Optimal. Leaf size=230 \[ -\frac{d^3 (c x-1)^5 (c x+1)^5 \left (a+b \cosh ^{-1}(c x)\right )}{10 c^4}-\frac{d^3 (c x-1)^4 (c x+1)^4 \left (a+b \cosh ^{-1}(c x)\right )}{8 c^4}+\frac{b d^3 x (c x-1)^{9/2} (c x+1)^{9/2}}{100 c^3}+\frac{7 b d^3 x (c x-1)^{7/2} (c x+1)^{7/2}}{1600 c^3}-\frac{49 b d^3 x (c x-1)^{5/2} (c x+1)^{5/2}}{9600 c^3}+\frac{49 b d^3 x (c x-1)^{3/2} (c x+1)^{3/2}}{7680 c^3}-\frac{49 b d^3 x \sqrt{c x-1} \sqrt{c x+1}}{5120 c^3}+\frac{49 b d^3 \cosh ^{-1}(c x)}{5120 c^4} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.281196, antiderivative size = 328, normalized size of antiderivative = 1.43, number of steps used = 11, number of rules used = 10, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {266, 43, 5731, 12, 566, 21, 388, 195, 217, 206} \[ \frac{d^3 \left (1-c^2 x^2\right )^5 \left (a+b \cosh ^{-1}(c x)\right )}{10 c^4}-\frac{d^3 \left (1-c^2 x^2\right )^4 \left (a+b \cosh ^{-1}(c x)\right )}{8 c^4}-\frac{b d^3 x \left (1-c^2 x^2\right )^5}{100 c^3 \sqrt{c x-1} \sqrt{c x+1}}+\frac{7 b d^3 x \left (1-c^2 x^2\right )^4}{1600 c^3 \sqrt{c x-1} \sqrt{c x+1}}+\frac{49 b d^3 x \left (1-c^2 x^2\right )^3}{9600 c^3 \sqrt{c x-1} \sqrt{c x+1}}+\frac{49 b d^3 x \left (1-c^2 x^2\right )^2}{7680 c^3 \sqrt{c x-1} \sqrt{c x+1}}+\frac{49 b d^3 x \left (1-c^2 x^2\right )}{5120 c^3 \sqrt{c x-1} \sqrt{c x+1}}+\frac{49 b d^3 \sqrt{c^2 x^2-1} \tanh ^{-1}\left (\frac{c x}{\sqrt{c^2 x^2-1}}\right )}{5120 c^4 \sqrt{c x-1} \sqrt{c x+1}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 266
Rule 43
Rule 5731
Rule 12
Rule 566
Rule 21
Rule 388
Rule 195
Rule 217
Rule 206
Rubi steps
\begin{align*} \int x^3 \left (d-c^2 d x^2\right )^3 \left (a+b \cosh ^{-1}(c x)\right ) \, dx &=-\frac{d^3 \left (1-c^2 x^2\right )^4 \left (a+b \cosh ^{-1}(c x)\right )}{8 c^4}+\frac{d^3 \left (1-c^2 x^2\right )^5 \left (a+b \cosh ^{-1}(c x)\right )}{10 c^4}-(b c) \int \frac{d^3 \left (-1-4 c^2 x^2\right ) \left (1-c^2 x^2\right )^4}{40 c^4 \sqrt{-1+c x} \sqrt{1+c x}} \, dx\\ &=-\frac{d^3 \left (1-c^2 x^2\right )^4 \left (a+b \cosh ^{-1}(c x)\right )}{8 c^4}+\frac{d^3 \left (1-c^2 x^2\right )^5 \left (a+b \cosh ^{-1}(c x)\right )}{10 c^4}-\frac{\left (b d^3\right ) \int \frac{\left (-1-4 c^2 x^2\right ) \left (1-c^2 x^2\right )^4}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{40 c^3}\\ &=-\frac{d^3 \left (1-c^2 x^2\right )^4 \left (a+b \cosh ^{-1}(c x)\right )}{8 c^4}+\frac{d^3 \left (1-c^2 x^2\right )^5 \left (a+b \cosh ^{-1}(c x)\right )}{10 c^4}-\frac{\left (b d^3 \sqrt{-1+c^2 x^2}\right ) \int \frac{\left (-1-4 c^2 x^2\right ) \left (1-c^2 x^2\right )^4}{\sqrt{-1+c^2 x^2}} \, dx}{40 c^3 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{d^3 \left (1-c^2 x^2\right )^4 \left (a+b \cosh ^{-1}(c x)\right )}{8 c^4}+\frac{d^3 \left (1-c^2 x^2\right )^5 \left (a+b \cosh ^{-1}(c x)\right )}{10 c^4}-\frac{\left (b d^3 \sqrt{-1+c^2 x^2}\right ) \int \left (-1-4 c^2 x^2\right ) \left (-1+c^2 x^2\right )^{7/2} \, dx}{40 c^3 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{b d^3 x \left (1-c^2 x^2\right )^5}{100 c^3 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{d^3 \left (1-c^2 x^2\right )^4 \left (a+b \cosh ^{-1}(c x)\right )}{8 c^4}+\frac{d^3 \left (1-c^2 x^2\right )^5 \left (a+b \cosh ^{-1}(c x)\right )}{10 c^4}+\frac{\left (7 b d^3 \sqrt{-1+c^2 x^2}\right ) \int \left (-1+c^2 x^2\right )^{7/2} \, dx}{200 c^3 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{7 b d^3 x \left (1-c^2 x^2\right )^4}{1600 c^3 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b d^3 x \left (1-c^2 x^2\right )^5}{100 c^3 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{d^3 \left (1-c^2 x^2\right )^4 \left (a+b \cosh ^{-1}(c x)\right )}{8 c^4}+\frac{d^3 \left (1-c^2 x^2\right )^5 \left (a+b \cosh ^{-1}(c x)\right )}{10 c^4}-\frac{\left (49 b d^3 \sqrt{-1+c^2 x^2}\right ) \int \left (-1+c^2 x^2\right )^{5/2} \, dx}{1600 c^3 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{49 b d^3 x \left (1-c^2 x^2\right )^3}{9600 c^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{7 b d^3 x \left (1-c^2 x^2\right )^4}{1600 c^3 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b d^3 x \left (1-c^2 x^2\right )^5}{100 c^3 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{d^3 \left (1-c^2 x^2\right )^4 \left (a+b \cosh ^{-1}(c x)\right )}{8 c^4}+\frac{d^3 \left (1-c^2 x^2\right )^5 \left (a+b \cosh ^{-1}(c x)\right )}{10 c^4}+\frac{\left (49 b d^3 \sqrt{-1+c^2 x^2}\right ) \int \left (-1+c^2 x^2\right )^{3/2} \, dx}{1920 c^3 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{49 b d^3 x \left (1-c^2 x^2\right )^2}{7680 c^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{49 b d^3 x \left (1-c^2 x^2\right )^3}{9600 c^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{7 b d^3 x \left (1-c^2 x^2\right )^4}{1600 c^3 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b d^3 x \left (1-c^2 x^2\right )^5}{100 c^3 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{d^3 \left (1-c^2 x^2\right )^4 \left (a+b \cosh ^{-1}(c x)\right )}{8 c^4}+\frac{d^3 \left (1-c^2 x^2\right )^5 \left (a+b \cosh ^{-1}(c x)\right )}{10 c^4}-\frac{\left (49 b d^3 \sqrt{-1+c^2 x^2}\right ) \int \sqrt{-1+c^2 x^2} \, dx}{2560 c^3 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{49 b d^3 x \left (1-c^2 x^2\right )}{5120 c^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{49 b d^3 x \left (1-c^2 x^2\right )^2}{7680 c^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{49 b d^3 x \left (1-c^2 x^2\right )^3}{9600 c^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{7 b d^3 x \left (1-c^2 x^2\right )^4}{1600 c^3 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b d^3 x \left (1-c^2 x^2\right )^5}{100 c^3 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{d^3 \left (1-c^2 x^2\right )^4 \left (a+b \cosh ^{-1}(c x)\right )}{8 c^4}+\frac{d^3 \left (1-c^2 x^2\right )^5 \left (a+b \cosh ^{-1}(c x)\right )}{10 c^4}+\frac{\left (49 b d^3 \sqrt{-1+c^2 x^2}\right ) \int \frac{1}{\sqrt{-1+c^2 x^2}} \, dx}{5120 c^3 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{49 b d^3 x \left (1-c^2 x^2\right )}{5120 c^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{49 b d^3 x \left (1-c^2 x^2\right )^2}{7680 c^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{49 b d^3 x \left (1-c^2 x^2\right )^3}{9600 c^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{7 b d^3 x \left (1-c^2 x^2\right )^4}{1600 c^3 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b d^3 x \left (1-c^2 x^2\right )^5}{100 c^3 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{d^3 \left (1-c^2 x^2\right )^4 \left (a+b \cosh ^{-1}(c x)\right )}{8 c^4}+\frac{d^3 \left (1-c^2 x^2\right )^5 \left (a+b \cosh ^{-1}(c x)\right )}{10 c^4}+\frac{\left (49 b d^3 \sqrt{-1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{1-c^2 x^2} \, dx,x,\frac{x}{\sqrt{-1+c^2 x^2}}\right )}{5120 c^3 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{49 b d^3 x \left (1-c^2 x^2\right )}{5120 c^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{49 b d^3 x \left (1-c^2 x^2\right )^2}{7680 c^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{49 b d^3 x \left (1-c^2 x^2\right )^3}{9600 c^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{7 b d^3 x \left (1-c^2 x^2\right )^4}{1600 c^3 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b d^3 x \left (1-c^2 x^2\right )^5}{100 c^3 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{d^3 \left (1-c^2 x^2\right )^4 \left (a+b \cosh ^{-1}(c x)\right )}{8 c^4}+\frac{d^3 \left (1-c^2 x^2\right )^5 \left (a+b \cosh ^{-1}(c x)\right )}{10 c^4}+\frac{49 b d^3 \sqrt{-1+c^2 x^2} \tanh ^{-1}\left (\frac{c x}{\sqrt{-1+c^2 x^2}}\right )}{5120 c^4 \sqrt{-1+c x} \sqrt{1+c x}}\\ \end{align*}
Mathematica [A] time = 0.342736, size = 162, normalized size = 0.7 \[ -\frac{d^3 \left (1920 a c^4 x^4 \left (4 c^6 x^6-15 c^4 x^4+20 c^2 x^2-10\right )+b c x \sqrt{c x-1} \sqrt{c x+1} \left (-768 c^8 x^8+2736 c^6 x^6-3208 c^4 x^4+790 c^2 x^2+1185\right )+1920 b c^4 x^4 \left (4 c^6 x^6-15 c^4 x^4+20 c^2 x^2-10\right ) \cosh ^{-1}(c x)+2370 b \tanh ^{-1}\left (\sqrt{\frac{c x-1}{c x+1}}\right )\right )}{76800 c^4} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.019, size = 284, normalized size = 1.2 \begin{align*} -{\frac{{c}^{6}{d}^{3}a{x}^{10}}{10}}+{\frac{3\,{c}^{4}{d}^{3}a{x}^{8}}{8}}-{\frac{{c}^{2}{d}^{3}a{x}^{6}}{2}}+{\frac{{d}^{3}a{x}^{4}}{4}}-{\frac{{c}^{6}{d}^{3}b{\rm arccosh} \left (cx\right ){x}^{10}}{10}}+{\frac{3\,{c}^{4}{d}^{3}b{\rm arccosh} \left (cx\right ){x}^{8}}{8}}-{\frac{{c}^{2}{d}^{3}b{\rm arccosh} \left (cx\right ){x}^{6}}{2}}+{\frac{{d}^{3}b{\rm arccosh} \left (cx\right ){x}^{4}}{4}}+{\frac{{d}^{3}b{c}^{5}{x}^{9}}{100}\sqrt{cx-1}\sqrt{cx+1}}-{\frac{57\,{d}^{3}b{c}^{3}{x}^{7}}{1600}\sqrt{cx-1}\sqrt{cx+1}}+{\frac{401\,{d}^{3}bc{x}^{5}}{9600}\sqrt{cx-1}\sqrt{cx+1}}-{\frac{79\,{d}^{3}b{x}^{3}}{7680\,c}\sqrt{cx-1}\sqrt{cx+1}}-{\frac{79\,{d}^{3}bx}{5120\,{c}^{3}}\sqrt{cx-1}\sqrt{cx+1}}-{\frac{79\,{d}^{3}b}{5120\,{c}^{4}}\sqrt{cx-1}\sqrt{cx+1}\ln \left ( cx+\sqrt{{c}^{2}{x}^{2}-1} \right ){\frac{1}{\sqrt{{c}^{2}{x}^{2}-1}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.40942, size = 725, normalized size = 3.15 \begin{align*} -\frac{1}{10} \, a c^{6} d^{3} x^{10} + \frac{3}{8} \, a c^{4} d^{3} x^{8} - \frac{1}{2} \, a c^{2} d^{3} x^{6} - \frac{1}{12800} \,{\left (1280 \, x^{10} \operatorname{arcosh}\left (c x\right ) -{\left (\frac{128 \, \sqrt{c^{2} x^{2} - 1} x^{9}}{c^{2}} + \frac{144 \, \sqrt{c^{2} x^{2} - 1} x^{7}}{c^{4}} + \frac{168 \, \sqrt{c^{2} x^{2} - 1} x^{5}}{c^{6}} + \frac{210 \, \sqrt{c^{2} x^{2} - 1} x^{3}}{c^{8}} + \frac{315 \, \sqrt{c^{2} x^{2} - 1} x}{c^{10}} + \frac{315 \, \log \left (2 \, c^{2} x + 2 \, \sqrt{c^{2} x^{2} - 1} \sqrt{c^{2}}\right )}{\sqrt{c^{2}} c^{10}}\right )} c\right )} b c^{6} d^{3} + \frac{1}{1024} \,{\left (384 \, x^{8} \operatorname{arcosh}\left (c x\right ) -{\left (\frac{48 \, \sqrt{c^{2} x^{2} - 1} x^{7}}{c^{2}} + \frac{56 \, \sqrt{c^{2} x^{2} - 1} x^{5}}{c^{4}} + \frac{70 \, \sqrt{c^{2} x^{2} - 1} x^{3}}{c^{6}} + \frac{105 \, \sqrt{c^{2} x^{2} - 1} x}{c^{8}} + \frac{105 \, \log \left (2 \, c^{2} x + 2 \, \sqrt{c^{2} x^{2} - 1} \sqrt{c^{2}}\right )}{\sqrt{c^{2}} c^{8}}\right )} c\right )} b c^{4} d^{3} + \frac{1}{4} \, a d^{3} x^{4} - \frac{1}{96} \,{\left (48 \, x^{6} \operatorname{arcosh}\left (c x\right ) -{\left (\frac{8 \, \sqrt{c^{2} x^{2} - 1} x^{5}}{c^{2}} + \frac{10 \, \sqrt{c^{2} x^{2} - 1} x^{3}}{c^{4}} + \frac{15 \, \sqrt{c^{2} x^{2} - 1} x}{c^{6}} + \frac{15 \, \log \left (2 \, c^{2} x + 2 \, \sqrt{c^{2} x^{2} - 1} \sqrt{c^{2}}\right )}{\sqrt{c^{2}} c^{6}}\right )} c\right )} b c^{2} d^{3} + \frac{1}{32} \,{\left (8 \, x^{4} \operatorname{arcosh}\left (c x\right ) -{\left (\frac{2 \, \sqrt{c^{2} x^{2} - 1} x^{3}}{c^{2}} + \frac{3 \, \sqrt{c^{2} x^{2} - 1} x}{c^{4}} + \frac{3 \, \log \left (2 \, c^{2} x + 2 \, \sqrt{c^{2} x^{2} - 1} \sqrt{c^{2}}\right )}{\sqrt{c^{2}} c^{4}}\right )} c\right )} b d^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.85556, size = 475, normalized size = 2.07 \begin{align*} -\frac{7680 \, a c^{10} d^{3} x^{10} - 28800 \, a c^{8} d^{3} x^{8} + 38400 \, a c^{6} d^{3} x^{6} - 19200 \, a c^{4} d^{3} x^{4} + 15 \,{\left (512 \, b c^{10} d^{3} x^{10} - 1920 \, b c^{8} d^{3} x^{8} + 2560 \, b c^{6} d^{3} x^{6} - 1280 \, b c^{4} d^{3} x^{4} + 79 \, b d^{3}\right )} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) -{\left (768 \, b c^{9} d^{3} x^{9} - 2736 \, b c^{7} d^{3} x^{7} + 3208 \, b c^{5} d^{3} x^{5} - 790 \, b c^{3} d^{3} x^{3} - 1185 \, b c d^{3} x\right )} \sqrt{c^{2} x^{2} - 1}}{76800 \, c^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 47.4026, size = 287, normalized size = 1.25 \begin{align*} \begin{cases} - \frac{a c^{6} d^{3} x^{10}}{10} + \frac{3 a c^{4} d^{3} x^{8}}{8} - \frac{a c^{2} d^{3} x^{6}}{2} + \frac{a d^{3} x^{4}}{4} - \frac{b c^{6} d^{3} x^{10} \operatorname{acosh}{\left (c x \right )}}{10} + \frac{b c^{5} d^{3} x^{9} \sqrt{c^{2} x^{2} - 1}}{100} + \frac{3 b c^{4} d^{3} x^{8} \operatorname{acosh}{\left (c x \right )}}{8} - \frac{57 b c^{3} d^{3} x^{7} \sqrt{c^{2} x^{2} - 1}}{1600} - \frac{b c^{2} d^{3} x^{6} \operatorname{acosh}{\left (c x \right )}}{2} + \frac{401 b c d^{3} x^{5} \sqrt{c^{2} x^{2} - 1}}{9600} + \frac{b d^{3} x^{4} \operatorname{acosh}{\left (c x \right )}}{4} - \frac{79 b d^{3} x^{3} \sqrt{c^{2} x^{2} - 1}}{7680 c} - \frac{79 b d^{3} x \sqrt{c^{2} x^{2} - 1}}{5120 c^{3}} - \frac{79 b d^{3} \operatorname{acosh}{\left (c x \right )}}{5120 c^{4}} & \text{for}\: c \neq 0 \\\frac{d^{3} x^{4} \left (a + \frac{i \pi b}{2}\right )}{4} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.75086, size = 633, normalized size = 2.75 \begin{align*} -\frac{1}{10} \, a c^{6} d^{3} x^{10} + \frac{3}{8} \, a c^{4} d^{3} x^{8} - \frac{1}{2} \, a c^{2} d^{3} x^{6} - \frac{1}{12800} \,{\left (1280 \, x^{10} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) -{\left (\sqrt{c^{2} x^{2} - 1}{\left (2 \,{\left (4 \,{\left (2 \, x^{2}{\left (\frac{8 \, x^{2}}{c^{2}} + \frac{9}{c^{4}}\right )} + \frac{21}{c^{6}}\right )} x^{2} + \frac{105}{c^{8}}\right )} x^{2} + \frac{315}{c^{10}}\right )} x - \frac{315 \, \log \left ({\left | -x{\left | c \right |} + \sqrt{c^{2} x^{2} - 1} \right |}\right )}{c^{10}{\left | c \right |}}\right )} c\right )} b c^{6} d^{3} + \frac{1}{1024} \,{\left (384 \, x^{8} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) -{\left (\sqrt{c^{2} x^{2} - 1}{\left (2 \,{\left (4 \, x^{2}{\left (\frac{6 \, x^{2}}{c^{2}} + \frac{7}{c^{4}}\right )} + \frac{35}{c^{6}}\right )} x^{2} + \frac{105}{c^{8}}\right )} x - \frac{105 \, \log \left ({\left | -x{\left | c \right |} + \sqrt{c^{2} x^{2} - 1} \right |}\right )}{c^{8}{\left | c \right |}}\right )} c\right )} b c^{4} d^{3} + \frac{1}{4} \, a d^{3} x^{4} - \frac{1}{96} \,{\left (48 \, x^{6} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) -{\left (\sqrt{c^{2} x^{2} - 1}{\left (2 \, x^{2}{\left (\frac{4 \, x^{2}}{c^{2}} + \frac{5}{c^{4}}\right )} + \frac{15}{c^{6}}\right )} x - \frac{15 \, \log \left ({\left | -x{\left | c \right |} + \sqrt{c^{2} x^{2} - 1} \right |}\right )}{c^{6}{\left | c \right |}}\right )} c\right )} b c^{2} d^{3} + \frac{1}{32} \,{\left (8 \, x^{4} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) -{\left (\sqrt{c^{2} x^{2} - 1} x{\left (\frac{2 \, x^{2}}{c^{2}} + \frac{3}{c^{4}}\right )} - \frac{3 \, \log \left ({\left | -x{\left | c \right |} + \sqrt{c^{2} x^{2} - 1} \right |}\right )}{c^{4}{\left | c \right |}}\right )} c\right )} b d^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]