Optimal. Leaf size=164 \[ -\frac {11}{5} b c d^3 \sqrt {1-c^2 x^2}-\frac {1}{5} b c d^3 \left (1-c^2 x^2\right )^{3/2}-\frac {1}{25} b c d^3 \left (1-c^2 x^2\right )^{5/2}-\frac {d^3 (a+b \text {ArcSin}(c x))}{x}-3 c^2 d^3 x (a+b \text {ArcSin}(c x))+c^4 d^3 x^3 (a+b \text {ArcSin}(c x))-\frac {1}{5} c^6 d^3 x^5 (a+b \text {ArcSin}(c x))-b c d^3 \tanh ^{-1}\left (\sqrt {1-c^2 x^2}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.16, antiderivative size = 164, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 7, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {276, 4777, 12,
1813, 1634, 65, 214} \begin {gather*} -\frac {1}{5} c^6 d^3 x^5 (a+b \text {ArcSin}(c x))+c^4 d^3 x^3 (a+b \text {ArcSin}(c x))-3 c^2 d^3 x (a+b \text {ArcSin}(c x))-\frac {d^3 (a+b \text {ArcSin}(c x))}{x}-\frac {1}{25} b c d^3 \left (1-c^2 x^2\right )^{5/2}-\frac {1}{5} b c d^3 \left (1-c^2 x^2\right )^{3/2}-\frac {11}{5} b c d^3 \sqrt {1-c^2 x^2}-b c d^3 \tanh ^{-1}\left (\sqrt {1-c^2 x^2}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 65
Rule 214
Rule 276
Rule 1634
Rule 1813
Rule 4777
Rubi steps
\begin {align*} \int \frac {\left (d-c^2 d x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )}{x^2} \, dx &=-\frac {d^3 \left (a+b \sin ^{-1}(c x)\right )}{x}-3 c^2 d^3 x \left (a+b \sin ^{-1}(c x)\right )+c^4 d^3 x^3 \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{5} c^6 d^3 x^5 \left (a+b \sin ^{-1}(c x)\right )-(b c) \int \frac {d^3 \left (-5-15 c^2 x^2+5 c^4 x^4-c^6 x^6\right )}{5 x \sqrt {1-c^2 x^2}} \, dx\\ &=-\frac {d^3 \left (a+b \sin ^{-1}(c x)\right )}{x}-3 c^2 d^3 x \left (a+b \sin ^{-1}(c x)\right )+c^4 d^3 x^3 \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{5} c^6 d^3 x^5 \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{5} \left (b c d^3\right ) \int \frac {-5-15 c^2 x^2+5 c^4 x^4-c^6 x^6}{x \sqrt {1-c^2 x^2}} \, dx\\ &=-\frac {d^3 \left (a+b \sin ^{-1}(c x)\right )}{x}-3 c^2 d^3 x \left (a+b \sin ^{-1}(c x)\right )+c^4 d^3 x^3 \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{5} c^6 d^3 x^5 \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{10} \left (b c d^3\right ) \text {Subst}\left (\int \frac {-5-15 c^2 x+5 c^4 x^2-c^6 x^3}{x \sqrt {1-c^2 x}} \, dx,x,x^2\right )\\ &=-\frac {d^3 \left (a+b \sin ^{-1}(c x)\right )}{x}-3 c^2 d^3 x \left (a+b \sin ^{-1}(c x)\right )+c^4 d^3 x^3 \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{5} c^6 d^3 x^5 \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{10} \left (b c d^3\right ) \text {Subst}\left (\int \left (-\frac {11 c^2}{\sqrt {1-c^2 x}}-\frac {5}{x \sqrt {1-c^2 x}}-3 c^2 \sqrt {1-c^2 x}-c^2 \left (1-c^2 x\right )^{3/2}\right ) \, dx,x,x^2\right )\\ &=-\frac {11}{5} b c d^3 \sqrt {1-c^2 x^2}-\frac {1}{5} b c d^3 \left (1-c^2 x^2\right )^{3/2}-\frac {1}{25} b c d^3 \left (1-c^2 x^2\right )^{5/2}-\frac {d^3 \left (a+b \sin ^{-1}(c x)\right )}{x}-3 c^2 d^3 x \left (a+b \sin ^{-1}(c x)\right )+c^4 d^3 x^3 \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{5} c^6 d^3 x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{2} \left (b c d^3\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {1-c^2 x}} \, dx,x,x^2\right )\\ &=-\frac {11}{5} b c d^3 \sqrt {1-c^2 x^2}-\frac {1}{5} b c d^3 \left (1-c^2 x^2\right )^{3/2}-\frac {1}{25} b c d^3 \left (1-c^2 x^2\right )^{5/2}-\frac {d^3 \left (a+b \sin ^{-1}(c x)\right )}{x}-3 c^2 d^3 x \left (a+b \sin ^{-1}(c x)\right )+c^4 d^3 x^3 \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{5} c^6 d^3 x^5 \left (a+b \sin ^{-1}(c x)\right )-\frac {\left (b d^3\right ) \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 {11}{5} b c d^3 \sqrt {1-c^2 x^2}-\frac {1}{5} b c d^3 \left (1-c^2 x^2\right )^{3/2}-\frac {1}{25} b c d^3 \left (1-c^2 x^2\right )^{5/2}-\frac {d^3 \left (a+b \sin ^{-1}(c x)\right )}{x}-3 c^2 d^3 x \left (a+b \sin ^{-1}(c x)\right )+c^4 d^3 x^3 \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{5} c^6 d^3 x^5 \left (a+b \sin ^{-1}(c x)\right )-b c d^3 \tanh ^{-1}\left (\sqrt {1-c^2 x^2}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.07, size = 166, normalized size = 1.01 \begin {gather*} -\frac {d^3 \left (25 a+75 a c^2 x^2-25 a c^4 x^4+5 a c^6 x^6+61 b c x \sqrt {1-c^2 x^2}-7 b c^3 x^3 \sqrt {1-c^2 x^2}+b c^5 x^5 \sqrt {1-c^2 x^2}+5 b \left (5+15 c^2 x^2-5 c^4 x^4+c^6 x^6\right ) \text {ArcSin}(c x)-25 b c x \log (x)+25 b c x \log \left (1+\sqrt {1-c^2 x^2}\right )\right )}{25 x} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.07, size = 155, normalized size = 0.95
method | result | size |
derivativedivides | \(c \left (-d^{3} a \left (\frac {c^{5} x^{5}}{5}-c^{3} x^{3}+3 c x +\frac {1}{c x}\right )-d^{3} b \left (\frac {\arcsin \left (c x \right ) c^{5} x^{5}}{5}-c^{3} x^{3} \arcsin \left (c x \right )+3 c x \arcsin \left (c x \right )+\frac {\arcsin \left (c x \right )}{c x}+\frac {c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{25}-\frac {7 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{25}+\frac {61 \sqrt {-c^{2} x^{2}+1}}{25}+\arctanh \left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right )\right )\right )\) | \(155\) |
default | \(c \left (-d^{3} a \left (\frac {c^{5} x^{5}}{5}-c^{3} x^{3}+3 c x +\frac {1}{c x}\right )-d^{3} b \left (\frac {\arcsin \left (c x \right ) c^{5} x^{5}}{5}-c^{3} x^{3} \arcsin \left (c x \right )+3 c x \arcsin \left (c x \right )+\frac {\arcsin \left (c x \right )}{c x}+\frac {c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{25}-\frac {7 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{25}+\frac {61 \sqrt {-c^{2} x^{2}+1}}{25}+\arctanh \left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right )\right )\right )\) | \(155\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.48, size = 250, normalized size = 1.52 \begin {gather*} -\frac {1}{5} \, a c^{6} d^{3} x^{5} - \frac {1}{75} \, {\left (15 \, x^{5} \arcsin \left (c x\right ) + {\left (\frac {3 \, \sqrt {-c^{2} x^{2} + 1} x^{4}}{c^{2}} + \frac {4 \, \sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {-c^{2} x^{2} + 1}}{c^{6}}\right )} c\right )} b c^{6} d^{3} + a c^{4} d^{3} x^{3} + \frac {1}{3} \, {\left (3 \, x^{3} \arcsin \left (c x\right ) + c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} b c^{4} d^{3} - 3 \, a c^{2} d^{3} x - 3 \, {\left (c x \arcsin \left (c x\right ) + \sqrt {-c^{2} x^{2} + 1}\right )} b c d^{3} - {\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^{3} - \frac {a d^{3}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 3.13, size = 188, normalized size = 1.15 \begin {gather*} -\frac {10 \, a c^{6} d^{3} x^{6} - 50 \, a c^{4} d^{3} x^{4} + 150 \, a c^{2} d^{3} x^{2} + 25 \, b c d^{3} x \log \left (\sqrt {-c^{2} x^{2} + 1} + 1\right ) - 25 \, b c d^{3} x \log \left (\sqrt {-c^{2} x^{2} + 1} - 1\right ) + 50 \, a d^{3} + 10 \, {\left (b c^{6} d^{3} x^{6} - 5 \, b c^{4} d^{3} x^{4} + 15 \, b c^{2} d^{3} x^{2} + 5 \, b d^{3}\right )} \arcsin \left (c x\right ) + 2 \, {\left (b c^{5} d^{3} x^{5} - 7 \, b c^{3} d^{3} x^{3} + 61 \, b c d^{3} x\right )} \sqrt {-c^{2} x^{2} + 1}}{50 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 4.95, size = 287, normalized size = 1.75 \begin {gather*} - \frac {a c^{6} d^{3} x^{5}}{5} + a c^{4} d^{3} x^{3} - 3 a c^{2} d^{3} x - \frac {a d^{3}}{x} + \frac {b c^{7} d^{3} \left (\begin {cases} - \frac {x^{4} \sqrt {- c^{2} x^{2} + 1}}{5 c^{2}} - \frac {4 x^{2} \sqrt {- c^{2} x^{2} + 1}}{15 c^{4}} - \frac {8 \sqrt {- c^{2} x^{2} + 1}}{15 c^{6}} & \text {for}\: c \neq 0 \\\frac {x^{6}}{6} & \text {otherwise} \end {cases}\right )}{5} - \frac {b c^{6} d^{3} x^{5} \operatorname {asin}{\left (c x \right )}}{5} - b c^{5} d^{3} \left (\begin {cases} - \frac {x^{2} \sqrt {- c^{2} x^{2} + 1}}{3 c^{2}} - \frac {2 \sqrt {- c^{2} x^{2} + 1}}{3 c^{4}} & \text {for}\: c \neq 0 \\\frac {x^{4}}{4} & \text {otherwise} \end {cases}\right ) + b c^{4} d^{3} x^{3} \operatorname {asin}{\left (c x \right )} - 3 b c^{2} d^{3} \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 ) + b c d^{3} \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^{3} \operatorname {asin}{\left (c x \right )}}{x} \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. 5513 vs.
\(2 (148) = 296\).
time = 28.59, size = 5513, normalized size = 33.62 \begin {gather*} \text {Too large to display} \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 {\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^3}{x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________