Optimal. Leaf size=66 \[ \frac {b e \sqrt {1-c^2 x^2}}{c}-\frac {d (a+b \text {ArcSin}(c x))}{x}+e x (a+b \text {ArcSin}(c x))-b c d \tanh ^{-1}\left (\sqrt {1-c^2 x^2}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.06, antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 6, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {14, 4815, 457,
81, 65, 214} \begin {gather*} -\frac {d (a+b \text {ArcSin}(c x))}{x}+e x (a+b \text {ArcSin}(c x))-b c d \tanh ^{-1}\left (\sqrt {1-c^2 x^2}\right )+\frac {b e \sqrt {1-c^2 x^2}}{c} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 14
Rule 65
Rule 81
Rule 214
Rule 457
Rule 4815
Rubi steps
\begin {align*} \int \frac {\left (d+e x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{x^2} \, dx &=-\frac {d \left (a+b \sin ^{-1}(c x)\right )}{x}+e x \left (a+b \sin ^{-1}(c x)\right )-(b c) \int \frac {-d+e x^2}{x \sqrt {1-c^2 x^2}} \, dx\\ &=-\frac {d \left (a+b \sin ^{-1}(c x)\right )}{x}+e x \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{2} (b c) \text {Subst}\left (\int \frac {-d+e x}{x \sqrt {1-c^2 x}} \, dx,x,x^2\right )\\ &=\frac {b e \sqrt {1-c^2 x^2}}{c}-\frac {d \left (a+b \sin ^{-1}(c x)\right )}{x}+e x \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{2} (b c d) \text {Subst}\left (\int \frac {1}{x \sqrt {1-c^2 x}} \, dx,x,x^2\right )\\ &=\frac {b e \sqrt {1-c^2 x^2}}{c}-\frac {d \left (a+b \sin ^{-1}(c x)\right )}{x}+e x \left (a+b \sin ^{-1}(c x)\right )-\frac {(b d) \text {Subst}\left (\int \frac {1}{\frac {1}{c^2}-\frac {x^2}{c^2}} \, dx,x,\sqrt {1-c^2 x^2}\right )}{c}\\ &=\frac {b e \sqrt {1-c^2 x^2}}{c}-\frac {d \left (a+b \sin ^{-1}(c x)\right )}{x}+e x \left (a+b \sin ^{-1}(c x)\right )-b c d \tanh ^{-1}\left (\sqrt {1-c^2 x^2}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.04, size = 71, normalized size = 1.08 \begin {gather*} -\frac {a d}{x}+a e x+\frac {b e \sqrt {1-c^2 x^2}}{c}-\frac {b d \text {ArcSin}(c x)}{x}+b e x \text {ArcSin}(c x)-b c d \tanh ^{-1}\left (\sqrt {1-c^2 x^2}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.01, size = 79, normalized size = 1.20
method | result | size |
derivativedivides | \(c \left (\frac {a \left (c e x -\frac {d c}{x}\right )}{c^{2}}+\frac {b \left (\arcsin \left (c x \right ) e c x -\frac {\arcsin \left (c x \right ) d c}{x}+e \sqrt {-c^{2} x^{2}+1}-d \,c^{2} \arctanh \left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right )\right )}{c^{2}}\right )\) | \(79\) |
default | \(c \left (\frac {a \left (c e x -\frac {d c}{x}\right )}{c^{2}}+\frac {b \left (\arcsin \left (c x \right ) e c x -\frac {\arcsin \left (c x \right ) d c}{x}+e \sqrt {-c^{2} x^{2}+1}-d \,c^{2} \arctanh \left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right )\right )}{c^{2}}\right )\) | \(79\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.50, size = 81, normalized size = 1.23 \begin {gather*} -{\left (c \log \left (\frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) + \frac {\arcsin \left (c x\right )}{x}\right )} b d + a x e + \frac {{\left (c x \arcsin \left (c x\right ) + \sqrt {-c^{2} x^{2} + 1}\right )} b e}{c} - \frac {a d}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 1.72, size = 106, normalized size = 1.61 \begin {gather*} -\frac {b c^{2} d x \log \left (\sqrt {-c^{2} x^{2} + 1} + 1\right ) - b c^{2} d x \log \left (\sqrt {-c^{2} x^{2} + 1} - 1\right ) - 2 \, a c x^{2} e - 2 \, \sqrt {-c^{2} x^{2} + 1} b x e + 2 \, a c d - 2 \, {\left (b c x^{2} e - b c d\right )} \arcsin \left (c x\right )}{2 \, c x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 2.44, size = 75, normalized size = 1.14 \begin {gather*} - \frac {a d}{x} + a e x + b c d \left (\begin {cases} - \operatorname {acosh}{\left (\frac {1}{c x} \right )} & \text {for}\: \frac {1}{\left |{c^{2} x^{2}}\right |} > 1 \\i \operatorname {asin}{\left (\frac {1}{c x} \right )} & \text {otherwise} \end {cases}\right ) - \frac {b d \operatorname {asin}{\left (c x \right )}}{x} + b e \left (\begin {cases} 0 & \text {for}\: c = 0 \\x \operatorname {asin}{\left (c x \right )} + \frac {\sqrt {- c^{2} x^{2} + 1}}{c} & \text {otherwise} \end {cases}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1032 vs.
\(2 (62) = 124\).
time = 0.57, size = 1032, normalized size = 15.64 \begin {gather*} -\frac {b c^{6} d x^{4} \arcsin \left (c x\right )}{2 \, {\left (\frac {c^{4} x^{3}}{{\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{3}} + \frac {c^{2} x}{\sqrt {-c^{2} x^{2} + 1} + 1}\right )} {\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{4}} - \frac {a c^{6} d x^{4}}{2 \, {\left (\frac {c^{4} x^{3}}{{\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{3}} + \frac {c^{2} x}{\sqrt {-c^{2} x^{2} + 1} + 1}\right )} {\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{4}} + \frac {b c^{5} d x^{3} \log \left ({\left | c \right |} {\left | x \right |}\right )}{{\left (\frac {c^{4} x^{3}}{{\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{3}} + \frac {c^{2} x}{\sqrt {-c^{2} x^{2} + 1} + 1}\right )} {\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{3}} - \frac {b c^{5} d x^{3} \log \left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}{{\left (\frac {c^{4} x^{3}}{{\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{3}} + \frac {c^{2} x}{\sqrt {-c^{2} x^{2} + 1} + 1}\right )} {\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{3}} - \frac {b c^{4} d x^{2} \arcsin \left (c x\right )}{{\left (\frac {c^{4} x^{3}}{{\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{3}} + \frac {c^{2} x}{\sqrt {-c^{2} x^{2} + 1} + 1}\right )} {\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{2}} - \frac {a c^{4} d x^{2}}{{\left (\frac {c^{4} x^{3}}{{\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{3}} + \frac {c^{2} x}{\sqrt {-c^{2} x^{2} + 1} + 1}\right )} {\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{2}} + \frac {b c^{3} d x \log \left ({\left | c \right |} {\left | x \right |}\right )}{{\left (\frac {c^{4} x^{3}}{{\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{3}} + \frac {c^{2} x}{\sqrt {-c^{2} x^{2} + 1} + 1}\right )} {\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}} - \frac {b c^{3} d x \log \left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}{{\left (\frac {c^{4} x^{3}}{{\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{3}} + \frac {c^{2} x}{\sqrt {-c^{2} x^{2} + 1} + 1}\right )} {\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}} - \frac {b c^{3} e x^{3}}{{\left (\frac {c^{4} x^{3}}{{\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{3}} + \frac {c^{2} x}{\sqrt {-c^{2} x^{2} + 1} + 1}\right )} {\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{3}} - \frac {b c^{2} d \arcsin \left (c x\right )}{2 \, {\left (\frac {c^{4} x^{3}}{{\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{3}} + \frac {c^{2} x}{\sqrt {-c^{2} x^{2} + 1} + 1}\right )}} + \frac {2 \, b c^{2} e x^{2} \arcsin \left (c x\right )}{{\left (\frac {c^{4} x^{3}}{{\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{3}} + \frac {c^{2} x}{\sqrt {-c^{2} x^{2} + 1} + 1}\right )} {\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{2}} - \frac {a c^{2} d}{2 \, {\left (\frac {c^{4} x^{3}}{{\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{3}} + \frac {c^{2} x}{\sqrt {-c^{2} x^{2} + 1} + 1}\right )}} + \frac {2 \, a c^{2} e x^{2}}{{\left (\frac {c^{4} x^{3}}{{\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{3}} + \frac {c^{2} x}{\sqrt {-c^{2} x^{2} + 1} + 1}\right )} {\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{2}} + \frac {b c e x}{{\left (\frac {c^{4} x^{3}}{{\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{3}} + \frac {c^{2} x}{\sqrt {-c^{2} x^{2} + 1} + 1}\right )} {\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.36, size = 70, normalized size = 1.06 \begin {gather*} \frac {b\,e\,\left (\sqrt {1-c^2\,x^2}+c\,x\,\mathrm {asin}\left (c\,x\right )\right )}{c}-\frac {b\,d\,\mathrm {asin}\left (c\,x\right )}{x}-b\,c\,d\,\mathrm {atanh}\left (\frac {1}{\sqrt {1-c^2\,x^2}}\right )-\frac {a\,\left (d-e\,x^2\right )}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________