3.2.7 \(\int (c e+d e x) (a+b \cosh ^{-1}(c+d x))^2 \, dx\) [107]

Optimal. Leaf size=110 \[ \frac {b^2 e (c+d x)^2}{4 d}-\frac {b e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )}{2 d}-\frac {e \left (a+b \cosh ^{-1}(c+d x)\right )^2}{4 d}+\frac {e (c+d x)^2 \left (a+b \cosh ^{-1}(c+d x)\right )^2}{2 d} \]

[Out]

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

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Rubi [A]
time = 0.16, antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {5996, 12, 5883, 5939, 5893, 30} \begin {gather*} \frac {e (c+d x)^2 \left (a+b \cosh ^{-1}(c+d x)\right )^2}{2 d}-\frac {b e \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x) \left (a+b \cosh ^{-1}(c+d x)\right )}{2 d}-\frac {e \left (a+b \cosh ^{-1}(c+d x)\right )^2}{4 d}+\frac {b^2 e (c+d x)^2}{4 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c*e + d*e*x)*(a + b*ArcCosh[c + d*x])^2,x]

[Out]

(b^2*e*(c + d*x)^2)/(4*d) - (b*e*Sqrt[-1 + c + d*x]*(c + d*x)*Sqrt[1 + c + d*x]*(a + b*ArcCosh[c + d*x]))/(2*d
) - (e*(a + b*ArcCosh[c + d*x])^2)/(4*d) + (e*(c + d*x)^2*(a + b*ArcCosh[c + d*x])^2)/(2*d)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 5883

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*ArcC
osh[c*x])^n/(d*(m + 1))), x] - Dist[b*c*(n/(d*(m + 1))), Int[(d*x)^(m + 1)*((a + b*ArcCosh[c*x])^(n - 1)/(Sqrt
[1 + c*x]*Sqrt[-1 + c*x])), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]

Rule 5893

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/(Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_.)*(x_)]), x_Symbol]
:> Simp[(1/(b*c*(n + 1)))*Simp[Sqrt[1 + c*x]/Sqrt[d1 + e1*x]]*Simp[Sqrt[-1 + c*x]/Sqrt[d2 + e2*x]]*(a + b*Arc
Cosh[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && EqQ[e1, c*d1] && EqQ[e2, (-c)*d2] && NeQ[n
, -1]

Rule 5939

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d1_) + (e1_.)*(x_))^(p_)*((d2_) + (e2_.)*(x_
))^(p_), x_Symbol] :> Simp[f*(f*x)^(m - 1)*(d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*((a + b*ArcCosh[c*x])^n/(e1
*e2*(m + 2*p + 1))), x] + (Dist[f^2*((m - 1)/(c^2*(m + 2*p + 1))), Int[(f*x)^(m - 2)*(d1 + e1*x)^p*(d2 + e2*x)
^p*(a + b*ArcCosh[c*x])^n, x], x] - Dist[b*f*(n/(c*(m + 2*p + 1)))*Simp[(d1 + e1*x)^p/(1 + c*x)^p]*Simp[(d2 +
e2*x)^p/(-1 + c*x)^p], Int[(f*x)^(m - 1)*(1 + c*x)^(p + 1/2)*(-1 + c*x)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n - 1)
, x], x]) /; FreeQ[{a, b, c, d1, e1, d2, e2, f, p}, x] && EqQ[e1, c*d1] && EqQ[e2, (-c)*d2] && GtQ[n, 0] && IG
tQ[m, 1] && NeQ[m + 2*p + 1, 0]

Rule 5996

Int[((a_.) + ArcCosh[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Dist[1/d, Subst[
Int[((d*e - c*f)/d + f*(x/d))^m*(a + b*ArcCosh[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x
]

Rubi steps

\begin {align*} \int (c e+d e x) \left (a+b \cosh ^{-1}(c+d x)\right )^2 \, dx &=\frac {\text {Subst}\left (\int e x \left (a+b \cosh ^{-1}(x)\right )^2 \, dx,x,c+d x\right )}{d}\\ &=\frac {e \text {Subst}\left (\int x \left (a+b \cosh ^{-1}(x)\right )^2 \, dx,x,c+d x\right )}{d}\\ &=\frac {e (c+d x)^2 \left (a+b \cosh ^{-1}(c+d x)\right )^2}{2 d}-\frac {(b e) \text {Subst}\left (\int \frac {x^2 \left (a+b \cosh ^{-1}(x)\right )}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{d}\\ &=-\frac {b e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )}{2 d}+\frac {e (c+d x)^2 \left (a+b \cosh ^{-1}(c+d x)\right )^2}{2 d}-\frac {(b e) \text {Subst}\left (\int \frac {a+b \cosh ^{-1}(x)}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{2 d}+\frac {\left (b^2 e\right ) \text {Subst}(\int x \, dx,x,c+d x)}{2 d}\\ &=\frac {b^2 e (c+d x)^2}{4 d}-\frac {b e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )}{2 d}-\frac {e \left (a+b \cosh ^{-1}(c+d x)\right )^2}{4 d}+\frac {e (c+d x)^2 \left (a+b \cosh ^{-1}(c+d x)\right )^2}{2 d}\\ \end {align*}

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Mathematica [A]
time = 0.14, size = 167, normalized size = 1.52 \begin {gather*} \frac {e \left ((c+d x) \left (2 a^2 (c+d x)+b^2 (c+d x)-2 a b \sqrt {-1+c+d x} \sqrt {1+c+d x}\right )-2 b (c+d x) \left (-2 a (c+d x)+b \sqrt {-1+c+d x} \sqrt {1+c+d x}\right ) \cosh ^{-1}(c+d x)+b^2 \left (-1+2 c^2+4 c d x+2 d^2 x^2\right ) \cosh ^{-1}(c+d x)^2-2 a b \log \left (c+d x+\sqrt {-1+c+d x} \sqrt {1+c+d x}\right )\right )}{4 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c*e + d*e*x)*(a + b*ArcCosh[c + d*x])^2,x]

[Out]

(e*((c + d*x)*(2*a^2*(c + d*x) + b^2*(c + d*x) - 2*a*b*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x]) - 2*b*(c + d*x)*(
-2*a*(c + d*x) + b*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x])*ArcCosh[c + d*x] + b^2*(-1 + 2*c^2 + 4*c*d*x + 2*d^2*
x^2)*ArcCosh[c + d*x]^2 - 2*a*b*Log[c + d*x + Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x]]))/(4*d)

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Maple [A]
time = 13.90, size = 182, normalized size = 1.65

method	result	size
derivativedivides	\(\frac {\frac {e \left (d x +c \right )^{2} a^{2}}{2}+e \,b^{2} \left (\frac {\mathrm {arccosh}\left (d x +c \right )^{2} \left (d x +c \right )^{2}}{2}-\frac {\mathrm {arccosh}\left (d x +c \right ) \left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{2}-\frac {\mathrm {arccosh}\left (d x +c \right )^{2}}{4}+\frac {\left (d x +c \right )^{2}}{4}\right )+e a b \left (d x +c \right )^{2} \mathrm {arccosh}\left (d x +c \right )-\frac {e a b \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (d x +c \right )}{2}-\frac {e a b \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \ln \left (d x +c +\sqrt {\left (d x +c \right )^{2}-1}\right )}{2 \sqrt {\left (d x +c \right )^{2}-1}}}{d}\)	\(182\)
default	\(\frac {\frac {e \left (d x +c \right )^{2} a^{2}}{2}+e \,b^{2} \left (\frac {\mathrm {arccosh}\left (d x +c \right )^{2} \left (d x +c \right )^{2}}{2}-\frac {\mathrm {arccosh}\left (d x +c \right ) \left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{2}-\frac {\mathrm {arccosh}\left (d x +c \right )^{2}}{4}+\frac {\left (d x +c \right )^{2}}{4}\right )+e a b \left (d x +c \right )^{2} \mathrm {arccosh}\left (d x +c \right )-\frac {e a b \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (d x +c \right )}{2}-\frac {e a b \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \ln \left (d x +c +\sqrt {\left (d x +c \right )^{2}-1}\right )}{2 \sqrt {\left (d x +c \right )^{2}-1}}}{d}\)	\(182\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*a^2*d*x^2*e + 1/2*(2*x^2*arccosh(d*x + c) - d*(3*c^2*log(2*d^2*x + 2*c*d + 2*sqrt(d^2*x^2 + 2*c*d*x + c^2
- 1)*d)/d^3 + sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*x/d^2 - (c^2 - 1)*log(2*d^2*x + 2*c*d + 2*sqrt(d^2*x^2 + 2*c*d
*x + c^2 - 1)*d)/d^3 - 3*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*c/d^3))*a*b*d*e + a^2*c*x*e + 2*((d*x + c)*arccosh(
d*x + c) - sqrt((d*x + c)^2 - 1))*a*b*c*e/d + 1/2*(b^2*d*x^2*e + 2*b^2*c*x*e)*log(d*x + sqrt(d*x + c + 1)*sqrt
(d*x + c - 1) + c)^2 - integrate((b^2*d^4*x^4*e + 4*b^2*c*d^3*x^3*e + (5*c^2*d^2 - d^2)*b^2*x^2*e + 2*(c^3*d -
c*d)*b^2*x*e + (b^2*d^3*x^3*e + 3*b^2*c*d^2*x^2*e + 2*b^2*c^2*d*x*e)*sqrt(d*x + c + 1)*sqrt(d*x + c - 1))*log
(d*x + sqrt(d*x + c + 1)*sqrt(d*x + c - 1) + c)/(d^3*x^3 + 3*c*d^2*x^2 + c^3 + (d^2*x^2 + 2*c*d*x + c^2 - 1)*s
qrt(d*x + c + 1)*sqrt(d*x + c - 1) + (3*c^2*d - d)*x - c), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 367 vs. \(2 (102) = 204\).
time = 0.38, size = 367, normalized size = 3.34 \begin {gather*} \frac {{\left ({\left (2 \, b^{2} d^{2} x^{2} + 4 \, b^{2} c d x + 2 \, b^{2} c^{2} - b^{2}\right )} \cosh \left (1\right ) + {\left (2 \, b^{2} d^{2} x^{2} + 4 \, b^{2} c d x + 2 \, b^{2} c^{2} - b^{2}\right )} \sinh \left (1\right )\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right )^{2} + {\left ({\left (2 \, a^{2} + b^{2}\right )} d^{2} x^{2} + 2 \, {\left (2 \, a^{2} + b^{2}\right )} c d x\right )} \cosh \left (1\right ) + 2 \, {\left ({\left (2 \, a b d^{2} x^{2} + 4 \, a b c d x + 2 \, a b c^{2} - a b\right )} \cosh \left (1\right ) + {\left (2 \, a b d^{2} x^{2} + 4 \, a b c d x + 2 \, a b c^{2} - a b\right )} \sinh \left (1\right ) - \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} {\left ({\left (b^{2} d x + b^{2} c\right )} \cosh \left (1\right ) + {\left (b^{2} d x + b^{2} c\right )} \sinh \left (1\right )\right )}\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right ) + {\left ({\left (2 \, a^{2} + b^{2}\right )} d^{2} x^{2} + 2 \, {\left (2 \, a^{2} + b^{2}\right )} c d x\right )} \sinh \left (1\right ) - 2 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} {\left ({\left (a b d x + a b c\right )} \cosh \left (1\right ) + {\left (a b d x + a b c\right )} \sinh \left (1\right )\right )}}{4 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 335 vs. \(2 (97) = 194\).
time = 0.22, size = 335, normalized size = 3.05 \begin {gather*} \begin {cases} a^{2} c e x + \frac {a^{2} d e x^{2}}{2} + \frac {a b c^{2} e \operatorname {acosh}{\left (c + d x \right )}}{d} + 2 a b c e x \operatorname {acosh}{\left (c + d x \right )} - \frac {a b c e \sqrt {c^{2} + 2 c d x + d^{2} x^{2} - 1}}{2 d} + a b d e x^{2} \operatorname {acosh}{\left (c + d x \right )} - \frac {a b e x \sqrt {c^{2} + 2 c d x + d^{2} x^{2} - 1}}{2} - \frac {a b e \operatorname {acosh}{\left (c + d x \right )}}{2 d} + \frac {b^{2} c^{2} e \operatorname {acosh}^{2}{\left (c + d x \right )}}{2 d} + b^{2} c e x \operatorname {acosh}^{2}{\left (c + d x \right )} + \frac {b^{2} c e x}{2} - \frac {b^{2} c e \sqrt {c^{2} + 2 c d x + d^{2} x^{2} - 1} \operatorname {acosh}{\left (c + d x \right )}}{2 d} + \frac {b^{2} d e x^{2} \operatorname {acosh}^{2}{\left (c + d x \right )}}{2} + \frac {b^{2} d e x^{2}}{4} - \frac {b^{2} e x \sqrt {c^{2} + 2 c d x + d^{2} x^{2} - 1} \operatorname {acosh}{\left (c + d x \right )}}{2} - \frac {b^{2} e \operatorname {acosh}^{2}{\left (c + d x \right )}}{4 d} & \text {for}\: d \neq 0 \\c e x \left (a + b \operatorname {acosh}{\left (c \right )}\right )^{2} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)*(a+b*acosh(d*x+c))**2,x)

[Out]

Piecewise((a**2*c*e*x + a**2*d*e*x**2/2 + a*b*c**2*e*acosh(c + d*x)/d + 2*a*b*c*e*x*acosh(c + d*x) - a*b*c*e*s
qrt(c**2 + 2*c*d*x + d**2*x**2 - 1)/(2*d) + a*b*d*e*x**2*acosh(c + d*x) - a*b*e*x*sqrt(c**2 + 2*c*d*x + d**2*x
**2 - 1)/2 - a*b*e*acosh(c + d*x)/(2*d) + b**2*c**2*e*acosh(c + d*x)**2/(2*d) + b**2*c*e*x*acosh(c + d*x)**2 +
b**2*c*e*x/2 - b**2*c*e*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)*acosh(c + d*x)/(2*d) + b**2*d*e*x**2*acosh(c + d
*x)**2/2 + b**2*d*e*x**2/4 - b**2*e*x*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)*acosh(c + d*x)/2 - b**2*e*acosh(c +
d*x)**2/(4*d), Ne(d, 0)), (c*e*x*(a + b*acosh(c))**2, True))

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((d*e*x + c*e)*(b*arccosh(d*x + c) + a)^2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*e + d*e*x)*(a + b*acosh(c + d*x))^2,x)

[Out]

int((c*e + d*e*x)*(a + b*acosh(c + d*x))^2, x)

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