\(\int \frac {\sinh ^3(a+b x)}{(c+d x)^{3/2}} \, dx\) [57]

Optimal result

Integrand size = 18, antiderivative size = 246 \[ \int \frac {\sinh ^3(a+b x)}{(c+d x)^{3/2}} \, dx=-\frac {3 \sqrt {b} e^{-a+\frac {b c}{d}} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{4 d^{3/2}}+\frac {\sqrt {b} e^{-3 a+\frac {3 b c}{d}} \sqrt {3 \pi } \text {erf}\left (\frac {\sqrt {3} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{4 d^{3/2}}-\frac {3 \sqrt {b} e^{a-\frac {b c}{d}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{4 d^{3/2}}+\frac {\sqrt {b} e^{3 a-\frac {3 b c}{d}} \sqrt {3 \pi } \text {erfi}\left (\frac {\sqrt {3} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{4 d^{3/2}}-\frac {2 \sinh ^3(a+b x)}{d \sqrt {c+d x}} \] Output:

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

Mathematica [A] (verified)

Time = 0.72 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.00 \[ \int \frac {\sinh ^3(a+b x)}{(c+d x)^{3/2}} \, dx=\frac {e^{-3 \left (a+b \left (\frac {c}{d}+x\right )\right )} \left (\sqrt {3} e^{6 a+3 b x} \sqrt {-\frac {b (c+d x)}{d}} \Gamma \left (\frac {1}{2},-\frac {3 b (c+d x)}{d}\right )-3 e^{4 a+\frac {2 b c}{d}+3 b x} \sqrt {-\frac {b (c+d x)}{d}} \Gamma \left (\frac {1}{2},-\frac {b (c+d x)}{d}\right )-e^{\frac {3 b c}{d}} \left (\left (-1+e^{2 (a+b x)}\right )^3-3 e^{2 a+\frac {b c}{d}+3 b x} \sqrt {\frac {b (c+d x)}{d}} \Gamma \left (\frac {1}{2},b \left (\frac {c}{d}+x\right )\right )+\sqrt {3} e^{\frac {3 b (c+d x)}{d}} \sqrt {\frac {b (c+d x)}{d}} \Gamma \left (\frac {1}{2},\frac {3 b (c+d x)}{d}\right )\right )\right )}{4 d \sqrt {c+d x}} \] Input:

Integrate[Sinh[a + b*x]^3/(c + d*x)^(3/2),x]
 

Output:

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

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.63 (sec) , antiderivative size = 265, normalized size of antiderivative = 1.08, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {3042, 26, 3794, 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.

\(\displaystyle \int \frac {\sinh ^3(a+b x)}{(c+d x)^{3/2}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {i \sin (i a+i b x)^3}{(c+d x)^{3/2}}dx\)

\(\Big \downarrow \) 26

\(\displaystyle i \int \frac {\sin (i a+i b x)^3}{(c+d x)^{3/2}}dx\)

\(\Big \downarrow \) 3794

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

\(\Big \downarrow \) 2009

\(\displaystyle i \left (\frac {6 i b \left (\frac {\sqrt {\pi } e^{\frac {b c}{d}-a} \text {erf}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{8 \sqrt {b} \sqrt {d}}-\frac {\sqrt {\frac {\pi }{3}} e^{\frac {3 b c}{d}-3 a} \text {erf}\left (\frac {\sqrt {3} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{8 \sqrt {b} \sqrt {d}}+\frac {\sqrt {\pi } e^{a-\frac {b c}{d}} \text {erfi}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{8 \sqrt {b} \sqrt {d}}-\frac {\sqrt {\frac {\pi }{3}} e^{3 a-\frac {3 b c}{d}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{8 \sqrt {b} \sqrt {d}}\right )}{d}+\frac {2 i \sinh ^3(a+b x)}{d \sqrt {c+d x}}\right )\)

Input:

Int[Sinh[a + b*x]^3/(c + d*x)^(3/2),x]
 

Output:

I*(((6*I)*b*((E^(-a + (b*c)/d)*Sqrt[Pi]*Erf[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[d 
]])/(8*Sqrt[b]*Sqrt[d]) - (E^(-3*a + (3*b*c)/d)*Sqrt[Pi/3]*Erf[(Sqrt[3]*Sq 
rt[b]*Sqrt[c + d*x])/Sqrt[d]])/(8*Sqrt[b]*Sqrt[d]) + (E^(a - (b*c)/d)*Sqrt 
[Pi]*Erfi[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(8*Sqrt[b]*Sqrt[d]) - (E^(3*a 
- (3*b*c)/d)*Sqrt[Pi/3]*Erfi[(Sqrt[3]*Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(8* 
Sqrt[b]*Sqrt[d])))/d + ((2*I)*Sinh[a + b*x]^3)/(d*Sqrt[c + d*x]))
 

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 3794

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Si 
mp[(c + d*x)^(m + 1)*(Sin[e + f*x]^n/(d*(m + 1))), x] - Simp[f*(n/(d*(m + 1 
)))   Int[ExpandTrigReduce[(c + d*x)^(m + 1), Cos[e + f*x]*Sin[e + f*x]^(n 
- 1), x], x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && GeQ[m, -2] & 
& LtQ[m, -1]
 

Maple [F]

\[\int \frac {\sinh \left (b x +a \right )^{3}}{\left (d x +c \right )^{\frac {3}{2}}}d x\]

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1346 vs. \(2 (182) = 364\).

Time = 0.12 (sec) , antiderivative size = 1346, normalized size of antiderivative = 5.47 \[ \int \frac {\sinh ^3(a+b x)}{(c+d x)^{3/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

1/4*(sqrt(3)*sqrt(pi)*((d*x + c)*cosh(b*x + a)^3*cosh(-3*(b*c - a*d)/d) - 
(d*x + c)*cosh(b*x + a)^3*sinh(-3*(b*c - a*d)/d) + ((d*x + c)*cosh(-3*(b*c 
 - a*d)/d) - (d*x + c)*sinh(-3*(b*c - a*d)/d))*sinh(b*x + a)^3 + 3*((d*x + 
 c)*cosh(b*x + a)*cosh(-3*(b*c - a*d)/d) - (d*x + c)*cosh(b*x + a)*sinh(-3 
*(b*c - a*d)/d))*sinh(b*x + a)^2 + 3*((d*x + c)*cosh(b*x + a)^2*cosh(-3*(b 
*c - a*d)/d) - (d*x + c)*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*x + c)*cosh(b*x + a)^3*cosh(-3*(b*c - a*d)/d) + (d*x + c)*cosh(b*x + a)^ 
3*sinh(-3*(b*c - a*d)/d) + ((d*x + c)*cosh(-3*(b*c - a*d)/d) + (d*x + c)*s 
inh(-3*(b*c - a*d)/d))*sinh(b*x + a)^3 + 3*((d*x + c)*cosh(b*x + a)*cosh(- 
3*(b*c - a*d)/d) + (d*x + c)*cosh(b*x + a)*sinh(-3*(b*c - a*d)/d))*sinh(b* 
x + a)^2 + 3*((d*x + c)*cosh(b*x + a)^2*cosh(-3*(b*c - a*d)/d) + (d*x + c) 
*cosh(b*x + a)^2*sinh(-3*(b*c - a*d)/d))*sinh(b*x + a))*sqrt(-b/d)*erf(sqr 
t(3)*sqrt(d*x + c)*sqrt(-b/d)) - 3*sqrt(pi)*((d*x + c)*cosh(b*x + a)^3*cos 
h(-(b*c - a*d)/d) - (d*x + c)*cosh(b*x + a)^3*sinh(-(b*c - a*d)/d) + ((d*x 
 + c)*cosh(-(b*c - a*d)/d) - (d*x + c)*sinh(-(b*c - a*d)/d))*sinh(b*x + a) 
^3 + 3*((d*x + c)*cosh(b*x + a)*cosh(-(b*c - a*d)/d) - (d*x + c)*cosh(b*x 
+ a)*sinh(-(b*c - a*d)/d))*sinh(b*x + a)^2 + 3*((d*x + c)*cosh(b*x + a)^2* 
cosh(-(b*c - a*d)/d) - (d*x + c)*cosh(b*x + a)^2*sinh(-(b*c - a*d)/d))*sin 
h(b*x + a))*sqrt(b/d)*erf(sqrt(d*x + c)*sqrt(b/d)) + 3*sqrt(pi)*((d*x +...
 

Sympy [F]

\[ \int \frac {\sinh ^3(a+b x)}{(c+d x)^{3/2}} \, dx=\int \frac {\sinh ^{3}{\left (a + b x \right )}}{\left (c + d x\right )^{\frac {3}{2}}}\, dx \] Input:

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 197, normalized size of antiderivative = 0.80 \[ \int \frac {\sinh ^3(a+b x)}{(c+d x)^{3/2}} \, dx=\frac {\frac {\sqrt {3} \sqrt {\frac {{\left (d x + c\right )} b}{d}} e^{\left (\frac {3 \, {\left (b c - a d\right )}}{d}\right )} \Gamma \left (-\frac {1}{2}, \frac {3 \, {\left (d x + c\right )} b}{d}\right )}{\sqrt {d x + c}} - \frac {\sqrt {3} \sqrt {-\frac {{\left (d x + c\right )} b}{d}} e^{\left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right )} \Gamma \left (-\frac {1}{2}, -\frac {3 \, {\left (d x + c\right )} b}{d}\right )}{\sqrt {d x + c}} - \frac {3 \, \sqrt {\frac {{\left (d x + c\right )} b}{d}} e^{\left (-a + \frac {b c}{d}\right )} \Gamma \left (-\frac {1}{2}, \frac {{\left (d x + c\right )} b}{d}\right )}{\sqrt {d x + c}} + \frac {3 \, \sqrt {-\frac {{\left (d x + c\right )} b}{d}} e^{\left (a - \frac {b c}{d}\right )} \Gamma \left (-\frac {1}{2}, -\frac {{\left (d x + c\right )} b}{d}\right )}{\sqrt {d x + c}}}{8 \, d} \] Input:

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

Output:

1/8*(sqrt(3)*sqrt((d*x + c)*b/d)*e^(3*(b*c - a*d)/d)*gamma(-1/2, 3*(d*x + 
c)*b/d)/sqrt(d*x + c) - sqrt(3)*sqrt(-(d*x + c)*b/d)*e^(-3*(b*c - a*d)/d)* 
gamma(-1/2, -3*(d*x + c)*b/d)/sqrt(d*x + c) - 3*sqrt((d*x + c)*b/d)*e^(-a 
+ b*c/d)*gamma(-1/2, (d*x + c)*b/d)/sqrt(d*x + c) + 3*sqrt(-(d*x + c)*b/d) 
*e^(a - b*c/d)*gamma(-1/2, -(d*x + c)*b/d)/sqrt(d*x + c))/d
 

Giac [F]

\[ \int \frac {\sinh ^3(a+b x)}{(c+d x)^{3/2}} \, dx=\int { \frac {\sinh \left (b x + a\right )^{3}}{{\left (d x + c\right )}^{\frac {3}{2}}} \,d x } \] Input:

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\sinh ^3(a+b x)}{(c+d x)^{3/2}} \, dx=\int \frac {{\mathrm {sinh}\left (a+b\,x\right )}^3}{{\left (c+d\,x\right )}^{3/2}} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {\sinh ^3(a+b x)}{(c+d x)^{3/2}} \, dx=\int \frac {\sinh \left (b x +a \right )^{3}}{\sqrt {d x +c}\, c +\sqrt {d x +c}\, d x}d x \] Input:

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

Output:

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