Optimal. Leaf size=568 \[ \frac {x^4 \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {8 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c^6 d^3}-\frac {22 b \sqrt {c x-1} \sqrt {c x+1} \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{3 c^6 d^2 \sqrt {d-c^2 d x^2}}-\frac {16 a b x \sqrt {c x-1} \sqrt {c x+1}}{3 c^5 d^2 \sqrt {d-c^2 d x^2}}+\frac {11 b x \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{3 c^5 d^2 \sqrt {d-c^2 d x^2}}-\frac {4 x^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {b x^3 \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{3 c^3 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}-\frac {11 b^2 \sqrt {c x-1} \sqrt {c x+1} \text {Li}_2\left (-e^{\cosh ^{-1}(c x)}\right )}{3 c^6 d^2 \sqrt {d-c^2 d x^2}}+\frac {11 b^2 \sqrt {c x-1} \sqrt {c x+1} \text {Li}_2\left (e^{\cosh ^{-1}(c x)}\right )}{3 c^6 d^2 \sqrt {d-c^2 d x^2}}-\frac {7 b^2 (1-c x) (c x+1)}{3 c^6 d^2 \sqrt {d-c^2 d x^2}}-\frac {16 b^2 x \sqrt {c x-1} \sqrt {c x+1} \cosh ^{-1}(c x)}{3 c^5 d^2 \sqrt {d-c^2 d x^2}}-\frac {b^2 x^2}{3 c^4 d^2 \sqrt {d-c^2 d x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 1.48, antiderivative size = 594, normalized size of antiderivative = 1.05, number of steps used = 27, number of rules used = 13, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.448, Rules used = {5798, 5752, 5718, 5654, 74, 5766, 5694, 4182, 2279, 2391, 5750, 98, 21} \[ -\frac {11 b^2 \sqrt {c x-1} \sqrt {c x+1} \text {PolyLog}\left (2,-e^{\cosh ^{-1}(c x)}\right )}{3 c^6 d^2 \sqrt {d-c^2 d x^2}}+\frac {11 b^2 \sqrt {c x-1} \sqrt {c x+1} \text {PolyLog}\left (2,e^{\cosh ^{-1}(c x)}\right )}{3 c^6 d^2 \sqrt {d-c^2 d x^2}}-\frac {16 a b x \sqrt {c x-1} \sqrt {c x+1}}{3 c^5 d^2 \sqrt {d-c^2 d x^2}}+\frac {x^4 \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c^2 d^2 (1-c x) (c x+1) \sqrt {d-c^2 d x^2}}+\frac {b x^3 \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{3 c^3 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}-\frac {4 x^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {11 b x \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{3 c^5 d^2 \sqrt {d-c^2 d x^2}}-\frac {8 (1-c x) (c x+1) \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c^6 d^2 \sqrt {d-c^2 d x^2}}-\frac {22 b \sqrt {c x-1} \sqrt {c x+1} \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{3 c^6 d^2 \sqrt {d-c^2 d x^2}}-\frac {b^2 x^2}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {7 b^2 (1-c x) (c x+1)}{3 c^6 d^2 \sqrt {d-c^2 d x^2}}-\frac {16 b^2 x \sqrt {c x-1} \sqrt {c x+1} \cosh ^{-1}(c x)}{3 c^5 d^2 \sqrt {d-c^2 d x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 21
Rule 74
Rule 98
Rule 2279
Rule 2391
Rule 4182
Rule 5654
Rule 5694
Rule 5718
Rule 5750
Rule 5752
Rule 5766
Rule 5798
Rubi steps
\begin {align*} \int \frac {x^5 \left (a+b \cosh ^{-1}(c x)\right )^2}{\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 )^2}{(-1+c x)^{5/2} (1+c x)^{5/2}} \, dx}{d^2 \sqrt {d-c^2 d x^2}}\\ &=\frac {x^4 \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c^2 d^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2}}+\frac {\left (4 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x^3 \left (a+b \cosh ^{-1}(c x)\right )^2}{(-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{3 c^2 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (2 b \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x^4 \left (a+b \cosh ^{-1}(c x)\right )}{\left (-1+c^2 x^2\right )^2} \, dx}{3 c d^2 \sqrt {d-c^2 d x^2}}\\ &=\frac {b x^3 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{3 c^3 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 )^2}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {x^4 \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c^2 d^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2}}+\frac {\left (8 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x \left (a+b \cosh ^{-1}(c x)\right )^2}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (b \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x^2 \left (a+b \cosh ^{-1}(c x)\right )}{-1+c^2 x^2} \, dx}{c^3 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (8 b \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x^2 \left (a+b \cosh ^{-1}(c x)\right )}{-1+c^2 x^2} \, dx}{3 c^3 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x^3}{(-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{3 c^2 d^2 \sqrt {d-c^2 d x^2}}\\ &=-\frac {b^2 x^2}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {11 b x \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{3 c^5 d^2 \sqrt {d-c^2 d x^2}}+\frac {b x^3 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{3 c^3 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 )^2}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {x^4 \left (a+b \cosh ^{-1}(c x)\right )^2}{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 )^2}{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 {a+b \cosh ^{-1}(c x)}{-1+c^2 x^2} \, dx}{c^5 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (8 b \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {a+b \cosh ^{-1}(c x)}{-1+c^2 x^2} \, dx}{3 c^5 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (16 b \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{3 c^5 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x (-2-2 c x)}{\sqrt {-1+c x} (1+c x)^{3/2}} \, dx}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (8 b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}\\ &=-\frac {b^2 x^2}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {16 a b x \sqrt {-1+c x} \sqrt {1+c x}}{3 c^5 d^2 \sqrt {d-c^2 d x^2}}+\frac {11 b^2 (1-c x) (1+c x)}{3 c^6 d^2 \sqrt {d-c^2 d x^2}}+\frac {11 b x \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{3 c^5 d^2 \sqrt {d-c^2 d x^2}}+\frac {b x^3 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{3 c^3 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 )^2}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {x^4 \left (a+b \cosh ^{-1}(c x)\right )^2}{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 )^2}{3 c^6 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (b \sqrt {-1+c x} \sqrt {1+c x}\right ) \operatorname {Subst}\left (\int (a+b x) \text {csch}(x) \, dx,x,\cosh ^{-1}(c x)\right )}{c^6 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (8 b \sqrt {-1+c x} \sqrt {1+c x}\right ) \operatorname {Subst}\left (\int (a+b x) \text {csch}(x) \, dx,x,\cosh ^{-1}(c x)\right )}{3 c^6 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (16 b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \cosh ^{-1}(c x) \, dx}{3 c^5 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (2 b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}\\ &=-\frac {b^2 x^2}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {16 a b x \sqrt {-1+c x} \sqrt {1+c x}}{3 c^5 d^2 \sqrt {d-c^2 d x^2}}+\frac {3 b^2 (1-c x) (1+c x)}{c^6 d^2 \sqrt {d-c^2 d x^2}}-\frac {16 b^2 x \sqrt {-1+c x} \sqrt {1+c x} \cosh ^{-1}(c x)}{3 c^5 d^2 \sqrt {d-c^2 d x^2}}+\frac {11 b x \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{3 c^5 d^2 \sqrt {d-c^2 d x^2}}+\frac {b x^3 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{3 c^3 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 )^2}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {x^4 \left (a+b \cosh ^{-1}(c x)\right )^2}{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 )^2}{3 c^6 d^2 \sqrt {d-c^2 d x^2}}-\frac {22 b \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{3 c^6 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \operatorname {Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{c^6 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \operatorname {Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{c^6 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (8 b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \operatorname {Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{3 c^6 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (8 b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \operatorname {Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{3 c^6 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (16 b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}\\ &=-\frac {b^2 x^2}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {16 a b x \sqrt {-1+c x} \sqrt {1+c x}}{3 c^5 d^2 \sqrt {d-c^2 d x^2}}-\frac {7 b^2 (1-c x) (1+c x)}{3 c^6 d^2 \sqrt {d-c^2 d x^2}}-\frac {16 b^2 x \sqrt {-1+c x} \sqrt {1+c x} \cosh ^{-1}(c x)}{3 c^5 d^2 \sqrt {d-c^2 d x^2}}+\frac {11 b x \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{3 c^5 d^2 \sqrt {d-c^2 d x^2}}+\frac {b x^3 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{3 c^3 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 )^2}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {x^4 \left (a+b \cosh ^{-1}(c x)\right )^2}{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 )^2}{3 c^6 d^2 \sqrt {d-c^2 d x^2}}-\frac {22 b \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{3 c^6 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{c^6 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{c^6 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (8 b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{3 c^6 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (8 b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{3 c^6 d^2 \sqrt {d-c^2 d x^2}}\\ &=-\frac {b^2 x^2}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {16 a b x \sqrt {-1+c x} \sqrt {1+c x}}{3 c^5 d^2 \sqrt {d-c^2 d x^2}}-\frac {7 b^2 (1-c x) (1+c x)}{3 c^6 d^2 \sqrt {d-c^2 d x^2}}-\frac {16 b^2 x \sqrt {-1+c x} \sqrt {1+c x} \cosh ^{-1}(c x)}{3 c^5 d^2 \sqrt {d-c^2 d x^2}}+\frac {11 b x \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{3 c^5 d^2 \sqrt {d-c^2 d x^2}}+\frac {b x^3 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{3 c^3 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 )^2}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {x^4 \left (a+b \cosh ^{-1}(c x)\right )^2}{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 )^2}{3 c^6 d^2 \sqrt {d-c^2 d x^2}}-\frac {22 b \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{3 c^6 d^2 \sqrt {d-c^2 d x^2}}-\frac {11 b^2 \sqrt {-1+c x} \sqrt {1+c x} \text {Li}_2\left (-e^{\cosh ^{-1}(c x)}\right )}{3 c^6 d^2 \sqrt {d-c^2 d x^2}}+\frac {11 b^2 \sqrt {-1+c x} \sqrt {1+c x} \text {Li}_2\left (e^{\cosh ^{-1}(c x)}\right )}{3 c^6 d^2 \sqrt {d-c^2 d x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 5.55, size = 437, normalized size = 0.77 \[ -\frac {8 a^2 \left (3 c^4 x^4-12 c^2 x^2+8\right )+2 a b \left (-36 \cosh \left (2 \cosh ^{-1}(c x)\right ) \cosh ^{-1}(c x)+3 \cosh \left (4 \cosh ^{-1}(c x)\right ) \cosh ^{-1}(c x)+25 \cosh ^{-1}(c x)+4 \sinh \left (2 \cosh ^{-1}(c x)\right )-3 \sinh \left (4 \cosh ^{-1}(c x)\right )-33 \sqrt {\frac {c x-1}{c x+1}} (c x+1) \log \left (\tanh \left (\frac {1}{2} \cosh ^{-1}(c x)\right )\right )+11 \sinh \left (3 \cosh ^{-1}(c x)\right ) \log \left (\tanh \left (\frac {1}{2} \cosh ^{-1}(c x)\right )\right )\right )+b^2 \left (88 \left (\frac {c x-1}{c x+1}\right )^{3/2} (c x+1)^3 \text {Li}_2\left (-e^{-\cosh ^{-1}(c x)}\right )-88 \left (\frac {c x-1}{c x+1}\right )^{3/2} (c x+1)^3 \text {Li}_2\left (e^{-\cosh ^{-1}(c x)}\right )+25 \cosh ^{-1}(c x)^2-4 \left (9 \cosh ^{-1}(c x)^2+7\right ) \cosh \left (2 \cosh ^{-1}(c x)\right )+3 \left (\cosh ^{-1}(c x)^2+2\right ) \cosh \left (4 \cosh ^{-1}(c x)\right )-66 \sqrt {\frac {c x-1}{c x+1}} (c x+1) \cosh ^{-1}(c x) \log \left (1-e^{-\cosh ^{-1}(c x)}\right )+66 \sqrt {\frac {c x-1}{c x+1}} (c x+1) \cosh ^{-1}(c x) \log \left (e^{-\cosh ^{-1}(c x)}+1\right )+8 \cosh ^{-1}(c x) \sinh \left (2 \cosh ^{-1}(c x)\right )-6 \cosh ^{-1}(c x) \sinh \left (4 \cosh ^{-1}(c x)\right )+22 \cosh ^{-1}(c x) \log \left (1-e^{-\cosh ^{-1}(c x)}\right ) \sinh \left (3 \cosh ^{-1}(c x)\right )-22 \cosh ^{-1}(c x) \log \left (e^{-\cosh ^{-1}(c x)}+1\right ) \sinh \left (3 \cosh ^{-1}(c x)\right )+22\right )}{24 c^6 d \left (d-c^2 d x^2\right )^{3/2}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.90, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (b^{2} x^{5} \operatorname {arcosh}\left (c x\right )^{2} + 2 \, a b x^{5} \operatorname {arcosh}\left (c x\right ) + a^{2} x^{5}\right )} \sqrt {-c^{2} d x^{2} + d}}{c^{6} d^{3} x^{6} - 3 \, c^{4} d^{3} x^{4} + 3 \, c^{2} d^{3} x^{2} - d^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.89, size = 1211, normalized size = 2.13 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {1}{3} \, a^{2} {\left (\frac {3 \, x^{4}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} c^{2} d} - \frac {12 \, x^{2}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} c^{4} d} + \frac {8}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} c^{6} d}\right )} - \frac {{\left (3 \, b^{2} c^{4} \sqrt {d} x^{4} - 12 \, b^{2} c^{2} \sqrt {d} x^{2} + 8 \, b^{2} \sqrt {d}\right )} \sqrt {c x + 1} \sqrt {-c x + 1} \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )^{2}}{3 \, {\left (c^{10} d^{3} x^{4} - 2 \, c^{8} d^{3} x^{2} + c^{6} d^{3}\right )}} - \int \frac {2 \, {\left ({\left (12 \, b^{2} c^{3} x^{3} + 3 \, {\left (a b c^{5} - b^{2} c^{5}\right )} x^{5} - 8 \, b^{2} c x\right )} {\left (c x + 1\right )} \sqrt {c x - 1} + {\left (15 \, b^{2} c^{4} x^{4} + 3 \, {\left (a b c^{6} - b^{2} c^{6}\right )} x^{6} - 20 \, b^{2} c^{2} x^{2} + 8 \, b^{2}\right )} \sqrt {c x + 1}\right )} \sqrt {-c x + 1} \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )}{3 \, {\left (c^{12} d^{\frac {5}{2}} x^{7} - 3 \, c^{10} d^{\frac {5}{2}} x^{5} + 3 \, c^{8} d^{\frac {5}{2}} x^{3} - c^{6} d^{\frac {5}{2}} x + {\left (c^{11} d^{\frac {5}{2}} x^{6} - 3 \, c^{9} d^{\frac {5}{2}} x^{4} + 3 \, c^{7} d^{\frac {5}{2}} x^{2} - c^{5} d^{\frac {5}{2}}\right )} \sqrt {c x + 1} \sqrt {c x - 1}\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^5\,{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2}{{\left (d-c^2\,d\,x^2\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{5} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2}}{\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________