Optimal. Leaf size=243 \[ -\frac{\sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{c^6 d^3}-\frac{2 \left (a+b \cosh ^{-1}(c x)\right )}{c^6 d^2 \sqrt{d-c^2 d x^2}}+\frac{a+b \cosh ^{-1}(c x)}{3 c^6 d \left (d-c^2 d x^2\right )^{3/2}}+\frac{b x \sqrt{d-c^2 d x^2}}{c^5 d^3 \sqrt{c x-1} \sqrt{c x+1}}-\frac{b x \sqrt{d-c^2 d x^2}}{6 c^5 d^3 \sqrt{c x-1} \sqrt{c x+1} \left (1-c^2 x^2\right )}+\frac{11 b \sqrt{d-c^2 d x^2} \tanh ^{-1}(c x)}{6 c^6 d^3 \sqrt{c x-1} \sqrt{c x+1}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.442693, antiderivative size = 280, normalized size of antiderivative = 1.15, number of steps used = 6, number of rules used = 9, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {5798, 98, 21, 74, 5733, 12, 1157, 388, 206} \[ \frac{x^4 \left (a+b \cosh ^{-1}(c x)\right )}{3 c^2 d^2 (1-c x) (c x+1) \sqrt{d-c^2 d x^2}}-\frac{4 x^2 \left (a+b \cosh ^{-1}(c x)\right )}{3 c^4 d^2 \sqrt{d-c^2 d x^2}}-\frac{8 (1-c x) (c x+1) \left (a+b \cosh ^{-1}(c x)\right )}{3 c^6 d^2 \sqrt{d-c^2 d x^2}}-\frac{b x \sqrt{c x-1} \sqrt{c x+1}}{c^5 d^2 \sqrt{d-c^2 d x^2}}+\frac{b x \sqrt{c x-1} \sqrt{c x+1}}{6 c^5 d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}-\frac{11 b \sqrt{c x-1} \sqrt{c x+1} \tanh ^{-1}(c x)}{6 c^6 d^2 \sqrt{d-c^2 d x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5798
Rule 98
Rule 21
Rule 74
Rule 5733
Rule 12
Rule 1157
Rule 388
Rule 206
Rubi steps
\begin{align*} \int \frac{x^5 \left (a+b \cosh ^{-1}(c x)\right )}{\left (d-c^2 d x^2\right )^{5/2}} \, dx &=\frac{\left (\sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{x^5 \left (a+b \cosh ^{-1}(c x)\right )}{(-1+c x)^{5/2} (1+c x)^{5/2}} \, dx}{d^2 \sqrt{d-c^2 d x^2}}\\ &=-\frac{4 x^2 \left (a+b \cosh ^{-1}(c x)\right )}{3 c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{x^4 \left (a+b \cosh ^{-1}(c x)\right )}{3 c^2 d^2 (1-c x) (1+c x) \sqrt{d-c^2 d x^2}}-\frac{8 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{3 c^6 d^2 \sqrt{d-c^2 d x^2}}-\frac{\left (b c \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{8-12 c^2 x^2+3 c^4 x^4}{3 c^6 \left (1-c^2 x^2\right )^2} \, dx}{d^2 \sqrt{d-c^2 d x^2}}\\ &=-\frac{4 x^2 \left (a+b \cosh ^{-1}(c x)\right )}{3 c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{x^4 \left (a+b \cosh ^{-1}(c x)\right )}{3 c^2 d^2 (1-c x) (1+c x) \sqrt{d-c^2 d x^2}}-\frac{8 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{3 c^6 d^2 \sqrt{d-c^2 d x^2}}-\frac{\left (b \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{8-12 c^2 x^2+3 c^4 x^4}{\left (1-c^2 x^2\right )^2} \, dx}{3 c^5 d^2 \sqrt{d-c^2 d x^2}}\\ &=\frac{b x \sqrt{-1+c x} \sqrt{1+c x}}{6 c^5 d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}-\frac{4 x^2 \left (a+b \cosh ^{-1}(c x)\right )}{3 c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{x^4 \left (a+b \cosh ^{-1}(c x)\right )}{3 c^2 d^2 (1-c x) (1+c x) \sqrt{d-c^2 d x^2}}-\frac{8 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{3 c^6 d^2 \sqrt{d-c^2 d x^2}}+\frac{\left (b \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{-17+6 c^2 x^2}{1-c^2 x^2} \, dx}{6 c^5 d^2 \sqrt{d-c^2 d x^2}}\\ &=-\frac{b x \sqrt{-1+c x} \sqrt{1+c x}}{c^5 d^2 \sqrt{d-c^2 d x^2}}+\frac{b x \sqrt{-1+c x} \sqrt{1+c x}}{6 c^5 d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}-\frac{4 x^2 \left (a+b \cosh ^{-1}(c x)\right )}{3 c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{x^4 \left (a+b \cosh ^{-1}(c x)\right )}{3 c^2 d^2 (1-c x) (1+c x) \sqrt{d-c^2 d x^2}}-\frac{8 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{3 c^6 d^2 \sqrt{d-c^2 d x^2}}-\frac{\left (11 b \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{1}{1-c^2 x^2} \, dx}{6 c^5 d^2 \sqrt{d-c^2 d x^2}}\\ &=-\frac{b x \sqrt{-1+c x} \sqrt{1+c x}}{c^5 d^2 \sqrt{d-c^2 d x^2}}+\frac{b x \sqrt{-1+c x} \sqrt{1+c x}}{6 c^5 d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}-\frac{4 x^2 \left (a+b \cosh ^{-1}(c x)\right )}{3 c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{x^4 \left (a+b \cosh ^{-1}(c x)\right )}{3 c^2 d^2 (1-c x) (1+c x) \sqrt{d-c^2 d x^2}}-\frac{8 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{3 c^6 d^2 \sqrt{d-c^2 d x^2}}-\frac{11 b \sqrt{-1+c x} \sqrt{1+c x} \tanh ^{-1}(c x)}{6 c^6 d^2 \sqrt{d-c^2 d x^2}}\\ \end{align*}
Mathematica [A] time = 0.165248, size = 167, normalized size = 0.69 \[ \frac{6 a c^4 x^4-24 a c^2 x^2+16 a-6 b c^3 x^3 \sqrt{c x-1} \sqrt{c x+1}+2 b \left (3 c^4 x^4-12 c^2 x^2+8\right ) \cosh ^{-1}(c x)-11 b \sqrt{c x-1} \sqrt{c x+1} \left (c^2 x^2-1\right ) \tanh ^{-1}(c x)+5 b c x \sqrt{c x-1} \sqrt{c x+1}}{6 c^6 d^2 \left (c^2 x^2-1\right ) \sqrt{d-c^2 d x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.273, size = 466, normalized size = 1.9 \begin{align*} -{\frac{{x}^{4}a}{{c}^{2}d} \left ( -{c}^{2}d{x}^{2}+d \right ) ^{-{\frac{3}{2}}}}+4\,{\frac{a{x}^{2}}{d{c}^{4} \left ( -{c}^{2}d{x}^{2}+d \right ) ^{3/2}}}-{\frac{8\,a}{3\,d{c}^{6}} \left ( -{c}^{2}d{x}^{2}+d \right ) ^{-{\frac{3}{2}}}}-{\frac{b{x}^{2}{\rm arccosh} \left (cx\right )}{{c}^{4}{d}^{3} \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }}+{\frac{bx}{{c}^{5}{d}^{3} \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{cx+1}\sqrt{cx-1}}+{\frac{b{\rm arccosh} \left (cx\right )}{{d}^{3}{c}^{6} \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }}+2\,{\frac{b\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }{\rm arccosh} \left (cx\right ){x}^{2}}{{d}^{3} \left ({c}^{2}{x}^{2}-1 \right ) ^{2}{c}^{4}}}+{\frac{bx}{6\,{d}^{3} \left ({c}^{2}{x}^{2}-1 \right ) ^{2}{c}^{5}}\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{cx+1}\sqrt{cx-1}}-{\frac{5\,b{\rm arccosh} \left (cx\right )}{3\,{d}^{3} \left ({c}^{2}{x}^{2}-1 \right ) ^{2}{c}^{6}}\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }}-{\frac{11\,b}{6\,{d}^{3}{c}^{6} \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{cx+1}\sqrt{cx-1}\ln \left ( cx+\sqrt{cx-1}\sqrt{cx+1}-1 \right ) }+{\frac{11\,b}{6\,{d}^{3}{c}^{6} \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{cx+1}\sqrt{cx-1}\ln \left ( 1+cx+\sqrt{cx-1}\sqrt{cx+1} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.6752, size = 1149, normalized size = 4.73 \begin{align*} \left [-\frac{8 \,{\left (3 \, b c^{4} x^{4} - 12 \, b c^{2} x^{2} + 8 \, b\right )} \sqrt{-c^{2} d x^{2} + d} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) + 11 \,{\left (b c^{4} x^{4} - 2 \, b c^{2} x^{2} + b\right )} \sqrt{-d} \log \left (-\frac{c^{6} d x^{6} + 5 \, c^{4} d x^{4} - 5 \, c^{2} d x^{2} + 4 \,{\left (c^{3} x^{3} + c x\right )} \sqrt{-c^{2} d x^{2} + d} \sqrt{c^{2} x^{2} - 1} \sqrt{-d} - d}{c^{6} x^{6} - 3 \, c^{4} x^{4} + 3 \, c^{2} x^{2} - 1}\right ) - 4 \,{\left (6 \, b c^{3} x^{3} - 5 \, b c x\right )} \sqrt{-c^{2} d x^{2} + d} \sqrt{c^{2} x^{2} - 1} + 8 \,{\left (3 \, a c^{4} x^{4} - 12 \, a c^{2} x^{2} + 8 \, a\right )} \sqrt{-c^{2} d x^{2} + d}}{24 \,{\left (c^{10} d^{3} x^{4} - 2 \, c^{8} d^{3} x^{2} + c^{6} d^{3}\right )}}, \frac{11 \,{\left (b c^{4} x^{4} - 2 \, b c^{2} x^{2} + b\right )} \sqrt{d} \arctan \left (\frac{2 \, \sqrt{-c^{2} d x^{2} + d} \sqrt{c^{2} x^{2} - 1} c \sqrt{d} x}{c^{4} d x^{4} - d}\right ) - 4 \,{\left (3 \, b c^{4} x^{4} - 12 \, b c^{2} x^{2} + 8 \, b\right )} \sqrt{-c^{2} d x^{2} + d} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) + 2 \,{\left (6 \, b c^{3} x^{3} - 5 \, b c x\right )} \sqrt{-c^{2} d x^{2} + d} \sqrt{c^{2} x^{2} - 1} - 4 \,{\left (3 \, a c^{4} x^{4} - 12 \, a c^{2} x^{2} + 8 \, a\right )} \sqrt{-c^{2} d x^{2} + d}}{12 \,{\left (c^{10} d^{3} x^{4} - 2 \, c^{8} d^{3} x^{2} + c^{6} d^{3}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \operatorname{arcosh}\left (c x\right ) + a\right )} x^{5}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]