Optimal. Leaf size=278 \[ \frac{1}{6} x^5 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac{x^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{24 c^2}-\frac{x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{16 c^4}-\frac{\sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{32 b c^5 \sqrt{c x-1} \sqrt{c x+1}}-\frac{b c x^6 \sqrt{d-c^2 d x^2}}{36 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b x^4 \sqrt{d-c^2 d x^2}}{96 c \sqrt{c x-1} \sqrt{c x+1}}+\frac{b x^2 \sqrt{d-c^2 d x^2}}{32 c^3 \sqrt{c x-1} \sqrt{c x+1}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.782305, antiderivative size = 278, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185, Rules used = {5798, 5743, 5759, 5676, 30} \[ \frac{1}{6} x^5 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac{x^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{24 c^2}-\frac{x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{16 c^4}-\frac{\sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{32 b c^5 \sqrt{c x-1} \sqrt{c x+1}}-\frac{b c x^6 \sqrt{d-c^2 d x^2}}{36 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b x^4 \sqrt{d-c^2 d x^2}}{96 c \sqrt{c x-1} \sqrt{c x+1}}+\frac{b x^2 \sqrt{d-c^2 d x^2}}{32 c^3 \sqrt{c x-1} \sqrt{c x+1}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5798
Rule 5743
Rule 5759
Rule 5676
Rule 30
Rubi steps
\begin{align*} \int x^4 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) \, dx &=\frac{\sqrt{d-c^2 d x^2} \int x^4 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{1}{6} x^5 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac{\sqrt{d-c^2 d x^2} \int \frac{x^4 \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{6 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (b c \sqrt{d-c^2 d x^2}\right ) \int x^5 \, dx}{6 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{b c x^6 \sqrt{d-c^2 d x^2}}{36 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{x^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{24 c^2}+\frac{1}{6} x^5 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac{\sqrt{d-c^2 d x^2} \int \frac{x^2 \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{8 c^2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (b \sqrt{d-c^2 d x^2}\right ) \int x^3 \, dx}{24 c \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{b x^4 \sqrt{d-c^2 d x^2}}{96 c \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b c x^6 \sqrt{d-c^2 d x^2}}{36 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{16 c^4}-\frac{x^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{24 c^2}+\frac{1}{6} x^5 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac{\sqrt{d-c^2 d x^2} \int \frac{a+b \cosh ^{-1}(c x)}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{16 c^4 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (b \sqrt{d-c^2 d x^2}\right ) \int x \, dx}{16 c^3 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{b x^2 \sqrt{d-c^2 d x^2}}{32 c^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b x^4 \sqrt{d-c^2 d x^2}}{96 c \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b c x^6 \sqrt{d-c^2 d x^2}}{36 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{16 c^4}-\frac{x^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{24 c^2}+\frac{1}{6} x^5 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac{\sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{32 b c^5 \sqrt{-1+c x} \sqrt{1+c x}}\\ \end{align*}
Mathematica [A] time = 1.14238, size = 198, normalized size = 0.71 \[ \frac{48 a c x \left (8 c^4 x^4-2 c^2 x^2-3\right ) \sqrt{d-c^2 d x^2}-144 a \sqrt{d} \tan ^{-1}\left (\frac{c x \sqrt{d-c^2 d x^2}}{\sqrt{d} \left (c^2 x^2-1\right )}\right )+\frac{b \sqrt{d-c^2 d x^2} \left (-72 \cosh ^{-1}(c x)^2+18 \cosh \left (2 \cosh ^{-1}(c x)\right )-9 \cosh \left (4 \cosh ^{-1}(c x)\right )-2 \cosh \left (6 \cosh ^{-1}(c x)\right )+12 \cosh ^{-1}(c x) \left (-3 \sinh \left (2 \cosh ^{-1}(c x)\right )+3 \sinh \left (4 \cosh ^{-1}(c x)\right )+\sinh \left (6 \cosh ^{-1}(c x)\right )\right )\right )}{\sqrt{\frac{c x-1}{c x+1}} (c x+1)}}{2304 c^5} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.514, size = 449, normalized size = 1.6 \begin{align*} -{\frac{{x}^{3}a}{6\,{c}^{2}d} \left ( -{c}^{2}d{x}^{2}+d \right ) ^{{\frac{3}{2}}}}-{\frac{ax}{8\,d{c}^{4}} \left ( -{c}^{2}d{x}^{2}+d \right ) ^{{\frac{3}{2}}}}+{\frac{ax}{16\,{c}^{4}}\sqrt{-{c}^{2}d{x}^{2}+d}}+{\frac{ad}{16\,{c}^{4}}\arctan \left ({x\sqrt{{c}^{2}d}{\frac{1}{\sqrt{-{c}^{2}d{x}^{2}+d}}}} \right ){\frac{1}{\sqrt{{c}^{2}d}}}}-{\frac{b \left ({\rm arccosh} \left (cx\right ) \right ) ^{2}}{32\,{c}^{5}}\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }{\frac{1}{\sqrt{cx-1}}}{\frac{1}{\sqrt{cx+1}}}}-{\frac{bc{x}^{6}}{36}\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }{\frac{1}{\sqrt{cx-1}}}{\frac{1}{\sqrt{cx+1}}}}+{\frac{b{c}^{2}{\rm arccosh} \left (cx\right ){x}^{7}}{ \left ( 6\,cx+6 \right ) \left ( cx-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }}-{\frac{5\,b{\rm arccosh} \left (cx\right ){x}^{5}}{ \left ( 24\,cx+24 \right ) \left ( cx-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }}-{\frac{b{\rm arccosh} \left (cx\right ){x}^{3}}{ \left ( 48\,cx+48 \right ){c}^{2} \left ( cx-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }}+{\frac{b{\rm arccosh} \left (cx\right )x}{ \left ( 16\,cx+16 \right ){c}^{4} \left ( cx-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }}-{\frac{25\,b}{2304\,{c}^{5}}\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }{\frac{1}{\sqrt{cx-1}}}{\frac{1}{\sqrt{cx+1}}}}+{\frac{b{x}^{4}}{96\,c}\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }{\frac{1}{\sqrt{cx-1}}}{\frac{1}{\sqrt{cx+1}}}}+{\frac{b{x}^{2}}{32\,{c}^{3}}\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }{\frac{1}{\sqrt{cx-1}}}{\frac{1}{\sqrt{cx+1}}}} \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 [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (b x^{4} \operatorname{arcosh}\left (c x\right ) + a x^{4}\right )} \sqrt{-c^{2} d x^{2} + d}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{4} \sqrt{- d \left (c x - 1\right ) \left (c x + 1\right )} \left (a + b \operatorname{acosh}{\left (c x \right )}\right )\, dx \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 \sqrt{-c^{2} d x^{2} + d}{\left (b \operatorname{arcosh}\left (c x\right ) + a\right )} x^{4}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]