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*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-b*c *cosh(2*b/d)*Shi(2*b*c/d/(d*x+c))/d^2
Time = 0.21 (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)
Result contains complex when optimal does not.
Time = 0.58 (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)
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]
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]
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]
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]
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]
Time = 2.25 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.50
| method | result | size |
| risch | \(-\frac {x}{2}+\frac {{\mathrm e}^{-\frac {2 b x}{d x +c}} \left (d x +c \right )}{4 d}+\frac {b c \,{\mathrm e}^{-\frac {2 b}{d}} \operatorname {expIntegral}_{1}\left (-\frac {2 b c}{d \left (d x +c \right )}\right )}{2 d^{2}}+\frac {{\mathrm e}^{\frac {2 b x}{d x +c}} x}{4}+\frac {c \,{\mathrm e}^{\frac {2 b x}{d x +c}}}{4 d}-\frac {b c \,{\mathrm e}^{\frac {2 b}{d}} \operatorname {expIntegral}_{1}\left (\frac {2 b c}{d \left (d x +c \right )}\right )}{2 d^{2}}\) | \(120\) |
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))
Leaf count of result is larger than twice the leaf count of optimal. 277 vs. \(2 (80) = 160\).
Time = 0.09 (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)
\[ \int \sinh ^2\left (\frac {b x}{c+d x}\right ) \, dx=\int \sinh ^{2}{\left (\frac {b x}{c + d x} \right )}\, dx \] Input:
integrate(sinh(b*x/(d*x+c))**2,x)
Output:
Integral(sinh(b*x/(c + d*x))**2, x)
\[ \int \sinh ^2\left (\frac {b x}{c+d x}\right ) \, dx=\int { \sinh \left (\frac {b x}{d x + c}\right )^{2} \,d x } \] 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
\[ \int \sinh ^2\left (\frac {b x}{c+d x}\right ) \, dx=\int { \sinh \left (\frac {b x}{d x + c}\right )^{2} \,d x } \] Input:
integrate(sinh(b*x/(d*x+c))^2,x, algorithm="giac")
Output:
integrate(sinh(b*x/(d*x + c))^2, x)
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)
\[ \int \sinh ^2\left (\frac {b x}{c+d x}\right ) \, dx=\frac {2 e^{\frac {4 b x}{d x +c}} b d \,x^{2}+e^{\frac {4 b x}{d x +c}} c^{2}+e^{\frac {4 b x}{d x +c}} c d x +4 e^{\frac {2 b x}{d x +c}} \left (\int \frac {x^{2}}{e^{\frac {2 b x}{d x +c}} c^{3}+3 e^{\frac {2 b x}{d x +c}} c^{2} d x +3 e^{\frac {2 b x}{d x +c}} c \,d^{2} x^{2}+e^{\frac {2 b x}{d x +c}} d^{3} x^{3}}d x \right ) b^{2} c^{2} d +4 e^{\frac {2 b x}{d x +c}} \left (\int \frac {x^{2}}{e^{\frac {2 b x}{d x +c}} c^{3}+3 e^{\frac {2 b x}{d x +c}} c^{2} d x +3 e^{\frac {2 b x}{d x +c}} c \,d^{2} x^{2}+e^{\frac {2 b x}{d x +c}} d^{3} x^{3}}d x \right ) b^{2} c \,d^{2} x -4 e^{\frac {2 b x}{d x +c}} \left (\int \frac {e^{\frac {2 b x}{d x +c}} x^{2}}{d^{3} x^{3}+3 c \,d^{2} x^{2}+3 c^{2} d x +c^{3}}d x \right ) b^{2} c^{2} d -4 e^{\frac {2 b x}{d x +c}} \left (\int \frac {e^{\frac {2 b x}{d x +c}} x^{2}}{d^{3} x^{3}+3 c \,d^{2} x^{2}+3 c^{2} d x +c^{3}}d x \right ) b^{2} c \,d^{2} x -4 e^{\frac {2 b x}{d x +c}} b c x -4 e^{\frac {2 b x}{d x +c}} b d \,x^{2}+2 b d \,x^{2}-c^{2}-c d x}{8 e^{\frac {2 b x}{d x +c}} b \left (d x +c \right )} \] Input:
int(sinh(b*x/(d*x+c))^2,x)
Output:
(2*e**((4*b*x)/(c + d*x))*b*d*x**2 + e**((4*b*x)/(c + d*x))*c**2 + e**((4* b*x)/(c + d*x))*c*d*x + 4*e**((2*b*x)/(c + d*x))*int(x**2/(e**((2*b*x)/(c + d*x))*c**3 + 3*e**((2*b*x)/(c + d*x))*c**2*d*x + 3*e**((2*b*x)/(c + d*x) )*c*d**2*x**2 + e**((2*b*x)/(c + d*x))*d**3*x**3),x)*b**2*c**2*d + 4*e**(( 2*b*x)/(c + d*x))*int(x**2/(e**((2*b*x)/(c + d*x))*c**3 + 3*e**((2*b*x)/(c + d*x))*c**2*d*x + 3*e**((2*b*x)/(c + d*x))*c*d**2*x**2 + e**((2*b*x)/(c + d*x))*d**3*x**3),x)*b**2*c*d**2*x - 4*e**((2*b*x)/(c + d*x))*int((e**((2 *b*x)/(c + d*x))*x**2)/(c**3 + 3*c**2*d*x + 3*c*d**2*x**2 + d**3*x**3),x)* b**2*c**2*d - 4*e**((2*b*x)/(c + d*x))*int((e**((2*b*x)/(c + d*x))*x**2)/( c**3 + 3*c**2*d*x + 3*c*d**2*x**2 + d**3*x**3),x)*b**2*c*d**2*x - 4*e**((2 *b*x)/(c + d*x))*b*c*x - 4*e**((2*b*x)/(c + d*x))*b*d*x**2 + 2*b*d*x**2 - c**2 - c*d*x)/(8*e**((2*b*x)/(c + d*x))*b*(c + d*x))