\(\int \frac {\sinh ^3(a+b x)}{\sqrt {c+d x}} \, dx\) [56]

Optimal result

Integrand size = 18, antiderivative size = 228 \[ \int \frac {\sinh ^3(a+b x)}{\sqrt {c+d x}} \, dx=\frac {3 e^{-a+\frac {b c}{d}} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{8 \sqrt {b} \sqrt {d}}-\frac {e^{-3 a+\frac {3 b c}{d}} \sqrt {\frac {\pi }{3}} \text {erf}\left (\frac {\sqrt {3} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{8 \sqrt {b} \sqrt {d}}-\frac {3 e^{a-\frac {b c}{d}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{8 \sqrt {b} \sqrt {d}}+\frac {e^{3 a-\frac {3 b c}{d}} \sqrt {\frac {\pi }{3}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{8 \sqrt {b} \sqrt {d}} \] Output:

3/8*exp(-a+b*c/d)*Pi^(1/2)*erf(b^(1/2)*(d*x+c)^(1/2)/d^(1/2))/b^(1/2)/d^(1 
/2)-1/24*exp(-3*a+3*b*c/d)*3^(1/2)*Pi^(1/2)*erf(3^(1/2)*b^(1/2)*(d*x+c)^(1 
/2)/d^(1/2))/b^(1/2)/d^(1/2)-3/8*exp(a-b*c/d)*Pi^(1/2)*erfi(b^(1/2)*(d*x+c 
)^(1/2)/d^(1/2))/b^(1/2)/d^(1/2)+1/24*exp(3*a-3*b*c/d)*3^(1/2)*Pi^(1/2)*er 
fi(3^(1/2)*b^(1/2)*(d*x+c)^(1/2)/d^(1/2))/b^(1/2)/d^(1/2)
 

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.84 \[ \int \frac {\sinh ^3(a+b x)}{\sqrt {c+d x}} \, dx=\frac {e^{-3 \left (a+\frac {b c}{d}\right )} \left (\sqrt {3} e^{6 a} \sqrt {-\frac {b (c+d x)}{d}} \Gamma \left (\frac {1}{2},-\frac {3 b (c+d x)}{d}\right )-9 e^{4 a+\frac {2 b c}{d}} \sqrt {-\frac {b (c+d x)}{d}} \Gamma \left (\frac {1}{2},-\frac {b (c+d x)}{d}\right )+e^{\frac {4 b c}{d}} \sqrt {\frac {b (c+d x)}{d}} \left (-9 e^{2 a} \Gamma \left (\frac {1}{2},\frac {b (c+d x)}{d}\right )+\sqrt {3} e^{\frac {2 b c}{d}} \Gamma \left (\frac {1}{2},\frac {3 b (c+d x)}{d}\right )\right )\right )}{24 b \sqrt {c+d x}} \] Input:

Integrate[Sinh[a + b*x]^3/Sqrt[c + d*x],x]
 

Output:

(Sqrt[3]*E^(6*a)*Sqrt[-((b*(c + d*x))/d)]*Gamma[1/2, (-3*b*(c + d*x))/d] - 
 9*E^(4*a + (2*b*c)/d)*Sqrt[-((b*(c + d*x))/d)]*Gamma[1/2, -((b*(c + d*x)) 
/d)] + E^((4*b*c)/d)*Sqrt[(b*(c + d*x))/d]*(-9*E^(2*a)*Gamma[1/2, (b*(c + 
d*x))/d] + Sqrt[3]*E^((2*b*c)/d)*Gamma[1/2, (3*b*(c + d*x))/d]))/(24*b*E^( 
3*(a + (b*c)/d))*Sqrt[c + d*x])
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.60 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.05, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {3042, 26, 3793, 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.

\(\displaystyle \int \frac {\sinh ^3(a+b x)}{\sqrt {c+d x}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {i \sin (i a+i b x)^3}{\sqrt {c+d x}}dx\)

\(\Big \downarrow \) 26

\(\displaystyle i \int \frac {\sin (i a+i b x)^3}{\sqrt {c+d x}}dx\)

\(\Big \downarrow \) 3793

\(\displaystyle i \int \left (\frac {3 i \sinh (a+b x)}{4 \sqrt {c+d x}}-\frac {i \sinh (3 a+3 b x)}{4 \sqrt {c+d x}}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle i \left (-\frac {3 i \sqrt {\pi } e^{\frac {b c}{d}-a} \text {erf}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{8 \sqrt {b} \sqrt {d}}+\frac {i \sqrt {\frac {\pi }{3}} e^{\frac {3 b c}{d}-3 a} \text {erf}\left (\frac {\sqrt {3} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{8 \sqrt {b} \sqrt {d}}+\frac {3 i \sqrt {\pi } e^{a-\frac {b c}{d}} \text {erfi}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{8 \sqrt {b} \sqrt {d}}-\frac {i \sqrt {\frac {\pi }{3}} e^{3 a-\frac {3 b c}{d}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{8 \sqrt {b} \sqrt {d}}\right )\)

Input:

Int[Sinh[a + b*x]^3/Sqrt[c + d*x],x]
 

Output:

I*((((-3*I)/8)*E^(-a + (b*c)/d)*Sqrt[Pi]*Erf[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[ 
d]])/(Sqrt[b]*Sqrt[d]) + ((I/8)*E^(-3*a + (3*b*c)/d)*Sqrt[Pi/3]*Erf[(Sqrt[ 
3]*Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(Sqrt[b]*Sqrt[d]) + (((3*I)/8)*E^(a - 
(b*c)/d)*Sqrt[Pi]*Erfi[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(Sqrt[b]*Sqrt[d]) 
 - ((I/8)*E^(3*a - (3*b*c)/d)*Sqrt[Pi/3]*Erfi[(Sqrt[3]*Sqrt[b]*Sqrt[c + d* 
x])/Sqrt[d]])/(Sqrt[b]*Sqrt[d]))
 

Defintions of rubi rules used

rule 26

Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a])   I 
nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
 

rule 2009

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

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3793

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> In 
t[ExpandTrigReduce[(c + d*x)^m, Sin[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f 
, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1]))
 

Maple [F]

\[\int \frac {\sinh \left (b x +a \right )^{3}}{\sqrt {d x +c}}d x\]

Input:

int(sinh(b*x+a)^3/(d*x+c)^(1/2),x)
 

Output:

int(sinh(b*x+a)^3/(d*x+c)^(1/2),x)
 

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.11 \[ \int \frac {\sinh ^3(a+b x)}{\sqrt {c+d x}} \, dx=-\frac {\sqrt {3} \sqrt {\pi } \sqrt {\frac {b}{d}} {\left (\cosh \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) - \sinh \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right )\right )} \operatorname {erf}\left (\sqrt {3} \sqrt {d x + c} \sqrt {\frac {b}{d}}\right ) + \sqrt {3} \sqrt {\pi } \sqrt {-\frac {b}{d}} {\left (\cosh \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) + \sinh \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right )\right )} \operatorname {erf}\left (\sqrt {3} \sqrt {d x + c} \sqrt {-\frac {b}{d}}\right ) - 9 \, \sqrt {\pi } \sqrt {\frac {b}{d}} {\left (\cosh \left (-\frac {b c - a d}{d}\right ) - \sinh \left (-\frac {b c - a d}{d}\right )\right )} \operatorname {erf}\left (\sqrt {d x + c} \sqrt {\frac {b}{d}}\right ) - 9 \, \sqrt {\pi } \sqrt {-\frac {b}{d}} {\left (\cosh \left (-\frac {b c - a d}{d}\right ) + \sinh \left (-\frac {b c - a d}{d}\right )\right )} \operatorname {erf}\left (\sqrt {d x + c} \sqrt {-\frac {b}{d}}\right )}{24 \, b} \] Input:

integrate(sinh(b*x+a)^3/(d*x+c)^(1/2),x, algorithm="fricas")
 

Output:

-1/24*(sqrt(3)*sqrt(pi)*sqrt(b/d)*(cosh(-3*(b*c - a*d)/d) - sinh(-3*(b*c - 
 a*d)/d))*erf(sqrt(3)*sqrt(d*x + c)*sqrt(b/d)) + sqrt(3)*sqrt(pi)*sqrt(-b/ 
d)*(cosh(-3*(b*c - a*d)/d) + sinh(-3*(b*c - a*d)/d))*erf(sqrt(3)*sqrt(d*x 
+ c)*sqrt(-b/d)) - 9*sqrt(pi)*sqrt(b/d)*(cosh(-(b*c - a*d)/d) - sinh(-(b*c 
 - a*d)/d))*erf(sqrt(d*x + c)*sqrt(b/d)) - 9*sqrt(pi)*sqrt(-b/d)*(cosh(-(b 
*c - a*d)/d) + sinh(-(b*c - a*d)/d))*erf(sqrt(d*x + c)*sqrt(-b/d)))/b
 

Sympy [F]

\[ \int \frac {\sinh ^3(a+b x)}{\sqrt {c+d x}} \, dx=\int \frac {\sinh ^{3}{\left (a + b x \right )}}{\sqrt {c + d x}}\, dx \] Input:

integrate(sinh(b*x+a)**3/(d*x+c)**(1/2),x)
 

Output:

Integral(sinh(a + b*x)**3/sqrt(c + d*x), x)
 

Maxima [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 178, normalized size of antiderivative = 0.78 \[ \int \frac {\sinh ^3(a+b x)}{\sqrt {c+d x}} \, dx=\frac {\frac {\sqrt {3} \sqrt {\pi } \operatorname {erf}\left (\sqrt {3} \sqrt {d x + c} \sqrt {-\frac {b}{d}}\right ) e^{\left (3 \, a - \frac {3 \, b c}{d}\right )}}{\sqrt {-\frac {b}{d}}} - \frac {\sqrt {3} \sqrt {\pi } \operatorname {erf}\left (\sqrt {3} \sqrt {d x + c} \sqrt {\frac {b}{d}}\right ) e^{\left (-3 \, a + \frac {3 \, b c}{d}\right )}}{\sqrt {\frac {b}{d}}} - \frac {9 \, \sqrt {\pi } \operatorname {erf}\left (\sqrt {d x + c} \sqrt {-\frac {b}{d}}\right ) e^{\left (a - \frac {b c}{d}\right )}}{\sqrt {-\frac {b}{d}}} + \frac {9 \, \sqrt {\pi } \operatorname {erf}\left (\sqrt {d x + c} \sqrt {\frac {b}{d}}\right ) e^{\left (-a + \frac {b c}{d}\right )}}{\sqrt {\frac {b}{d}}}}{24 \, d} \] Input:

integrate(sinh(b*x+a)^3/(d*x+c)^(1/2),x, algorithm="maxima")
 

Output:

1/24*(sqrt(3)*sqrt(pi)*erf(sqrt(3)*sqrt(d*x + c)*sqrt(-b/d))*e^(3*a - 3*b* 
c/d)/sqrt(-b/d) - sqrt(3)*sqrt(pi)*erf(sqrt(3)*sqrt(d*x + c)*sqrt(b/d))*e^ 
(-3*a + 3*b*c/d)/sqrt(b/d) - 9*sqrt(pi)*erf(sqrt(d*x + c)*sqrt(-b/d))*e^(a 
 - b*c/d)/sqrt(-b/d) + 9*sqrt(pi)*erf(sqrt(d*x + c)*sqrt(b/d))*e^(-a + b*c 
/d)/sqrt(b/d))/d
 

Giac [F]

\[ \int \frac {\sinh ^3(a+b x)}{\sqrt {c+d x}} \, dx=\int { \frac {\sinh \left (b x + a\right )^{3}}{\sqrt {d x + c}} \,d x } \] Input:

integrate(sinh(b*x+a)^3/(d*x+c)^(1/2),x, algorithm="giac")
 

Output:

integrate(sinh(b*x + a)^3/sqrt(d*x + c), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sinh ^3(a+b x)}{\sqrt {c+d x}} \, dx=\int \frac {{\mathrm {sinh}\left (a+b\,x\right )}^3}{\sqrt {c+d\,x}} \,d x \] Input:

int(sinh(a + b*x)^3/(c + d*x)^(1/2),x)
 

Output:

int(sinh(a + b*x)^3/(c + d*x)^(1/2), x)
 

Reduce [F]

\[ \int \frac {\sinh ^3(a+b x)}{\sqrt {c+d x}} \, dx=\int \frac {\sinh \left (b x +a \right )^{3}}{\sqrt {d x +c}}d x \] Input:

int(sinh(b*x+a)^3/(d*x+c)^(1/2),x)
 

Output:

int(sinh(a + b*x)**3/sqrt(c + d*x),x)