\(\int \sinh ^3(\frac {a}{c+d x}) \, dx\) [291]

Optimal result

Integrand size = 12, antiderivative size = 59 \[ \int \sinh ^3\left (\frac {a}{c+d x}\right ) \, dx=\frac {3 a \text {Chi}\left (\frac {a}{c+d x}\right )}{4 d}-\frac {3 a \text {Chi}\left (\frac {3 a}{c+d x}\right )}{4 d}+\frac {(c+d x) \sinh ^3\left (\frac {a}{c+d x}\right )}{d} \] Output:

3/4*a*Chi(a/(d*x+c))/d-3/4*a*Chi(3*a/(d*x+c))/d+(d*x+c)*sinh(a/(d*x+c))^3/ 
d
 

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.92 \[ \int \sinh ^3\left (\frac {a}{c+d x}\right ) \, dx=\frac {3 a \text {Chi}\left (\frac {a}{c+d x}\right )-3 a \text {Chi}\left (\frac {3 a}{c+d x}\right )+4 (c+d x) \sinh ^3\left (\frac {a}{c+d x}\right )}{4 d} \] Input:

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

Output:

(3*a*CoshIntegral[a/(c + d*x)] - 3*a*CoshIntegral[(3*a)/(c + d*x)] + 4*(c 
+ d*x)*Sinh[a/(c + d*x)]^3)/(4*d)
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.40 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.08, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5833, 5825, 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.

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

\(\Big \downarrow \) 5833

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

\(\Big \downarrow \) 5825

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3794

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

((-I)*((3*I)*a*(CoshIntegral[a/(c + d*x)]/4 - CoshIntegral[(3*a)/(c + d*x) 
]/4) + I*(c + d*x)*Sinh[a/(c + d*x)]^3))/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 5825

Int[((a_.) + (b_.)*Sinh[(c_.) + (d_.)*(x_)^(n_)])^(p_.), x_Symbol] :> -Subs 
t[Int[(a + b*Sinh[c + d/x^n])^p/x^2, x], x, 1/x] /; FreeQ[{a, b, c, d}, x] 
&& ILtQ[n, 0] && IntegerQ[p]
 

rule 5833

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

Maple [A] (verified)

Time = 0.52 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.25

method	result	size
derivativedivides	\(-\frac {a \left (\frac {3 \left (d x +c \right ) \sinh \left (\frac {a}{d x +c}\right )}{4 a}-\frac {3 \,\operatorname {Chi}\left (\frac {a}{d x +c}\right )}{4}-\frac {\left (d x +c \right ) \sinh \left (\frac {3 a}{d x +c}\right )}{4 a}+\frac {3 \,\operatorname {Chi}\left (\frac {3 a}{d x +c}\right )}{4}\right )}{d}\)	\(74\)
default	\(-\frac {a \left (\frac {3 \left (d x +c \right ) \sinh \left (\frac {a}{d x +c}\right )}{4 a}-\frac {3 \,\operatorname {Chi}\left (\frac {a}{d x +c}\right )}{4}-\frac {\left (d x +c \right ) \sinh \left (\frac {3 a}{d x +c}\right )}{4 a}+\frac {3 \,\operatorname {Chi}\left (\frac {3 a}{d x +c}\right )}{4}\right )}{d}\)	\(74\)
risch	\(-\frac {{\mathrm e}^{-\frac {3 a}{d x +c}} x}{8}-\frac {{\mathrm e}^{-\frac {3 a}{d x +c}} c}{8 d}+\frac {3 a \,\operatorname {expIntegral}_{1}\left (\frac {3 a}{d x +c}\right )}{8 d}+\frac {3 \,{\mathrm e}^{-\frac {a}{d x +c}} x}{8}+\frac {3 \,{\mathrm e}^{-\frac {a}{d x +c}} c}{8 d}-\frac {3 a \,\operatorname {expIntegral}_{1}\left (\frac {a}{d x +c}\right )}{8 d}+\frac {{\mathrm e}^{\frac {3 a}{d x +c}} x}{8}+\frac {{\mathrm e}^{\frac {3 a}{d x +c}} c}{8 d}+\frac {3 a \,\operatorname {expIntegral}_{1}\left (-\frac {3 a}{d x +c}\right )}{8 d}-\frac {3 \,{\mathrm e}^{\frac {a}{d x +c}} x}{8}-\frac {3 \,{\mathrm e}^{\frac {a}{d x +c}} c}{8 d}-\frac {3 a \,\operatorname {expIntegral}_{1}\left (-\frac {a}{d x +c}\right )}{8 d}\)	\(201\)

Input:

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

Output:

-1/d*a*(3/4/a*(d*x+c)*sinh(a/(d*x+c))-3/4*Chi(a/(d*x+c))-1/4/a*(d*x+c)*sin 
h(3*a/(d*x+c))+3/4*Chi(3*a/(d*x+c)))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 118 vs. \(2 (55) = 110\).

Time = 0.10 (sec) , antiderivative size = 118, normalized size of antiderivative = 2.00 \[ \int \sinh ^3\left (\frac {a}{c+d x}\right ) \, dx=\frac {2 \, {\left (d x + c\right )} \sinh \left (\frac {a}{d x + c}\right )^{3} - 3 \, a {\rm Ei}\left (\frac {3 \, a}{d x + c}\right ) + 3 \, a {\rm Ei}\left (\frac {a}{d x + c}\right ) + 3 \, a {\rm Ei}\left (-\frac {a}{d x + c}\right ) - 3 \, a {\rm Ei}\left (-\frac {3 \, a}{d x + c}\right ) + 6 \, {\left ({\left (d x + c\right )} \cosh \left (\frac {a}{d x + c}\right )^{2} - d x - c\right )} \sinh \left (\frac {a}{d x + c}\right )}{8 \, d} \] Input:

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

Output:

1/8*(2*(d*x + c)*sinh(a/(d*x + c))^3 - 3*a*Ei(3*a/(d*x + c)) + 3*a*Ei(a/(d 
*x + c)) + 3*a*Ei(-a/(d*x + c)) - 3*a*Ei(-3*a/(d*x + c)) + 6*((d*x + c)*co 
sh(a/(d*x + c))^2 - d*x - c)*sinh(a/(d*x + c)))/d
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

3/8*a*d*integrate(x*e^(3*a/(d*x + c))/(d^2*x^2 + 2*c*d*x + c^2), x) - 3/8* 
a*d*integrate(x*e^(a/(d*x + c))/(d^2*x^2 + 2*c*d*x + c^2), x) - 3/8*a*d*in 
tegrate(x*e^(-a/(d*x + c))/(d^2*x^2 + 2*c*d*x + c^2), x) + 3/8*a*d*integra 
te(x*e^(-3*a/(d*x + c))/(d^2*x^2 + 2*c*d*x + c^2), x) + 1/8*x*e^(3*a/(d*x 
+ c)) - 3/8*x*e^(a/(d*x + c)) + 3/8*x*e^(-a/(d*x + c)) - 1/8*x*e^(-3*a/(d* 
x + c))
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 211 vs. \(2 (55) = 110\).

Time = 0.17 (sec) , antiderivative size = 211, normalized size of antiderivative = 3.58 \[ \int \sinh ^3\left (\frac {a}{c+d x}\right ) \, dx=-\frac {{\left (\frac {3 \, a^{3} {\rm Ei}\left (\frac {3 \, a}{d x + c}\right ) e^{\left (\frac {3 \, a}{d x + c}\right )}}{d x + c} - \frac {3 \, a^{3} {\rm Ei}\left (\frac {a}{d x + c}\right ) e^{\left (\frac {3 \, a}{d x + c}\right )}}{d x + c} - \frac {3 \, a^{3} {\rm Ei}\left (-\frac {a}{d x + c}\right ) e^{\left (\frac {3 \, a}{d x + c}\right )}}{d x + c} + \frac {3 \, a^{3} {\rm Ei}\left (-\frac {3 \, a}{d x + c}\right ) e^{\left (\frac {3 \, a}{d x + c}\right )}}{d x + c} - a^{2} e^{\left (\frac {6 \, a}{d x + c}\right )} + 3 \, a^{2} e^{\left (\frac {4 \, a}{d x + c}\right )} - 3 \, a^{2} e^{\left (\frac {2 \, a}{d x + c}\right )} + a^{2}\right )} {\left (d x + c\right )} e^{\left (-\frac {3 \, a}{d x + c}\right )}}{8 \, a^{2} d} \] Input:

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

Output:

-1/8*(3*a^3*Ei(3*a/(d*x + c))*e^(3*a/(d*x + c))/(d*x + c) - 3*a^3*Ei(a/(d* 
x + c))*e^(3*a/(d*x + c))/(d*x + c) - 3*a^3*Ei(-a/(d*x + c))*e^(3*a/(d*x + 
 c))/(d*x + c) + 3*a^3*Ei(-3*a/(d*x + c))*e^(3*a/(d*x + c))/(d*x + c) - a^ 
2*e^(6*a/(d*x + c)) + 3*a^2*e^(4*a/(d*x + c)) - 3*a^2*e^(2*a/(d*x + c)) + 
a^2)*(d*x + c)*e^(-3*a/(d*x + c))/(a^2*d)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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