Optimal. Leaf size=336 \[ -\frac{d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^3}{8 b c \sqrt{c x-1} \sqrt{c x+1}}+\frac{1}{4} x \left (d-c^2 d x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac{3}{8} d x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac{b d \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{8 c \sqrt{c x-1} \sqrt{c x+1}}-\frac{3 b c d x^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{8 \sqrt{c x-1} \sqrt{c x+1}}+\frac{15}{64} b^2 d x \sqrt{d-c^2 d x^2}+\frac{1}{32} b^2 d x (1-c x) (c x+1) \sqrt{d-c^2 d x^2}+\frac{9 b^2 d \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x)}{64 c \sqrt{c x-1} \sqrt{c x+1}} \]
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Rubi [A] time = 0.608932, antiderivative size = 348, normalized size of antiderivative = 1.04, number of steps used = 11, number of rules used = 9, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.346, Rules used = {5713, 5685, 5683, 5676, 5662, 90, 52, 5716, 38} \[ -\frac{d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^3}{8 b c \sqrt{c x-1} \sqrt{c x+1}}+\frac{3}{8} d x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac{1}{4} d x (1-c x) (c x+1) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac{b d \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{8 c \sqrt{c x-1} \sqrt{c x+1}}-\frac{3 b c d x^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{8 \sqrt{c x-1} \sqrt{c x+1}}+\frac{15}{64} b^2 d x \sqrt{d-c^2 d x^2}+\frac{1}{32} b^2 d x (1-c x) (c x+1) \sqrt{d-c^2 d x^2}+\frac{9 b^2 d \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x)}{64 c \sqrt{c x-1} \sqrt{c x+1}} \]
Antiderivative was successfully verified.
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Rule 5713
Rule 5685
Rule 5683
Rule 5676
Rule 5662
Rule 90
Rule 52
Rule 5716
Rule 38
Rubi steps
\begin{align*} \int \left (d-c^2 d x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )^2 \, dx &=-\frac{\left (d \sqrt{d-c^2 d x^2}\right ) \int (-1+c x)^{3/2} (1+c x)^{3/2} \left (a+b \cosh ^{-1}(c x)\right )^2 \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{1}{4} d x (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac{\left (3 d \sqrt{d-c^2 d x^2}\right ) \int \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2 \, dx}{4 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (b c d \sqrt{d-c^2 d x^2}\right ) \int x \left (-1+c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{2 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{b d \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{8 c \sqrt{-1+c x} \sqrt{1+c x}}+\frac{3}{8} d x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac{1}{4} d x (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac{\left (3 d \sqrt{d-c^2 d x^2}\right ) \int \frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{8 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (b^2 d \sqrt{d-c^2 d x^2}\right ) \int (-1+c x)^{3/2} (1+c x)^{3/2} \, dx}{8 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (3 b c d \sqrt{d-c^2 d x^2}\right ) \int x \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{4 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{1}{32} b^2 d x (1-c x) (1+c x) \sqrt{d-c^2 d x^2}-\frac{3 b c d x^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{8 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b d \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{8 c \sqrt{-1+c x} \sqrt{1+c x}}+\frac{3}{8} d x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac{1}{4} d x (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac{d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^3}{8 b c \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (3 b^2 d \sqrt{d-c^2 d x^2}\right ) \int \sqrt{-1+c x} \sqrt{1+c x} \, dx}{32 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (3 b^2 c^2 d \sqrt{d-c^2 d x^2}\right ) \int \frac{x^2}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{8 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{15}{64} b^2 d x \sqrt{d-c^2 d x^2}+\frac{1}{32} b^2 d x (1-c x) (1+c x) \sqrt{d-c^2 d x^2}-\frac{3 b c d x^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{8 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b d \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{8 c \sqrt{-1+c x} \sqrt{1+c x}}+\frac{3}{8} d x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac{1}{4} d x (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac{d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^3}{8 b c \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (3 b^2 d \sqrt{d-c^2 d x^2}\right ) \int \frac{1}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{64 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (3 b^2 d \sqrt{d-c^2 d x^2}\right ) \int \frac{1}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{16 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{15}{64} b^2 d x \sqrt{d-c^2 d x^2}+\frac{1}{32} b^2 d x (1-c x) (1+c x) \sqrt{d-c^2 d x^2}+\frac{9 b^2 d \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x)}{64 c \sqrt{-1+c x} \sqrt{1+c x}}-\frac{3 b c d x^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{8 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b d \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{8 c \sqrt{-1+c x} \sqrt{1+c x}}+\frac{3}{8} d x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac{1}{4} d x (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac{d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^3}{8 b c \sqrt{-1+c x} \sqrt{1+c x}}\\ \end{align*}
Mathematica [A] time = 2.92394, size = 374, normalized size = 1.11 \[ \frac{-288 a^2 d^{3/2} \sqrt{\frac{c x-1}{c x+1}} (c x+1) \tan ^{-1}\left (\frac{c x \sqrt{d-c^2 d x^2}}{\sqrt{d} \left (c^2 x^2-1\right )}\right )-96 a^2 c d x \sqrt{\frac{c x-1}{c x+1}} (c x+1) \left (2 c^2 x^2-5\right ) \sqrt{d-c^2 d x^2}-192 a b d \sqrt{d-c^2 d x^2} \left (\cosh \left (2 \cosh ^{-1}(c x)\right )+2 \cosh ^{-1}(c x) \left (\cosh ^{-1}(c x)-\sinh \left (2 \cosh ^{-1}(c x)\right )\right )\right )+12 a b d \sqrt{d-c^2 d x^2} \left (8 \cosh ^{-1}(c x)^2+\cosh \left (4 \cosh ^{-1}(c x)\right )-4 \cosh ^{-1}(c x) \sinh \left (4 \cosh ^{-1}(c x)\right )\right )-32 b^2 d \sqrt{d-c^2 d x^2} \left (4 \cosh ^{-1}(c x)^3+6 \cosh \left (2 \cosh ^{-1}(c x)\right ) \cosh ^{-1}(c x)-3 \left (2 \cosh ^{-1}(c x)^2+1\right ) \sinh \left (2 \cosh ^{-1}(c x)\right )\right )+b^2 d \sqrt{d-c^2 d x^2} \left (32 \cosh ^{-1}(c x)^3+12 \cosh \left (4 \cosh ^{-1}(c x)\right ) \cosh ^{-1}(c x)-3 \left (8 \cosh ^{-1}(c x)^2+1\right ) \sinh \left (4 \cosh ^{-1}(c x)\right )\right )}{768 c \sqrt{\frac{c x-1}{c x+1}} (c x+1)} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.263, size = 775, normalized size = 2.3 \begin{align*} \text{result too large to display} \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^{2} c^{2} d x^{2} - a^{2} d +{\left (b^{2} c^{2} d x^{2} - b^{2} d\right )} \operatorname{arcosh}\left (c x\right )^{2} + 2 \,{\left (a b c^{2} d x^{2} - a b d\right )} \operatorname{arcosh}\left (c x\right )\right )} \sqrt{-c^{2} d x^{2} + d}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac{3}{2}} \left (a + b \operatorname{acosh}{\left (c x \right )}\right )^{2}\, dx \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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