3.3.93 \(\int \sinh ^2(\frac {b x}{c+d x}) \, dx\) [293]

3.3.93.1 Optimal result

Integrand size = 13, antiderivative size = 80 \[ \int \sinh ^2\left (\frac {b x}{c+d x}\right ) \, dx=\frac {b c \text {Chi}\left (\frac {2 b c}{d (c+d x)}\right ) \sinh \left (\frac {2 b}{d}\right )}{d^2}+\frac {(c+d x) \sinh ^2\left (\frac {b x}{c+d x}\right )}{d}-\frac {b c \cosh \left (\frac {2 b}{d}\right ) \text {Shi}\left (\frac {2 b c}{d (c+d x)}\right )}{d^2} \]

output

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

3.3.93.2 Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.44 \[ \int \sinh ^2\left (\frac {b x}{c+d x}\right ) \, dx=\frac {d e^{-\frac {2 b x}{c+d x}} \left (c \left (1+e^{\frac {4 b x}{c+d x}}\right )+d \left (-1+e^{\frac {2 b x}{c+d x}}\right )^2 x\right )+4 b c \text {Chi}\left (\frac {2 b c}{c d+d^2 x}\right ) \sinh \left (\frac {2 b}{d}\right )-4 b c \cosh \left (\frac {2 b}{d}\right ) \text {Shi}\left (\frac {2 b c}{c d+d^2 x}\right )}{4 d^2} \]

input

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

output

((d*(c*(1 + E^((4*b*x)/(c + d*x))) + d*(-1 + E^((2*b*x)/(c + d*x)))^2*x))/ 
E^((2*b*x)/(c + d*x)) + 4*b*c*CoshIntegral[(2*b*c)/(c*d + d^2*x)]*Sinh[(2* 
b)/d] - 4*b*c*Cosh[(2*b)/d]*SinhIntegral[(2*b*c)/(c*d + d^2*x)])/(4*d^2)
 

3.3.93.3 Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.56 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.22, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {6141, 3042, 25, 3794, 27, 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.

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

\(\Big \downarrow \) 6141

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3794

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3784

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3779

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

\(\Big \downarrow \) 3782

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

input

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

output

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

3.3.93.3.1 Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 3042

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

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 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 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]
 

3.3.93.4 Maple [A] (verified)

Time = 6.92 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.50

method	result	size
risch	\(-\frac {x}{2}+\frac {{\mathrm e}^{-\frac {2 b x}{d x +c}} \left (d x +c \right )}{4 d}+\frac {b c \,{\mathrm e}^{-\frac {2 b}{d}} \operatorname {Ei}_{1}\left (-\frac {2 b c}{d \left (d x +c \right )}\right )}{2 d^{2}}+\frac {{\mathrm e}^{\frac {2 b x}{d x +c}} x}{4}+\frac {c \,{\mathrm e}^{\frac {2 b x}{d x +c}}}{4 d}-\frac {b c \,{\mathrm e}^{\frac {2 b}{d}} \operatorname {Ei}_{1}\left (\frac {2 b c}{d \left (d x +c \right )}\right )}{2 d^{2}}\)	\(120\)

input

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

output

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

3.3.93.5 Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 277 vs. \(2 (80) = 160\).

Time = 0.27 (sec) , antiderivative size = 277, normalized size of antiderivative = 3.46 \[ \int \sinh ^2\left (\frac {b x}{c+d x}\right ) \, dx=-\frac {d^{2} x - {\left (d^{2} x + c d\right )} \cosh \left (\frac {b x}{d x + c}\right )^{2} + {\left (b c {\rm Ei}\left (-\frac {2 \, b c}{d^{2} x + c d}\right ) \cosh \left (\frac {2 \, b}{d}\right ) - d^{2} x - c d\right )} \sinh \left (\frac {b x}{d x + c}\right )^{2} - {\left (b c {\rm Ei}\left (-\frac {2 \, b c}{d^{2} x + c d}\right ) \cosh \left (\frac {b x}{d x + c}\right )^{2} - b c {\rm Ei}\left (\frac {2 \, b c}{d^{2} x + c d}\right )\right )} \cosh \left (\frac {2 \, b}{d}\right ) - {\left (b c {\rm Ei}\left (-\frac {2 \, b c}{d^{2} x + c d}\right ) \cosh \left (\frac {b x}{d x + c}\right )^{2} - b c {\rm Ei}\left (-\frac {2 \, b c}{d^{2} x + c d}\right ) \sinh \left (\frac {b x}{d x + c}\right )^{2} + b c {\rm Ei}\left (\frac {2 \, b c}{d^{2} x + c d}\right )\right )} \sinh \left (\frac {2 \, b}{d}\right )}{2 \, {\left (d^{2} \cosh \left (\frac {b x}{d x + c}\right )^{2} - d^{2} \sinh \left (\frac {b x}{d x + c}\right )^{2}\right )}} \]

input

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

output

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

3.3.93.6 Sympy [F]

\[ \int \sinh ^2\left (\frac {b x}{c+d x}\right ) \, dx=\int \sinh ^{2}{\left (\frac {b x}{c + d x} \right )}\, dx \]

input

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

output

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

3.3.93.7 Maxima [F]

\[ \int \sinh ^2\left (\frac {b x}{c+d x}\right ) \, dx=\int { \sinh \left (\frac {b x}{d x + c}\right )^{2} \,d x } \]

input

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

output

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

3.3.93.8 Giac [F]

\[ \int \sinh ^2\left (\frac {b x}{c+d x}\right ) \, dx=\int { \sinh \left (\frac {b x}{d x + c}\right )^{2} \,d x } \]

input

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

output

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

3.3.93.9 Mupad [F(-1)]

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

input

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

output

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