Integrand size = 17, antiderivative size = 121 \[ \int \sinh \left (e+\frac {f (a+b x)}{c+d x}\right ) \, dx=\frac {(b c-a d) f \cosh \left (e+\frac {b f}{d}\right ) \text {Chi}\left (\frac {(b c-a d) f}{d (c+d x)}\right )}{d^2}+\frac {(c+d x) \sinh \left (\frac {c e+a f+d e x+b f x}{c+d x}\right )}{d}-\frac {(b c-a d) f \sinh \left (e+\frac {b f}{d}\right ) \text {Shi}\left (\frac {(b c-a d) f}{d (c+d x)}\right )}{d^2} \]
(-a*d+b*c)*f*Chi((-a*d+b*c)*f/d/(d*x+c))*cosh(e+b*f/d)/d^2-(-a*d+b*c)*f*Sh i((-a*d+b*c)*f/d/(d*x+c))*sinh(e+b*f/d)/d^2+(d*x+c)*sinh((b*f*x+d*e*x+a*f+ c*e)/(d*x+c))/d
Leaf count is larger than twice the leaf count of optimal. \(363\) vs. \(2(121)=242\).
Time = 1.59 (sec) , antiderivative size = 363, normalized size of antiderivative = 3.00 \[ \int \sinh \left (e+\frac {f (a+b x)}{c+d x}\right ) \, dx=\frac {-c d e^{-\frac {c e+a f+d e x+b f x}{c+d x}}+c d e^{\frac {c e+a f+d e x+b f x}{c+d x}}+2 d^2 x \cosh \left (\frac {-b c f+a d f}{d (c+d x)}\right ) \sinh \left (e+\frac {b f}{d}\right )+2 d^2 x \cosh \left (e+\frac {b f}{d}\right ) \sinh \left (\frac {-b c f+a d f}{d (c+d x)}\right )+(b c-a d) f \left (\text {Chi}\left (\frac {(b c-a d) f}{d (c+d x)}\right ) \left (\cosh \left (e+\frac {b f}{d}\right )-\sinh \left (e+\frac {b f}{d}\right )\right )+\text {Chi}\left (\frac {-b c f+a d f}{d (c+d x)}\right ) \left (\cosh \left (e+\frac {b f}{d}\right )+\sinh \left (e+\frac {b f}{d}\right )\right )+\cosh \left (e+\frac {b f}{d}\right ) \text {Shi}\left (\frac {(b c-a d) f}{d (c+d x)}\right )-\sinh \left (e+\frac {b f}{d}\right ) \text {Shi}\left (\frac {(b c-a d) f}{d (c+d x)}\right )+\cosh \left (e+\frac {b f}{d}\right ) \text {Shi}\left (\frac {-b c f+a d f}{d (c+d x)}\right )+\sinh \left (e+\frac {b f}{d}\right ) \text {Shi}\left (\frac {-b c f+a d f}{d (c+d x)}\right )\right )}{2 d^2} \]
(-((c*d)/E^((c*e + a*f + d*e*x + b*f*x)/(c + d*x))) + c*d*E^((c*e + a*f + d*e*x + b*f*x)/(c + d*x)) + 2*d^2*x*Cosh[(-(b*c*f) + a*d*f)/(d*(c + d*x))] *Sinh[e + (b*f)/d] + 2*d^2*x*Cosh[e + (b*f)/d]*Sinh[(-(b*c*f) + a*d*f)/(d* (c + d*x))] + (b*c - a*d)*f*(CoshIntegral[((b*c - a*d)*f)/(d*(c + d*x))]*( Cosh[e + (b*f)/d] - Sinh[e + (b*f)/d]) + CoshIntegral[(-(b*c*f) + a*d*f)/( d*(c + d*x))]*(Cosh[e + (b*f)/d] + Sinh[e + (b*f)/d]) + Cosh[e + (b*f)/d]* SinhIntegral[((b*c - a*d)*f)/(d*(c + d*x))] - Sinh[e + (b*f)/d]*SinhIntegr al[((b*c - a*d)*f)/(d*(c + d*x))] + Cosh[e + (b*f)/d]*SinhIntegral[(-(b*c* f) + a*d*f)/(d*(c + d*x))] + Sinh[e + (b*f)/d]*SinhIntegral[(-(b*c*f) + a* d*f)/(d*(c + d*x))]))/(2*d^2)
Result contains complex when optimal does not.
Time = 0.70 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.05, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.706, Rules used = {6143, 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.
| \(\displaystyle \int \sinh \left (\frac {f (a+b x)}{c+d x}+e\right ) \, dx\) |
| \(\Big \downarrow \) 6143 |
| \(\displaystyle \int \sinh \left (\frac {a f+x (b f+d e)+c e}{c+d x}\right )dx\) |
| \(\Big \downarrow \) 6141 |
| \(\displaystyle -\frac {\int (c+d x)^2 \sinh \left (e+\frac {b f}{d}-\frac {(b c-a d) f}{d (c+d x)}\right )d\frac {1}{c+d x}}{d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\int -i (c+d x)^2 \sin \left (i \left (e+\frac {b f}{d}\right )-\frac {i (b c-a d) f}{d (c+d x)}\right )d\frac {1}{c+d x}}{d}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {i \int (c+d x)^2 \sin \left (i \left (e+\frac {b f}{d}\right )-\frac {i (b c-a d) f}{d (c+d x)}\right )d\frac {1}{c+d x}}{d}\) |
| \(\Big \downarrow \) 3778 |
| \(\displaystyle \frac {i \left (-\frac {i f (b c-a d) \int (c+d x) \cosh \left (e+\frac {b f}{d}-\frac {(b c-a d) f}{d (c+d x)}\right )d\frac {1}{c+d x}}{d}-i (c+d x) \sinh \left (-\frac {f (b c-a d)}{d (c+d x)}+\frac {b f}{d}+e\right )\right )}{d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {i \left (-\frac {i f (b c-a d) \int (c+d x) \sin \left (-\frac {i (b c-a d) f}{d (c+d x)}+i \left (e+\frac {b f}{d}\right )+\frac {\pi }{2}\right )d\frac {1}{c+d x}}{d}-i (c+d x) \sinh \left (-\frac {f (b c-a d)}{d (c+d x)}+\frac {b f}{d}+e\right )\right )}{d}\) |
| \(\Big \downarrow \) 3784 |
| \(\displaystyle \frac {i \left (-\frac {i f (b c-a d) \left (\cosh \left (\frac {b f}{d}+e\right ) \int (c+d x) \cosh \left (\frac {(b c-a d) f}{d (c+d x)}\right )d\frac {1}{c+d x}-i \sinh \left (\frac {b f}{d}+e\right ) \int -i (c+d x) \sinh \left (\frac {(b c-a d) f}{d (c+d x)}\right )d\frac {1}{c+d x}\right )}{d}-i (c+d x) \sinh \left (-\frac {f (b c-a d)}{d (c+d x)}+\frac {b f}{d}+e\right )\right )}{d}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {i \left (-\frac {i f (b c-a d) \left (\cosh \left (\frac {b f}{d}+e\right ) \int (c+d x) \cosh \left (\frac {(b c-a d) f}{d (c+d x)}\right )d\frac {1}{c+d x}-\sinh \left (\frac {b f}{d}+e\right ) \int (c+d x) \sinh \left (\frac {(b c-a d) f}{d (c+d x)}\right )d\frac {1}{c+d x}\right )}{d}-i (c+d x) \sinh \left (-\frac {f (b c-a d)}{d (c+d x)}+\frac {b f}{d}+e\right )\right )}{d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {i \left (-\frac {i f (b c-a d) \left (\cosh \left (\frac {b f}{d}+e\right ) \int (c+d x) \sin \left (\frac {i (b c-a d) f}{d (c+d x)}+\frac {\pi }{2}\right )d\frac {1}{c+d x}-\sinh \left (\frac {b f}{d}+e\right ) \int -i (c+d x) \sin \left (\frac {i (b c-a d) f}{d (c+d x)}\right )d\frac {1}{c+d x}\right )}{d}-i (c+d x) \sinh \left (-\frac {f (b c-a d)}{d (c+d x)}+\frac {b f}{d}+e\right )\right )}{d}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {i \left (-\frac {i f (b c-a d) \left (i \sinh \left (\frac {b f}{d}+e\right ) \int (c+d x) \sin \left (\frac {i (b c-a d) f}{d (c+d x)}\right )d\frac {1}{c+d x}+\cosh \left (\frac {b f}{d}+e\right ) \int (c+d x) \sin \left (\frac {i (b c-a d) f}{d (c+d x)}+\frac {\pi }{2}\right )d\frac {1}{c+d x}\right )}{d}-i (c+d x) \sinh \left (-\frac {f (b c-a d)}{d (c+d x)}+\frac {b f}{d}+e\right )\right )}{d}\) |
| \(\Big \downarrow \) 3779 |
| \(\displaystyle \frac {i \left (-\frac {i f (b c-a d) \left (-\sinh \left (\frac {b f}{d}+e\right ) \text {Shi}\left (\frac {(b c-a d) f}{d (c+d x)}\right )+\cosh \left (\frac {b f}{d}+e\right ) \int (c+d x) \sin \left (\frac {i (b c-a d) f}{d (c+d x)}+\frac {\pi }{2}\right )d\frac {1}{c+d x}\right )}{d}-i (c+d x) \sinh \left (-\frac {f (b c-a d)}{d (c+d x)}+\frac {b f}{d}+e\right )\right )}{d}\) |
| \(\Big \downarrow \) 3782 |
| \(\displaystyle \frac {i \left (-\frac {i f (b c-a d) \left (\cosh \left (\frac {b f}{d}+e\right ) \text {Chi}\left (\frac {(b c-a d) f}{d (c+d x)}\right )-\sinh \left (\frac {b f}{d}+e\right ) \text {Shi}\left (\frac {(b c-a d) f}{d (c+d x)}\right )\right )}{d}-i (c+d x) \sinh \left (-\frac {f (b c-a d)}{d (c+d x)}+\frac {b f}{d}+e\right )\right )}{d}\) |
(I*((-I)*(c + d*x)*Sinh[e + (b*f)/d - ((b*c - a*d)*f)/(d*(c + d*x))] - (I* (b*c - a*d)*f*(Cosh[e + (b*f)/d]*CoshIntegral[((b*c - a*d)*f)/(d*(c + d*x) )] - Sinh[e + (b*f)/d]*SinhIntegral[((b*c - a*d)*f)/(d*(c + d*x))]))/d))/d
3.3.98.3.1 Defintions of rubi rules used
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[((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]
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[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]
Int[Sinh[u_]^(n_.), x_Symbol] :> With[{lst = QuotientOfLinearsParts[u, x]}, Int[Sinh[(lst[[1]] + lst[[2]]*x)/(lst[[3]] + lst[[4]]*x)]^n, x]] /; IGtQ[n , 0] && QuotientOfLinearsQ[u, x]
Leaf count of result is larger than twice the leaf count of optimal. \(462\) vs. \(2(121)=242\).
Time = 1.66 (sec) , antiderivative size = 463, normalized size of antiderivative = 3.83
| method | result | size |
| risch | \(-\frac {{\mathrm e}^{-\frac {b f x +d e x +a f +c e}{d x +c}} a f}{2 \left (\frac {d f a}{d x +c}-\frac {b c f}{d x +c}\right )}+\frac {{\mathrm e}^{-\frac {b f x +d e x +a f +c e}{d x +c}} b c f}{2 d \left (\frac {d f a}{d x +c}-\frac {b c f}{d x +c}\right )}+\frac {{\mathrm e}^{-\frac {f b +d e}{d}} \operatorname {Ei}_{1}\left (\frac {a d f -b c f}{d \left (d x +c \right )}\right ) a f}{2 d}-\frac {{\mathrm e}^{-\frac {f b +d e}{d}} \operatorname {Ei}_{1}\left (\frac {a d f -b c f}{d \left (d x +c \right )}\right ) b c f}{2 d^{2}}+\frac {{\mathrm e}^{\frac {b f x +d e x +a f +c e}{d x +c}} a f}{2 d \left (\frac {f a}{d x +c}-\frac {b c f}{\left (d x +c \right ) d}\right )}-\frac {{\mathrm e}^{\frac {b f x +d e x +a f +c e}{d x +c}} b c f}{2 d^{2} \left (\frac {f a}{d x +c}-\frac {b c f}{\left (d x +c \right ) d}\right )}+\frac {{\mathrm e}^{\frac {f b +d e}{d}} \operatorname {Ei}_{1}\left (-\frac {a d f -b c f}{d \left (d x +c \right )}-\frac {f b +d e}{d}-\frac {-f b -d e}{d}\right ) a f}{2 d}-\frac {{\mathrm e}^{\frac {f b +d e}{d}} \operatorname {Ei}_{1}\left (-\frac {a d f -b c f}{d \left (d x +c \right )}-\frac {f b +d e}{d}-\frac {-f b -d e}{d}\right ) b c f}{2 d^{2}}\) | \(463\) |
-1/2*exp(-(b*f*x+d*e*x+a*f+c*e)/(d*x+c))/(d*f/(d*x+c)*a-1/(d*x+c)*b*c*f)*a *f+1/2/d*exp(-(b*f*x+d*e*x+a*f+c*e)/(d*x+c))/(d*f/(d*x+c)*a-1/(d*x+c)*b*c* f)*b*c*f+1/2/d*exp(-(b*f+d*e)/d)*Ei(1,1/d*(a*d*f-b*c*f)/(d*x+c))*a*f-1/2/d ^2*exp(-(b*f+d*e)/d)*Ei(1,1/d*(a*d*f-b*c*f)/(d*x+c))*b*c*f+1/2/d*exp((b*f* x+d*e*x+a*f+c*e)/(d*x+c))/(f/(d*x+c)*a-1/(d*x+c)/d*b*c*f)*a*f-1/2/d^2*exp( (b*f*x+d*e*x+a*f+c*e)/(d*x+c))/(f/(d*x+c)*a-1/(d*x+c)/d*b*c*f)*b*c*f+1/2/d *exp((b*f+d*e)/d)*Ei(1,-1/d*(a*d*f-b*c*f)/(d*x+c)-(b*f+d*e)/d-(-b*f-d*e)/d )*a*f-1/2/d^2*exp((b*f+d*e)/d)*Ei(1,-1/d*(a*d*f-b*c*f)/(d*x+c)-(b*f+d*e)/d -(-b*f-d*e)/d)*b*c*f
Time = 0.29 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.67 \[ \int \sinh \left (e+\frac {f (a+b x)}{c+d x}\right ) \, dx=\frac {{\left ({\left (b c - a d\right )} f {\rm Ei}\left (\frac {{\left (b c - a d\right )} f}{d^{2} x + c d}\right ) + {\left (b c - a d\right )} f {\rm Ei}\left (-\frac {{\left (b c - a d\right )} f}{d^{2} x + c d}\right )\right )} \cosh \left (\frac {d e + b f}{d}\right ) + 2 \, {\left (d^{2} x + c d\right )} \sinh \left (\frac {c e + a f + {\left (d e + b f\right )} x}{d x + c}\right ) - {\left ({\left (b c - a d\right )} f {\rm Ei}\left (\frac {{\left (b c - a d\right )} f}{d^{2} x + c d}\right ) - {\left (b c - a d\right )} f {\rm Ei}\left (-\frac {{\left (b c - a d\right )} f}{d^{2} x + c d}\right )\right )} \sinh \left (\frac {d e + b f}{d}\right )}{2 \, d^{2}} \]
1/2*(((b*c - a*d)*f*Ei((b*c - a*d)*f/(d^2*x + c*d)) + (b*c - a*d)*f*Ei(-(b *c - a*d)*f/(d^2*x + c*d)))*cosh((d*e + b*f)/d) + 2*(d^2*x + c*d)*sinh((c* e + a*f + (d*e + b*f)*x)/(d*x + c)) - ((b*c - a*d)*f*Ei((b*c - a*d)*f/(d^2 *x + c*d)) - (b*c - a*d)*f*Ei(-(b*c - a*d)*f/(d^2*x + c*d)))*sinh((d*e + b *f)/d))/d^2
\[ \int \sinh \left (e+\frac {f (a+b x)}{c+d x}\right ) \, dx=\int \sinh {\left (e + \frac {f \left (a + b x\right )}{c + d x} \right )}\, dx \]
\[ \int \sinh \left (e+\frac {f (a+b x)}{c+d x}\right ) \, dx=\int { \sinh \left (e + \frac {{\left (b x + a\right )} f}{d x + c}\right ) \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 1624 vs. \(2 (121) = 242\).
Time = 5.02 (sec) , antiderivative size = 1624, normalized size of antiderivative = 13.42 \[ \int \sinh \left (e+\frac {f (a+b x)}{c+d x}\right ) \, dx=\text {Too large to display} \]
1/2*(b^2*c^2*d*e*f^2*Ei(-(d*e + b*f - (d*e*x + b*f*x + c*e + a*f)*d/(d*x + c))/d)*e^((d*e + b*f)/d) - 2*a*b*c*d^2*e*f^2*Ei(-(d*e + b*f - (d*e*x + b* f*x + c*e + a*f)*d/(d*x + c))/d)*e^((d*e + b*f)/d) + a^2*d^3*e*f^2*Ei(-(d* e + b*f - (d*e*x + b*f*x + c*e + a*f)*d/(d*x + c))/d)*e^((d*e + b*f)/d) + b^3*c^2*f^3*Ei(-(d*e + b*f - (d*e*x + b*f*x + c*e + a*f)*d/(d*x + c))/d)*e ^((d*e + b*f)/d) - 2*a*b^2*c*d*f^3*Ei(-(d*e + b*f - (d*e*x + b*f*x + c*e + a*f)*d/(d*x + c))/d)*e^((d*e + b*f)/d) + a^2*b*d^2*f^3*Ei(-(d*e + b*f - ( d*e*x + b*f*x + c*e + a*f)*d/(d*x + c))/d)*e^((d*e + b*f)/d) - (d*e*x + b* f*x + c*e + a*f)*b^2*c^2*d*f^2*Ei(-(d*e + b*f - (d*e*x + b*f*x + c*e + a*f )*d/(d*x + c))/d)*e^((d*e + b*f)/d)/(d*x + c) + 2*(d*e*x + b*f*x + c*e + a *f)*a*b*c*d^2*f^2*Ei(-(d*e + b*f - (d*e*x + b*f*x + c*e + a*f)*d/(d*x + c) )/d)*e^((d*e + b*f)/d)/(d*x + c) - (d*e*x + b*f*x + c*e + a*f)*a^2*d^3*f^2 *Ei(-(d*e + b*f - (d*e*x + b*f*x + c*e + a*f)*d/(d*x + c))/d)*e^((d*e + b* f)/d)/(d*x + c) + b^2*c^2*d*f^2*e^((d*e*x + b*f*x + c*e + a*f)/(d*x + c)) - 2*a*b*c*d^2*f^2*e^((d*e*x + b*f*x + c*e + a*f)/(d*x + c)) + a^2*d^3*f^2* e^((d*e*x + b*f*x + c*e + a*f)/(d*x + c)))*((d*e + b*f)*c/(b*c*f - a*d*f)^ 2 - (c*e + a*f)*d/(b*c*f - a*d*f)^2)/(d^3*e + b*d^2*f - (d*e*x + b*f*x + c *e + a*f)*d^3/(d*x + c)) + 1/2*(b^2*c^2*d*e*f^2*Ei((d*e + b*f - (d*e*x + b *f*x + c*e + a*f)*d/(d*x + c))/d)*e^(-(d*e + b*f)/d) - 2*a*b*c*d^2*e*f^2*E i((d*e + b*f - (d*e*x + b*f*x + c*e + a*f)*d/(d*x + c))/d)*e^(-(d*e + b...
Timed out. \[ \int \sinh \left (e+\frac {f (a+b x)}{c+d x}\right ) \, dx=\int \mathrm {sinh}\left (e+\frac {f\,\left (a+b\,x\right )}{c+d\,x}\right ) \,d x \]