\(\int \sinh ^3(\frac {b x}{c+d x}) \, dx\) [294]

Optimal result

Integrand size = 13, antiderivative size = 143 \[ \int \sinh ^3\left (\frac {b x}{c+d x}\right ) \, dx=-\frac {3 b c \cosh \left (\frac {b}{d}\right ) \text {Chi}\left (\frac {b c}{d (c+d x)}\right )}{4 d^2}+\frac {3 b c \cosh \left (\frac {3 b}{d}\right ) \text {Chi}\left (\frac {3 b c}{d (c+d x)}\right )}{4 d^2}+\frac {(c+d x) \sinh ^3\left (\frac {b x}{c+d x}\right )}{d}+\frac {3 b c \sinh \left (\frac {b}{d}\right ) \text {Shi}\left (\frac {b c}{d (c+d x)}\right )}{4 d^2}-\frac {3 b c \sinh \left (\frac {3 b}{d}\right ) \text {Shi}\left (\frac {3 b c}{d (c+d x)}\right )}{4 d^2} \] Output:

-3/4*b*c*cosh(b/d)*Chi(b*c/d/(d*x+c))/d^2+3/4*b*c*cosh(3*b/d)*Chi(3*b*c/d/ 
(d*x+c))/d^2+(d*x+c)*sinh(b*x/(d*x+c))^3/d+3/4*b*c*sinh(b/d)*Shi(b*c/d/(d* 
x+c))/d^2-3/4*b*c*sinh(3*b/d)*Shi(3*b*c/d/(d*x+c))/d^2
 

Mathematica [A] (verified)

Time = 0.40 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.62 \[ \int \sinh ^3\left (\frac {b x}{c+d x}\right ) \, dx=\frac {-c d e^{-\frac {3 b x}{c+d x}}+3 c d e^{-\frac {b x}{c+d x}}-3 c d e^{\frac {b x}{c+d x}}+c d e^{\frac {3 b x}{c+d x}}-d^2 e^{-\frac {3 b x}{c+d x}} x+d^2 e^{\frac {3 b x}{c+d x}} x-6 b c \cosh \left (\frac {b}{d}\right ) \text {Chi}\left (\frac {b c}{c d+d^2 x}\right )+6 b c \cosh \left (\frac {3 b}{d}\right ) \text {Chi}\left (\frac {3 b c}{c d+d^2 x}\right )-6 d^2 x \sinh \left (\frac {b x}{c+d x}\right )+6 b c \sinh \left (\frac {b}{d}\right ) \text {Shi}\left (\frac {b c}{c d+d^2 x}\right )-6 b c \sinh \left (\frac {3 b}{d}\right ) \text {Shi}\left (\frac {3 b c}{c d+d^2 x}\right )}{8 d^2} \] Input:

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

Output:

(-((c*d)/E^((3*b*x)/(c + d*x))) + (3*c*d)/E^((b*x)/(c + d*x)) - 3*c*d*E^(( 
b*x)/(c + d*x)) + c*d*E^((3*b*x)/(c + d*x)) - (d^2*x)/E^((3*b*x)/(c + d*x) 
) + d^2*E^((3*b*x)/(c + d*x))*x - 6*b*c*Cosh[b/d]*CoshIntegral[(b*c)/(c*d 
+ d^2*x)] + 6*b*c*Cosh[(3*b)/d]*CoshIntegral[(3*b*c)/(c*d + d^2*x)] - 6*d^ 
2*x*Sinh[(b*x)/(c + d*x)] + 6*b*c*Sinh[b/d]*SinhIntegral[(b*c)/(c*d + d^2* 
x)] - 6*b*c*Sinh[(3*b)/d]*SinhIntegral[(3*b*c)/(c*d + d^2*x)])/(8*d^2)
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.48 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.05, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {6141, 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.

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

\(\Big \downarrow \) 6141

\(\displaystyle -\frac {\int (c+d x)^2 \sinh ^3\left (\frac {b}{d}-\frac {b c}{d (c+d x)}\right )d\frac {1}{c+d x}}{d}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\int i (c+d x)^2 \sin \left (\frac {i b}{d}-\frac {i b c}{d (c+d x)}\right )^3d\frac {1}{c+d x}}{d}\)

\(\Big \downarrow \) 26

\(\displaystyle -\frac {i \int (c+d x)^2 \sin \left (\frac {i b}{d}-\frac {i b c}{d (c+d x)}\right )^3d\frac {1}{c+d x}}{d}\)

\(\Big \downarrow \) 3794

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {i \left (i (c+d x) \sinh ^3\left (\frac {b}{d}-\frac {b c}{d (c+d x)}\right )-\frac {3 i b c \left (\frac {1}{4} \cosh \left (\frac {b}{d}\right ) \text {Chi}\left (\frac {b c}{d (c+d x)}\right )-\frac {1}{4} \cosh \left (\frac {3 b}{d}\right ) \text {Chi}\left (\frac {3 b c}{d (c+d x)}\right )-\frac {1}{4} \sinh \left (\frac {b}{d}\right ) \text {Shi}\left (\frac {b c}{d (c+d x)}\right )+\frac {1}{4} \sinh \left (\frac {3 b}{d}\right ) \text {Shi}\left (\frac {3 b c}{d (c+d x)}\right )\right )}{d}\right )}{d}\)

Input:

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

Output:

((-I)*(I*(c + d*x)*Sinh[b/d - (b*c)/(d*(c + d*x))]^3 - ((3*I)*b*c*((Cosh[b 
/d]*CoshIntegral[(b*c)/(d*(c + d*x))])/4 - (Cosh[(3*b)/d]*CoshIntegral[(3* 
b*c)/(d*(c + d*x))])/4 - (Sinh[b/d]*SinhIntegral[(b*c)/(d*(c + d*x))])/4 + 
 (Sinh[(3*b)/d]*SinhIntegral[(3*b*c)/(d*(c + d*x))])/4))/d))/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 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]
 

rule 6141

Int[Sinh[((e_.)*((a_.) + (b_.)*(x_)))/((c_.) + (d_.)*(x_))]^(n_.), x_Symbol 
] :> Simp[-d^(-1)   Subst[Int[Sinh[b*(e/d) - e*(b*c - a*d)*(x/d)]^n/x^2, x] 
, x, 1/(c + d*x)], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[n, 0] && NeQ[b*c - 
a*d, 0]
 

Maple [A] (verified)

Time = 1.84 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.75

method	result	size
risch	\(\frac {3 b c \,{\mathrm e}^{\frac {b}{d}} \operatorname {expIntegral}_{1}\left (\frac {b c}{d \left (d x +c \right )}\right )}{8 d^{2}}+\frac {3 \,{\mathrm e}^{-\frac {b x}{d x +c}} x}{8}+\frac {3 b c \,{\mathrm e}^{-\frac {b}{d}} \operatorname {expIntegral}_{1}\left (-\frac {b c}{d \left (d x +c \right )}\right )}{8 d^{2}}-\frac {3 \,{\mathrm e}^{\frac {b x}{d x +c}} x}{8}+\frac {{\mathrm e}^{\frac {3 b x}{d x +c}} x}{8}-\frac {3 \,{\mathrm e}^{\frac {3 b}{d}} \operatorname {expIntegral}_{1}\left (\frac {3 b c}{d \left (d x +c \right )}\right ) b c}{8 d^{2}}-\frac {{\mathrm e}^{-\frac {3 b x}{d x +c}} x}{8}-\frac {3 \,{\mathrm e}^{-\frac {3 b}{d}} \operatorname {expIntegral}_{1}\left (-\frac {3 b c}{d \left (d x +c \right )}\right ) b c}{8 d^{2}}+\frac {3 \,{\mathrm e}^{-\frac {b x}{d x +c}} c}{8 d}-\frac {3 c \,{\mathrm e}^{\frac {b x}{d x +c}}}{8 d}+\frac {{\mathrm e}^{\frac {3 b x}{d x +c}} c}{8 d}-\frac {{\mathrm e}^{-\frac {3 b x}{d x +c}} c}{8 d}\)	\(250\)

Input:

int(sinh(b*x/(d*x+c))^3,x,method=_RETURNVERBOSE)
 

Output:

3/8*b*c/d^2*exp(b/d)*Ei(1,b*c/d/(d*x+c))+3/8*exp(-b*x/(d*x+c))*x+3/8*b*c/d 
^2*exp(-b/d)*Ei(1,-b*c/d/(d*x+c))-3/8*exp(b*x/(d*x+c))*x+1/8*exp(3*b*x/(d* 
x+c))*x-3/8/d^2*exp(3*b/d)*Ei(1,3*b*c/d/(d*x+c))*b*c-1/8*exp(-3*b*x/(d*x+c 
))*x-3/8/d^2*exp(-3*b/d)*Ei(1,-3*b*c/d/(d*x+c))*b*c+3/8/d*exp(-b*x/(d*x+c) 
)*c-3/8*c/d*exp(b*x/(d*x+c))+1/8/d*exp(3*b*x/(d*x+c))*c-1/8/d*exp(-3*b*x/( 
d*x+c))*c
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 701 vs. \(2 (135) = 270\).

Time = 0.10 (sec) , antiderivative size = 701, normalized size of antiderivative = 4.90 \[ \int \sinh ^3\left (\frac {b x}{c+d x}\right ) \, dx =\text {Too large to display} \] Input:

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

Output:

1/8*(3*(b*c*Ei(-3*b*c/(d^2*x + c*d))*cosh(3*b/d) - b*c*Ei(-b*c/(d^2*x + c* 
d))*cosh(b/d))*sinh(b*x/(d*x + c))^4 + 2*(d^2*x + c*d)*sinh(b*x/(d*x + c)) 
^3 - 6*(b*c*Ei(-3*b*c/(d^2*x + c*d))*cosh(b*x/(d*x + c))^2*cosh(3*b/d) - b 
*c*Ei(-b*c/(d^2*x + c*d))*cosh(b*x/(d*x + c))^2*cosh(b/d))*sinh(b*x/(d*x + 
 c))^2 + 3*(b*c*Ei(-3*b*c/(d^2*x + c*d))*cosh(b*x/(d*x + c))^4 + b*c*Ei(3* 
b*c/(d^2*x + c*d)))*cosh(3*b/d) - 3*(b*c*Ei(-b*c/(d^2*x + c*d))*cosh(b*x/( 
d*x + c))^4 + b*c*Ei(b*c/(d^2*x + c*d)))*cosh(b/d) - 6*(d^2*x - (d^2*x + c 
*d)*cosh(b*x/(d*x + c))^2 + c*d)*sinh(b*x/(d*x + c)) + 3*(b*c*Ei(-3*b*c/(d 
^2*x + c*d))*cosh(b*x/(d*x + c))^4 - 2*b*c*Ei(-3*b*c/(d^2*x + c*d))*cosh(b 
*x/(d*x + c))^2*sinh(b*x/(d*x + c))^2 + b*c*Ei(-3*b*c/(d^2*x + c*d))*sinh( 
b*x/(d*x + c))^4 - b*c*Ei(3*b*c/(d^2*x + c*d)))*sinh(3*b/d) - 3*(b*c*Ei(-b 
*c/(d^2*x + c*d))*cosh(b*x/(d*x + c))^4 - 2*b*c*Ei(-b*c/(d^2*x + c*d))*cos 
h(b*x/(d*x + c))^2*sinh(b*x/(d*x + c))^2 + b*c*Ei(-b*c/(d^2*x + c*d))*sinh 
(b*x/(d*x + c))^4 - b*c*Ei(b*c/(d^2*x + c*d)))*sinh(b/d))/(d^2*cosh(b*x/(d 
*x + c))^4 - 2*d^2*cosh(b*x/(d*x + c))^2*sinh(b*x/(d*x + c))^2 + d^2*sinh( 
b*x/(d*x + c))^4)
                                                                                    
                                                                                    
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

-3/8*b*c*integrate(x*e^(3*b*c/(d^2*x + c*d))/(d^2*x^2*e^(3*b/d) + 2*c*d*x* 
e^(3*b/d) + c^2*e^(3*b/d)), x) + 3/8*b*c*integrate(x*e^(b*c/(d^2*x + c*d)) 
/(d^2*x^2*e^(b/d) + 2*c*d*x*e^(b/d) + c^2*e^(b/d)), x) + 3/8*b*c*integrate 
(x*e^(-b*c/(d^2*x + c*d) + b/d)/(d^2*x^2 + 2*c*d*x + c^2), x) - 3/8*b*c*in 
tegrate(x*e^(-3*b*c/(d^2*x + c*d) + 3*b/d)/(d^2*x^2 + 2*c*d*x + c^2), x) - 
 1/8*(x*e^(3*b*c/(d^2*x + c*d)) - 3*x*e^(b*c/(d^2*x + c*d) + 2*b/d) + 3*x* 
e^(-b*c/(d^2*x + c*d) + 4*b/d) - x*e^(-3*b*c/(d^2*x + c*d) + 6*b/d))*e^(-3 
*b/d)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \sinh ^3\left (\frac {b x}{c+d x}\right ) \, dx=\frac {3 e^{\frac {6 b x}{d x +c}} b d \,x^{2}+e^{\frac {6 b x}{d x +c}} c^{2}+e^{\frac {6 b x}{d x +c}} c d x -9 e^{\frac {4 b x}{d x +c}} b d \,x^{2}-9 e^{\frac {4 b x}{d x +c}} c^{2}-9 e^{\frac {4 b x}{d x +c}} c d x +9 e^{\frac {3 b x}{d x +c}} \left (\int \frac {x^{2}}{e^{\frac {b x}{d x +c}} c^{3}+3 e^{\frac {b x}{d x +c}} c^{2} d x +3 e^{\frac {b x}{d x +c}} c \,d^{2} x^{2}+e^{\frac {b x}{d x +c}} d^{3} x^{3}}d x \right ) b^{2} c^{2} d +9 e^{\frac {3 b x}{d x +c}} \left (\int \frac {x^{2}}{e^{\frac {b x}{d x +c}} c^{3}+3 e^{\frac {b x}{d x +c}} c^{2} d x +3 e^{\frac {b x}{d x +c}} c \,d^{2} x^{2}+e^{\frac {b x}{d x +c}} d^{3} x^{3}}d x \right ) b^{2} c \,d^{2} x -9 e^{\frac {3 b x}{d x +c}} \left (\int \frac {x^{2}}{e^{\frac {3 b x}{d x +c}} c^{3}+3 e^{\frac {3 b x}{d x +c}} c^{2} d x +3 e^{\frac {3 b x}{d x +c}} c \,d^{2} x^{2}+e^{\frac {3 b x}{d x +c}} d^{3} x^{3}}d x \right ) b^{2} c^{2} d -9 e^{\frac {3 b x}{d x +c}} \left (\int \frac {x^{2}}{e^{\frac {3 b x}{d x +c}} c^{3}+3 e^{\frac {3 b x}{d x +c}} c^{2} d x +3 e^{\frac {3 b x}{d x +c}} c \,d^{2} x^{2}+e^{\frac {3 b x}{d x +c}} d^{3} x^{3}}d x \right ) b^{2} c \,d^{2} x +9 e^{\frac {3 b x}{d x +c}} \left (\int \frac {e^{\frac {b x}{d x +c}} x^{2}}{d^{3} x^{3}+3 c \,d^{2} x^{2}+3 c^{2} d x +c^{3}}d x \right ) b^{2} c^{2} d +9 e^{\frac {3 b x}{d x +c}} \left (\int \frac {e^{\frac {b x}{d x +c}} x^{2}}{d^{3} x^{3}+3 c \,d^{2} x^{2}+3 c^{2} d x +c^{3}}d x \right ) b^{2} c \,d^{2} x -9 e^{\frac {3 b x}{d x +c}} \left (\int \frac {e^{\frac {3 b x}{d x +c}} x^{2}}{d^{3} x^{3}+3 c \,d^{2} x^{2}+3 c^{2} d x +c^{3}}d x \right ) b^{2} c^{2} d -9 e^{\frac {3 b x}{d x +c}} \left (\int \frac {e^{\frac {3 b x}{d x +c}} x^{2}}{d^{3} x^{3}+3 c \,d^{2} x^{2}+3 c^{2} d x +c^{3}}d x \right ) b^{2} c \,d^{2} x +9 e^{\frac {2 b x}{d x +c}} b d \,x^{2}-9 e^{\frac {2 b x}{d x +c}} c^{2}-9 e^{\frac {2 b x}{d x +c}} c d x -3 b d \,x^{2}+c^{2}+c d x}{24 e^{\frac {3 b x}{d x +c}} b \left (d x +c \right )} \] Input:

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

Output:

(3*e**((6*b*x)/(c + d*x))*b*d*x**2 + e**((6*b*x)/(c + d*x))*c**2 + e**((6* 
b*x)/(c + d*x))*c*d*x - 9*e**((4*b*x)/(c + d*x))*b*d*x**2 - 9*e**((4*b*x)/ 
(c + d*x))*c**2 - 9*e**((4*b*x)/(c + d*x))*c*d*x + 9*e**((3*b*x)/(c + d*x) 
)*int(x**2/(e**((b*x)/(c + d*x))*c**3 + 3*e**((b*x)/(c + d*x))*c**2*d*x + 
3*e**((b*x)/(c + d*x))*c*d**2*x**2 + e**((b*x)/(c + d*x))*d**3*x**3),x)*b* 
*2*c**2*d + 9*e**((3*b*x)/(c + d*x))*int(x**2/(e**((b*x)/(c + d*x))*c**3 + 
 3*e**((b*x)/(c + d*x))*c**2*d*x + 3*e**((b*x)/(c + d*x))*c*d**2*x**2 + e* 
*((b*x)/(c + d*x))*d**3*x**3),x)*b**2*c*d**2*x - 9*e**((3*b*x)/(c + d*x))* 
int(x**2/(e**((3*b*x)/(c + d*x))*c**3 + 3*e**((3*b*x)/(c + d*x))*c**2*d*x 
+ 3*e**((3*b*x)/(c + d*x))*c*d**2*x**2 + e**((3*b*x)/(c + d*x))*d**3*x**3) 
,x)*b**2*c**2*d - 9*e**((3*b*x)/(c + d*x))*int(x**2/(e**((3*b*x)/(c + d*x) 
)*c**3 + 3*e**((3*b*x)/(c + d*x))*c**2*d*x + 3*e**((3*b*x)/(c + d*x))*c*d* 
*2*x**2 + e**((3*b*x)/(c + d*x))*d**3*x**3),x)*b**2*c*d**2*x + 9*e**((3*b* 
x)/(c + d*x))*int((e**((b*x)/(c + d*x))*x**2)/(c**3 + 3*c**2*d*x + 3*c*d** 
2*x**2 + d**3*x**3),x)*b**2*c**2*d + 9*e**((3*b*x)/(c + d*x))*int((e**((b* 
x)/(c + d*x))*x**2)/(c**3 + 3*c**2*d*x + 3*c*d**2*x**2 + d**3*x**3),x)*b** 
2*c*d**2*x - 9*e**((3*b*x)/(c + d*x))*int((e**((3*b*x)/(c + d*x))*x**2)/(c 
**3 + 3*c**2*d*x + 3*c*d**2*x**2 + d**3*x**3),x)*b**2*c**2*d - 9*e**((3*b* 
x)/(c + d*x))*int((e**((3*b*x)/(c + d*x))*x**2)/(c**3 + 3*c**2*d*x + 3*c*d 
**2*x**2 + d**3*x**3),x)*b**2*c*d**2*x + 9*e**((2*b*x)/(c + d*x))*b*d*x...