3.12 \(\int \frac{\sinh ^2(a+b x)}{c+d x} \, dx\)

Optimal. Leaf size=78 \[ \frac{\cosh \left (2 a-\frac{2 b c}{d}\right ) \text{Chi}\left (\frac{2 b c}{d}+2 b x\right )}{2 d}+\frac{\sinh \left (2 a-\frac{2 b c}{d}\right ) \text{Shi}\left (\frac{2 b c}{d}+2 b x\right )}{2 d}-\frac{\log (c+d x)}{2 d} \]

[Out]

(Cosh[2*a - (2*b*c)/d]*CoshIntegral[(2*b*c)/d + 2*b*x])/(2*d) - Log[c + d*x]/(2*d) + (Sinh[2*a - (2*b*c)/d]*Si
nhIntegral[(2*b*c)/d + 2*b*x])/(2*d)

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Rubi [A]  time = 0.16549, antiderivative size = 78, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3312, 3303, 3298, 3301} \[ \frac{\cosh \left (2 a-\frac{2 b c}{d}\right ) \text{Chi}\left (\frac{2 b c}{d}+2 b x\right )}{2 d}+\frac{\sinh \left (2 a-\frac{2 b c}{d}\right ) \text{Shi}\left (\frac{2 b c}{d}+2 b x\right )}{2 d}-\frac{\log (c+d x)}{2 d} \]

Antiderivative was successfully verified.

[In]

Int[Sinh[a + b*x]^2/(c + d*x),x]

[Out]

(Cosh[2*a - (2*b*c)/d]*CoshIntegral[(2*b*c)/d + 2*b*x])/(2*d) - Log[c + d*x]/(2*d) + (Sinh[2*a - (2*b*c)/d]*Si
nhIntegral[(2*b*c)/d + 2*b*x])/(2*d)

Rule 3312

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin
[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1])
)

Rule 3303

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 3298

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 3301

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]

Rubi steps

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

Mathematica [A]  time = 0.121673, size = 66, normalized size = 0.85 \[ \frac{\cosh \left (2 a-\frac{2 b c}{d}\right ) \text{Chi}\left (\frac{2 b (c+d x)}{d}\right )+\sinh \left (2 a-\frac{2 b c}{d}\right ) \text{Shi}\left (\frac{2 b (c+d x)}{d}\right )-\log (c+d x)}{2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.074, size = 97, normalized size = 1.2 \begin{align*} -{\frac{\ln \left ( dx+c \right ) }{2\,d}}-{\frac{1}{4\,d}{{\rm e}^{-2\,{\frac{da-cb}{d}}}}{\it Ei} \left ( 1,2\,bx+2\,a-2\,{\frac{da-cb}{d}} \right ) }-{\frac{1}{4\,d}{{\rm e}^{2\,{\frac{da-cb}{d}}}}{\it Ei} \left ( 1,-2\,bx-2\,a-2\,{\frac{-da+cb}{d}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*ln(d*x+c)/d-1/4/d*exp(-2*(a*d-b*c)/d)*Ei(1,2*b*x+2*a-2*(a*d-b*c)/d)-1/4/d*exp(2*(a*d-b*c)/d)*Ei(1,-2*b*x-
2*a-2*(-a*d+b*c)/d)

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Maxima [A]  time = 1.38425, size = 97, normalized size = 1.24 \begin{align*} -\frac{e^{\left (-2 \, a + \frac{2 \, b c}{d}\right )} E_{1}\left (\frac{2 \,{\left (d x + c\right )} b}{d}\right )}{4 \, d} - \frac{e^{\left (2 \, a - \frac{2 \, b c}{d}\right )} E_{1}\left (-\frac{2 \,{\left (d x + c\right )} b}{d}\right )}{4 \, d} - \frac{\log \left (d x + c\right )}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*e^(-2*a + 2*b*c/d)*exp_integral_e(1, 2*(d*x + c)*b/d)/d - 1/4*e^(2*a - 2*b*c/d)*exp_integral_e(1, -2*(d*x
+ c)*b/d)/d - 1/2*log(d*x + c)/d

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Fricas [A]  time = 2.63968, size = 232, normalized size = 2.97 \begin{align*} \frac{{\left ({\rm Ei}\left (\frac{2 \,{\left (b d x + b c\right )}}{d}\right ) +{\rm Ei}\left (-\frac{2 \,{\left (b d x + b c\right )}}{d}\right )\right )} \cosh \left (-\frac{2 \,{\left (b c - a d\right )}}{d}\right ) +{\left ({\rm Ei}\left (\frac{2 \,{\left (b d x + b c\right )}}{d}\right ) -{\rm Ei}\left (-\frac{2 \,{\left (b d x + b c\right )}}{d}\right )\right )} \sinh \left (-\frac{2 \,{\left (b c - a d\right )}}{d}\right ) - 2 \, \log \left (d x + c\right )}{4 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*((Ei(2*(b*d*x + b*c)/d) + Ei(-2*(b*d*x + b*c)/d))*cosh(-2*(b*c - a*d)/d) + (Ei(2*(b*d*x + b*c)/d) - Ei(-2*
(b*d*x + b*c)/d))*sinh(-2*(b*c - a*d)/d) - 2*log(d*x + c))/d

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sinh ^{2}{\left (a + b x \right )}}{c + d x}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.19975, size = 92, normalized size = 1.18 \begin{align*} \frac{{\rm Ei}\left (\frac{2 \,{\left (b d x + b c\right )}}{d}\right ) e^{\left (2 \, a - \frac{2 \, b c}{d}\right )} +{\rm Ei}\left (-\frac{2 \,{\left (b d x + b c\right )}}{d}\right ) e^{\left (-2 \, a + \frac{2 \, b c}{d}\right )} - 2 \, \log \left (d x + c\right )}{4 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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