Optimal. Leaf size=89 \[ \frac{1}{6} x^6 \left (a+b \csc ^{-1}(c x)\right )+\frac{b x^5 \sqrt{1-\frac{1}{c^2 x^2}}}{30 c}+\frac{2 b x^3 \sqrt{1-\frac{1}{c^2 x^2}}}{45 c^3}+\frac{4 b x \sqrt{1-\frac{1}{c^2 x^2}}}{45 c^5} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0418436, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {5221, 271, 191} \[ \frac{1}{6} x^6 \left (a+b \csc ^{-1}(c x)\right )+\frac{b x^5 \sqrt{1-\frac{1}{c^2 x^2}}}{30 c}+\frac{2 b x^3 \sqrt{1-\frac{1}{c^2 x^2}}}{45 c^3}+\frac{4 b x \sqrt{1-\frac{1}{c^2 x^2}}}{45 c^5} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5221
Rule 271
Rule 191
Rubi steps
\begin{align*} \int x^5 \left (a+b \csc ^{-1}(c x)\right ) \, dx &=\frac{1}{6} x^6 \left (a+b \csc ^{-1}(c x)\right )+\frac{b \int \frac{x^4}{\sqrt{1-\frac{1}{c^2 x^2}}} \, dx}{6 c}\\ &=\frac{b \sqrt{1-\frac{1}{c^2 x^2}} x^5}{30 c}+\frac{1}{6} x^6 \left (a+b \csc ^{-1}(c x)\right )+\frac{(2 b) \int \frac{x^2}{\sqrt{1-\frac{1}{c^2 x^2}}} \, dx}{15 c^3}\\ &=\frac{2 b \sqrt{1-\frac{1}{c^2 x^2}} x^3}{45 c^3}+\frac{b \sqrt{1-\frac{1}{c^2 x^2}} x^5}{30 c}+\frac{1}{6} x^6 \left (a+b \csc ^{-1}(c x)\right )+\frac{(4 b) \int \frac{1}{\sqrt{1-\frac{1}{c^2 x^2}}} \, dx}{45 c^5}\\ &=\frac{4 b \sqrt{1-\frac{1}{c^2 x^2}} x}{45 c^5}+\frac{2 b \sqrt{1-\frac{1}{c^2 x^2}} x^3}{45 c^3}+\frac{b \sqrt{1-\frac{1}{c^2 x^2}} x^5}{30 c}+\frac{1}{6} x^6 \left (a+b \csc ^{-1}(c x)\right )\\ \end{align*}
Mathematica [A] time = 0.131926, size = 72, normalized size = 0.81 \[ \frac{a x^6}{6}+b \sqrt{\frac{c^2 x^2-1}{c^2 x^2}} \left (\frac{2 x^3}{45 c^3}+\frac{4 x}{45 c^5}+\frac{x^5}{30 c}\right )+\frac{1}{6} b x^6 \csc ^{-1}(c x) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.174, size = 83, normalized size = 0.9 \begin{align*}{\frac{1}{{c}^{6}} \left ({\frac{{c}^{6}{x}^{6}a}{6}}+b \left ({\frac{{c}^{6}{x}^{6}{\rm arccsc} \left (cx\right )}{6}}+{\frac{ \left ({c}^{2}{x}^{2}-1 \right ) \left ( 3\,{c}^{4}{x}^{4}+4\,{c}^{2}{x}^{2}+8 \right ) }{90\,cx}{\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}}}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.957131, size = 108, normalized size = 1.21 \begin{align*} \frac{1}{6} \, a x^{6} + \frac{1}{90} \,{\left (15 \, x^{6} \operatorname{arccsc}\left (c x\right ) + \frac{3 \, c^{4} x^{5}{\left (-\frac{1}{c^{2} x^{2}} + 1\right )}^{\frac{5}{2}} + 10 \, c^{2} x^{3}{\left (-\frac{1}{c^{2} x^{2}} + 1\right )}^{\frac{3}{2}} + 15 \, x \sqrt{-\frac{1}{c^{2} x^{2}} + 1}}{c^{5}}\right )} b \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.4637, size = 143, normalized size = 1.61 \begin{align*} \frac{15 \, b c^{6} x^{6} \operatorname{arccsc}\left (c x\right ) + 15 \, a c^{6} x^{6} +{\left (3 \, b c^{4} x^{4} + 4 \, b c^{2} x^{2} + 8 \, b\right )} \sqrt{c^{2} x^{2} - 1}}{90 \, c^{6}} \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^{5} \left (a + b \operatorname{acsc}{\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{\left (b \operatorname{arccsc}\left (c x\right ) + a\right )} x^{5}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]