\(\int x \sinh (a+b \sqrt {c+d x}) \, dx\) [94]

Optimal result

Integrand size = 16, antiderivative size = 167 \[ \int x \sinh \left (a+b \sqrt {c+d x}\right ) \, dx=\frac {12 \sqrt {c+d x} \cosh \left (a+b \sqrt {c+d x}\right )}{b^3 d^2}-\frac {2 c \sqrt {c+d x} \cosh \left (a+b \sqrt {c+d x}\right )}{b d^2}+\frac {2 (c+d x)^{3/2} \cosh \left (a+b \sqrt {c+d x}\right )}{b d^2}-\frac {12 \sinh \left (a+b \sqrt {c+d x}\right )}{b^4 d^2}+\frac {2 c \sinh \left (a+b \sqrt {c+d x}\right )}{b^2 d^2}-\frac {6 (c+d x) \sinh \left (a+b \sqrt {c+d x}\right )}{b^2 d^2} \] Output:

12*(d*x+c)^(1/2)*cosh(a+b*(d*x+c)^(1/2))/b^3/d^2-2*c*(d*x+c)^(1/2)*cosh(a+ 
b*(d*x+c)^(1/2))/b/d^2+2*(d*x+c)^(3/2)*cosh(a+b*(d*x+c)^(1/2))/b/d^2-12*si 
nh(a+b*(d*x+c)^(1/2))/b^4/d^2+2*c*sinh(a+b*(d*x+c)^(1/2))/b^2/d^2-6*(d*x+c 
)*sinh(a+b*(d*x+c)^(1/2))/b^2/d^2
 

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.43 \[ \int x \sinh \left (a+b \sqrt {c+d x}\right ) \, dx=\frac {2 b \sqrt {c+d x} \left (6+b^2 d x\right ) \cosh \left (a+b \sqrt {c+d x}\right )-2 \left (6+b^2 (2 c+3 d x)\right ) \sinh \left (a+b \sqrt {c+d x}\right )}{b^4 d^2} \] Input:

Integrate[x*Sinh[a + b*Sqrt[c + d*x]],x]
 

Output:

(2*b*Sqrt[c + d*x]*(6 + b^2*d*x)*Cosh[a + b*Sqrt[c + d*x]] - 2*(6 + b^2*(2 
*c + 3*d*x))*Sinh[a + b*Sqrt[c + d*x]])/(b^4*d^2)
 

Rubi [A] (verified)

Time = 0.52 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.92, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {5887, 25, 7267, 5809, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int x \sinh \left (a+b \sqrt {c+d x}\right ) \, dx\)

\(\Big \downarrow \) 5887

\(\displaystyle \frac {\int d x \sinh \left (a+b \sqrt {c+d x}\right )d(c+d x)}{d^2}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\int -d x \sinh \left (a+b \sqrt {c+d x}\right )d(c+d x)}{d^2}\)

\(\Big \downarrow \) 7267

\(\displaystyle -\frac {2 \int -d x \sqrt {c+d x} \sinh \left (a+b \sqrt {c+d x}\right )d\sqrt {c+d x}}{d^2}\)

\(\Big \downarrow \) 5809

\(\displaystyle -\frac {2 \int \left (c \sqrt {c+d x} \sinh \left (a+b \sqrt {c+d x}\right )-(c+d x)^{3/2} \sinh \left (a+b \sqrt {c+d x}\right )\right )d\sqrt {c+d x}}{d^2}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {2 \left (\frac {6 \sinh \left (a+b \sqrt {c+d x}\right )}{b^4}-\frac {6 \sqrt {c+d x} \cosh \left (a+b \sqrt {c+d x}\right )}{b^3}+\frac {3 (c+d x) \sinh \left (a+b \sqrt {c+d x}\right )}{b^2}-\frac {c \sinh \left (a+b \sqrt {c+d x}\right )}{b^2}-\frac {(c+d x)^{3/2} \cosh \left (a+b \sqrt {c+d x}\right )}{b}+\frac {c \sqrt {c+d x} \cosh \left (a+b \sqrt {c+d x}\right )}{b}\right )}{d^2}\)

Input:

Int[x*Sinh[a + b*Sqrt[c + d*x]],x]
 

Output:

(-2*((-6*Sqrt[c + d*x]*Cosh[a + b*Sqrt[c + d*x]])/b^3 + (c*Sqrt[c + d*x]*C 
osh[a + b*Sqrt[c + d*x]])/b - ((c + d*x)^(3/2)*Cosh[a + b*Sqrt[c + d*x]])/ 
b + (6*Sinh[a + b*Sqrt[c + d*x]])/b^4 - (c*Sinh[a + b*Sqrt[c + d*x]])/b^2 
+ (3*(c + d*x)*Sinh[a + b*Sqrt[c + d*x]])/b^2))/d^2
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 5809

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*Sinh[(c_.) + (d_.)*(x 
_)], x_Symbol] :> Int[ExpandIntegrand[Sinh[c + d*x], (e*x)^m*(a + b*x^n)^p, 
 x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && IGtQ[p, 0]
 

rule 5887

Int[(x_)^(m_.)*((a_.) + (b_.)*Sinh[(c_.) + (d_.)*(u_)^(n_)])^(p_.), x_Symbo 
l] :> Simp[1/Coefficient[u, x, 1]^(m + 1)   Subst[Int[(x - Coefficient[u, x 
, 0])^m*(a + b*Sinh[c + d*x^n])^p, x], x, u], x] /; FreeQ[{a, b, c, d, n, p 
}, x] && LinearQ[u, x] && NeQ[u, x] && IntegerQ[m]
 

rule 7267

Int[u_, x_Symbol] :> With[{lst = SubstForFractionalPowerOfLinear[u, x]}, Si 
mp[lst[[2]]*lst[[4]]   Subst[Int[lst[[1]], x], x, lst[[3]]^(1/lst[[2]])], x 
] /;  !FalseQ[lst] && SubstForFractionalPowerQ[u, lst[[3]], x]]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(300\) vs. \(2(149)=298\).

Time = 0.30 (sec) , antiderivative size = 301, normalized size of antiderivative = 1.80

method	result	size
parts	\(\frac {2 x \sqrt {d x +c}\, \cosh \left (a +b \sqrt {d x +c}\right )}{d b}-\frac {2 x \sinh \left (a +b \sqrt {d x +c}\right )}{d \,b^{2}}-\frac {2 \left (\frac {2 \left (a +b \sqrt {d x +c}\right )^{2} \sinh \left (a +b \sqrt {d x +c}\right )-4 \left (a +b \sqrt {d x +c}\right ) \cosh \left (a +b \sqrt {d x +c}\right )+4 \sinh \left (a +b \sqrt {d x +c}\right )-2 a \left (\left (a +b \sqrt {d x +c}\right ) \sinh \left (a +b \sqrt {d x +c}\right )-\cosh \left (a +b \sqrt {d x +c}\right )\right )}{b^{2} d}-\frac {2 a \left (\left (a +b \sqrt {d x +c}\right ) \sinh \left (a +b \sqrt {d x +c}\right )-\cosh \left (a +b \sqrt {d x +c}\right )-a \sinh \left (a +b \sqrt {d x +c}\right )\right )}{d \,b^{2}}-\frac {2 \left (\left (a +b \sqrt {d x +c}\right ) \cosh \left (a +b \sqrt {d x +c}\right )-\sinh \left (a +b \sqrt {d x +c}\right )-a \cosh \left (a +b \sqrt {d x +c}\right )\right )}{d \,b^{2}}\right )}{d \,b^{2}}\)	\(301\)
derivativedivides	\(\frac {\frac {6 a^{2} \left (\left (a +b \sqrt {d x +c}\right ) \cosh \left (a +b \sqrt {d x +c}\right )-\sinh \left (a +b \sqrt {d x +c}\right )\right )}{b^{2}}-\frac {2 a^{3} \cosh \left (a +b \sqrt {d x +c}\right )}{b^{2}}-\frac {6 a \left (\left (a +b \sqrt {d x +c}\right )^{2} \cosh \left (a +b \sqrt {d x +c}\right )-2 \left (a +b \sqrt {d x +c}\right ) \sinh \left (a +b \sqrt {d x +c}\right )+2 \cosh \left (a +b \sqrt {d x +c}\right )\right )}{b^{2}}+\frac {2 \left (\left (a +b \sqrt {d x +c}\right )^{3} \cosh \left (a +b \sqrt {d x +c}\right )-3 \left (a +b \sqrt {d x +c}\right )^{2} \sinh \left (a +b \sqrt {d x +c}\right )+6 \left (a +b \sqrt {d x +c}\right ) \cosh \left (a +b \sqrt {d x +c}\right )-6 \sinh \left (a +b \sqrt {d x +c}\right )\right )}{b^{2}}-2 c \left (\left (a +b \sqrt {d x +c}\right ) \cosh \left (a +b \sqrt {d x +c}\right )-\sinh \left (a +b \sqrt {d x +c}\right )\right )+2 c a \cosh \left (a +b \sqrt {d x +c}\right )}{b^{2} d^{2}}\)	\(303\)
default	\(\frac {\frac {6 a^{2} \left (\left (a +b \sqrt {d x +c}\right ) \cosh \left (a +b \sqrt {d x +c}\right )-\sinh \left (a +b \sqrt {d x +c}\right )\right )}{b^{2}}-\frac {2 a^{3} \cosh \left (a +b \sqrt {d x +c}\right )}{b^{2}}-\frac {6 a \left (\left (a +b \sqrt {d x +c}\right )^{2} \cosh \left (a +b \sqrt {d x +c}\right )-2 \left (a +b \sqrt {d x +c}\right ) \sinh \left (a +b \sqrt {d x +c}\right )+2 \cosh \left (a +b \sqrt {d x +c}\right )\right )}{b^{2}}+\frac {2 \left (\left (a +b \sqrt {d x +c}\right )^{3} \cosh \left (a +b \sqrt {d x +c}\right )-3 \left (a +b \sqrt {d x +c}\right )^{2} \sinh \left (a +b \sqrt {d x +c}\right )+6 \left (a +b \sqrt {d x +c}\right ) \cosh \left (a +b \sqrt {d x +c}\right )-6 \sinh \left (a +b \sqrt {d x +c}\right )\right )}{b^{2}}-2 c \left (\left (a +b \sqrt {d x +c}\right ) \cosh \left (a +b \sqrt {d x +c}\right )-\sinh \left (a +b \sqrt {d x +c}\right )\right )+2 c a \cosh \left (a +b \sqrt {d x +c}\right )}{b^{2} d^{2}}\)	\(303\)

Input:

int(x*sinh(a+b*(d*x+c)^(1/2)),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.41 \[ \int x \sinh \left (a+b \sqrt {c+d x}\right ) \, dx=\frac {2 \, {\left ({\left (b^{3} d x + 6 \, b\right )} \sqrt {d x + c} \cosh \left (\sqrt {d x + c} b + a\right ) - {\left (3 \, b^{2} d x + 2 \, b^{2} c + 6\right )} \sinh \left (\sqrt {d x + c} b + a\right )\right )}}{b^{4} d^{2}} \] Input:

integrate(x*sinh(a+b*(d*x+c)^(1/2)),x, algorithm="fricas")
 

Output:

2*((b^3*d*x + 6*b)*sqrt(d*x + c)*cosh(sqrt(d*x + c)*b + a) - (3*b^2*d*x + 
2*b^2*c + 6)*sinh(sqrt(d*x + c)*b + a))/(b^4*d^2)
 

Sympy [A] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.90 \[ \int x \sinh \left (a+b \sqrt {c+d x}\right ) \, dx=\begin {cases} \frac {x^{2} \sinh {\left (a \right )}}{2} & \text {for}\: b = 0 \wedge \left (b = 0 \vee d = 0\right ) \\\frac {x^{2} \sinh {\left (a + b \sqrt {c} \right )}}{2} & \text {for}\: d = 0 \\\frac {2 x \sqrt {c + d x} \cosh {\left (a + b \sqrt {c + d x} \right )}}{b d} - \frac {4 c \sinh {\left (a + b \sqrt {c + d x} \right )}}{b^{2} d^{2}} - \frac {6 x \sinh {\left (a + b \sqrt {c + d x} \right )}}{b^{2} d} + \frac {12 \sqrt {c + d x} \cosh {\left (a + b \sqrt {c + d x} \right )}}{b^{3} d^{2}} - \frac {12 \sinh {\left (a + b \sqrt {c + d x} \right )}}{b^{4} d^{2}} & \text {otherwise} \end {cases} \] Input:

integrate(x*sinh(a+b*(d*x+c)**(1/2)),x)
 

Output:

Piecewise((x**2*sinh(a)/2, Eq(b, 0) & (Eq(b, 0) | Eq(d, 0))), (x**2*sinh(a 
 + b*sqrt(c))/2, Eq(d, 0)), (2*x*sqrt(c + d*x)*cosh(a + b*sqrt(c + d*x))/( 
b*d) - 4*c*sinh(a + b*sqrt(c + d*x))/(b**2*d**2) - 6*x*sinh(a + b*sqrt(c + 
 d*x))/(b**2*d) + 12*sqrt(c + d*x)*cosh(a + b*sqrt(c + d*x))/(b**3*d**2) - 
 12*sinh(a + b*sqrt(c + d*x))/(b**4*d**2), True))
 

Maxima [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 293, normalized size of antiderivative = 1.75 \[ \int x \sinh \left (a+b \sqrt {c+d x}\right ) \, dx=\frac {2 \, d^{2} x^{2} \sinh \left (\sqrt {d x + c} b + a\right ) - {\left (\frac {c^{2} e^{\left (\sqrt {d x + c} b + a\right )}}{b} - \frac {c^{2} e^{\left (-\sqrt {d x + c} b - a\right )}}{b} - \frac {2 \, {\left ({\left (d x + c\right )} b^{2} e^{a} - 2 \, \sqrt {d x + c} b e^{a} + 2 \, e^{a}\right )} c e^{\left (\sqrt {d x + c} b\right )}}{b^{3}} + \frac {2 \, {\left ({\left (d x + c\right )} b^{2} + 2 \, \sqrt {d x + c} b + 2\right )} c e^{\left (-\sqrt {d x + c} b - a\right )}}{b^{3}} + \frac {{\left ({\left (d x + c\right )}^{2} b^{4} e^{a} - 4 \, {\left (d x + c\right )}^{\frac {3}{2}} b^{3} e^{a} + 12 \, {\left (d x + c\right )} b^{2} e^{a} - 24 \, \sqrt {d x + c} b e^{a} + 24 \, e^{a}\right )} e^{\left (\sqrt {d x + c} b\right )}}{b^{5}} - \frac {{\left ({\left (d x + c\right )}^{2} b^{4} + 4 \, {\left (d x + c\right )}^{\frac {3}{2}} b^{3} + 12 \, {\left (d x + c\right )} b^{2} + 24 \, \sqrt {d x + c} b + 24\right )} e^{\left (-\sqrt {d x + c} b - a\right )}}{b^{5}}\right )} b}{4 \, d^{2}} \] Input:

integrate(x*sinh(a+b*(d*x+c)^(1/2)),x, algorithm="maxima")
 

Output:

1/4*(2*d^2*x^2*sinh(sqrt(d*x + c)*b + a) - (c^2*e^(sqrt(d*x + c)*b + a)/b 
- c^2*e^(-sqrt(d*x + c)*b - a)/b - 2*((d*x + c)*b^2*e^a - 2*sqrt(d*x + c)* 
b*e^a + 2*e^a)*c*e^(sqrt(d*x + c)*b)/b^3 + 2*((d*x + c)*b^2 + 2*sqrt(d*x + 
 c)*b + 2)*c*e^(-sqrt(d*x + c)*b - a)/b^3 + ((d*x + c)^2*b^4*e^a - 4*(d*x 
+ c)^(3/2)*b^3*e^a + 12*(d*x + c)*b^2*e^a - 24*sqrt(d*x + c)*b*e^a + 24*e^ 
a)*e^(sqrt(d*x + c)*b)/b^5 - ((d*x + c)^2*b^4 + 4*(d*x + c)^(3/2)*b^3 + 12 
*(d*x + c)*b^2 + 24*sqrt(d*x + c)*b + 24)*e^(-sqrt(d*x + c)*b - a)/b^5)*b) 
/d^2
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 299 vs. \(2 (149) = 298\).

Time = 0.13 (sec) , antiderivative size = 299, normalized size of antiderivative = 1.79 \[ \int x \sinh \left (a+b \sqrt {c+d x}\right ) \, dx=-\frac {\frac {{\left ({\left (\sqrt {d x + c} b + a\right )} b^{2} c - a b^{2} c - {\left (\sqrt {d x + c} b + a\right )}^{3} + 3 \, {\left (\sqrt {d x + c} b + a\right )}^{2} a - 3 \, {\left (\sqrt {d x + c} b + a\right )} a^{2} + a^{3} - b^{2} c + 3 \, {\left (\sqrt {d x + c} b + a\right )}^{2} - 6 \, {\left (\sqrt {d x + c} b + a\right )} a + 3 \, a^{2} - 6 \, \sqrt {d x + c} b + 6\right )} e^{\left (\sqrt {d x + c} b + a\right )}}{b^{3} d} + \frac {{\left ({\left (\sqrt {d x + c} b + a\right )} b^{2} c - a b^{2} c - {\left (\sqrt {d x + c} b + a\right )}^{3} + 3 \, {\left (\sqrt {d x + c} b + a\right )}^{2} a - 3 \, {\left (\sqrt {d x + c} b + a\right )} a^{2} + a^{3} + b^{2} c - 3 \, {\left (\sqrt {d x + c} b + a\right )}^{2} + 6 \, {\left (\sqrt {d x + c} b + a\right )} a - 3 \, a^{2} - 6 \, \sqrt {d x + c} b - 6\right )} e^{\left (-\sqrt {d x + c} b - a\right )}}{b^{3} d}}{b d} \] Input:

integrate(x*sinh(a+b*(d*x+c)^(1/2)),x, algorithm="giac")
 

Output:

-(((sqrt(d*x + c)*b + a)*b^2*c - a*b^2*c - (sqrt(d*x + c)*b + a)^3 + 3*(sq 
rt(d*x + c)*b + a)^2*a - 3*(sqrt(d*x + c)*b + a)*a^2 + a^3 - b^2*c + 3*(sq 
rt(d*x + c)*b + a)^2 - 6*(sqrt(d*x + c)*b + a)*a + 3*a^2 - 6*sqrt(d*x + c) 
*b + 6)*e^(sqrt(d*x + c)*b + a)/(b^3*d) + ((sqrt(d*x + c)*b + a)*b^2*c - a 
*b^2*c - (sqrt(d*x + c)*b + a)^3 + 3*(sqrt(d*x + c)*b + a)^2*a - 3*(sqrt(d 
*x + c)*b + a)*a^2 + a^3 + b^2*c - 3*(sqrt(d*x + c)*b + a)^2 + 6*(sqrt(d*x 
 + c)*b + a)*a - 3*a^2 - 6*sqrt(d*x + c)*b - 6)*e^(-sqrt(d*x + c)*b - a)/( 
b^3*d))/(b*d)
 

Mupad [F(-1)]

Timed out. \[ \int x \sinh \left (a+b \sqrt {c+d x}\right ) \, dx=\int x\,\mathrm {sinh}\left (a+b\,\sqrt {c+d\,x}\right ) \,d x \] Input:

int(x*sinh(a + b*(c + d*x)^(1/2)),x)
 

Output:

int(x*sinh(a + b*(c + d*x)^(1/2)), x)
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.60 \[ \int x \sinh \left (a+b \sqrt {c+d x}\right ) \, dx=\frac {2 \sqrt {d x +c}\, \cosh \left (\sqrt {d x +c}\, b +a \right ) b^{3} d x +12 \sqrt {d x +c}\, \cosh \left (\sqrt {d x +c}\, b +a \right ) b -4 \sinh \left (\sqrt {d x +c}\, b +a \right ) b^{2} c -6 \sinh \left (\sqrt {d x +c}\, b +a \right ) b^{2} d x -12 \sinh \left (\sqrt {d x +c}\, b +a \right )}{b^{4} d^{2}} \] Input:

int(x*sinh(a+b*(d*x+c)^(1/2)),x)
 

Output:

(2*(sqrt(c + d*x)*cosh(sqrt(c + d*x)*b + a)*b**3*d*x + 6*sqrt(c + d*x)*cos 
h(sqrt(c + d*x)*b + a)*b - 2*sinh(sqrt(c + d*x)*b + a)*b**2*c - 3*sinh(sqr 
t(c + d*x)*b + a)*b**2*d*x - 6*sinh(sqrt(c + d*x)*b + a)))/(b**4*d**2)