3.14 \(\int \frac{\cosh ^2(a+b x)}{(c+d x)^3} \, dx\)

Optimal. Leaf size=112 \[ \frac{b^2 \cosh \left (2 a-\frac{2 b c}{d}\right ) \text{Chi}\left (\frac{2 b c}{d}+2 b x\right )}{d^3}+\frac{b^2 \sinh \left (2 a-\frac{2 b c}{d}\right ) \text{Shi}\left (\frac{2 b c}{d}+2 b x\right )}{d^3}-\frac{b \sinh (a+b x) \cosh (a+b x)}{d^2 (c+d x)}-\frac{\cosh ^2(a+b x)}{2 d (c+d x)^2} \]

[Out]

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

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Rubi [A]  time = 0.187065, antiderivative size = 112, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {3314, 31, 3312, 3303, 3298, 3301} \[ \frac{b^2 \cosh \left (2 a-\frac{2 b c}{d}\right ) \text{Chi}\left (\frac{2 b c}{d}+2 b x\right )}{d^3}+\frac{b^2 \sinh \left (2 a-\frac{2 b c}{d}\right ) \text{Shi}\left (\frac{2 b c}{d}+2 b x\right )}{d^3}-\frac{b \sinh (a+b x) \cosh (a+b x)}{d^2 (c+d x)}-\frac{\cosh ^2(a+b x)}{2 d (c+d x)^2} \]

Antiderivative was successfully verified.

[In]

Int[Cosh[a + b*x]^2/(c + d*x)^3,x]

[Out]

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

Rule 3314

Int[((c_.) + (d_.)*(x_))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[((c + d*x)^(m + 1)*(b*Si
n[e + f*x])^n)/(d*(m + 1)), x] + (Dist[(b^2*f^2*n*(n - 1))/(d^2*(m + 1)*(m + 2)), Int[(c + d*x)^(m + 2)*(b*Sin
[e + f*x])^(n - 2), x], x] - Dist[(f^2*n^2)/(d^2*(m + 1)*(m + 2)), Int[(c + d*x)^(m + 2)*(b*Sin[e + f*x])^n, x
], x] - Simp[(b*f*n*(c + d*x)^(m + 2)*Cos[e + f*x]*(b*Sin[e + f*x])^(n - 1))/(d^2*(m + 1)*(m + 2)), x]) /; Fre
eQ[{b, c, d, e, f}, x] && GtQ[n, 1] && LtQ[m, -2]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

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{\cosh ^2(a+b x)}{(c+d x)^3} \, dx &=-\frac{\cosh ^2(a+b x)}{2 d (c+d x)^2}-\frac{b \cosh (a+b x) \sinh (a+b x)}{d^2 (c+d x)}-\frac{b^2 \int \frac{1}{c+d x} \, dx}{d^2}+\frac{\left (2 b^2\right ) \int \frac{\cosh ^2(a+b x)}{c+d x} \, dx}{d^2}\\ &=-\frac{\cosh ^2(a+b x)}{2 d (c+d x)^2}-\frac{b^2 \log (c+d x)}{d^3}-\frac{b \cosh (a+b x) \sinh (a+b x)}{d^2 (c+d x)}+\frac{\left (2 b^2\right ) \int \left (\frac{1}{2 (c+d x)}+\frac{\cosh (2 a+2 b x)}{2 (c+d x)}\right ) \, dx}{d^2}\\ &=-\frac{\cosh ^2(a+b x)}{2 d (c+d x)^2}-\frac{b \cosh (a+b x) \sinh (a+b x)}{d^2 (c+d x)}+\frac{b^2 \int \frac{\cosh (2 a+2 b x)}{c+d x} \, dx}{d^2}\\ &=-\frac{\cosh ^2(a+b x)}{2 d (c+d x)^2}-\frac{b \cosh (a+b x) \sinh (a+b x)}{d^2 (c+d x)}+\frac{\left (b^2 \cosh \left (2 a-\frac{2 b c}{d}\right )\right ) \int \frac{\cosh \left (\frac{2 b c}{d}+2 b x\right )}{c+d x} \, dx}{d^2}+\frac{\left (b^2 \sinh \left (2 a-\frac{2 b c}{d}\right )\right ) \int \frac{\sinh \left (\frac{2 b c}{d}+2 b x\right )}{c+d x} \, dx}{d^2}\\ &=-\frac{\cosh ^2(a+b x)}{2 d (c+d x)^2}+\frac{b^2 \cosh \left (2 a-\frac{2 b c}{d}\right ) \text{Chi}\left (\frac{2 b c}{d}+2 b x\right )}{d^3}-\frac{b \cosh (a+b x) \sinh (a+b x)}{d^2 (c+d x)}+\frac{b^2 \sinh \left (2 a-\frac{2 b c}{d}\right ) \text{Shi}\left (\frac{2 b c}{d}+2 b x\right )}{d^3}\\ \end{align*}

Mathematica [A]  time = 0.879828, size = 102, normalized size = 0.91 \[ \frac{2 b^2 \cosh \left (2 a-\frac{2 b c}{d}\right ) \text{Chi}\left (\frac{2 b (c+d x)}{d}\right )+2 b^2 \sinh \left (2 a-\frac{2 b c}{d}\right ) \text{Shi}\left (\frac{2 b (c+d x)}{d}\right )-\frac{d \left (b (c+d x) \sinh (2 (a+b x))+d \cosh ^2(a+b x)\right )}{(c+d x)^2}}{2 d^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.081, size = 299, normalized size = 2.7 \begin{align*} -{\frac{1}{4\,d \left ( dx+c \right ) ^{2}}}+{\frac{{b}^{3}{{\rm e}^{-2\,bx-2\,a}}x}{4\,d \left ({b}^{2}{d}^{2}{x}^{2}+2\,{b}^{2}cdx+{c}^{2}{b}^{2} \right ) }}+{\frac{{b}^{3}{{\rm e}^{-2\,bx-2\,a}}c}{4\,{d}^{2} \left ({b}^{2}{d}^{2}{x}^{2}+2\,{b}^{2}cdx+{c}^{2}{b}^{2} \right ) }}-{\frac{{b}^{2}{{\rm e}^{-2\,bx-2\,a}}}{8\,d \left ({b}^{2}{d}^{2}{x}^{2}+2\,{b}^{2}cdx+{c}^{2}{b}^{2} \right ) }}-{\frac{{b}^{2}}{2\,{d}^{3}}{{\rm e}^{-2\,{\frac{da-cb}{d}}}}{\it Ei} \left ( 1,2\,bx+2\,a-2\,{\frac{da-cb}{d}} \right ) }-{\frac{{b}^{2}{{\rm e}^{2\,bx+2\,a}}}{8\,{d}^{3}} \left ({\frac{cb}{d}}+bx \right ) ^{-2}}-{\frac{{b}^{2}{{\rm e}^{2\,bx+2\,a}}}{4\,{d}^{3}} \left ({\frac{cb}{d}}+bx \right ) ^{-1}}-{\frac{{b}^{2}}{2\,{d}^{3}}{{\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(cosh(b*x+a)^2/(d*x+c)^3,x)

[Out]

-1/4/d/(d*x+c)^2+1/4*b^3*exp(-2*b*x-2*a)/d/(b^2*d^2*x^2+2*b^2*c*d*x+b^2*c^2)*x+1/4*b^3*exp(-2*b*x-2*a)/d^2/(b^
2*d^2*x^2+2*b^2*c*d*x+b^2*c^2)*c-1/8*b^2*exp(-2*b*x-2*a)/d/(b^2*d^2*x^2+2*b^2*c*d*x+b^2*c^2)-1/2*b^2/d^3*exp(-
2*(a*d-b*c)/d)*Ei(1,2*b*x+2*a-2*(a*d-b*c)/d)-1/8*b^2/d^3*exp(2*b*x+2*a)/(b*c/d+b*x)^2-1/4*b^2/d^3*exp(2*b*x+2*
a)/(b*c/d+b*x)-1/2*b^2/d^3*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 = 1.28828, size = 134, normalized size = 1.2 \begin{align*} -\frac{1}{4 \,{\left (d^{3} x^{2} + 2 \, c d^{2} x + c^{2} d\right )}} - \frac{e^{\left (-2 \, a + \frac{2 \, b c}{d}\right )} E_{3}\left (\frac{2 \,{\left (d x + c\right )} b}{d}\right )}{4 \,{\left (d x + c\right )}^{2} d} - \frac{e^{\left (2 \, a - \frac{2 \, b c}{d}\right )} E_{3}\left (-\frac{2 \,{\left (d x + c\right )} b}{d}\right )}{4 \,{\left (d x + c\right )}^{2} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4/(d^3*x^2 + 2*c*d^2*x + c^2*d) - 1/4*e^(-2*a + 2*b*c/d)*exp_integral_e(3, 2*(d*x + c)*b/d)/((d*x + c)^2*d)
- 1/4*e^(2*a - 2*b*c/d)*exp_integral_e(3, -2*(d*x + c)*b/d)/((d*x + c)^2*d)

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Fricas [B]  time = 1.8496, size = 597, normalized size = 5.33 \begin{align*} -\frac{d^{2} \cosh \left (b x + a\right )^{2} + d^{2} \sinh \left (b x + a\right )^{2} + 4 \,{\left (b d^{2} x + b c d\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + d^{2} - 2 \,{\left ({\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )}{\rm Ei}\left (\frac{2 \,{\left (b d x + b c\right )}}{d}\right ) +{\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\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 ) - 2 \,{\left ({\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )}{\rm Ei}\left (\frac{2 \,{\left (b d x + b c\right )}}{d}\right ) -{\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\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 )}{4 \,{\left (d^{5} x^{2} + 2 \, c d^{4} x + c^{2} d^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cosh ^{2}{\left (a + b x \right )}}{\left (c + d x\right )^{3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.40771, size = 446, normalized size = 3.98 \begin{align*} \frac{4 \, b^{2} d^{2} x^{2}{\rm Ei}\left (\frac{2 \,{\left (b d x + b c\right )}}{d}\right ) e^{\left (2 \, a - \frac{2 \, b c}{d}\right )} + 4 \, b^{2} d^{2} x^{2}{\rm Ei}\left (-\frac{2 \,{\left (b d x + b c\right )}}{d}\right ) e^{\left (-2 \, a + \frac{2 \, b c}{d}\right )} + 8 \, b^{2} c d x{\rm Ei}\left (\frac{2 \,{\left (b d x + b c\right )}}{d}\right ) e^{\left (2 \, a - \frac{2 \, b c}{d}\right )} + 8 \, b^{2} c d x{\rm Ei}\left (-\frac{2 \,{\left (b d x + b c\right )}}{d}\right ) e^{\left (-2 \, a + \frac{2 \, b c}{d}\right )} + 4 \, b^{2} c^{2}{\rm Ei}\left (\frac{2 \,{\left (b d x + b c\right )}}{d}\right ) e^{\left (2 \, a - \frac{2 \, b c}{d}\right )} + 4 \, b^{2} c^{2}{\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 \, b d^{2} x e^{\left (2 \, b x + 2 \, a\right )} + 2 \, b d^{2} x e^{\left (-2 \, b x - 2 \, a\right )} - 2 \, b c d e^{\left (2 \, b x + 2 \, a\right )} + 2 \, b c d e^{\left (-2 \, b x - 2 \, a\right )} - d^{2} e^{\left (2 \, b x + 2 \, a\right )} - d^{2} e^{\left (-2 \, b x - 2 \, a\right )} - 2 \, d^{2}}{8 \,{\left (d^{5} x^{2} + 2 \, c d^{4} x + c^{2} d^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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