Optimal. Leaf size=224 \[ \frac {x^3 \left (a+b \cosh ^{-1}(c x)\right )}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {\sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c^5 d^2 \sqrt {d-c^2 d x^2}}-\frac {x \left (a+b \cosh ^{-1}(c x)\right )}{c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \log \left (1-c^2 x^2\right )}{3 c^5 d^2 \sqrt {d-c^2 d x^2}}+\frac {b \sqrt {c x-1} \sqrt {c x+1}}{6 c^5 d \left (d-c^2 d x^2\right )^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.76, antiderivative size = 251, normalized size of antiderivative = 1.12, number of steps used = 8, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {5798, 5752, 5676, 260, 266, 43} \[ \frac {x^3 \left (a+b \cosh ^{-1}(c x)\right )}{3 c^2 d^2 (1-c x) (c x+1) \sqrt {d-c^2 d x^2}}-\frac {x \left (a+b \cosh ^{-1}(c x)\right )}{c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {\sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c^5 d^2 \sqrt {d-c^2 d x^2}}+\frac {b \sqrt {c x-1} \sqrt {c x+1}}{6 c^5 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \log \left (1-c^2 x^2\right )}{3 c^5 d^2 \sqrt {d-c^2 d x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rule 260
Rule 266
Rule 5676
Rule 5752
Rule 5798
Rubi steps
\begin {align*} \int \frac {x^4 \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^4 \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 {x^3 \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 {\left (\sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x^2 \left (a+b \cosh ^{-1}(c x)\right )}{(-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{c^2 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (b \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x^3}{\left (-1+c^2 x^2\right )^2} \, dx}{3 c d^2 \sqrt {d-c^2 d x^2}}\\ &=-\frac {x \left (a+b \cosh ^{-1}(c x)\right )}{c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {x^3 \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 {\left (\sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {a+b \cosh ^{-1}(c x)}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{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}{-1+c^2 x^2} \, dx}{c^3 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 \frac {x}{\left (-1+c^2 x\right )^2} \, dx,x,x^2\right )}{6 c d^2 \sqrt {d-c^2 d x^2}}\\ &=-\frac {x \left (a+b \cosh ^{-1}(c x)\right )}{c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {x^3 \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 {\sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c^5 d^2 \sqrt {d-c^2 d x^2}}+\frac {b \sqrt {-1+c x} \sqrt {1+c x} \log \left (1-c^2 x^2\right )}{2 c^5 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 \left (\frac {1}{c^2 \left (-1+c^2 x\right )^2}+\frac {1}{c^2 \left (-1+c^2 x\right )}\right ) \, dx,x,x^2\right )}{6 c d^2 \sqrt {d-c^2 d x^2}}\\ &=\frac {b \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 {x \left (a+b \cosh ^{-1}(c x)\right )}{c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {x^3 \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 {\sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c^5 d^2 \sqrt {d-c^2 d x^2}}+\frac {2 b \sqrt {-1+c x} \sqrt {1+c x} \log \left (1-c^2 x^2\right )}{3 c^5 d^2 \sqrt {d-c^2 d x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.76, size = 225, normalized size = 1.00 \[ \frac {\frac {2 a c x \left (4 c^2 x^2-3\right ) \sqrt {d-c^2 d x^2}}{\left (c^2 x^2-1\right )^2}-6 a \sqrt {d} \tan ^{-1}\left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (c^2 x^2-1\right )}\right )+\frac {b d \left (-\frac {\sqrt {\frac {c x-1}{c x+1}} (c x+1)+2 c x \cosh ^{-1}(c x)}{c^2 x^2-1}-8 c x \cosh ^{-1}(c x)+\sqrt {\frac {c x-1}{c x+1}} (c x+1) \left (8 \log \left (\sqrt {\frac {c x-1}{c x+1}} (c x+1)\right )+3 \cosh ^{-1}(c x)^2\right )\right )}{\sqrt {d-c^2 d x^2}}}{6 c^5 d^3} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.66, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (b x^{4} \operatorname {arcosh}\left (c x\right ) + a x^{4}\right )} \sqrt {-c^{2} d x^{2} + d}}{c^{6} d^{3} x^{6} - 3 \, c^{4} d^{3} x^{4} + 3 \, c^{2} d^{3} x^{2} - d^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} x^{4}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.84, size = 1519, normalized size = 6.78 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{3} \, {\left (x {\left (\frac {3 \, x^{2}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} c^{2} d} - \frac {2}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} c^{4} d}\right )} - \frac {x}{\sqrt {-c^{2} d x^{2} + d} c^{4} d^{2}} + \frac {3 \, \arcsin \left (c x\right )}{c^{5} d^{\frac {5}{2}}}\right )} a + b \int \frac {x^{4} \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^4\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}{{\left (d-c^2\,d\,x^2\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{4} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )}{\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________