\(\int \sinh (\frac {a+b x}{c+d x}) \, dx\) [295]

Optimal result

Integrand size = 14, antiderivative size = 101 \[ \int \sinh \left (\frac {a+b x}{c+d x}\right ) \, dx=\frac {(b c-a d) \cosh \left (\frac {b}{d}\right ) \text {Chi}\left (\frac {b c-a d}{d (c+d x)}\right )}{d^2}+\frac {(c+d x) \sinh \left (\frac {a+b x}{c+d x}\right )}{d}-\frac {(b c-a d) \sinh \left (\frac {b}{d}\right ) \text {Shi}\left (\frac {b c-a d}{d (c+d x)}\right )}{d^2} \] Output:

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(302\) vs. \(2(101)=202\).

Time = 0.45 (sec) , antiderivative size = 302, normalized size of antiderivative = 2.99 \[ \int \sinh \left (\frac {a+b x}{c+d x}\right ) \, dx=\frac {-c d e^{-\frac {a+b x}{c+d x}}+c d e^{\frac {a+b x}{c+d x}}+2 d^2 x \cosh \left (\frac {-b c+a d}{d (c+d x)}\right ) \sinh \left (\frac {b}{d}\right )+2 d^2 x \cosh \left (\frac {b}{d}\right ) \sinh \left (\frac {-b c+a d}{d (c+d x)}\right )+(b c-a d) \left (\text {Chi}\left (\frac {b c-a d}{c d+d^2 x}\right ) \left (\cosh \left (\frac {b}{d}\right )-\sinh \left (\frac {b}{d}\right )\right )+\text {Chi}\left (\frac {-b c+a d}{d (c+d x)}\right ) \left (\cosh \left (\frac {b}{d}\right )+\sinh \left (\frac {b}{d}\right )\right )+\cosh \left (\frac {b}{d}\right ) \text {Shi}\left (\frac {-b c+a d}{d (c+d x)}\right )+\sinh \left (\frac {b}{d}\right ) \text {Shi}\left (\frac {-b c+a d}{d (c+d x)}\right )+\cosh \left (\frac {b}{d}\right ) \text {Shi}\left (\frac {b c-a d}{c d+d^2 x}\right )-\sinh \left (\frac {b}{d}\right ) \text {Shi}\left (\frac {b c-a d}{c d+d^2 x}\right )\right )}{2 d^2} \] Input:

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

Output:

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

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.64 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.14, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.786, Rules used = {6141, 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.

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

\(\Big \downarrow \) 6141

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3778

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3784

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3042

\(\displaystyle \frac {i \left (-\frac {i (b c-a d) \left (\cosh \left (\frac {b}{d}\right ) \int (c+d x) \sin \left (\frac {i (b c-a d)}{d (c+d x)}+\frac {\pi }{2}\right )d\frac {1}{c+d x}-\sinh \left (\frac {b}{d}\right ) \int -i (c+d x) \sin \left (\frac {i (b c-a d)}{d (c+d x)}\right )d\frac {1}{c+d x}\right )}{d}-i (c+d x) \sinh \left (\frac {b}{d}-\frac {b c-a d}{d (c+d x)}\right )\right )}{d}\)

\(\Big \downarrow \) 26

\(\displaystyle \frac {i \left (-\frac {i (b c-a d) \left (i \sinh \left (\frac {b}{d}\right ) \int (c+d x) \sin \left (\frac {i (b c-a d)}{d (c+d x)}\right )d\frac {1}{c+d x}+\cosh \left (\frac {b}{d}\right ) \int (c+d x) \sin \left (\frac {i (b c-a d)}{d (c+d x)}+\frac {\pi }{2}\right )d\frac {1}{c+d x}\right )}{d}-i (c+d x) \sinh \left (\frac {b}{d}-\frac {b c-a d}{d (c+d x)}\right )\right )}{d}\)

\(\Big \downarrow \) 3779

\(\displaystyle \frac {i \left (-\frac {i (b c-a d) \left (-\sinh \left (\frac {b}{d}\right ) \text {Shi}\left (\frac {b c-a d}{d (c+d x)}\right )+\cosh \left (\frac {b}{d}\right ) \int (c+d x) \sin \left (\frac {i (b c-a d)}{d (c+d x)}+\frac {\pi }{2}\right )d\frac {1}{c+d x}\right )}{d}-i (c+d x) \sinh \left (\frac {b}{d}-\frac {b c-a d}{d (c+d x)}\right )\right )}{d}\)

\(\Big \downarrow \) 3782

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

Input:

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

Output:

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

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 [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(346\) vs. \(2(101)=202\).

Time = 0.39 (sec) , antiderivative size = 347, normalized size of antiderivative = 3.44

method	result	size
risch	\(-\frac {{\mathrm e}^{-\frac {b x +a}{d x +c}} a}{2 \left (\frac {d a}{d x +c}-\frac {b c}{d x +c}\right )}+\frac {{\mathrm e}^{-\frac {b x +a}{d x +c}} b c}{2 d \left (\frac {d a}{d x +c}-\frac {b c}{d x +c}\right )}+\frac {{\mathrm e}^{-\frac {b}{d}} \operatorname {expIntegral}_{1}\left (\frac {d a -b c}{d \left (d x +c \right )}\right ) a}{2 d}-\frac {{\mathrm e}^{-\frac {b}{d}} \operatorname {expIntegral}_{1}\left (\frac {d a -b c}{d \left (d x +c \right )}\right ) b c}{2 d^{2}}+\frac {d \,{\mathrm e}^{\frac {b x +a}{d x +c}} x a}{2 d a -2 b c}-\frac {{\mathrm e}^{\frac {b x +a}{d x +c}} x b c}{2 \left (d a -b c \right )}+\frac {{\mathrm e}^{\frac {b x +a}{d x +c}} c a}{2 d a -2 b c}-\frac {{\mathrm e}^{\frac {b x +a}{d x +c}} c^{2} b}{2 d \left (d a -b c \right )}+\frac {{\mathrm e}^{\frac {b}{d}} \operatorname {expIntegral}_{1}\left (-\frac {d a -b c}{d \left (d x +c \right )}\right ) a}{2 d}-\frac {{\mathrm e}^{\frac {b}{d}} \operatorname {expIntegral}_{1}\left (-\frac {d a -b c}{d \left (d x +c \right )}\right ) b c}{2 d^{2}}\)	\(347\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.69 \[ \int \sinh \left (\frac {a+b x}{c+d x}\right ) \, dx=\frac {{\left ({\left (b c - a d\right )} {\rm Ei}\left (\frac {b c - a d}{d^{2} x + c d}\right ) + {\left (b c - a d\right )} {\rm Ei}\left (-\frac {b c - a d}{d^{2} x + c d}\right )\right )} \cosh \left (\frac {b}{d}\right ) + 2 \, {\left (d^{2} x + c d\right )} \sinh \left (\frac {b x + a}{d x + c}\right ) - {\left ({\left (b c - a d\right )} {\rm Ei}\left (\frac {b c - a d}{d^{2} x + c d}\right ) - {\left (b c - a d\right )} {\rm Ei}\left (-\frac {b c - a d}{d^{2} x + c d}\right )\right )} \sinh \left (\frac {b}{d}\right )}{2 \, d^{2}} \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 764 vs. \(2 (101) = 202\).

Time = 1.73 (sec) , antiderivative size = 764, normalized size of antiderivative = 7.56 \[ \int \sinh \left (\frac {a+b x}{c+d x}\right ) \, dx =\text {Too large to display} \] Input:

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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