3.164 \(\int (d x)^m (a+b \cosh ^{-1}(c x))^2 \, dx\)

Optimal. Leaf size=181 \[ -\frac{2 b^2 c^2 (d x)^{m+3} \text{HypergeometricPFQ}\left (\left \{1,\frac{m}{2}+\frac{3}{2},\frac{m}{2}+\frac{3}{2}\right \},\left \{\frac{m}{2}+2,\frac{m}{2}+\frac{5}{2}\right \},c^2 x^2\right )}{d^3 (m+1) (m+2) (m+3)}-\frac{2 b c \sqrt{1-c x} (d x)^{m+2} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{m+2}{2},\frac{m+4}{2},c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )}{d^2 (m+1) (m+2) \sqrt{c x-1}}+\frac{(d x)^{m+1} \left (a+b \cosh ^{-1}(c x)\right )^2}{d (m+1)} \]

[Out]

((d*x)^(1 + m)*(a + b*ArcCosh[c*x])^2)/(d*(1 + m)) - (2*b*c*(d*x)^(2 + m)*Sqrt[1 - c*x]*(a + b*ArcCosh[c*x])*H
ypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, c^2*x^2])/(d^2*(1 + m)*(2 + m)*Sqrt[-1 + c*x]) - (2*b^2*c^2*(d*x)^
(3 + m)*HypergeometricPFQ[{1, 3/2 + m/2, 3/2 + m/2}, {2 + m/2, 5/2 + m/2}, c^2*x^2])/(d^3*(1 + m)*(2 + m)*(3 +
m))

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Rubi [A]  time = 0.309833, antiderivative size = 194, normalized size of antiderivative = 1.07, number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {5662, 5763} \[ -\frac{2 b^2 c^2 (d x)^{m+3} \, _3F_2\left (1,\frac{m}{2}+\frac{3}{2},\frac{m}{2}+\frac{3}{2};\frac{m}{2}+2,\frac{m}{2}+\frac{5}{2};c^2 x^2\right )}{d^3 (m+1) (m+2) (m+3)}-\frac{2 b c \sqrt{1-c^2 x^2} (d x)^{m+2} \, _2F_1\left (\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )}{d^2 (m+1) (m+2) \sqrt{c x-1} \sqrt{c x+1}}+\frac{(d x)^{m+1} \left (a+b \cosh ^{-1}(c x)\right )^2}{d (m+1)} \]

Antiderivative was successfully verified.

[In]

Int[(d*x)^m*(a + b*ArcCosh[c*x])^2,x]

[Out]

((d*x)^(1 + m)*(a + b*ArcCosh[c*x])^2)/(d*(1 + m)) - (2*b*c*(d*x)^(2 + m)*Sqrt[1 - c^2*x^2]*(a + b*ArcCosh[c*x
])*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, c^2*x^2])/(d^2*(1 + m)*(2 + m)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) -
(2*b^2*c^2*(d*x)^(3 + m)*HypergeometricPFQ[{1, 3/2 + m/2, 3/2 + m/2}, {2 + m/2, 5/2 + m/2}, c^2*x^2])/(d^3*(1
+ m)*(2 + m)*(3 + m))

Rule 5662

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))/(Sqr
t[-1 + c*x]*Sqrt[1 + c*x]), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]

Rule 5763

Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_))/(Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_.)*(x_
)]), x_Symbol] :> Simp[((f*x)^(m + 1)*Sqrt[1 - c^2*x^2]*(a + b*ArcCosh[c*x])*Hypergeometric2F1[1/2, (1 + m)/2,
(3 + m)/2, c^2*x^2])/(f*(m + 1)*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x]), x] + Simp[(b*c*(f*x)^(m + 2)*Hypergeometric
PFQ[{1, 1 + m/2, 1 + m/2}, {3/2 + m/2, 2 + m/2}, c^2*x^2])/(Sqrt[-(d1*d2)]*f^2*(m + 1)*(m + 2)), x] /; FreeQ[{
a, b, c, d1, e1, d2, e2, f, m}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && GtQ[d1, 0] && LtQ[d2, 0] &&  !
IntegerQ[m]

Rubi steps

\begin{align*} \int (d x)^m \left (a+b \cosh ^{-1}(c x)\right )^2 \, dx &=\frac{(d x)^{1+m} \left (a+b \cosh ^{-1}(c x)\right )^2}{d (1+m)}-\frac{(2 b c) \int \frac{(d x)^{1+m} \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{d (1+m)}\\ &=\frac{(d x)^{1+m} \left (a+b \cosh ^{-1}(c x)\right )^2}{d (1+m)}-\frac{2 b c (d x)^{2+m} \sqrt{1-c^2 x^2} \left (a+b \cosh ^{-1}(c x)\right ) \, _2F_1\left (\frac{1}{2},\frac{2+m}{2};\frac{4+m}{2};c^2 x^2\right )}{d^2 (1+m) (2+m) \sqrt{-1+c x} \sqrt{1+c x}}-\frac{2 b^2 c^2 (d x)^{3+m} \, _3F_2\left (1,\frac{3}{2}+\frac{m}{2},\frac{3}{2}+\frac{m}{2};2+\frac{m}{2},\frac{5}{2}+\frac{m}{2};c^2 x^2\right )}{d^3 (1+m) (2+m) (3+m)}\\ \end{align*}

Mathematica [A]  time = 0.246303, size = 164, normalized size = 0.91 \[ \frac{x (d x)^m \left (-\frac{2 b^2 c^2 x^2 \text{HypergeometricPFQ}\left (\left \{1,\frac{m}{2}+\frac{3}{2},\frac{m}{2}+\frac{3}{2}\right \},\left \{\frac{m}{2}+2,\frac{m}{2}+\frac{5}{2}\right \},c^2 x^2\right )}{m^2+5 m+6}-\frac{2 b c x \sqrt{1-c^2 x^2} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{m+2}{2},\frac{m+4}{2},c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )}{(m+2) \sqrt{c x-1} \sqrt{c x+1}}+\left (a+b \cosh ^{-1}(c x)\right )^2\right )}{m+1} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(d*x)^m*((a + b*ArcCosh[c*x])^2 - (2*b*c*x*Sqrt[1 - c^2*x^2]*(a + b*ArcCosh[c*x])*Hypergeometric2F1[1/2, (2
+ m)/2, (4 + m)/2, c^2*x^2])/((2 + m)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (2*b^2*c^2*x^2*HypergeometricPFQ[{1, 3/
2 + m/2, 3/2 + m/2}, {2 + m/2, 5/2 + m/2}, c^2*x^2])/(6 + 5*m + m^2)))/(1 + m)

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Maple [F]  time = 2.246, size = 0, normalized size = 0. \begin{align*} \int \left ( dx \right ) ^{m} \left ( a+b{\rm arccosh} \left (cx\right ) \right ) ^{2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x)^m*(a+b*arccosh(c*x))^2,x)

[Out]

int((d*x)^m*(a+b*arccosh(c*x))^2,x)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (b^{2} \operatorname{arcosh}\left (c x\right )^{2} + 2 \, a b \operatorname{arcosh}\left (c x\right ) + a^{2}\right )} \left (d x\right )^{m}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b^2*arccosh(c*x)^2 + 2*a*b*arccosh(c*x) + a^2)*(d*x)^m, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((d*x)**m*(a + b*acosh(c*x))**2, x)

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Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out