\(\int \frac {\sinh (a+b x)}{(c+d x)^2} \, dx\) [6]

Optimal result

Integrand size = 14, antiderivative size = 71 \[ \int \frac {\sinh (a+b x)}{(c+d x)^2} \, dx=\frac {b \cosh \left (a-\frac {b c}{d}\right ) \text {Chi}\left (\frac {b c}{d}+b x\right )}{d^2}-\frac {\sinh (a+b x)}{d (c+d x)}+\frac {b \sinh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (\frac {b c}{d}+b x\right )}{d^2} \] Output:

b*cosh(a-b*c/d)*Chi(b*c/d+b*x)/d^2-sinh(b*x+a)/d/(d*x+c)+b*sinh(a-b*c/d)*S 
hi(b*c/d+b*x)/d^2
 

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.92 \[ \int \frac {\sinh (a+b x)}{(c+d x)^2} \, dx=\frac {b \cosh \left (a-\frac {b c}{d}\right ) \text {Chi}\left (b \left (\frac {c}{d}+x\right )\right )-\frac {d \sinh (a+b x)}{c+d x}+b \sinh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (b \left (\frac {c}{d}+x\right )\right )}{d^2} \] Input:

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

Output:

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

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.53 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.18, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.714, Rules used = {3042, 26, 3778, 3042, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3778

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3784

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3779

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

\(\Big \downarrow \) 3782

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

Input:

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

Output:

(-I)*(((-I)*Sinh[a + b*x])/(d*(c + d*x)) + (I*b*((Cosh[a - (b*c)/d]*CoshIn 
tegral[(b*c)/d + b*x])/d + (Sinh[a - (b*c)/d]*SinhIntegral[(b*c)/d + b*x]) 
/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 [A] (verified)

Time = 0.18 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.87

method	result	size
risch	\(\frac {b \,{\mathrm e}^{-b x -a}}{2 d \left (d x b +b c \right )}-\frac {b \,{\mathrm e}^{-\frac {d a -b c}{d}} \operatorname {expIntegral}_{1}\left (b x +a -\frac {d a -b c}{d}\right )}{2 d^{2}}-\frac {b \,{\mathrm e}^{b x +a}}{2 d^{2} \left (\frac {b c}{d}+b x \right )}-\frac {b \,{\mathrm e}^{\frac {d a -b c}{d}} \operatorname {expIntegral}_{1}\left (-b x -a -\frac {-d a +b c}{d}\right )}{2 d^{2}}\)	\(133\)

Input:

int(sinh(b*x+a)/(d*x+c)^2,x,method=_RETURNVERBOSE)
 

Output:

1/2*b*exp(-b*x-a)/d/(b*d*x+b*c)-1/2*b/d^2*exp(-(a*d-b*c)/d)*Ei(1,b*x+a-(a* 
d-b*c)/d)-1/2*b/d^2*exp(b*x+a)/(b*c/d+b*x)-1/2*b/d^2*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. 148 vs. \(2 (71) = 142\).

Time = 0.09 (sec) , antiderivative size = 148, normalized size of antiderivative = 2.08 \[ \int \frac {\sinh (a+b x)}{(c+d x)^2} \, dx=\frac {{\left ({\left (b d x + b c\right )} {\rm Ei}\left (\frac {b d x + b c}{d}\right ) + {\left (b d x + b c\right )} {\rm Ei}\left (-\frac {b d x + b c}{d}\right )\right )} \cosh \left (-\frac {b c - a d}{d}\right ) - 2 \, d \sinh \left (b x + a\right ) + {\left ({\left (b d x + b c\right )} {\rm Ei}\left (\frac {b d x + b c}{d}\right ) - {\left (b d x + b c\right )} {\rm Ei}\left (-\frac {b d x + b c}{d}\right )\right )} \sinh \left (-\frac {b c - a d}{d}\right )}{2 \, {\left (d^{3} x + c d^{2}\right )}} \] Input:

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.13 \[ \int \frac {\sinh (a+b x)}{(c+d x)^2} \, dx=-\frac {b {\left (\frac {e^{\left (-a + \frac {b c}{d}\right )} E_{1}\left (\frac {{\left (d x + c\right )} b}{d}\right )}{d} + \frac {e^{\left (a - \frac {b c}{d}\right )} E_{1}\left (-\frac {{\left (d x + c\right )} b}{d}\right )}{d}\right )}}{2 \, d} - \frac {\sinh \left (b x + a\right )}{{\left (d x + c\right )} d} \] Input:

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 615 vs. \(2 (71) = 142\).

Time = 0.15 (sec) , antiderivative size = 615, normalized size of antiderivative = 8.66 \[ \int \frac {\sinh (a+b x)}{(c+d x)^2} \, dx=\frac {{\left ({\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )} b^{2} {\rm Ei}\left (-\frac {{\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )} + b c - a d}{d}\right ) e^{\left (\frac {b c - a d}{d}\right )} + b^{3} c {\rm Ei}\left (-\frac {{\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )} + b c - a d}{d}\right ) e^{\left (\frac {b c - a d}{d}\right )} - a b^{2} d {\rm Ei}\left (-\frac {{\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )} + b c - a d}{d}\right ) e^{\left (\frac {b c - a d}{d}\right )} + b^{2} d e^{\left (-\frac {{\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )}}{d}\right )}\right )} d^{2}}{2 \, {\left ({\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )} d^{4} + b c d^{4} - a d^{5}\right )} b} + \frac {{\left ({\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )} b^{2} {\rm Ei}\left (\frac {{\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )} + b c - a d}{d}\right ) e^{\left (-\frac {b c - a d}{d}\right )} + b^{3} c {\rm Ei}\left (\frac {{\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )} + b c - a d}{d}\right ) e^{\left (-\frac {b c - a d}{d}\right )} - a b^{2} d {\rm Ei}\left (\frac {{\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )} + b c - a d}{d}\right ) e^{\left (-\frac {b c - a d}{d}\right )} - b^{2} d e^{\left (\frac {{\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )}}{d}\right )}\right )} d^{2}}{2 \, {\left ({\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )} d^{4} + b c d^{4} - a d^{5}\right )} b} \] Input:

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\sinh (a+b x)}{(c+d x)^2} \, dx=\frac {e^{2 a} \left (\int \frac {e^{b x}}{d^{2} x^{2}+2 c d x +c^{2}}d x \right )-\left (\int \frac {1}{e^{b x} c^{2}+2 e^{b x} c d x +e^{b x} d^{2} x^{2}}d x \right )}{2 e^{a}} \] Input:

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

Output:

(e**(2*a)*int(e**(b*x)/(c**2 + 2*c*d*x + d**2*x**2),x) - int(1/(e**(b*x)*c 
**2 + 2*e**(b*x)*c*d*x + e**(b*x)*d**2*x**2),x))/(2*e**a)