\(\int \sinh (\frac {a+b x}{c+d x}) \, dx\) [295]

Optimal result

Integrand size = 14, antiderivative size = 101 \[ \int \sinh \left (\frac {a+b x}{c+d x}\right ) \, dx=\frac {(b c-a d) \cosh \left (\frac {b}{d}\right ) \text {Chi}\left (\frac {b c-a d}{d (c+d x)}\right )}{d^2}+\frac {(c+d x) \sinh \left (\frac {a+b x}{c+d x}\right )}{d}-\frac {(b c-a d) \sinh \left (\frac {b}{d}\right ) \text {Shi}\left (\frac {b c-a d}{d (c+d x)}\right )}{d^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.13 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {5726, 3378, 3384, 3379, 3382} \[ \int \sinh \left (\frac {a+b x}{c+d x}\right ) \, dx=\frac {\cosh \left (\frac {b}{d}\right ) (b c-a d) \text {Chi}\left (\frac {b c-a d}{d (c+d x)}\right )}{d^2}-\frac {\sinh \left (\frac {b}{d}\right ) (b c-a d) \text {Shi}\left (\frac {b c-a d}{d (c+d x)}\right )}{d^2}+\frac {(c+d x) \sinh \left (\frac {a+b x}{c+d x}\right )}{d} \]

[In]

Int[Sinh[(a + b*x)/(c + d*x)],x]

[Out]

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

Rule 3378

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] - Dist[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 3379

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> 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 3382

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> 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 3384

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Dist[Cos[(d*e - c*f)/d], Int[Sin[c*(f/d) + f*x]
/(c + d*x), x], x] + Dist[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 5726

Int[Sinh[((e_.)*((a_.) + (b_.)*(x_)))/((c_.) + (d_.)*(x_))]^(n_.), x_Symbol] :> Dist[-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]

Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \frac {\sinh \left (\frac {b}{d}-\frac {(b c-a d) x}{d}\right )}{x^2} \, dx,x,\frac {1}{c+d x}\right )}{d} \\ & = \frac {(c+d x) \sinh \left (\frac {a+b x}{c+d x}\right )}{d}+\frac {(b c-a d) \text {Subst}\left (\int \frac {\cosh \left (\frac {b}{d}-\frac {(b c-a d) x}{d}\right )}{x} \, dx,x,\frac {1}{c+d x}\right )}{d^2} \\ & = \frac {(c+d x) \sinh \left (\frac {a+b x}{c+d x}\right )}{d}+\frac {\left ((b c-a d) \cosh \left (\frac {b}{d}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {(b c-a d) x}{d}\right )}{x} \, dx,x,\frac {1}{c+d x}\right )}{d^2}-\frac {\left ((b c-a d) \sinh \left (\frac {b}{d}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {(b c-a d) x}{d}\right )}{x} \, dx,x,\frac {1}{c+d x}\right )}{d^2} \\ & = \frac {(b c-a d) \cosh \left (\frac {b}{d}\right ) \text {Chi}\left (\frac {b c-a d}{d (c+d x)}\right )}{d^2}+\frac {(c+d x) \sinh \left (\frac {a+b x}{c+d x}\right )}{d}-\frac {(b c-a d) \sinh \left (\frac {b}{d}\right ) \text {Shi}\left (\frac {b c-a d}{d (c+d x)}\right )}{d^2} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(302\) vs. \(2(101)=202\).

Time = 0.51 (sec) , antiderivative size = 302, normalized size of antiderivative = 2.99 \[ \int \sinh \left (\frac {a+b x}{c+d x}\right ) \, dx=\frac {-c d e^{-\frac {a+b x}{c+d x}}+c d e^{\frac {a+b x}{c+d x}}+2 d^2 x \cosh \left (\frac {-b c+a d}{d (c+d x)}\right ) \sinh \left (\frac {b}{d}\right )+2 d^2 x \cosh \left (\frac {b}{d}\right ) \sinh \left (\frac {-b c+a d}{d (c+d x)}\right )+(b c-a d) \left (\text {Chi}\left (\frac {b c-a d}{c d+d^2 x}\right ) \left (\cosh \left (\frac {b}{d}\right )-\sinh \left (\frac {b}{d}\right )\right )+\text {Chi}\left (\frac {-b c+a d}{d (c+d x)}\right ) \left (\cosh \left (\frac {b}{d}\right )+\sinh \left (\frac {b}{d}\right )\right )+\cosh \left (\frac {b}{d}\right ) \text {Shi}\left (\frac {-b c+a d}{d (c+d x)}\right )+\sinh \left (\frac {b}{d}\right ) \text {Shi}\left (\frac {-b c+a d}{d (c+d x)}\right )+\cosh \left (\frac {b}{d}\right ) \text {Shi}\left (\frac {b c-a d}{c d+d^2 x}\right )-\sinh \left (\frac {b}{d}\right ) \text {Shi}\left (\frac {b c-a d}{c d+d^2 x}\right )\right )}{2 d^2} \]

[In]

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

[Out]

(-((c*d)/E^((a + b*x)/(c + d*x))) + c*d*E^((a + b*x)/(c + d*x)) + 2*d^2*x*Cosh[(-(b*c) + a*d)/(d*(c + d*x))]*S
inh[b/d] + 2*d^2*x*Cosh[b/d]*Sinh[(-(b*c) + a*d)/(d*(c + d*x))] + (b*c - a*d)*(CoshIntegral[(b*c - a*d)/(c*d +
d^2*x)]*(Cosh[b/d] - Sinh[b/d]) + CoshIntegral[(-(b*c) + a*d)/(d*(c + d*x))]*(Cosh[b/d] + Sinh[b/d]) + Cosh[b
/d]*SinhIntegral[(-(b*c) + a*d)/(d*(c + d*x))] + Sinh[b/d]*SinhIntegral[(-(b*c) + a*d)/(d*(c + d*x))] + Cosh[b
/d]*SinhIntegral[(b*c - a*d)/(c*d + d^2*x)] - Sinh[b/d]*SinhIntegral[(b*c - a*d)/(c*d + d^2*x)]))/(2*d^2)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(346\) vs. \(2(101)=202\).

Time = 1.08 (sec) , antiderivative size = 347, normalized size of antiderivative = 3.44

method	result	size
risch	\(-\frac {{\mathrm e}^{-\frac {b x +a}{d x +c}} a}{2 \left (\frac {d a}{d x +c}-\frac {b c}{d x +c}\right )}+\frac {{\mathrm e}^{-\frac {b x +a}{d x +c}} b c}{2 d \left (\frac {d a}{d x +c}-\frac {b c}{d x +c}\right )}+\frac {{\mathrm e}^{-\frac {b}{d}} \operatorname {Ei}_{1}\left (\frac {a d -b c}{d \left (d x +c \right )}\right ) a}{2 d}-\frac {{\mathrm e}^{-\frac {b}{d}} \operatorname {Ei}_{1}\left (\frac {a d -b c}{d \left (d x +c \right )}\right ) b c}{2 d^{2}}+\frac {d \,{\mathrm e}^{\frac {b x +a}{d x +c}} x a}{2 a d -2 b c}-\frac {{\mathrm e}^{\frac {b x +a}{d x +c}} x b c}{2 \left (a d -b c \right )}+\frac {{\mathrm e}^{\frac {b x +a}{d x +c}} c a}{2 a d -2 b c}-\frac {{\mathrm e}^{\frac {b x +a}{d x +c}} c^{2} b}{2 d \left (a d -b c \right )}+\frac {{\mathrm e}^{\frac {b}{d}} \operatorname {Ei}_{1}\left (-\frac {a d -b c}{d \left (d x +c \right )}\right ) a}{2 d}-\frac {{\mathrm e}^{\frac {b}{d}} \operatorname {Ei}_{1}\left (-\frac {a d -b c}{d \left (d x +c \right )}\right ) b c}{2 d^{2}}\)	\(347\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.69 \[ \int \sinh \left (\frac {a+b x}{c+d x}\right ) \, dx=\frac {{\left ({\left (b c - a d\right )} {\rm Ei}\left (\frac {b c - a d}{d^{2} x + c d}\right ) + {\left (b c - a d\right )} {\rm Ei}\left (-\frac {b c - a d}{d^{2} x + c d}\right )\right )} \cosh \left (\frac {b}{d}\right ) + 2 \, {\left (d^{2} x + c d\right )} \sinh \left (\frac {b x + a}{d x + c}\right ) - {\left ({\left (b c - a d\right )} {\rm Ei}\left (\frac {b c - a d}{d^{2} x + c d}\right ) - {\left (b c - a d\right )} {\rm Ei}\left (-\frac {b c - a d}{d^{2} x + c d}\right )\right )} \sinh \left (\frac {b}{d}\right )}{2 \, d^{2}} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 764 vs. \(2 (101) = 202\).

Time = 1.90 (sec) , antiderivative size = 764, normalized size of antiderivative = 7.56 \[ \int \sinh \left (\frac {a+b x}{c+d x}\right ) \, dx=\frac {{\left (b^{3} c^{2} {\rm Ei}\left (-\frac {b - \frac {{\left (b x + a\right )} d}{d x + c}}{d}\right ) e^{\frac {b}{d}} - 2 \, a b^{2} c d {\rm Ei}\left (-\frac {b - \frac {{\left (b x + a\right )} d}{d x + c}}{d}\right ) e^{\frac {b}{d}} - \frac {{\left (b x + a\right )} b^{2} c^{2} d {\rm Ei}\left (-\frac {b - \frac {{\left (b x + a\right )} d}{d x + c}}{d}\right ) e^{\frac {b}{d}}}{d x + c} + a^{2} b d^{2} {\rm Ei}\left (-\frac {b - \frac {{\left (b x + a\right )} d}{d x + c}}{d}\right ) e^{\frac {b}{d}} + \frac {2 \, {\left (b x + a\right )} a b c d^{2} {\rm Ei}\left (-\frac {b - \frac {{\left (b x + a\right )} d}{d x + c}}{d}\right ) e^{\frac {b}{d}}}{d x + c} - \frac {{\left (b x + a\right )} a^{2} d^{3} {\rm Ei}\left (-\frac {b - \frac {{\left (b x + a\right )} d}{d x + c}}{d}\right ) e^{\frac {b}{d}}}{d x + c} + b^{2} c^{2} d e^{\left (\frac {b x + a}{d x + c}\right )} - 2 \, a b c d^{2} e^{\left (\frac {b x + a}{d x + c}\right )} + a^{2} d^{3} e^{\left (\frac {b x + a}{d x + c}\right )}\right )} {\left (\frac {b c}{{\left (b c - a d\right )}^{2}} - \frac {a d}{{\left (b c - a d\right )}^{2}}\right )}}{2 \, {\left (b d^{2} - \frac {{\left (b x + a\right )} d^{3}}{d x + c}\right )}} + \frac {{\left (b^{3} c^{2} {\rm Ei}\left (\frac {b - \frac {{\left (b x + a\right )} d}{d x + c}}{d}\right ) e^{\left (-\frac {b}{d}\right )} - 2 \, a b^{2} c d {\rm Ei}\left (\frac {b - \frac {{\left (b x + a\right )} d}{d x + c}}{d}\right ) e^{\left (-\frac {b}{d}\right )} - \frac {{\left (b x + a\right )} b^{2} c^{2} d {\rm Ei}\left (\frac {b - \frac {{\left (b x + a\right )} d}{d x + c}}{d}\right ) e^{\left (-\frac {b}{d}\right )}}{d x + c} + a^{2} b d^{2} {\rm Ei}\left (\frac {b - \frac {{\left (b x + a\right )} d}{d x + c}}{d}\right ) e^{\left (-\frac {b}{d}\right )} + \frac {2 \, {\left (b x + a\right )} a b c d^{2} {\rm Ei}\left (\frac {b - \frac {{\left (b x + a\right )} d}{d x + c}}{d}\right ) e^{\left (-\frac {b}{d}\right )}}{d x + c} - \frac {{\left (b x + a\right )} a^{2} d^{3} {\rm Ei}\left (\frac {b - \frac {{\left (b x + a\right )} d}{d x + c}}{d}\right ) e^{\left (-\frac {b}{d}\right )}}{d x + c} - b^{2} c^{2} d e^{\left (-\frac {b x + a}{d x + c}\right )} + 2 \, a b c d^{2} e^{\left (-\frac {b x + a}{d x + c}\right )} - a^{2} d^{3} e^{\left (-\frac {b x + a}{d x + c}\right )}\right )} {\left (\frac {b c}{{\left (b c - a d\right )}^{2}} - \frac {a d}{{\left (b c - a d\right )}^{2}}\right )}}{2 \, {\left (b d^{2} - \frac {{\left (b x + a\right )} d^{3}}{d x + c}\right )}} \]

[In]

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

[Out]

1/2*(b^3*c^2*Ei(-(b - (b*x + a)*d/(d*x + c))/d)*e^(b/d) - 2*a*b^2*c*d*Ei(-(b - (b*x + a)*d/(d*x + c))/d)*e^(b/
d) - (b*x + a)*b^2*c^2*d*Ei(-(b - (b*x + a)*d/(d*x + c))/d)*e^(b/d)/(d*x + c) + a^2*b*d^2*Ei(-(b - (b*x + a)*d
/(d*x + c))/d)*e^(b/d) + 2*(b*x + a)*a*b*c*d^2*Ei(-(b - (b*x + a)*d/(d*x + c))/d)*e^(b/d)/(d*x + c) - (b*x + a
)*a^2*d^3*Ei(-(b - (b*x + a)*d/(d*x + c))/d)*e^(b/d)/(d*x + c) + b^2*c^2*d*e^((b*x + a)/(d*x + c)) - 2*a*b*c*d
^2*e^((b*x + a)/(d*x + c)) + a^2*d^3*e^((b*x + a)/(d*x + c)))*(b*c/(b*c - a*d)^2 - a*d/(b*c - a*d)^2)/(b*d^2 -
(b*x + a)*d^3/(d*x + c)) + 1/2*(b^3*c^2*Ei((b - (b*x + a)*d/(d*x + c))/d)*e^(-b/d) - 2*a*b^2*c*d*Ei((b - (b*x
+ a)*d/(d*x + c))/d)*e^(-b/d) - (b*x + a)*b^2*c^2*d*Ei((b - (b*x + a)*d/(d*x + c))/d)*e^(-b/d)/(d*x + c) + a^
2*b*d^2*Ei((b - (b*x + a)*d/(d*x + c))/d)*e^(-b/d) + 2*(b*x + a)*a*b*c*d^2*Ei((b - (b*x + a)*d/(d*x + c))/d)*e
^(-b/d)/(d*x + c) - (b*x + a)*a^2*d^3*Ei((b - (b*x + a)*d/(d*x + c))/d)*e^(-b/d)/(d*x + c) - b^2*c^2*d*e^(-(b*
x + a)/(d*x + c)) + 2*a*b*c*d^2*e^(-(b*x + a)/(d*x + c)) - a^2*d^3*e^(-(b*x + a)/(d*x + c)))*(b*c/(b*c - a*d)^
2 - a*d/(b*c - a*d)^2)/(b*d^2 - (b*x + a)*d^3/(d*x + c))

Mupad [F(-1)]

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

[In]

int(sinh((a + b*x)/(c + d*x)),x)

[Out]

int(sinh((a + b*x)/(c + d*x)), x)