right )}}{b^{2}} + \frac {2 d^{2} \cosh {\left (a + b x \right )}}{b^{3}} & \text {for}\: b \neq 0 \\\left (c^{2} x + c d x^{2} + \frac {d^{2} x^{3}}{3}\right ) \sinh {\left (a \right )} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 134 vs. \(2 (49) = 98\).

Time = 0.20 (sec) , antiderivative size = 134, normalized size of antiderivative = 2.73 \[ \int (c+d x)^2 \sinh (a+b x) \, dx=\frac {c^{2} e^{\left (b x + a\right )}}{2 \, b} + \frac {{\left (b x e^{a} - e^{a}\right )} c d e^{\left (b x\right )}}{b^{2}} + \frac {c^{2} e^{\left (-b x - a\right )}}{2 \, b} + \frac {{\left (b x + 1\right )} c d e^{\left (-b x - a\right )}}{b^{2}} + \frac {{\left (b^{2} x^{2} e^{a} - 2 \, b x e^{a} + 2 \, e^{a}\right )} d^{2} e^{\left (b x\right )}}{2 \, b^{3}} + \frac {{\left (b^{2} x^{2} + 2 \, b x + 2\right )} d^{2} e^{\left (-b x - a\right )}}{2 \, b^{3}} \]

[In]

integrate((d*x+c)^2*sinh(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 = 0.28 (sec) , antiderivative size = 112, normalized size of antiderivative = 2.29 \[ \int (c+d x)^2 \sinh (a+b x) \, dx=\frac {{\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2} - 2 \, b d^{2} x - 2 \, b c d + 2 \, d^{2}\right )} e^{\left (b x + a\right )}}{2 \, b^{3}} + \frac {{\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2} + 2 \, b d^{2} x + 2 \, b c d + 2 \, d^{2}\right )} e^{\left (-b x - a\right )}}{2 \, b^{3}} \]

[In]

integrate((d*x+c)^2*sinh(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 = 0.10 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.67 \[ \int (c+d x)^2 \sinh (a+b x) \, dx=\frac {\mathrm {cosh}\left (a+b\,x\right )\,\left (b^2\,c^2+2\,d^2\right )}{b^3}+\frac {d^2\,x^2\,\mathrm {cosh}\left (a+b\,x\right )}{b}-\frac {2\,c\,d\,\mathrm {sinh}\left (a+b\,x\right )}{b^2}-\frac {2\,d^2\,x\,\mathrm {sinh}\left (a+b\,x\right )}{b^2}+\frac {2\,c\,d\,x\,\mathrm {cosh}\left (a+b\,x\right )}{b} \]

[In]

int(sinh(a + b*x)*(c + d*x)^2,x)

[Out]

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