\(\int \sqrt {c+d x} \sinh ^3(a+b x) \, dx\) [55]

Optimal result

Integrand size = 18, antiderivative size = 275 \[ \int \sqrt {c+d x} \sinh ^3(a+b x) \, dx=-\frac {3 \sqrt {c+d x} \cosh (a+b x)}{4 b}+\frac {\sqrt {c+d x} \cosh (3 a+3 b x)}{12 b}+\frac {3 \sqrt {d} e^{-a+\frac {b c}{d}} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{16 b^{3/2}}-\frac {\sqrt {d} e^{-3 a+\frac {3 b c}{d}} \sqrt {\frac {\pi }{3}} \text {erf}\left (\frac {\sqrt {3} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{48 b^{3/2}}+\frac {3 \sqrt {d} e^{a-\frac {b c}{d}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{16 b^{3/2}}-\frac {\sqrt {d} e^{3 a-\frac {3 b c}{d}} \sqrt {\frac {\pi }{3}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{48 b^{3/2}} \] Output:

-3/4*(d*x+c)^(1/2)*cosh(b*x+a)/b+1/12*(d*x+c)^(1/2)*cosh(3*b*x+3*a)/b+3/16 
*d^(1/2)*exp(-a+b*c/d)*Pi^(1/2)*erf(b^(1/2)*(d*x+c)^(1/2)/d^(1/2))/b^(3/2) 
-1/144*d^(1/2)*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^(3/2)+3/16*d^(1/2)*exp(a-b*c/d)*Pi^(1/2)*erfi(b^(1/2) 
*(d*x+c)^(1/2)/d^(1/2))/b^(3/2)-1/144*d^(1/2)*exp(3*a-3*b*c/d)*3^(1/2)*Pi^ 
(1/2)*erfi(3^(1/2)*b^(1/2)*(d*x+c)^(1/2)/d^(1/2))/b^(3/2)
 

Mathematica [A] (verified)

Time = 0.21 (sec) , antiderivative size = 209, normalized size of antiderivative = 0.76 \[ \int \sqrt {c+d x} \sinh ^3(a+b x) \, dx=\frac {e^{-3 \left (a+\frac {b c}{d}\right )} \sqrt {c+d x} \left (\sqrt {3} e^{6 a} \sqrt {\frac {b (c+d x)}{d}} \Gamma \left (\frac {3}{2},-\frac {3 b (c+d x)}{d}\right )-27 e^{4 a+\frac {2 b c}{d}} \sqrt {\frac {b (c+d x)}{d}} \Gamma \left (\frac {3}{2},-\frac {b (c+d x)}{d}\right )+e^{\frac {4 b c}{d}} \sqrt {-\frac {b (c+d x)}{d}} \left (-27 e^{2 a} \Gamma \left (\frac {3}{2},\frac {b (c+d x)}{d}\right )+\sqrt {3} e^{\frac {2 b c}{d}} \Gamma \left (\frac {3}{2},\frac {3 b (c+d x)}{d}\right )\right )\right )}{72 b \sqrt {-\frac {b^2 (c+d x)^2}{d^2}}} \] Input:

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

Output:

(Sqrt[c + d*x]*(Sqrt[3]*E^(6*a)*Sqrt[(b*(c + d*x))/d]*Gamma[3/2, (-3*b*(c 
+ d*x))/d] - 27*E^(4*a + (2*b*c)/d)*Sqrt[(b*(c + d*x))/d]*Gamma[3/2, -((b* 
(c + d*x))/d)] + E^((4*b*c)/d)*Sqrt[-((b*(c + d*x))/d)]*(-27*E^(2*a)*Gamma 
[3/2, (b*(c + d*x))/d] + Sqrt[3]*E^((2*b*c)/d)*Gamma[3/2, (3*b*(c + d*x))/ 
d])))/(72*b*E^(3*(a + (b*c)/d))*Sqrt[-((b^2*(c + d*x)^2)/d^2)])
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.70 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.06, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3793

\(\displaystyle i \int \left (\frac {3}{4} i \sqrt {c+d x} \sinh (a+b x)-\frac {1}{4} i \sqrt {c+d x} \sinh (3 a+3 b x)\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle i \left (-\frac {3 i \sqrt {\pi } \sqrt {d} e^{\frac {b c}{d}-a} \text {erf}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{16 b^{3/2}}+\frac {i \sqrt {\frac {\pi }{3}} \sqrt {d} e^{\frac {3 b c}{d}-3 a} \text {erf}\left (\frac {\sqrt {3} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{48 b^{3/2}}-\frac {3 i \sqrt {\pi } \sqrt {d} e^{a-\frac {b c}{d}} \text {erfi}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{16 b^{3/2}}+\frac {i \sqrt {\frac {\pi }{3}} \sqrt {d} e^{3 a-\frac {3 b c}{d}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{48 b^{3/2}}+\frac {3 i \sqrt {c+d x} \cosh (a+b x)}{4 b}-\frac {i \sqrt {c+d x} \cosh (3 a+3 b x)}{12 b}\right )\)

Input:

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

Output:

I*((((3*I)/4)*Sqrt[c + d*x]*Cosh[a + b*x])/b - ((I/12)*Sqrt[c + d*x]*Cosh[ 
3*a + 3*b*x])/b - (((3*I)/16)*Sqrt[d]*E^(-a + (b*c)/d)*Sqrt[Pi]*Erf[(Sqrt[ 
b]*Sqrt[c + d*x])/Sqrt[d]])/b^(3/2) + ((I/48)*Sqrt[d]*E^(-3*a + (3*b*c)/d) 
*Sqrt[Pi/3]*Erf[(Sqrt[3]*Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/b^(3/2) - (((3*I 
)/16)*Sqrt[d]*E^(a - (b*c)/d)*Sqrt[Pi]*Erfi[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[d 
]])/b^(3/2) + ((I/48)*Sqrt[d]*E^(3*a - (3*b*c)/d)*Sqrt[Pi/3]*Erfi[(Sqrt[3] 
*Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/b^(3/2))
 

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]

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

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1216 vs. \(2 (201) = 402\).

Time = 0.12 (sec) , antiderivative size = 1216, normalized size of antiderivative = 4.42 \[ \int \sqrt {c+d x} \sinh ^3(a+b x) \, dx=\text {Too large to display} \] Input:

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

Output:

-1/144*(sqrt(3)*sqrt(pi)*(d*cosh(b*x + a)^3*cosh(-3*(b*c - a*d)/d) - d*cos 
h(b*x + a)^3*sinh(-3*(b*c - a*d)/d) + (d*cosh(-3*(b*c - a*d)/d) - d*sinh(- 
3*(b*c - a*d)/d))*sinh(b*x + a)^3 + 3*(d*cosh(b*x + a)*cosh(-3*(b*c - a*d) 
/d) - d*cosh(b*x + a)*sinh(-3*(b*c - a*d)/d))*sinh(b*x + a)^2 + 3*(d*cosh( 
b*x + a)^2*cosh(-3*(b*c - a*d)/d) - d*cosh(b*x + a)^2*sinh(-3*(b*c - a*d)/ 
d))*sinh(b*x + a))*sqrt(b/d)*erf(sqrt(3)*sqrt(d*x + c)*sqrt(b/d)) - sqrt(3 
)*sqrt(pi)*(d*cosh(b*x + a)^3*cosh(-3*(b*c - a*d)/d) + d*cosh(b*x + a)^3*s 
inh(-3*(b*c - a*d)/d) + (d*cosh(-3*(b*c - a*d)/d) + d*sinh(-3*(b*c - a*d)/ 
d))*sinh(b*x + a)^3 + 3*(d*cosh(b*x + a)*cosh(-3*(b*c - a*d)/d) + d*cosh(b 
*x + a)*sinh(-3*(b*c - a*d)/d))*sinh(b*x + a)^2 + 3*(d*cosh(b*x + a)^2*cos 
h(-3*(b*c - a*d)/d) + d*cosh(b*x + a)^2*sinh(-3*(b*c - a*d)/d))*sinh(b*x + 
 a))*sqrt(-b/d)*erf(sqrt(3)*sqrt(d*x + c)*sqrt(-b/d)) - 27*sqrt(pi)*(d*cos 
h(b*x + a)^3*cosh(-(b*c - a*d)/d) - d*cosh(b*x + a)^3*sinh(-(b*c - a*d)/d) 
 + (d*cosh(-(b*c - a*d)/d) - d*sinh(-(b*c - a*d)/d))*sinh(b*x + a)^3 + 3*( 
d*cosh(b*x + a)*cosh(-(b*c - a*d)/d) - d*cosh(b*x + a)*sinh(-(b*c - a*d)/d 
))*sinh(b*x + a)^2 + 3*(d*cosh(b*x + a)^2*cosh(-(b*c - a*d)/d) - d*cosh(b* 
x + a)^2*sinh(-(b*c - a*d)/d))*sinh(b*x + a))*sqrt(b/d)*erf(sqrt(d*x + c)* 
sqrt(b/d)) + 27*sqrt(pi)*(d*cosh(b*x + a)^3*cosh(-(b*c - a*d)/d) + d*cosh( 
b*x + a)^3*sinh(-(b*c - a*d)/d) + (d*cosh(-(b*c - a*d)/d) + d*sinh(-(b*c - 
 a*d)/d))*sinh(b*x + a)^3 + 3*(d*cosh(b*x + a)*cosh(-(b*c - a*d)/d) + d...
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 333, normalized size of antiderivative = 1.21 \[ \int \sqrt {c+d x} \sinh ^3(a+b x) \, dx=-\frac {\frac {\sqrt {3} \sqrt {\pi } d \operatorname {erf}\left (\sqrt {3} \sqrt {d x + c} \sqrt {-\frac {b}{d}}\right ) e^{\left (3 \, a - \frac {3 \, b c}{d}\right )}}{b \sqrt {-\frac {b}{d}}} + \frac {\sqrt {3} \sqrt {\pi } d \operatorname {erf}\left (\sqrt {3} \sqrt {d x + c} \sqrt {\frac {b}{d}}\right ) e^{\left (-3 \, a + \frac {3 \, b c}{d}\right )}}{b \sqrt {\frac {b}{d}}} - \frac {27 \, \sqrt {\pi } d \operatorname {erf}\left (\sqrt {d x + c} \sqrt {-\frac {b}{d}}\right ) e^{\left (a - \frac {b c}{d}\right )}}{b \sqrt {-\frac {b}{d}}} - \frac {27 \, \sqrt {\pi } d \operatorname {erf}\left (\sqrt {d x + c} \sqrt {\frac {b}{d}}\right ) e^{\left (-a + \frac {b c}{d}\right )}}{b \sqrt {\frac {b}{d}}} - \frac {6 \, \sqrt {d x + c} d e^{\left (3 \, a + \frac {3 \, {\left (d x + c\right )} b}{d} - \frac {3 \, b c}{d}\right )}}{b} + \frac {54 \, \sqrt {d x + c} d e^{\left (a + \frac {{\left (d x + c\right )} b}{d} - \frac {b c}{d}\right )}}{b} + \frac {54 \, \sqrt {d x + c} d e^{\left (-a - \frac {{\left (d x + c\right )} b}{d} + \frac {b c}{d}\right )}}{b} - \frac {6 \, \sqrt {d x + c} d e^{\left (-3 \, a - \frac {3 \, {\left (d x + c\right )} b}{d} + \frac {3 \, b c}{d}\right )}}{b}}{144 \, d} \] Input:

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

Output:

-1/144*(sqrt(3)*sqrt(pi)*d*erf(sqrt(3)*sqrt(d*x + c)*sqrt(-b/d))*e^(3*a - 
3*b*c/d)/(b*sqrt(-b/d)) + sqrt(3)*sqrt(pi)*d*erf(sqrt(3)*sqrt(d*x + c)*sqr 
t(b/d))*e^(-3*a + 3*b*c/d)/(b*sqrt(b/d)) - 27*sqrt(pi)*d*erf(sqrt(d*x + c) 
*sqrt(-b/d))*e^(a - b*c/d)/(b*sqrt(-b/d)) - 27*sqrt(pi)*d*erf(sqrt(d*x + c 
)*sqrt(b/d))*e^(-a + b*c/d)/(b*sqrt(b/d)) - 6*sqrt(d*x + c)*d*e^(3*a + 3*( 
d*x + c)*b/d - 3*b*c/d)/b + 54*sqrt(d*x + c)*d*e^(a + (d*x + c)*b/d - b*c/ 
d)/b + 54*sqrt(d*x + c)*d*e^(-a - (d*x + c)*b/d + b*c/d)/b - 6*sqrt(d*x + 
c)*d*e^(-3*a - 3*(d*x + c)*b/d + 3*b*c/d)/b)/d
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

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

Output:

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