\(\int (c+d x) \sinh (a+b x) \, dx\) [4]

Optimal result

Integrand size = 12, antiderivative size = 28 \[ \int (c+d x) \sinh (a+b x) \, dx=\frac {(c+d x) \cosh (a+b x)}{b}-\frac {d \sinh (a+b x)}{b^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3377, 2717} \[ \int (c+d x) \sinh (a+b x) \, dx=\frac {(c+d x) \cosh (a+b x)}{b}-\frac {d \sinh (a+b x)}{b^2} \]

[In]

Int[(c + d*x)*Sinh[a + b*x],x]

[Out]

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

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

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.96 \[ \int (c+d x) \sinh (a+b x) \, dx=\frac {b (c+d x) \cosh (a+b x)-d \sinh (a+b x)}{b^2} \]

[In]

Integrate[(c + d*x)*Sinh[a + b*x],x]

[Out]

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

Maple [A] (verified)

Time = 0.58 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.32

method	result	size
parts	\(\frac {\cosh \left (b x +a \right ) d x}{b}+\frac {\cosh \left (b x +a \right ) c}{b}-\frac {d \sinh \left (b x +a \right )}{b^{2}}\)	\(37\)
risch	\(\frac {\left (b d x +b c -d \right ) {\mathrm e}^{b x +a}}{2 b^{2}}+\frac {\left (b d x +b c +d \right ) {\mathrm e}^{-b x -a}}{2 b^{2}}\)	\(47\)
derivativedivides	\(\frac {\frac {d \left (\left (b x +a \right ) \cosh \left (b x +a \right )-\sinh \left (b x +a \right )\right )}{b}-\frac {d a \cosh \left (b x +a \right )}{b}+c \cosh \left (b x +a \right )}{b}\)	\(53\)
default	\(\frac {\frac {d \left (\left (b x +a \right ) \cosh \left (b x +a \right )-\sinh \left (b x +a \right )\right )}{b}-\frac {d a \cosh \left (b x +a \right )}{b}+c \cosh \left (b x +a \right )}{b}\)	\(53\)
parallelrisch	\(\frac {-x \tanh \left (\frac {b x}{2}+\frac {a}{2}\right )^{2} b d +2 d \tanh \left (\frac {b x}{2}+\frac {a}{2}\right )-2 b \left (\frac {d x}{2}+c \right )}{b^{2} \left (\tanh \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}-1\right )}\)	\(58\)
meijerg	\(-\frac {2 d \sinh \left (a \right ) \sqrt {\pi }\, \left (-\frac {1}{2 \sqrt {\pi }}+\frac {\cosh \left (b x \right )}{2 \sqrt {\pi }}-\frac {x b \sinh \left (b x \right )}{2 \sqrt {\pi }}\right )}{b^{2}}-\frac {d \cosh \left (a \right ) \left (-\cosh \left (b x \right ) b x +\sinh \left (b x \right )\right )}{b^{2}}+\frac {c \sinh \left (a \right ) \sinh \left (b x \right )}{b}-\frac {c \cosh \left (a \right ) \sqrt {\pi }\, \left (\frac {1}{\sqrt {\pi }}-\frac {\cosh \left (b x \right )}{\sqrt {\pi }}\right )}{b}\)	\(95\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04 \[ \int (c+d x) \sinh (a+b x) \, dx=\frac {{\left (b d x + b c\right )} \cosh \left (b x + a\right ) - d \sinh \left (b x + a\right )}{b^{2}} \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.64 \[ \int (c+d x) \sinh (a+b x) \, dx=\begin {cases} \frac {c \cosh {\left (a + b x \right )}}{b} + \frac {d x \cosh {\left (a + b x \right )}}{b} - \frac {d \sinh {\left (a + b x \right )}}{b^{2}} & \text {for}\: b \neq 0 \\\left (c x + \frac {d x^{2}}{2}\right ) \sinh {\left (a \right )} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 68 vs. \(2 (28) = 56\).

Time = 0.20 (sec) , antiderivative size = 68, normalized size of antiderivative = 2.43 \[ \int (c+d x) \sinh (a+b x) \, dx=\frac {c e^{\left (b x + a\right )}}{2 \, b} + \frac {{\left (b x e^{a} - e^{a}\right )} d e^{\left (b x\right )}}{2 \, b^{2}} + \frac {c e^{\left (-b x - a\right )}}{2 \, b} + \frac {{\left (b x + 1\right )} d e^{\left (-b x - a\right )}}{2 \, b^{2}} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.64 \[ \int (c+d x) \sinh (a+b x) \, dx=\frac {{\left (b d x + b c - d\right )} e^{\left (b x + a\right )}}{2 \, b^{2}} + \frac {{\left (b d x + b c + d\right )} e^{\left (-b x - a\right )}}{2 \, b^{2}} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.79 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.25 \[ \int (c+d x) \sinh (a+b x) \, dx=\frac {c\,\mathrm {cosh}\left (a+b\,x\right )+d\,x\,\mathrm {cosh}\left (a+b\,x\right )}{b}-\frac {d\,\mathrm {sinh}\left (a+b\,x\right )}{b^2} \]

[In]

int(sinh(a + b*x)*(c + d*x),x)

[Out]

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