3.3.92 \(\int \sinh (\frac {b x}{c+d x}) \, dx\) [292]

3.3.92.1 Optimal result

Integrand size = 11, antiderivative size = 74 \[ \int \sinh \left (\frac {b x}{c+d x}\right ) \, dx=\frac {b c \cosh \left (\frac {b}{d}\right ) \text {Chi}\left (\frac {b c}{d (c+d x)}\right )}{d^2}+\frac {(c+d x) \sinh \left (\frac {b x}{c+d x}\right )}{d}-\frac {b c \sinh \left (\frac {b}{d}\right ) \text {Shi}\left (\frac {b c}{d (c+d x)}\right )}{d^2} \]

output

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

3.3.92.2 Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.26 \[ \int \sinh \left (\frac {b x}{c+d x}\right ) \, dx=\frac {d e^{-\frac {b x}{c+d x}} \left (-1+e^{\frac {2 b x}{c+d x}}\right ) (c+d x)+2 b c \cosh \left (\frac {b}{d}\right ) \text {Chi}\left (\frac {b c}{c d+d^2 x}\right )-2 b c \sinh \left (\frac {b}{d}\right ) \text {Shi}\left (\frac {b c}{c d+d^2 x}\right )}{2 d^2} \]

input

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

output

((d*(-1 + E^((2*b*x)/(c + d*x)))*(c + d*x))/E^((b*x)/(c + d*x)) + 2*b*c*Co 
sh[b/d]*CoshIntegral[(b*c)/(c*d + d^2*x)] - 2*b*c*Sinh[b/d]*SinhIntegral[( 
b*c)/(c*d + d^2*x)])/(2*d^2)
 

3.3.92.3 Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.53 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.23, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, 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.

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

\(\Big \downarrow \) 6141

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3778

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3784

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3779

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

\(\Big \downarrow \) 3782

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

input

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

output

(I*((-I)*(c + d*x)*Sinh[b/d - (b*c)/(d*(c + d*x))] - (I*b*c*(Cosh[b/d]*Cos 
hIntegral[(b*c)/(d*(c + d*x))] - Sinh[b/d]*SinhIntegral[(b*c)/(d*(c + d*x) 
)]))/d))/d
 

3.3.92.3.1 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]
 

3.3.92.4 Maple [A] (verified)

Time = 1.09 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.53

method	result	size
risch	\(-\frac {{\mathrm e}^{-\frac {b x}{d x +c}} \left (d x +c \right )}{2 d}-\frac {b c \,{\mathrm e}^{-\frac {b}{d}} \operatorname {Ei}_{1}\left (-\frac {b c}{d \left (d x +c \right )}\right )}{2 d^{2}}+\frac {{\mathrm e}^{\frac {b x}{d x +c}} x}{2}+\frac {c \,{\mathrm e}^{\frac {b x}{d x +c}}}{2 d}-\frac {b c \,{\mathrm e}^{\frac {b}{d}} \operatorname {Ei}_{1}\left (\frac {b c}{d \left (d x +c \right )}\right )}{2 d^{2}}\)	\(113\)

input

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

output

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

3.3.92.5 Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 253 vs. \(2 (74) = 148\).

Time = 0.27 (sec) , antiderivative size = 253, normalized size of antiderivative = 3.42 \[ \int \sinh \left (\frac {b x}{c+d x}\right ) \, dx=-\frac {b c {\rm Ei}\left (-\frac {b c}{d^{2} x + c d}\right ) \cosh \left (\frac {b}{d}\right ) \sinh \left (\frac {b x}{d x + c}\right )^{2} - {\left (b c {\rm Ei}\left (-\frac {b c}{d^{2} x + c d}\right ) \cosh \left (\frac {b x}{d x + c}\right )^{2} + b c {\rm Ei}\left (\frac {b c}{d^{2} x + c d}\right )\right )} \cosh \left (\frac {b}{d}\right ) - 2 \, {\left (d^{2} x + c d\right )} \sinh \left (\frac {b x}{d x + c}\right ) - {\left (b c {\rm Ei}\left (-\frac {b c}{d^{2} x + c d}\right ) \cosh \left (\frac {b x}{d x + c}\right )^{2} - b c {\rm Ei}\left (-\frac {b c}{d^{2} x + c d}\right ) \sinh \left (\frac {b x}{d x + c}\right )^{2} - b c {\rm Ei}\left (\frac {b c}{d^{2} x + c d}\right )\right )} \sinh \left (\frac {b}{d}\right )}{2 \, {\left (d^{2} \cosh \left (\frac {b x}{d x + c}\right )^{2} - d^{2} \sinh \left (\frac {b x}{d x + c}\right )^{2}\right )}} \]

input

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

output

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

3.3.92.6 Sympy [F]

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

input

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

output

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

3.3.92.7 Maxima [F]

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

input

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

output

-1/2*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) - 1/2*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) - 1/2*(x*e^(b*c/(d^2*x + c*d)) - x*e^(-b*c/( 
d^2*x + c*d) + 2*b/d))*e^(-b/d)
 

3.3.92.8 Giac [F]

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

input

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

output

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

3.3.92.9 Mupad [F(-1)]

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

input

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

output

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