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

Optimal. Leaf size=242 \[ -\frac {a^2}{2 d (c+d x)^2}+\frac {a b f^2 \text {Chi}\left (x f+\frac {c f}{d}\right ) \cosh \left (e-\frac {c f}{d}\right )}{d^3}+\frac {a b f^2 \sinh \left (e-\frac {c f}{d}\right ) \text {Shi}\left (x f+\frac {c f}{d}\right )}{d^3}-\frac {a b f \sinh (e+f x)}{d^2 (c+d x)}-\frac {a b \cosh (e+f x)}{d (c+d x)^2}+\frac {b^2 f^2 \text {Chi}\left (2 x f+\frac {2 c f}{d}\right ) \cosh \left (2 e-\frac {2 c f}{d}\right )}{d^3}+\frac {b^2 f^2 \sinh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (2 x f+\frac {2 c f}{d}\right )}{d^3}-\frac {b^2 f \sinh (e+f x) \cosh (e+f x)}{d^2 (c+d x)}-\frac {b^2 \cosh ^2(e+f x)}{2 d (c+d x)^2} \]

[Out]

-1/2*a^2/d/(d*x+c)^2+b^2*f^2*Chi(2*c*f/d+2*f*x)*cosh(-2*e+2*c*f/d)/d^3+a*b*f^2*Chi(c*f/d+f*x)*cosh(-e+c*f/d)/d
^3-a*b*cosh(f*x+e)/d/(d*x+c)^2-1/2*b^2*cosh(f*x+e)^2/d/(d*x+c)^2-b^2*f^2*Shi(2*c*f/d+2*f*x)*sinh(-2*e+2*c*f/d)
/d^3-a*b*f^2*Shi(c*f/d+f*x)*sinh(-e+c*f/d)/d^3-a*b*f*sinh(f*x+e)/d^2/(d*x+c)-b^2*f*cosh(f*x+e)*sinh(f*x+e)/d^2
/(d*x+c)

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Rubi [A]  time = 0.43, antiderivative size = 242, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 8, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {3317, 3297, 3303, 3298, 3301, 3314, 31, 3312} \[ -\frac {a^2}{2 d (c+d x)^2}+\frac {a b f^2 \text {Chi}\left (x f+\frac {c f}{d}\right ) \cosh \left (e-\frac {c f}{d}\right )}{d^3}+\frac {a b f^2 \sinh \left (e-\frac {c f}{d}\right ) \text {Shi}\left (x f+\frac {c f}{d}\right )}{d^3}-\frac {a b f \sinh (e+f x)}{d^2 (c+d x)}-\frac {a b \cosh (e+f x)}{d (c+d x)^2}+\frac {b^2 f^2 \text {Chi}\left (2 x f+\frac {2 c f}{d}\right ) \cosh \left (2 e-\frac {2 c f}{d}\right )}{d^3}+\frac {b^2 f^2 \sinh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (2 x f+\frac {2 c f}{d}\right )}{d^3}-\frac {b^2 f \sinh (e+f x) \cosh (e+f x)}{d^2 (c+d x)}-\frac {b^2 \cosh ^2(e+f x)}{2 d (c+d x)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 31

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

Rule 3297

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 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]

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 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 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 3317

Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Int[ExpandIntegrand[
(c + d*x)^m, (a + b*Sin[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[n, 0] && (EqQ[n, 1] ||
IGtQ[m, 0] || NeQ[a^2 - b^2, 0])

Rubi steps

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

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Mathematica [A]  time = 1.32, size = 394, normalized size = 1.63 \[ -\frac {2 a^2 d^2-4 a b c^2 f^2 \sinh \left (e-\frac {c f}{d}\right ) \text {Shi}\left (f \left (\frac {c}{d}+x\right )\right )-4 a b f^2 (c+d x)^2 \text {Chi}\left (f \left (\frac {c}{d}+x\right )\right ) \cosh \left (e-\frac {c f}{d}\right )-4 a b d^2 f^2 x^2 \sinh \left (e-\frac {c f}{d}\right ) \text {Shi}\left (f \left (\frac {c}{d}+x\right )\right )-8 a b c d f^2 x \sinh \left (e-\frac {c f}{d}\right ) \text {Shi}\left (f \left (\frac {c}{d}+x\right )\right )+4 a b c d f \sinh (e+f x)+4 a b d^2 f x \sinh (e+f x)+4 a b d^2 \cosh (e+f x)-4 b^2 c^2 f^2 \sinh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (\frac {2 f (c+d x)}{d}\right )-4 b^2 f^2 (c+d x)^2 \text {Chi}\left (\frac {2 f (c+d x)}{d}\right ) \cosh \left (2 e-\frac {2 c f}{d}\right )-4 b^2 d^2 f^2 x^2 \sinh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (\frac {2 f (c+d x)}{d}\right )-8 b^2 c d f^2 x \sinh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (\frac {2 f (c+d x)}{d}\right )+2 b^2 c d f \sinh (2 (e+f x))+2 b^2 d^2 f x \sinh (2 (e+f x))+b^2 d^2 \cosh (2 (e+f x))+b^2 d^2}{4 d^3 (c+d x)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/4*(2*a^2*d^2 + b^2*d^2 + 4*a*b*d^2*Cosh[e + f*x] + b^2*d^2*Cosh[2*(e + f*x)] - 4*a*b*f^2*(c + d*x)^2*Cosh[e
- (c*f)/d]*CoshIntegral[f*(c/d + x)] - 4*b^2*f^2*(c + d*x)^2*Cosh[2*e - (2*c*f)/d]*CoshIntegral[(2*f*(c + d*x
))/d] + 4*a*b*c*d*f*Sinh[e + f*x] + 4*a*b*d^2*f*x*Sinh[e + f*x] + 2*b^2*c*d*f*Sinh[2*(e + f*x)] + 2*b^2*d^2*f*
x*Sinh[2*(e + f*x)] - 4*a*b*c^2*f^2*Sinh[e - (c*f)/d]*SinhIntegral[f*(c/d + x)] - 8*a*b*c*d*f^2*x*Sinh[e - (c*
f)/d]*SinhIntegral[f*(c/d + x)] - 4*a*b*d^2*f^2*x^2*Sinh[e - (c*f)/d]*SinhIntegral[f*(c/d + x)] - 4*b^2*c^2*f^
2*Sinh[2*e - (2*c*f)/d]*SinhIntegral[(2*f*(c + d*x))/d] - 8*b^2*c*d*f^2*x*Sinh[2*e - (2*c*f)/d]*SinhIntegral[(
2*f*(c + d*x))/d] - 4*b^2*d^2*f^2*x^2*Sinh[2*e - (2*c*f)/d]*SinhIntegral[(2*f*(c + d*x))/d])/(d^3*(c + d*x)^2)

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fricas [B]  time = 0.52, size = 586, normalized size = 2.42 \[ -\frac {b^{2} d^{2} \cosh \left (f x + e\right )^{2} + b^{2} d^{2} \sinh \left (f x + e\right )^{2} + 4 \, a b d^{2} \cosh \left (f x + e\right ) + {\left (2 \, a^{2} + b^{2}\right )} d^{2} - 2 \, {\left ({\left (a b d^{2} f^{2} x^{2} + 2 \, a b c d f^{2} x + a b c^{2} f^{2}\right )} {\rm Ei}\left (\frac {d f x + c f}{d}\right ) + {\left (a b d^{2} f^{2} x^{2} + 2 \, a b c d f^{2} x + a b c^{2} f^{2}\right )} {\rm Ei}\left (-\frac {d f x + c f}{d}\right )\right )} \cosh \left (-\frac {d e - c f}{d}\right ) - 2 \, {\left ({\left (b^{2} d^{2} f^{2} x^{2} + 2 \, b^{2} c d f^{2} x + b^{2} c^{2} f^{2}\right )} {\rm Ei}\left (\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) + {\left (b^{2} d^{2} f^{2} x^{2} + 2 \, b^{2} c d f^{2} x + b^{2} c^{2} f^{2}\right )} {\rm Ei}\left (-\frac {2 \, {\left (d f x + c f\right )}}{d}\right )\right )} \cosh \left (-\frac {2 \, {\left (d e - c f\right )}}{d}\right ) + 4 \, {\left (a b d^{2} f x + a b c d f + {\left (b^{2} d^{2} f x + b^{2} c d f\right )} \cosh \left (f x + e\right )\right )} \sinh \left (f x + e\right ) + 2 \, {\left ({\left (a b d^{2} f^{2} x^{2} + 2 \, a b c d f^{2} x + a b c^{2} f^{2}\right )} {\rm Ei}\left (\frac {d f x + c f}{d}\right ) - {\left (a b d^{2} f^{2} x^{2} + 2 \, a b c d f^{2} x + a b c^{2} f^{2}\right )} {\rm Ei}\left (-\frac {d f x + c f}{d}\right )\right )} \sinh \left (-\frac {d e - c f}{d}\right ) + 2 \, {\left ({\left (b^{2} d^{2} f^{2} x^{2} + 2 \, b^{2} c d f^{2} x + b^{2} c^{2} f^{2}\right )} {\rm Ei}\left (\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) - {\left (b^{2} d^{2} f^{2} x^{2} + 2 \, b^{2} c d f^{2} x + b^{2} c^{2} f^{2}\right )} {\rm Ei}\left (-\frac {2 \, {\left (d f x + c f\right )}}{d}\right )\right )} \sinh \left (-\frac {2 \, {\left (d e - c f\right )}}{d}\right )}{4 \, {\left (d^{5} x^{2} + 2 \, c d^{4} x + c^{2} d^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [B]  time = 0.15, size = 702, normalized size = 2.90 \[ \frac {4 \, b^{2} d^{2} f^{2} x^{2} {\rm Ei}\left (-\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) e^{\left (\frac {2 \, c f}{d} - 2 \, e\right )} + 4 \, a b d^{2} f^{2} x^{2} {\rm Ei}\left (-\frac {d f x + c f}{d}\right ) e^{\left (\frac {c f}{d} - e\right )} + 4 \, a b d^{2} f^{2} x^{2} {\rm Ei}\left (\frac {d f x + c f}{d}\right ) e^{\left (-\frac {c f}{d} + e\right )} + 4 \, b^{2} d^{2} f^{2} x^{2} {\rm Ei}\left (\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) e^{\left (-\frac {2 \, c f}{d} + 2 \, e\right )} + 8 \, b^{2} c d f^{2} x {\rm Ei}\left (-\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) e^{\left (\frac {2 \, c f}{d} - 2 \, e\right )} + 8 \, a b c d f^{2} x {\rm Ei}\left (-\frac {d f x + c f}{d}\right ) e^{\left (\frac {c f}{d} - e\right )} + 8 \, a b c d f^{2} x {\rm Ei}\left (\frac {d f x + c f}{d}\right ) e^{\left (-\frac {c f}{d} + e\right )} + 8 \, b^{2} c d f^{2} x {\rm Ei}\left (\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) e^{\left (-\frac {2 \, c f}{d} + 2 \, e\right )} + 4 \, b^{2} c^{2} f^{2} {\rm Ei}\left (-\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) e^{\left (\frac {2 \, c f}{d} - 2 \, e\right )} + 4 \, a b c^{2} f^{2} {\rm Ei}\left (-\frac {d f x + c f}{d}\right ) e^{\left (\frac {c f}{d} - e\right )} + 4 \, a b c^{2} f^{2} {\rm Ei}\left (\frac {d f x + c f}{d}\right ) e^{\left (-\frac {c f}{d} + e\right )} + 4 \, b^{2} c^{2} f^{2} {\rm Ei}\left (\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) e^{\left (-\frac {2 \, c f}{d} + 2 \, e\right )} - 2 \, b^{2} d^{2} f x e^{\left (2 \, f x + 2 \, e\right )} - 4 \, a b d^{2} f x e^{\left (f x + e\right )} + 4 \, a b d^{2} f x e^{\left (-f x - e\right )} + 2 \, b^{2} d^{2} f x e^{\left (-2 \, f x - 2 \, e\right )} - 2 \, b^{2} c d f e^{\left (2 \, f x + 2 \, e\right )} - 4 \, a b c d f e^{\left (f x + e\right )} + 4 \, a b c d f e^{\left (-f x - e\right )} + 2 \, b^{2} c d f e^{\left (-2 \, f x - 2 \, e\right )} - b^{2} d^{2} e^{\left (2 \, f x + 2 \, e\right )} - 4 \, a b d^{2} e^{\left (f x + e\right )} - 4 \, a b d^{2} e^{\left (-f x - e\right )} - b^{2} d^{2} e^{\left (-2 \, f x - 2 \, e\right )} - 4 \, a^{2} d^{2} - 2 \, b^{2} d^{2}}{8 \, {\left (d^{5} x^{2} + 2 \, c d^{4} x + c^{2} d^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.38, size = 626, normalized size = 2.59 \[ \frac {f^{3} a b \,{\mathrm e}^{-f x -e} x}{2 d \left (d^{2} f^{2} x^{2}+2 c d \,f^{2} x +c^{2} f^{2}\right )}+\frac {f^{3} a b \,{\mathrm e}^{-f x -e} c}{2 d^{2} \left (d^{2} f^{2} x^{2}+2 c d \,f^{2} x +c^{2} f^{2}\right )}-\frac {f^{2} a b \,{\mathrm e}^{-f x -e}}{2 d \left (d^{2} f^{2} x^{2}+2 c d \,f^{2} x +c^{2} f^{2}\right )}-\frac {f^{2} a b \,{\mathrm e}^{\frac {c f -d e}{d}} \Ei \left (1, f x +e +\frac {c f -d e}{d}\right )}{2 d^{3}}-\frac {a b \,f^{2} {\mathrm e}^{f x +e}}{2 d^{3} \left (\frac {c f}{d}+f x \right )^{2}}-\frac {a b \,f^{2} {\mathrm e}^{f x +e}}{2 d^{3} \left (\frac {c f}{d}+f x \right )}-\frac {a b \,f^{2} {\mathrm e}^{-\frac {c f -d e}{d}} \Ei \left (1, -f x -e -\frac {c f -d e}{d}\right )}{2 d^{3}}-\frac {a^{2}}{2 d \left (d x +c \right )^{2}}-\frac {b^{2}}{4 d \left (d x +c \right )^{2}}+\frac {f^{3} b^{2} {\mathrm e}^{-2 f x -2 e} x}{4 d \left (d^{2} f^{2} x^{2}+2 c d \,f^{2} x +c^{2} f^{2}\right )}+\frac {f^{3} b^{2} {\mathrm e}^{-2 f x -2 e} c}{4 d^{2} \left (d^{2} f^{2} x^{2}+2 c d \,f^{2} x +c^{2} f^{2}\right )}-\frac {f^{2} b^{2} {\mathrm e}^{-2 f x -2 e}}{8 d \left (d^{2} f^{2} x^{2}+2 c d \,f^{2} x +c^{2} f^{2}\right )}-\frac {f^{2} b^{2} {\mathrm e}^{\frac {2 c f -2 d e}{d}} \Ei \left (1, 2 f x +2 e +\frac {2 c f -2 d e}{d}\right )}{2 d^{3}}-\frac {b^{2} f^{2} {\mathrm e}^{2 f x +2 e}}{8 d^{3} \left (\frac {c f}{d}+f x \right )^{2}}-\frac {b^{2} f^{2} {\mathrm e}^{2 f x +2 e}}{4 d^{3} \left (\frac {c f}{d}+f x \right )}-\frac {b^{2} f^{2} {\mathrm e}^{-\frac {2 \left (c f -d e \right )}{d}} \Ei \left (1, -2 f x -2 e -\frac {2 \left (c f -d e \right )}{d}\right )}{2 d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*cosh(f*x+e))^2/(d*x+c)^3,x)

[Out]

1/2*f^3*a*b*exp(-f*x-e)/d/(d^2*f^2*x^2+2*c*d*f^2*x+c^2*f^2)*x+1/2*f^3*a*b*exp(-f*x-e)/d^2/(d^2*f^2*x^2+2*c*d*f
^2*x+c^2*f^2)*c-1/2*f^2*a*b*exp(-f*x-e)/d/(d^2*f^2*x^2+2*c*d*f^2*x+c^2*f^2)-1/2*f^2*a*b/d^3*exp((c*f-d*e)/d)*E
i(1,f*x+e+(c*f-d*e)/d)-1/2*a*b*f^2/d^3*exp(f*x+e)/(c*f/d+f*x)^2-1/2*a*b*f^2/d^3*exp(f*x+e)/(c*f/d+f*x)-1/2*a*b
*f^2/d^3*exp(-(c*f-d*e)/d)*Ei(1,-f*x-e-(c*f-d*e)/d)-1/2*a^2/d/(d*x+c)^2-1/4*b^2/d/(d*x+c)^2+1/4*f^3*b^2*exp(-2
*f*x-2*e)/d/(d^2*f^2*x^2+2*c*d*f^2*x+c^2*f^2)*x+1/4*f^3*b^2*exp(-2*f*x-2*e)/d^2/(d^2*f^2*x^2+2*c*d*f^2*x+c^2*f
^2)*c-1/8*f^2*b^2*exp(-2*f*x-2*e)/d/(d^2*f^2*x^2+2*c*d*f^2*x+c^2*f^2)-1/2*f^2*b^2/d^3*exp(2*(c*f-d*e)/d)*Ei(1,
2*f*x+2*e+2*(c*f-d*e)/d)-1/8*b^2*f^2/d^3*exp(2*f*x+2*e)/(c*f/d+f*x)^2-1/4*b^2*f^2/d^3*exp(2*f*x+2*e)/(c*f/d+f*
x)-1/2*b^2*f^2/d^3*exp(-2*(c*f-d*e)/d)*Ei(1,-2*f*x-2*e-2*(c*f-d*e)/d)

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maxima [A]  time = 0.45, size = 201, normalized size = 0.83 \[ -\frac {1}{4} \, b^{2} {\left (\frac {1}{d^{3} x^{2} + 2 \, c d^{2} x + c^{2} d} + \frac {e^{\left (-2 \, e + \frac {2 \, c f}{d}\right )} E_{3}\left (\frac {2 \, {\left (d x + c\right )} f}{d}\right )}{{\left (d x + c\right )}^{2} d} + \frac {e^{\left (2 \, e - \frac {2 \, c f}{d}\right )} E_{3}\left (-\frac {2 \, {\left (d x + c\right )} f}{d}\right )}{{\left (d x + c\right )}^{2} d}\right )} - a b {\left (\frac {e^{\left (-e + \frac {c f}{d}\right )} E_{3}\left (\frac {{\left (d x + c\right )} f}{d}\right )}{{\left (d x + c\right )}^{2} d} + \frac {e^{\left (e - \frac {c f}{d}\right )} E_{3}\left (-\frac {{\left (d x + c\right )} f}{d}\right )}{{\left (d x + c\right )}^{2} d}\right )} - \frac {a^{2}}{2 \, {\left (d^{3} x^{2} + 2 \, c d^{2} x + c^{2} d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (a+b\,\mathrm {cosh}\left (e+f\,x\right )\right )}^2}{{\left (c+d\,x\right )}^3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*cosh(e + f*x))^2/(c + d*x)^3,x)

[Out]

int((a + b*cosh(e + f*x))^2/(c + d*x)^3, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b \cosh {\left (e + f x \right )}\right )^{2}}{\left (c + d x\right )^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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