\(\int \frac {\sinh (a+b x)}{(c+d x)^3} \, dx\) [7]

Optimal result

Integrand size = 14, antiderivative size = 104 \[ \int \frac {\sinh (a+b x)}{(c+d x)^3} \, dx=-\frac {b \cosh (a+b x)}{2 d^2 (c+d x)}+\frac {b^2 \text {Chi}\left (\frac {b c}{d}+b x\right ) \sinh \left (a-\frac {b c}{d}\right )}{2 d^3}-\frac {\sinh (a+b x)}{2 d (c+d x)^2}+\frac {b^2 \cosh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (\frac {b c}{d}+b x\right )}{2 d^3} \] Output:

-1/2*b*cosh(b*x+a)/d^2/(d*x+c)+1/2*b^2*Chi(b*c/d+b*x)*sinh(a-b*c/d)/d^3-1/ 
2*sinh(b*x+a)/d/(d*x+c)^2+1/2*b^2*cosh(a-b*c/d)*Shi(b*c/d+b*x)/d^3
 

Mathematica [A] (verified)

Time = 0.32 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.85 \[ \int \frac {\sinh (a+b x)}{(c+d x)^3} \, dx=\frac {b^2 \text {Chi}\left (b \left (\frac {c}{d}+x\right )\right ) \sinh \left (a-\frac {b c}{d}\right )-\frac {d (b (c+d x) \cosh (a+b x)+d \sinh (a+b x))}{(c+d x)^2}+b^2 \cosh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (b \left (\frac {c}{d}+x\right )\right )}{2 d^3} \] Input:

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

Output:

(b^2*CoshIntegral[b*(c/d + x)]*Sinh[a - (b*c)/d] - (d*(b*(c + d*x)*Cosh[a 
+ b*x] + d*Sinh[a + b*x]))/(c + d*x)^2 + b^2*Cosh[a - (b*c)/d]*SinhIntegra 
l[b*(c/d + x)])/(2*d^3)
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.67 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.16, number of steps used = 14, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {3042, 26, 3778, 3042, 3778, 26, 3042, 26, 3784, 26, 3042, 26, 3779, 3782}

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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3778

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

\(\Big \downarrow \) 3042

\(\displaystyle -i \left (\frac {i b \int \frac {\sin \left (i a+i b x+\frac {\pi }{2}\right )}{(c+d x)^2}dx}{2 d}-\frac {i \sinh (a+b x)}{2 d (c+d x)^2}\right )\)

\(\Big \downarrow \) 3778

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3784

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3042

\(\displaystyle -i \left (\frac {i b \left (-\frac {\cosh (a+b x)}{d (c+d x)}-\frac {i b \left (i \sinh \left (a-\frac {b c}{d}\right ) \int \frac {\sin \left (\frac {i b c}{d}+i b x+\frac {\pi }{2}\right )}{c+d x}dx+i \cosh \left (a-\frac {b c}{d}\right ) \int -\frac {i \sin \left (\frac {i b c}{d}+i b x\right )}{c+d x}dx\right )}{d}\right )}{2 d}-\frac {i \sinh (a+b x)}{2 d (c+d x)^2}\right )\)

\(\Big \downarrow \) 26

\(\displaystyle -i \left (\frac {i b \left (-\frac {\cosh (a+b x)}{d (c+d x)}-\frac {i b \left (i \sinh \left (a-\frac {b c}{d}\right ) \int \frac {\sin \left (\frac {i b c}{d}+i b x+\frac {\pi }{2}\right )}{c+d x}dx+\cosh \left (a-\frac {b c}{d}\right ) \int \frac {\sin \left (\frac {i b c}{d}+i b x\right )}{c+d x}dx\right )}{d}\right )}{2 d}-\frac {i \sinh (a+b x)}{2 d (c+d x)^2}\right )\)

\(\Big \downarrow \) 3779

\(\displaystyle -i \left (\frac {i b \left (-\frac {\cosh (a+b x)}{d (c+d x)}-\frac {i b \left (i \sinh \left (a-\frac {b c}{d}\right ) \int \frac {\sin \left (\frac {i b c}{d}+i b x+\frac {\pi }{2}\right )}{c+d x}dx+\frac {i \cosh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (\frac {b c}{d}+b x\right )}{d}\right )}{d}\right )}{2 d}-\frac {i \sinh (a+b x)}{2 d (c+d x)^2}\right )\)

\(\Big \downarrow \) 3782

\(\displaystyle -i \left (\frac {i b \left (-\frac {\cosh (a+b x)}{d (c+d x)}-\frac {i b \left (\frac {i \sinh \left (a-\frac {b c}{d}\right ) \text {Chi}\left (\frac {b c}{d}+b x\right )}{d}+\frac {i \cosh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (\frac {b c}{d}+b x\right )}{d}\right )}{d}\right )}{2 d}-\frac {i \sinh (a+b x)}{2 d (c+d x)^2}\right )\)

Input:

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

Output:

(-I)*(((-1/2*I)*Sinh[a + b*x])/(d*(c + d*x)^2) + ((I/2)*b*(-(Cosh[a + b*x] 
/(d*(c + d*x))) - (I*b*((I*CoshIntegral[(b*c)/d + b*x]*Sinh[a - (b*c)/d])/ 
d + (I*Cosh[a - (b*c)/d]*SinhIntegral[(b*c)/d + b*x])/d))/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 3042

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

rule 3778

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

rule 3779

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbo 
l] :> Simp[I*(SinhIntegral[c*f*(fz/d) + f*fz*x]/d), x] /; FreeQ[{c, d, e, f 
, fz}, x] && EqQ[d*e - c*f*fz*I, 0]
 

rule 3782

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbo 
l] :> Simp[CoshIntegral[c*f*(fz/d) + f*fz*x]/d, x] /; FreeQ[{c, d, e, f, fz 
}, x] && EqQ[d*(e - Pi/2) - c*f*fz*I, 0]
 

rule 3784

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[Cos[(d* 
e - c*f)/d]   Int[Sin[c*(f/d) + f*x]/(c + d*x), x], x] + Simp[Sin[(d*e - c* 
f)/d]   Int[Cos[c*(f/d) + f*x]/(c + d*x), x], x] /; FreeQ[{c, d, e, f}, x] 
&& NeQ[d*e - c*f, 0]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(276\) vs. \(2(96)=192\).

Time = 0.24 (sec) , antiderivative size = 277, normalized size of antiderivative = 2.66

method	result	size
risch	\(-\frac {b^{3} {\mathrm e}^{-b x -a} x}{4 d \left (d^{2} x^{2} b^{2}+2 b^{2} c d x +b^{2} c^{2}\right )}-\frac {b^{3} {\mathrm e}^{-b x -a} c}{4 d^{2} \left (d^{2} x^{2} b^{2}+2 b^{2} c d x +b^{2} c^{2}\right )}+\frac {b^{2} {\mathrm e}^{-b x -a}}{4 d \left (d^{2} x^{2} b^{2}+2 b^{2} c d x +b^{2} c^{2}\right )}+\frac {b^{2} {\mathrm e}^{-\frac {d a -b c}{d}} \operatorname {expIntegral}_{1}\left (b x +a -\frac {d a -b c}{d}\right )}{4 d^{3}}-\frac {b^{2} {\mathrm e}^{b x +a}}{4 d^{3} \left (\frac {b c}{d}+b x \right )^{2}}-\frac {b^{2} {\mathrm e}^{b x +a}}{4 d^{3} \left (\frac {b c}{d}+b x \right )}-\frac {b^{2} {\mathrm e}^{\frac {d a -b c}{d}} \operatorname {expIntegral}_{1}\left (-b x -a -\frac {-d a +b c}{d}\right )}{4 d^{3}}\)	\(277\)

Input:

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

Output:

-1/4*b^3*exp(-b*x-a)/d/(b^2*d^2*x^2+2*b^2*c*d*x+b^2*c^2)*x-1/4*b^3*exp(-b* 
x-a)/d^2/(b^2*d^2*x^2+2*b^2*c*d*x+b^2*c^2)*c+1/4*b^2*exp(-b*x-a)/d/(b^2*d^ 
2*x^2+2*b^2*c*d*x+b^2*c^2)+1/4*b^2/d^3*exp(-(a*d-b*c)/d)*Ei(1,b*x+a-(a*d-b 
*c)/d)-1/4*b^2/d^3*exp(b*x+a)/(b*c/d+b*x)^2-1/4*b^2/d^3*exp(b*x+a)/(b*c/d+ 
b*x)-1/4*b^2/d^3*exp((a*d-b*c)/d)*Ei(1,-b*x-a-(-a*d+b*c)/d)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 254 vs. \(2 (96) = 192\).

Time = 0.07 (sec) , antiderivative size = 254, normalized size of antiderivative = 2.44 \[ \int \frac {\sinh (a+b x)}{(c+d x)^3} \, dx=-\frac {2 \, d^{2} \sinh \left (b x + a\right ) + 2 \, {\left (b d^{2} x + b c d\right )} \cosh \left (b x + a\right ) - {\left ({\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} {\rm Ei}\left (\frac {b d x + b c}{d}\right ) - {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} {\rm Ei}\left (-\frac {b d x + b c}{d}\right )\right )} \cosh \left (-\frac {b c - a d}{d}\right ) - {\left ({\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} {\rm Ei}\left (\frac {b d x + b c}{d}\right ) + {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} {\rm Ei}\left (-\frac {b d x + b c}{d}\right )\right )} \sinh \left (-\frac {b c - a d}{d}\right )}{4 \, {\left (d^{5} x^{2} + 2 \, c d^{4} x + c^{2} d^{3}\right )}} \] Input:

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

Output:

-1/4*(2*d^2*sinh(b*x + a) + 2*(b*d^2*x + b*c*d)*cosh(b*x + a) - ((b^2*d^2* 
x^2 + 2*b^2*c*d*x + b^2*c^2)*Ei((b*d*x + b*c)/d) - (b^2*d^2*x^2 + 2*b^2*c* 
d*x + b^2*c^2)*Ei(-(b*d*x + b*c)/d))*cosh(-(b*c - a*d)/d) - ((b^2*d^2*x^2 
+ 2*b^2*c*d*x + b^2*c^2)*Ei((b*d*x + b*c)/d) + (b^2*d^2*x^2 + 2*b^2*c*d*x 
+ b^2*c^2)*Ei(-(b*d*x + b*c)/d))*sinh(-(b*c - a*d)/d))/(d^5*x^2 + 2*c*d^4* 
x + c^2*d^3)
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.90 \[ \int \frac {\sinh (a+b x)}{(c+d x)^3} \, dx=-\frac {b {\left (\frac {e^{\left (-a + \frac {b c}{d}\right )} E_{2}\left (\frac {{\left (d x + c\right )} b}{d}\right )}{{\left (d x + c\right )} d} + \frac {e^{\left (a - \frac {b c}{d}\right )} E_{2}\left (-\frac {{\left (d x + c\right )} b}{d}\right )}{{\left (d x + c\right )} d}\right )}}{4 \, d} - \frac {\sinh \left (b x + a\right )}{2 \, {\left (d x + c\right )}^{2} d} \] Input:

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

Output:

-1/4*b*(e^(-a + b*c/d)*exp_integral_e(2, (d*x + c)*b/d)/((d*x + c)*d) + e^ 
(a - b*c/d)*exp_integral_e(2, -(d*x + c)*b/d)/((d*x + c)*d))/d - 1/2*sinh( 
b*x + a)/((d*x + c)^2*d)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 301 vs. \(2 (96) = 192\).

Time = 0.18 (sec) , antiderivative size = 301, normalized size of antiderivative = 2.89 \[ \int \frac {\sinh (a+b x)}{(c+d x)^3} \, dx=\frac {b^{2} d^{2} x^{2} {\rm Ei}\left (\frac {b d x + b c}{d}\right ) e^{\left (a - \frac {b c}{d}\right )} - b^{2} d^{2} x^{2} {\rm Ei}\left (-\frac {b d x + b c}{d}\right ) e^{\left (-a + \frac {b c}{d}\right )} + 2 \, b^{2} c d x {\rm Ei}\left (\frac {b d x + b c}{d}\right ) e^{\left (a - \frac {b c}{d}\right )} - 2 \, b^{2} c d x {\rm Ei}\left (-\frac {b d x + b c}{d}\right ) e^{\left (-a + \frac {b c}{d}\right )} + b^{2} c^{2} {\rm Ei}\left (\frac {b d x + b c}{d}\right ) e^{\left (a - \frac {b c}{d}\right )} - b^{2} c^{2} {\rm Ei}\left (-\frac {b d x + b c}{d}\right ) e^{\left (-a + \frac {b c}{d}\right )} - b d^{2} x e^{\left (b x + a\right )} - b d^{2} x e^{\left (-b x - a\right )} - b c d e^{\left (b x + a\right )} - b c d e^{\left (-b x - a\right )} - d^{2} e^{\left (b x + a\right )} + d^{2} e^{\left (-b x - a\right )}}{4 \, {\left (d^{5} x^{2} + 2 \, c d^{4} x + c^{2} d^{3}\right )}} \] Input:

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

Output:

1/4*(b^2*d^2*x^2*Ei((b*d*x + b*c)/d)*e^(a - b*c/d) - b^2*d^2*x^2*Ei(-(b*d* 
x + b*c)/d)*e^(-a + b*c/d) + 2*b^2*c*d*x*Ei((b*d*x + b*c)/d)*e^(a - b*c/d) 
 - 2*b^2*c*d*x*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) + b^2*c^2*Ei((b*d*x + b 
*c)/d)*e^(a - b*c/d) - b^2*c^2*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) - b*d^2 
*x*e^(b*x + a) - b*d^2*x*e^(-b*x - a) - b*c*d*e^(b*x + a) - b*c*d*e^(-b*x 
- a) - d^2*e^(b*x + a) + d^2*e^(-b*x - a))/(d^5*x^2 + 2*c*d^4*x + c^2*d^3)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\sinh (a+b x)}{(c+d x)^3} \, dx=\text {too large to display} \] Input:

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

Output:

(2*e**(a + b*x)*cosh(a + b*x)*b**3*c**3 + 2*e**(a + b*x)*cosh(a + b*x)*b** 
3*c**2*d*x - 8*e**(a + b*x)*cosh(a + b*x)*b*c*d**2 - 8*e**(a + b*x)*cosh(a 
 + b*x)*b*d**3*x - e**(2*a + 2*b*x)*b**3*c**2*d*x + e**(2*a + 2*b*x)*b**2* 
c**2*d + 2*e**(2*a + 2*b*x)*b*c*d**2 + 4*e**(2*a + 2*b*x)*b*d**3*x - e**(2 
*a + b*x)*int((e**(b*x)*x)/(b**2*c**5 + 3*b**2*c**4*d*x + 3*b**2*c**3*d**2 
*x**2 + b**2*c**2*d**3*x**3 - 4*c**3*d**2 - 12*c**2*d**3*x - 12*c*d**4*x** 
2 - 4*d**5*x**3),x)*b**6*c**7*d - 2*e**(2*a + b*x)*int((e**(b*x)*x)/(b**2* 
c**5 + 3*b**2*c**4*d*x + 3*b**2*c**3*d**2*x**2 + b**2*c**2*d**3*x**3 - 4*c 
**3*d**2 - 12*c**2*d**3*x - 12*c*d**4*x**2 - 4*d**5*x**3),x)*b**6*c**6*d** 
2*x - e**(2*a + b*x)*int((e**(b*x)*x)/(b**2*c**5 + 3*b**2*c**4*d*x + 3*b** 
2*c**3*d**2*x**2 + b**2*c**2*d**3*x**3 - 4*c**3*d**2 - 12*c**2*d**3*x - 12 
*c*d**4*x**2 - 4*d**5*x**3),x)*b**6*c**5*d**3*x**2 - 2*e**(2*a + b*x)*int( 
(e**(b*x)*x)/(b**2*c**5 + 3*b**2*c**4*d*x + 3*b**2*c**3*d**2*x**2 + b**2*c 
**2*d**3*x**3 - 4*c**3*d**2 - 12*c**2*d**3*x - 12*c*d**4*x**2 - 4*d**5*x** 
3),x)*b**5*c**6*d**2 - 4*e**(2*a + b*x)*int((e**(b*x)*x)/(b**2*c**5 + 3*b* 
*2*c**4*d*x + 3*b**2*c**3*d**2*x**2 + b**2*c**2*d**3*x**3 - 4*c**3*d**2 - 
12*c**2*d**3*x - 12*c*d**4*x**2 - 4*d**5*x**3),x)*b**5*c**5*d**3*x - 2*e** 
(2*a + b*x)*int((e**(b*x)*x)/(b**2*c**5 + 3*b**2*c**4*d*x + 3*b**2*c**3*d* 
*2*x**2 + b**2*c**2*d**3*x**3 - 4*c**3*d**2 - 12*c**2*d**3*x - 12*c*d**4*x 
**2 - 4*d**5*x**3),x)*b**5*c**4*d**4*x**2 + 6*e**(2*a + b*x)*int((e**(b...