Integrand size = 10, antiderivative size = 36 \[ \int \sinh \left (\frac {a}{c+d x}\right ) \, dx=-\frac {a \text {Chi}\left (\frac {a}{c+d x}\right )}{d}+\frac {(c+d x) \sinh \left (\frac {a}{c+d x}\right )}{d} \] Output:
-a*Chi(a/(d*x+c))/d+(d*x+c)*sinh(a/(d*x+c))/d
Time = 0.02 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00 \[ \int \sinh \left (\frac {a}{c+d x}\right ) \, dx=-\frac {a \text {Chi}\left (\frac {a}{c+d x}\right )}{d}+\frac {(c+d x) \sinh \left (\frac {a}{c+d x}\right )}{d} \] Input:
Integrate[Sinh[a/(c + d*x)],x]
Output:
-((a*CoshIntegral[a/(c + d*x)])/d) + ((c + d*x)*Sinh[a/(c + d*x)])/d
Result contains complex when optimal does not.
Time = 0.35 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.17, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {5833, 5825, 3042, 26, 3778, 3042, 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.
| \(\displaystyle \int \sinh \left (\frac {a}{c+d x}\right ) \, dx\) |
| \(\Big \downarrow \) 5833 |
| \(\displaystyle \frac {\int \sinh \left (\frac {a}{c+d x}\right )d(c+d x)}{d}\) |
| \(\Big \downarrow \) 5825 |
| \(\displaystyle -\frac {\int (c+d x)^2 \sinh \left (\frac {a}{c+d x}\right )d\frac {1}{c+d x}}{d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\int -i (c+d x)^2 \sin \left (\frac {i a}{c+d x}\right )d\frac {1}{c+d x}}{d}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {i \int (c+d x)^2 \sin \left (\frac {i a}{c+d x}\right )d\frac {1}{c+d x}}{d}\) |
| \(\Big \downarrow \) 3778 |
| \(\displaystyle \frac {i \left (i a \int (c+d x) \cosh \left (\frac {a}{c+d x}\right )d\frac {1}{c+d x}-i (c+d x) \sinh \left (\frac {a}{c+d x}\right )\right )}{d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {i \left (i a \int (c+d x) \sin \left (\frac {i a}{c+d x}+\frac {\pi }{2}\right )d\frac {1}{c+d x}-i (c+d x) \sinh \left (\frac {a}{c+d x}\right )\right )}{d}\) |
| \(\Big \downarrow \) 3782 |
| \(\displaystyle \frac {i \left (i a \text {Chi}\left (\frac {a}{c+d x}\right )-i (c+d x) \sinh \left (\frac {a}{c+d x}\right )\right )}{d}\) |
Input:
Int[Sinh[a/(c + d*x)],x]
Output:
(I*(I*a*CoshIntegral[a/(c + d*x)] - I*(c + d*x)*Sinh[a/(c + d*x)]))/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[((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[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[((a_.) + (b_.)*Sinh[(c_.) + (d_.)*(x_)^(n_)])^(p_.), x_Symbol] :> -Subs t[Int[(a + b*Sinh[c + d/x^n])^p/x^2, x], x, 1/x] /; FreeQ[{a, b, c, d}, x] && ILtQ[n, 0] && IntegerQ[p]
Int[((a_.) + (b_.)*Sinh[(c_.) + (d_.)*(u_)^(n_)])^(p_.), x_Symbol] :> Simp[ 1/Coefficient[u, x, 1] Subst[Int[(a + b*Sinh[c + d*x^n])^p, x], x, u], x] /; FreeQ[{a, b, c, d, n}, x] && IntegerQ[p] && LinearQ[u, x] && NeQ[u, x]
Time = 0.35 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.06
| method | result | size |
| derivativedivides | \(-\frac {a \left (-\frac {\left (d x +c \right ) \sinh \left (\frac {a}{d x +c}\right )}{a}+\operatorname {Chi}\left (\frac {a}{d x +c}\right )\right )}{d}\) | \(38\) |
| default | \(-\frac {a \left (-\frac {\left (d x +c \right ) \sinh \left (\frac {a}{d x +c}\right )}{a}+\operatorname {Chi}\left (\frac {a}{d x +c}\right )\right )}{d}\) | \(38\) |
| risch | \(-\frac {{\mathrm e}^{-\frac {a}{d x +c}} x}{2}-\frac {{\mathrm e}^{-\frac {a}{d x +c}} c}{2 d}+\frac {a \,\operatorname {expIntegral}_{1}\left (\frac {a}{d x +c}\right )}{2 d}+\frac {{\mathrm e}^{\frac {a}{d x +c}} x}{2}+\frac {{\mathrm e}^{\frac {a}{d x +c}} c}{2 d}+\frac {a \,\operatorname {expIntegral}_{1}\left (-\frac {a}{d x +c}\right )}{2 d}\) | \(99\) |
Input:
int(sinh(a/(d*x+c)),x,method=_RETURNVERBOSE)
Output:
-1/d*a*(-1/a*(d*x+c)*sinh(a/(d*x+c))+Chi(a/(d*x+c)))
Time = 0.11 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.33 \[ \int \sinh \left (\frac {a}{c+d x}\right ) \, dx=-\frac {a {\rm Ei}\left (\frac {a}{d x + c}\right ) + a {\rm Ei}\left (-\frac {a}{d x + c}\right ) - 2 \, {\left (d x + c\right )} \sinh \left (\frac {a}{d x + c}\right )}{2 \, d} \] Input:
integrate(sinh(a/(d*x+c)),x, algorithm="fricas")
Output:
-1/2*(a*Ei(a/(d*x + c)) + a*Ei(-a/(d*x + c)) - 2*(d*x + c)*sinh(a/(d*x + c )))/d
\[ \int \sinh \left (\frac {a}{c+d x}\right ) \, dx=\int \sinh {\left (\frac {a}{c + d x} \right )}\, dx \] Input:
integrate(sinh(a/(d*x+c)),x)
Output:
Integral(sinh(a/(c + d*x)), x)
\[ \int \sinh \left (\frac {a}{c+d x}\right ) \, dx=\int { \sinh \left (\frac {a}{d x + c}\right ) \,d x } \] Input:
integrate(sinh(a/(d*x+c)),x, algorithm="maxima")
Output:
1/2*a*d*integrate(x*e^(a/(d*x + c))/(d^2*x^2 + 2*c*d*x + c^2), x) + 1/2*a* d*integrate(x*e^(-a/(d*x + c))/(d^2*x^2 + 2*c*d*x + c^2), x) + 1/2*x*e^(a/ (d*x + c)) - 1/2*x*e^(-a/(d*x + c))
Leaf count of result is larger than twice the leaf count of optimal. 102 vs. \(2 (36) = 72\).
Time = 0.14 (sec) , antiderivative size = 102, normalized size of antiderivative = 2.83 \[ \int \sinh \left (\frac {a}{c+d x}\right ) \, dx=-\frac {{\left (\frac {a^{3} {\rm Ei}\left (\frac {a}{d x + c}\right )}{d x + c} - a^{2} e^{\left (\frac {a}{d x + c}\right )}\right )} {\left (d x + c\right )}}{2 \, a^{2} d} - \frac {{\left (\frac {a^{3} {\rm Ei}\left (-\frac {a}{d x + c}\right )}{d x + c} + a^{2} e^{\left (-\frac {a}{d x + c}\right )}\right )} {\left (d x + c\right )}}{2 \, a^{2} d} \] Input:
integrate(sinh(a/(d*x+c)),x, algorithm="giac")
Output:
-1/2*(a^3*Ei(a/(d*x + c))/(d*x + c) - a^2*e^(a/(d*x + c)))*(d*x + c)/(a^2* d) - 1/2*(a^3*Ei(-a/(d*x + c))/(d*x + c) + a^2*e^(-a/(d*x + c)))*(d*x + c) /(a^2*d)
Timed out. \[ \int \sinh \left (\frac {a}{c+d x}\right ) \, dx=\int \mathrm {sinh}\left (\frac {a}{c+d\,x}\right ) \,d x \] Input:
int(sinh(a/(c + d*x)),x)
Output:
int(sinh(a/(c + d*x)), x)
\[ \int \sinh \left (\frac {a}{c+d x}\right ) \, dx=\frac {e^{\frac {2 a}{d x +c}} a \,d^{2} x^{2}-e^{\frac {2 a}{d x +c}} c^{3}-e^{\frac {2 a}{d x +c}} c^{2} d x +e^{\frac {a}{d x +c}} \left (\int \frac {x^{2}}{e^{\frac {a}{d x +c}} c^{3}+3 e^{\frac {a}{d x +c}} c^{2} d x +3 e^{\frac {a}{d x +c}} c \,d^{2} x^{2}+e^{\frac {a}{d x +c}} d^{3} x^{3}}d x \right ) a^{2} c \,d^{3}+e^{\frac {a}{d x +c}} \left (\int \frac {x^{2}}{e^{\frac {a}{d x +c}} c^{3}+3 e^{\frac {a}{d x +c}} c^{2} d x +3 e^{\frac {a}{d x +c}} c \,d^{2} x^{2}+e^{\frac {a}{d x +c}} d^{3} x^{3}}d x \right ) a^{2} d^{4} x +e^{\frac {a}{d x +c}} \left (\int \frac {e^{\frac {a}{d x +c}} x^{2}}{d^{3} x^{3}+3 c \,d^{2} x^{2}+3 c^{2} d x +c^{3}}d x \right ) a^{2} c \,d^{3}+e^{\frac {a}{d x +c}} \left (\int \frac {e^{\frac {a}{d x +c}} x^{2}}{d^{3} x^{3}+3 c \,d^{2} x^{2}+3 c^{2} d x +c^{3}}d x \right ) a^{2} d^{4} x -a \,d^{2} x^{2}-c^{3}-c^{2} d x}{2 e^{\frac {a}{d x +c}} a d \left (d x +c \right )} \] Input:
int(sinh(a/(d*x+c)),x)
Output:
(e**((2*a)/(c + d*x))*a*d**2*x**2 - e**((2*a)/(c + d*x))*c**3 - e**((2*a)/ (c + d*x))*c**2*d*x + e**(a/(c + d*x))*int(x**2/(e**(a/(c + d*x))*c**3 + 3 *e**(a/(c + d*x))*c**2*d*x + 3*e**(a/(c + d*x))*c*d**2*x**2 + e**(a/(c + d *x))*d**3*x**3),x)*a**2*c*d**3 + e**(a/(c + d*x))*int(x**2/(e**(a/(c + d*x ))*c**3 + 3*e**(a/(c + d*x))*c**2*d*x + 3*e**(a/(c + d*x))*c*d**2*x**2 + e **(a/(c + d*x))*d**3*x**3),x)*a**2*d**4*x + e**(a/(c + d*x))*int((e**(a/(c + d*x))*x**2)/(c**3 + 3*c**2*d*x + 3*c*d**2*x**2 + d**3*x**3),x)*a**2*c*d **3 + e**(a/(c + d*x))*int((e**(a/(c + d*x))*x**2)/(c**3 + 3*c**2*d*x + 3* c*d**2*x**2 + d**3*x**3),x)*a**2*d**4*x - a*d**2*x**2 - c**3 - c**2*d*x)/( 2*e**(a/(c + d*x))*a*d*(c + d*x))