3.2.59 \(\int (c+d x) (a+b \sinh (e+f x)) \, dx\) [159]

Optimal. Leaf size=45 \[ \frac {a (c+d x)^2}{2 d}+\frac {b (c+d x) \cosh (e+f x)}{f}-\frac {b d \sinh (e+f x)}{f^2} \]

[Out]

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

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Rubi [A]
time = 0.03, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {3398, 3377, 2717} \begin {gather*} \frac {a (c+d x)^2}{2 d}+\frac {b (c+d x) \cosh (e+f x)}{f}-\frac {b d \sinh (e+f x)}{f^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

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Mathematica [A]
time = 0.09, size = 43, normalized size = 0.96 \begin {gather*} \frac {1}{2} a x (2 c+d x)+\frac {b (c+d x) \cosh (e+f x)}{f}-\frac {b d \sinh (e+f x)}{f^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(90\) vs. \(2(43)=86\).
time = 0.37, size = 91, normalized size = 2.02

method	result	size
risch	\(\frac {a d \,x^{2}}{2}+a c x +\frac {b \left (d x f +c f -d \right ) {\mathrm e}^{f x +e}}{2 f^{2}}+\frac {b \left (d x f +c f +d \right ) {\mathrm e}^{-f x -e}}{2 f^{2}}\)	\(60\)
derivativedivides	\(\frac {\frac {d a \left (f x +e \right )^{2}}{2 f}+\frac {d b \left (\left (f x +e \right ) \cosh \left (f x +e \right )-\sinh \left (f x +e \right )\right )}{f}-\frac {d e a \left (f x +e \right )}{f}-\frac {d e b \cosh \left (f x +e \right )}{f}+a c \left (f x +e \right )+b c \cosh \left (f x +e \right )}{f}\)	\(91\)
default	\(\frac {\frac {d a \left (f x +e \right )^{2}}{2 f}+\frac {d b \left (\left (f x +e \right ) \cosh \left (f x +e \right )-\sinh \left (f x +e \right )\right )}{f}-\frac {d e a \left (f x +e \right )}{f}-\frac {d e b \cosh \left (f x +e \right )}{f}+a c \left (f x +e \right )+b c \cosh \left (f x +e \right )}{f}\)	\(91\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.26, size = 69, normalized size = 1.53 \begin {gather*} \frac {1}{2} \, a d x^{2} + a c x + \frac {1}{2} \, b 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 {b c \cosh \left (f x + e\right )}{f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.33, size = 57, normalized size = 1.27 \begin {gather*} \frac {a d f^{2} x^{2} + 2 \, a c f^{2} x - 2 \, b d \sinh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) + 2 \, {\left (b d f x + b c f\right )} \cosh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right )}{2 \, f^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]
time = 0.10, size = 68, normalized size = 1.51 \begin {gather*} \begin {cases} a c x + \frac {a d x^{2}}{2} + \frac {b c \cosh {\left (e + f x \right )}}{f} + \frac {b d x \cosh {\left (e + f x \right )}}{f} - \frac {b d \sinh {\left (e + f x \right )}}{f^{2}} & \text {for}\: f \neq 0 \\\left (a + b \sinh {\left (e \right )}\right ) \left (c x + \frac {d x^{2}}{2}\right ) & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 0.45, size = 64, normalized size = 1.42 \begin {gather*} \frac {1}{2} \, a d x^{2} + a c x + \frac {{\left (b d f x + b c f - b d\right )} e^{\left (f x + e\right )}}{2 \, f^{2}} + \frac {{\left (b d f x + b c f + b d\right )} e^{\left (-f x - e\right )}}{2 \, f^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.14, size = 49, normalized size = 1.09 \begin {gather*} \frac {f\,\left (b\,c\,\mathrm {cosh}\left (e+f\,x\right )+b\,d\,x\,\mathrm {cosh}\left (e+f\,x\right )\right )-b\,d\,\mathrm {sinh}\left (e+f\,x\right )}{f^2}+a\,c\,x+\frac {a\,d\,x^2}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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