Optimal. Leaf size=108 \[ \frac {x \left (a+b \csc ^{-1}(c x)\right )}{d \sqrt {d+e x^2}}+\frac {b x \sqrt {1-c^2 x^2} \sqrt {1+\frac {e x^2}{d}} F\left (\text {ArcSin}(c x)\left |-\frac {e}{c^2 d}\right .\right )}{d \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.06, antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {197, 5337, 12,
432, 430} \begin {gather*} \frac {x \left (a+b \csc ^{-1}(c x)\right )}{d \sqrt {d+e x^2}}+\frac {b x \sqrt {1-c^2 x^2} \sqrt {\frac {e x^2}{d}+1} F\left (\text {ArcSin}(c x)\left |-\frac {e}{c^2 d}\right .\right )}{d \sqrt {c^2 x^2} \sqrt {c^2 x^2-1} \sqrt {d+e x^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 197
Rule 430
Rule 432
Rule 5337
Rubi steps
\begin {align*} \int \frac {a+b \csc ^{-1}(c x)}{\left (d+e x^2\right )^{3/2}} \, dx &=\frac {x \left (a+b \csc ^{-1}(c x)\right )}{d \sqrt {d+e x^2}}+\frac {(b c x) \int \frac {1}{d \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \, dx}{\sqrt {c^2 x^2}}\\ &=\frac {x \left (a+b \csc ^{-1}(c x)\right )}{d \sqrt {d+e x^2}}+\frac {(b c x) \int \frac {1}{\sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \, dx}{d \sqrt {c^2 x^2}}\\ &=\frac {x \left (a+b \csc ^{-1}(c x)\right )}{d \sqrt {d+e x^2}}+\frac {\left (b c x \sqrt {1+\frac {e x^2}{d}}\right ) \int \frac {1}{\sqrt {-1+c^2 x^2} \sqrt {1+\frac {e x^2}{d}}} \, dx}{d \sqrt {c^2 x^2} \sqrt {d+e x^2}}\\ &=\frac {x \left (a+b \csc ^{-1}(c x)\right )}{d \sqrt {d+e x^2}}+\frac {\left (b c x \sqrt {1-c^2 x^2} \sqrt {1+\frac {e x^2}{d}}\right ) \int \frac {1}{\sqrt {1-c^2 x^2} \sqrt {1+\frac {e x^2}{d}}} \, dx}{d \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}\\ &=\frac {x \left (a+b \csc ^{-1}(c x)\right )}{d \sqrt {d+e x^2}}+\frac {b x \sqrt {1-c^2 x^2} \sqrt {1+\frac {e x^2}{d}} F\left (\sin ^{-1}(c x)|-\frac {e}{c^2 d}\right )}{d \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.13, size = 112, normalized size = 1.04 \begin {gather*} \frac {x \left (a+b \csc ^{-1}(c x)\right )}{d \sqrt {d+e x^2}}+\frac {b c \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {1-c^2 x^2} \sqrt {1+\frac {e x^2}{d}} F\left (\text {ArcSin}(c x)\left |-\frac {e}{c^2 d}\right .\right )}{d \left (-c+c^3 x^2\right ) \sqrt {d+e x^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {a +b \,\mathrm {arccsc}\left (c x \right )}{\left (e \,x^{2}+d \right )^{\frac {3}{2}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.12, size = 79, normalized size = 0.73 \begin {gather*} -\frac {{\left (b x^{2} e + b d\right )} \sqrt {-d} {\rm ellipticF}\left (c x, -\frac {e}{c^{2} d}\right ) - {\left (b c d x \operatorname {arccsc}\left (c x\right ) + a c d x\right )} \sqrt {x^{2} e + d}}{c d^{2} x^{2} e + c d^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a + b \operatorname {acsc}{\left (c x \right )}}{\left (d + e x^{2}\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )}{{\left (e\,x^2+d\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________