3.1.6 \(\int \frac {\sinh (a+b x)}{(c+d x)^2} \, dx\) [6]

Optimal. Leaf size=71 \[ \frac {b \cosh \left (a-\frac {b c}{d}\right ) \text {Chi}\left (\frac {b c}{d}+b x\right )}{d^2}-\frac {\sinh (a+b x)}{d (c+d x)}+\frac {b \sinh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (\frac {b c}{d}+b x\right )}{d^2} \]

[Out]

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

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Rubi [A]
time = 0.09, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3378, 3384, 3379, 3382} \begin {gather*} \frac {b \cosh \left (a-\frac {b c}{d}\right ) \text {Chi}\left (\frac {b c}{d}+b x\right )}{d^2}+\frac {b \sinh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (\frac {b c}{d}+b x\right )}{d^2}-\frac {\sinh (a+b x)}{d (c+d x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sinh[a + b*x]/(c + d*x)^2,x]

[Out]

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

Rubi steps

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

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Mathematica [A]
time = 0.15, size = 65, normalized size = 0.92 \begin {gather*} \frac {b \cosh \left (a-\frac {b c}{d}\right ) \text {Chi}\left (b \left (\frac {c}{d}+x\right )\right )-\frac {d \sinh (a+b x)}{c+d x}+b \sinh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (b \left (\frac {c}{d}+x\right )\right )}{d^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.38, size = 133, normalized size = 1.87

method	result	size
risch	\(\frac {b \,{\mathrm e}^{-b x -a}}{2 d \left (b d x +b c \right )}-\frac {b \,{\mathrm e}^{-\frac {a d -b c}{d}} \expIntegral \left (1, b x +a -\frac {a d -b c}{d}\right )}{2 d^{2}}-\frac {b \,{\mathrm e}^{b x +a}}{2 d^{2} \left (\frac {b c}{d}+b x \right )}-\frac {b \,{\mathrm e}^{\frac {a d -b c}{d}} \expIntegral \left (1, -b x -a -\frac {-a d +b c}{d}\right )}{2 d^{2}}\)	\(133\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.30, size = 80, normalized size = 1.13 \begin {gather*} -\frac {b {\left (\frac {e^{\left (-a + \frac {b c}{d}\right )} E_{1}\left (\frac {{\left (d x + c\right )} b}{d}\right )}{d} + \frac {e^{\left (a - \frac {b c}{d}\right )} E_{1}\left (-\frac {{\left (d x + c\right )} b}{d}\right )}{d}\right )}}{2 \, d} - \frac {\sinh \left (b x + a\right )}{{\left (d x + c\right )} d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 148 vs. \(2 (71) = 142\).
time = 0.37, size = 148, normalized size = 2.08 \begin {gather*} \frac {{\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 ) - 2 \, d \sinh \left (b x + a\right ) + {\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 )}{2 \, {\left (d^{3} x + c d^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 615 vs. \(2 (71) = 142\).
time = 0.45, size = 615, normalized size = 8.66 \begin {gather*} \frac {{\left ({\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )} b^{2} {\rm Ei}\left (-\frac {{\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )} + b c - a d}{d}\right ) e^{\left (\frac {b c - a d}{d}\right )} + b^{3} c {\rm Ei}\left (-\frac {{\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )} + b c - a d}{d}\right ) e^{\left (\frac {b c - a d}{d}\right )} - a b^{2} d {\rm Ei}\left (-\frac {{\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )} + b c - a d}{d}\right ) e^{\left (\frac {b c - a d}{d}\right )} + b^{2} d e^{\left (-\frac {{\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )}}{d}\right )}\right )} d^{2}}{2 \, {\left ({\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )} d^{4} + b c d^{4} - a d^{5}\right )} b} + \frac {{\left ({\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )} b^{2} {\rm Ei}\left (\frac {{\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )} + b c - a d}{d}\right ) e^{\left (-\frac {b c - a d}{d}\right )} + b^{3} c {\rm Ei}\left (\frac {{\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )} + b c - a d}{d}\right ) e^{\left (-\frac {b c - a d}{d}\right )} - a b^{2} d {\rm Ei}\left (\frac {{\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )} + b c - a d}{d}\right ) e^{\left (-\frac {b c - a d}{d}\right )} - b^{2} d e^{\left (\frac {{\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )}}{d}\right )}\right )} d^{2}}{2 \, {\left ({\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )} d^{4} + b c d^{4} - a d^{5}\right )} b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*((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) + 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) - 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) +
b^2*d*e^(-(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) + 1/2*((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) + 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) - 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) - b^2*d*e^((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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\mathrm {sinh}\left (a+b\,x\right )}{{\left (c+d\,x\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(a + b*x)/(c + d*x)^2,x)

[Out]

int(sinh(a + b*x)/(c + d*x)^2, x)

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