Optimal. Leaf size=307 \[ -\frac{b c d^2 \left (15 m^2+100 m+149\right ) \sqrt{1-c^2 x^2} (f x)^{m+2} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{m+2}{2},\frac{m+4}{2},c^2 x^2\right )}{f^2 (m+1) (m+2) (m+3)^2 (m+5)^2 \sqrt{c x-1} \sqrt{c x+1}}-\frac{2 c^2 d^2 (f x)^{m+3} \left (a+b \cosh ^{-1}(c x)\right )}{f^3 (m+3)}+\frac{c^4 d^2 (f x)^{m+5} \left (a+b \cosh ^{-1}(c x)\right )}{f^5 (m+5)}+\frac{d^2 (f x)^{m+1} \left (a+b \cosh ^{-1}(c x)\right )}{f (m+1)}-\frac{b c d^2 \left (m^2+13 m+38\right ) \left (1-c^2 x^2\right ) (f x)^{m+2}}{f^2 (m+3)^2 (m+5)^2 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b c^3 d^2 \left (1-c^2 x^2\right ) (f x)^{m+4}}{f^4 (m+5)^2 \sqrt{c x-1} \sqrt{c x+1}} \]
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Rubi [A] time = 0.500995, antiderivative size = 307, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 8, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.296, Rules used = {270, 5731, 12, 520, 1267, 459, 365, 364} \[ -\frac{2 c^2 d^2 (f x)^{m+3} \left (a+b \cosh ^{-1}(c x)\right )}{f^3 (m+3)}+\frac{c^4 d^2 (f x)^{m+5} \left (a+b \cosh ^{-1}(c x)\right )}{f^5 (m+5)}+\frac{d^2 (f x)^{m+1} \left (a+b \cosh ^{-1}(c x)\right )}{f (m+1)}-\frac{b c d^2 \left (15 m^2+100 m+149\right ) \sqrt{1-c^2 x^2} (f x)^{m+2} \, _2F_1\left (\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};c^2 x^2\right )}{f^2 (m+1) (m+2) (m+3)^2 (m+5)^2 \sqrt{c x-1} \sqrt{c x+1}}-\frac{b c d^2 \left (m^2+13 m+38\right ) \left (1-c^2 x^2\right ) (f x)^{m+2}}{f^2 (m+3)^2 (m+5)^2 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b c^3 d^2 \left (1-c^2 x^2\right ) (f x)^{m+4}}{f^4 (m+5)^2 \sqrt{c x-1} \sqrt{c x+1}} \]
Antiderivative was successfully verified.
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Rule 270
Rule 5731
Rule 12
Rule 520
Rule 1267
Rule 459
Rule 365
Rule 364
Rubi steps
\begin{align*} \int (f x)^m \left (d-c^2 d x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right ) \, dx &=\frac{d^2 (f x)^{1+m} \left (a+b \cosh ^{-1}(c x)\right )}{f (1+m)}-\frac{2 c^2 d^2 (f x)^{3+m} \left (a+b \cosh ^{-1}(c x)\right )}{f^3 (3+m)}+\frac{c^4 d^2 (f x)^{5+m} \left (a+b \cosh ^{-1}(c x)\right )}{f^5 (5+m)}-(b c) \int \frac{d^2 (f x)^{1+m} \left (\frac{1}{1+m}-\frac{2 c^2 x^2}{3+m}+\frac{c^4 x^4}{5+m}\right )}{f \sqrt{-1+c x} \sqrt{1+c x}} \, dx\\ &=\frac{d^2 (f x)^{1+m} \left (a+b \cosh ^{-1}(c x)\right )}{f (1+m)}-\frac{2 c^2 d^2 (f x)^{3+m} \left (a+b \cosh ^{-1}(c x)\right )}{f^3 (3+m)}+\frac{c^4 d^2 (f x)^{5+m} \left (a+b \cosh ^{-1}(c x)\right )}{f^5 (5+m)}-\frac{\left (b c d^2\right ) \int \frac{(f x)^{1+m} \left (\frac{1}{1+m}-\frac{2 c^2 x^2}{3+m}+\frac{c^4 x^4}{5+m}\right )}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{f}\\ &=\frac{d^2 (f x)^{1+m} \left (a+b \cosh ^{-1}(c x)\right )}{f (1+m)}-\frac{2 c^2 d^2 (f x)^{3+m} \left (a+b \cosh ^{-1}(c x)\right )}{f^3 (3+m)}+\frac{c^4 d^2 (f x)^{5+m} \left (a+b \cosh ^{-1}(c x)\right )}{f^5 (5+m)}-\frac{\left (b c d^2 \sqrt{-1+c^2 x^2}\right ) \int \frac{(f x)^{1+m} \left (\frac{1}{1+m}-\frac{2 c^2 x^2}{3+m}+\frac{c^4 x^4}{5+m}\right )}{\sqrt{-1+c^2 x^2}} \, dx}{f \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{b c^3 d^2 (f x)^{4+m} \left (1-c^2 x^2\right )}{f^4 (5+m)^2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{d^2 (f x)^{1+m} \left (a+b \cosh ^{-1}(c x)\right )}{f (1+m)}-\frac{2 c^2 d^2 (f x)^{3+m} \left (a+b \cosh ^{-1}(c x)\right )}{f^3 (3+m)}+\frac{c^4 d^2 (f x)^{5+m} \left (a+b \cosh ^{-1}(c x)\right )}{f^5 (5+m)}-\frac{\left (b d^2 \sqrt{-1+c^2 x^2}\right ) \int \frac{(f x)^{1+m} \left (\frac{c^2 (5+m)}{1+m}-\frac{c^4 \left (38+13 m+m^2\right ) x^2}{(3+m) (5+m)}\right )}{\sqrt{-1+c^2 x^2}} \, dx}{c f (5+m) \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{b c d^2 \left (38+13 m+m^2\right ) (f x)^{2+m} \left (1-c^2 x^2\right )}{f^2 (3+m)^2 (5+m)^2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b c^3 d^2 (f x)^{4+m} \left (1-c^2 x^2\right )}{f^4 (5+m)^2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{d^2 (f x)^{1+m} \left (a+b \cosh ^{-1}(c x)\right )}{f (1+m)}-\frac{2 c^2 d^2 (f x)^{3+m} \left (a+b \cosh ^{-1}(c x)\right )}{f^3 (3+m)}+\frac{c^4 d^2 (f x)^{5+m} \left (a+b \cosh ^{-1}(c x)\right )}{f^5 (5+m)}-\frac{\left (b c d^2 \left (149+100 m+15 m^2\right ) \sqrt{-1+c^2 x^2}\right ) \int \frac{(f x)^{1+m}}{\sqrt{-1+c^2 x^2}} \, dx}{f (1+m) (3+m)^2 (5+m)^2 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{b c d^2 \left (38+13 m+m^2\right ) (f x)^{2+m} \left (1-c^2 x^2\right )}{f^2 (3+m)^2 (5+m)^2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b c^3 d^2 (f x)^{4+m} \left (1-c^2 x^2\right )}{f^4 (5+m)^2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{d^2 (f x)^{1+m} \left (a+b \cosh ^{-1}(c x)\right )}{f (1+m)}-\frac{2 c^2 d^2 (f x)^{3+m} \left (a+b \cosh ^{-1}(c x)\right )}{f^3 (3+m)}+\frac{c^4 d^2 (f x)^{5+m} \left (a+b \cosh ^{-1}(c x)\right )}{f^5 (5+m)}-\frac{\left (b c d^2 \left (149+100 m+15 m^2\right ) \sqrt{1-c^2 x^2}\right ) \int \frac{(f x)^{1+m}}{\sqrt{1-c^2 x^2}} \, dx}{f (1+m) (3+m)^2 (5+m)^2 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{b c d^2 \left (38+13 m+m^2\right ) (f x)^{2+m} \left (1-c^2 x^2\right )}{f^2 (3+m)^2 (5+m)^2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b c^3 d^2 (f x)^{4+m} \left (1-c^2 x^2\right )}{f^4 (5+m)^2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{d^2 (f x)^{1+m} \left (a+b \cosh ^{-1}(c x)\right )}{f (1+m)}-\frac{2 c^2 d^2 (f x)^{3+m} \left (a+b \cosh ^{-1}(c x)\right )}{f^3 (3+m)}+\frac{c^4 d^2 (f x)^{5+m} \left (a+b \cosh ^{-1}(c x)\right )}{f^5 (5+m)}-\frac{b c d^2 \left (149+100 m+15 m^2\right ) (f x)^{2+m} \sqrt{1-c^2 x^2} \, _2F_1\left (\frac{1}{2},\frac{2+m}{2};\frac{4+m}{2};c^2 x^2\right )}{f^2 (1+m) (2+m) (3+m)^2 (5+m)^2 \sqrt{-1+c x} \sqrt{1+c x}}\\ \end{align*}
Mathematica [A] time = 0.436913, size = 290, normalized size = 0.94 \[ d^2 x (f x)^m \left (\frac{2 b c^3 x^3 \sqrt{1-c^2 x^2} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{m+4}{2},\frac{m+6}{2},c^2 x^2\right )}{\left (m^2+7 m+12\right ) \sqrt{c x-1} \sqrt{c x+1}}-\frac{b c x \sqrt{1-c^2 x^2} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{m+2}{2},\frac{m+4}{2},c^2 x^2\right )}{\left (m^2+3 m+2\right ) \sqrt{c x-1} \sqrt{c x+1}}-\frac{b c^5 x^5 \sqrt{1-c^2 x^2} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{m+6}{2},\frac{m+8}{2},c^2 x^2\right )}{(m+5) (m+6) \sqrt{c x-1} \sqrt{c x+1}}+\frac{c^4 x^4 \left (a+b \cosh ^{-1}(c x)\right )}{m+5}-\frac{2 c^2 x^2 \left (a+b \cosh ^{-1}(c x)\right )}{m+3}+\frac{a+b \cosh ^{-1}(c x)}{m+1}\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 2.434, size = 0, normalized size = 0. \begin{align*} \int \left ( fx \right ) ^{m} \left ( -{c}^{2}d{x}^{2}+d \right ) ^{2} \left ( a+b{\rm arccosh} \left (cx\right ) \right ) \, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (a c^{4} d^{2} x^{4} - 2 \, a c^{2} d^{2} x^{2} + a d^{2} +{\left (b c^{4} d^{2} x^{4} - 2 \, b c^{2} d^{2} x^{2} + b d^{2}\right )} \operatorname{arcosh}\left (c x\right )\right )} \left (f x\right )^{m}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [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.
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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.
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