\(\int \frac {\sinh ^3(a+b x)}{(c+d x)^2} \, dx\) [21]

Optimal result

Integrand size = 16, antiderivative size = 145 \[ \int \frac {\sinh ^3(a+b x)}{(c+d x)^2} \, dx=-\frac {3 b \cosh \left (a-\frac {b c}{d}\right ) \text {Chi}\left (\frac {b c}{d}+b x\right )}{4 d^2}+\frac {3 b \cosh \left (3 a-\frac {3 b c}{d}\right ) \text {Chi}\left (\frac {3 b c}{d}+3 b x\right )}{4 d^2}-\frac {\sinh ^3(a+b x)}{d (c+d x)}-\frac {3 b \sinh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (\frac {b c}{d}+b x\right )}{4 d^2}+\frac {3 b \sinh \left (3 a-\frac {3 b c}{d}\right ) \text {Shi}\left (\frac {3 b c}{d}+3 b x\right )}{4 d^2} \]

[Out]

3/4*b*Chi(3*b*c/d+3*b*x)*cosh(3*a-3*b*c/d)/d^2-3/4*b*Chi(b*c/d+b*x)*cosh(a-b*c/d)/d^2+3/4*b*Shi(3*b*c/d+3*b*x)
*sinh(3*a-3*b*c/d)/d^2-3/4*b*Shi(b*c/d+b*x)*sinh(a-b*c/d)/d^2-sinh(b*x+a)^3/d/(d*x+c)

Rubi [A] (verified)

Time = 0.19 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3394, 3384, 3379, 3382} \[ \int \frac {\sinh ^3(a+b x)}{(c+d x)^2} \, dx=-\frac {3 b \cosh \left (a-\frac {b c}{d}\right ) \text {Chi}\left (\frac {b c}{d}+b x\right )}{4 d^2}+\frac {3 b \cosh \left (3 a-\frac {3 b c}{d}\right ) \text {Chi}\left (\frac {3 b c}{d}+3 b x\right )}{4 d^2}-\frac {3 b \sinh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (\frac {b c}{d}+b x\right )}{4 d^2}+\frac {3 b \sinh \left (3 a-\frac {3 b c}{d}\right ) \text {Shi}\left (\frac {3 b c}{d}+3 b x\right )}{4 d^2}-\frac {\sinh ^3(a+b x)}{d (c+d x)} \]

[In]

Int[Sinh[a + b*x]^3/(c + d*x)^2,x]

[Out]

(-3*b*Cosh[a - (b*c)/d]*CoshIntegral[(b*c)/d + b*x])/(4*d^2) + (3*b*Cosh[3*a - (3*b*c)/d]*CoshIntegral[(3*b*c)
/d + 3*b*x])/(4*d^2) - Sinh[a + b*x]^3/(d*(c + d*x)) - (3*b*Sinh[a - (b*c)/d]*SinhIntegral[(b*c)/d + b*x])/(4*
d^2) + (3*b*Sinh[3*a - (3*b*c)/d]*SinhIntegral[(3*b*c)/d + 3*b*x])/(4*d^2)

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 3394

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Simp[(c + d*x)^(m + 1)*(Sin[e + f*x]^
n/(d*(m + 1))), x] - Dist[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]

Rubi steps \begin{align*} \text {integral}& = -\frac {\sinh ^3(a+b x)}{d (c+d x)}-\frac {(3 b) \int \left (\frac {\cosh (a+b x)}{4 (c+d x)}-\frac {\cosh (3 a+3 b x)}{4 (c+d x)}\right ) \, dx}{d} \\ & = -\frac {\sinh ^3(a+b x)}{d (c+d x)}-\frac {(3 b) \int \frac {\cosh (a+b x)}{c+d x} \, dx}{4 d}+\frac {(3 b) \int \frac {\cosh (3 a+3 b x)}{c+d x} \, dx}{4 d} \\ & = -\frac {\sinh ^3(a+b x)}{d (c+d x)}+\frac {\left (3 b \cosh \left (3 a-\frac {3 b c}{d}\right )\right ) \int \frac {\cosh \left (\frac {3 b c}{d}+3 b x\right )}{c+d x} \, dx}{4 d}-\frac {\left (3 b \cosh \left (a-\frac {b c}{d}\right )\right ) \int \frac {\cosh \left (\frac {b c}{d}+b x\right )}{c+d x} \, dx}{4 d}+\frac {\left (3 b \sinh \left (3 a-\frac {3 b c}{d}\right )\right ) \int \frac {\sinh \left (\frac {3 b c}{d}+3 b x\right )}{c+d x} \, dx}{4 d}-\frac {\left (3 b \sinh \left (a-\frac {b c}{d}\right )\right ) \int \frac {\sinh \left (\frac {b c}{d}+b x\right )}{c+d x} \, dx}{4 d} \\ & = -\frac {3 b \cosh \left (a-\frac {b c}{d}\right ) \text {Chi}\left (\frac {b c}{d}+b x\right )}{4 d^2}+\frac {3 b \cosh \left (3 a-\frac {3 b c}{d}\right ) \text {Chi}\left (\frac {3 b c}{d}+3 b x\right )}{4 d^2}-\frac {\sinh ^3(a+b x)}{d (c+d x)}-\frac {3 b \sinh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (\frac {b c}{d}+b x\right )}{4 d^2}+\frac {3 b \sinh \left (3 a-\frac {3 b c}{d}\right ) \text {Shi}\left (\frac {3 b c}{d}+3 b x\right )}{4 d^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.73 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.10 \[ \int \frac {\sinh ^3(a+b x)}{(c+d x)^2} \, dx=\frac {6 d \cosh (b x) \sinh (a)-2 d \cosh (3 b x) \sinh (3 a)+6 d \cosh (a) \sinh (b x)-2 d \cosh (3 a) \sinh (3 b x)+6 b (c+d x) \left (-\cosh \left (a-\frac {b c}{d}\right ) \text {Chi}\left (b \left (\frac {c}{d}+x\right )\right )+\cosh \left (3 a-\frac {3 b c}{d}\right ) \text {Chi}\left (\frac {3 b (c+d x)}{d}\right )-\sinh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (b \left (\frac {c}{d}+x\right )\right )+\sinh \left (3 a-\frac {3 b c}{d}\right ) \text {Shi}\left (\frac {3 b (c+d x)}{d}\right )\right )}{8 d^2 (c+d x)} \]

[In]

Integrate[Sinh[a + b*x]^3/(c + d*x)^2,x]

[Out]

(6*d*Cosh[b*x]*Sinh[a] - 2*d*Cosh[3*b*x]*Sinh[3*a] + 6*d*Cosh[a]*Sinh[b*x] - 2*d*Cosh[3*a]*Sinh[3*b*x] + 6*b*(
c + d*x)*(-(Cosh[a - (b*c)/d]*CoshIntegral[b*(c/d + x)]) + Cosh[3*a - (3*b*c)/d]*CoshIntegral[(3*b*(c + d*x))/
d] - Sinh[a - (b*c)/d]*SinhIntegral[b*(c/d + x)] + Sinh[3*a - (3*b*c)/d]*SinhIntegral[(3*b*(c + d*x))/d]))/(8*
d^2*(c + d*x))

Maple [A] (verified)

Time = 2.37 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.87

method	result	size
risch	\(\frac {b \,{\mathrm e}^{-3 b x -3 a}}{8 d \left (b d x +b c \right )}-\frac {3 b \,{\mathrm e}^{-\frac {3 \left (a d -b c \right )}{d}} \operatorname {Ei}_{1}\left (3 b x +3 a -\frac {3 \left (a d -b c \right )}{d}\right )}{8 d^{2}}-\frac {3 b \,{\mathrm e}^{-b x -a}}{8 d \left (b d x +b c \right )}+\frac {3 b \,{\mathrm e}^{-\frac {a d -b c}{d}} \operatorname {Ei}_{1}\left (b x +a -\frac {a d -b c}{d}\right )}{8 d^{2}}+\frac {3 b \,{\mathrm e}^{b x +a}}{8 d^{2} \left (\frac {b c}{d}+b x \right )}+\frac {3 b \,{\mathrm e}^{\frac {a d -b c}{d}} \operatorname {Ei}_{1}\left (-b x -a -\frac {-a d +b c}{d}\right )}{8 d^{2}}-\frac {b \,{\mathrm e}^{3 b x +3 a}}{8 d^{2} \left (\frac {b c}{d}+b x \right )}-\frac {3 b \,{\mathrm e}^{\frac {3 a d -3 b c}{d}} \operatorname {Ei}_{1}\left (-3 b x -3 a -\frac {3 \left (-a d +b c \right )}{d}\right )}{8 d^{2}}\)	\(271\)

[In]

int(sinh(b*x+a)^3/(d*x+c)^2,x,method=_RETURNVERBOSE)

[Out]

1/8*b*exp(-3*b*x-3*a)/d/(b*d*x+b*c)-3/8*b/d^2*exp(-3*(a*d-b*c)/d)*Ei(1,3*b*x+3*a-3*(a*d-b*c)/d)-3/8*b*exp(-b*x
-a)/d/(b*d*x+b*c)+3/8*b/d^2*exp(-(a*d-b*c)/d)*Ei(1,b*x+a-(a*d-b*c)/d)+3/8*b/d^2*exp(b*x+a)/(b*c/d+b*x)+3/8*b/d
^2*exp((a*d-b*c)/d)*Ei(1,-b*x-a-(-a*d+b*c)/d)-1/8*b/d^2*exp(3*b*x+3*a)/(b*c/d+b*x)-3/8*b/d^2*exp(3*(a*d-b*c)/d
)*Ei(1,-3*b*x-3*a-3*(-a*d+b*c)/d)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 301 vs. \(2 (137) = 274\).

Time = 0.25 (sec) , antiderivative size = 301, normalized size of antiderivative = 2.08 \[ \int \frac {\sinh ^3(a+b x)}{(c+d x)^2} \, dx=-\frac {2 \, d \sinh \left (b x + a\right )^{3} + 3 \, {\left ({\left (b d x + b c\right )} {\rm Ei}\left (\frac {b d x + b c}{d}\right ) + {\left (b d x + b c\right )} {\rm Ei}\left (-\frac {b d x + b c}{d}\right )\right )} \cosh \left (-\frac {b c - a d}{d}\right ) - 3 \, {\left ({\left (b d x + b c\right )} {\rm Ei}\left (\frac {3 \, {\left (b d x + b c\right )}}{d}\right ) + {\left (b d x + b c\right )} {\rm Ei}\left (-\frac {3 \, {\left (b d x + b c\right )}}{d}\right )\right )} \cosh \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) + 6 \, {\left (d \cosh \left (b x + a\right )^{2} - d\right )} \sinh \left (b x + a\right ) + 3 \, {\left ({\left (b d x + b c\right )} {\rm Ei}\left (\frac {b d x + b c}{d}\right ) - {\left (b d x + b c\right )} {\rm Ei}\left (-\frac {b d x + b c}{d}\right )\right )} \sinh \left (-\frac {b c - a d}{d}\right ) - 3 \, {\left ({\left (b d x + b c\right )} {\rm Ei}\left (\frac {3 \, {\left (b d x + b c\right )}}{d}\right ) - {\left (b d x + b c\right )} {\rm Ei}\left (-\frac {3 \, {\left (b d x + b c\right )}}{d}\right )\right )} \sinh \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right )}{8 \, {\left (d^{3} x + c d^{2}\right )}} \]

[In]

integrate(sinh(b*x+a)^3/(d*x+c)^2,x, algorithm="fricas")

[Out]

-1/8*(2*d*sinh(b*x + a)^3 + 3*((b*d*x + b*c)*Ei((b*d*x + b*c)/d) + (b*d*x + b*c)*Ei(-(b*d*x + b*c)/d))*cosh(-(
b*c - a*d)/d) - 3*((b*d*x + b*c)*Ei(3*(b*d*x + b*c)/d) + (b*d*x + b*c)*Ei(-3*(b*d*x + b*c)/d))*cosh(-3*(b*c -
a*d)/d) + 6*(d*cosh(b*x + a)^2 - d)*sinh(b*x + a) + 3*((b*d*x + b*c)*Ei((b*d*x + b*c)/d) - (b*d*x + b*c)*Ei(-(
b*d*x + b*c)/d))*sinh(-(b*c - a*d)/d) - 3*((b*d*x + b*c)*Ei(3*(b*d*x + b*c)/d) - (b*d*x + b*c)*Ei(-3*(b*d*x +
b*c)/d))*sinh(-3*(b*c - a*d)/d))/(d^3*x + c*d^2)

Sympy [F]

\[ \int \frac {\sinh ^3(a+b x)}{(c+d x)^2} \, dx=\int \frac {\sinh ^{3}{\left (a + b x \right )}}{\left (c + d x\right )^{2}}\, dx \]

[In]

integrate(sinh(b*x+a)**3/(d*x+c)**2,x)

[Out]

Integral(sinh(a + b*x)**3/(c + d*x)**2, x)

Maxima [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.00 \[ \int \frac {\sinh ^3(a+b x)}{(c+d x)^2} \, dx=\frac {e^{\left (-3 \, a + \frac {3 \, b c}{d}\right )} E_{2}\left (\frac {3 \, {\left (d x + c\right )} b}{d}\right )}{8 \, {\left (d x + c\right )} d} - \frac {3 \, e^{\left (-a + \frac {b c}{d}\right )} E_{2}\left (\frac {{\left (d x + c\right )} b}{d}\right )}{8 \, {\left (d x + c\right )} d} + \frac {3 \, e^{\left (a - \frac {b c}{d}\right )} E_{2}\left (-\frac {{\left (d x + c\right )} b}{d}\right )}{8 \, {\left (d x + c\right )} d} - \frac {e^{\left (3 \, a - \frac {3 \, b c}{d}\right )} E_{2}\left (-\frac {3 \, {\left (d x + c\right )} b}{d}\right )}{8 \, {\left (d x + c\right )} d} \]

[In]

integrate(sinh(b*x+a)^3/(d*x+c)^2,x, algorithm="maxima")

[Out]

1/8*e^(-3*a + 3*b*c/d)*exp_integral_e(2, 3*(d*x + c)*b/d)/((d*x + c)*d) - 3/8*e^(-a + b*c/d)*exp_integral_e(2,
(d*x + c)*b/d)/((d*x + c)*d) + 3/8*e^(a - b*c/d)*exp_integral_e(2, -(d*x + c)*b/d)/((d*x + c)*d) - 1/8*e^(3*a
- 3*b*c/d)*exp_integral_e(2, -3*(d*x + c)*b/d)/((d*x + c)*d)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1076 vs. \(2 (137) = 274\).

Time = 0.33 (sec) , antiderivative size = 1076, normalized size of antiderivative = 7.42 \[ \int \frac {\sinh ^3(a+b x)}{(c+d x)^2} \, dx=\text {Too large to display} \]

[In]

integrate(sinh(b*x+a)^3/(d*x+c)^2,x, algorithm="giac")

[Out]

1/8*(3*(d*x + c)*(b - b*c/(d*x + c) + a*d/(d*x + c))*b^2*Ei(-3*((d*x + c)*(b - b*c/(d*x + c) + a*d/(d*x + c))
+ b*c - a*d)/d)*e^(3*(b*c - a*d)/d) + 3*b^3*c*Ei(-3*((d*x + c)*(b - b*c/(d*x + c) + a*d/(d*x + c)) + b*c - a*d
)/d)*e^(3*(b*c - a*d)/d) - 3*a*b^2*d*Ei(-3*((d*x + c)*(b - b*c/(d*x + c) + a*d/(d*x + c)) + b*c - a*d)/d)*e^(3
*(b*c - a*d)/d) - 3*(d*x + c)*(b - b*c/(d*x + c) + a*d/(d*x + c))*b^2*Ei(-((d*x + c)*(b - b*c/(d*x + c) + a*d/
(d*x + c)) + b*c - a*d)/d)*e^((b*c - a*d)/d) - 3*b^3*c*Ei(-((d*x + c)*(b - b*c/(d*x + c) + a*d/(d*x + c)) + b*
c - a*d)/d)*e^((b*c - a*d)/d) + 3*a*b^2*d*Ei(-((d*x + c)*(b - b*c/(d*x + c) + a*d/(d*x + c)) + b*c - a*d)/d)*e
^((b*c - a*d)/d) - 3*(d*x + c)*(b - b*c/(d*x + c) + a*d/(d*x + c))*b^2*Ei(((d*x + c)*(b - b*c/(d*x + c) + a*d/
(d*x + c)) + b*c - a*d)/d)*e^(-(b*c - a*d)/d) - 3*b^3*c*Ei(((d*x + c)*(b - b*c/(d*x + c) + a*d/(d*x + c)) + b*
c - a*d)/d)*e^(-(b*c - a*d)/d) + 3*a*b^2*d*Ei(((d*x + c)*(b - b*c/(d*x + c) + a*d/(d*x + c)) + b*c - a*d)/d)*e
^(-(b*c - a*d)/d) + 3*(d*x + c)*(b - b*c/(d*x + c) + a*d/(d*x + c))*b^2*Ei(3*((d*x + c)*(b - b*c/(d*x + c) + a
*d/(d*x + c)) + b*c - a*d)/d)*e^(-3*(b*c - a*d)/d) + 3*b^3*c*Ei(3*((d*x + c)*(b - b*c/(d*x + c) + a*d/(d*x + c
)) + b*c - a*d)/d)*e^(-3*(b*c - a*d)/d) - 3*a*b^2*d*Ei(3*((d*x + c)*(b - b*c/(d*x + c) + a*d/(d*x + c)) + b*c
- a*d)/d)*e^(-3*(b*c - a*d)/d) - b^2*d*e^(3*(d*x + c)*(b - b*c/(d*x + c) + a*d/(d*x + c))/d) + 3*b^2*d*e^((d*x
+ c)*(b - b*c/(d*x + c) + a*d/(d*x + c))/d) - 3*b^2*d*e^(-(d*x + c)*(b - b*c/(d*x + c) + a*d/(d*x + c))/d) +
b^2*d*e^(-3*(d*x + c)*(b - b*c/(d*x + c) + a*d/(d*x + c))/d))*d^2/(((d*x + c)*(b - b*c/(d*x + c) + a*d/(d*x +
c))*d^4 + b*c*d^4 - a*d^5)*b)

Mupad [F(-1)]

Timed out. \[ \int \frac {\sinh ^3(a+b x)}{(c+d x)^2} \, dx=\int \frac {{\mathrm {sinh}\left (a+b\,x\right )}^3}{{\left (c+d\,x\right )}^2} \,d x \]

[In]

int(sinh(a + b*x)^3/(c + d*x)^2,x)

[Out]

int(sinh(a + b*x)^3/(c + d*x)^2, x)