Optimal. Leaf size=284 \[ -\frac {a+b \sin ^{-1}\left (c+d x^2\right )}{3 x^3}-\frac {2 b d \sqrt {-c^2-2 c d x^2-d^2 x^4+1}}{3 \left (1-c^2\right ) x}+\frac {2 b d^{3/2} \sqrt {1-\frac {d x^2}{1-c}} \sqrt {\frac {d x^2}{c+1}+1} F\left (\sin ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {1-c}}\right )|-\frac {1-c}{c+1}\right )}{3 \sqrt {1-c} \sqrt {-c^2-2 c d x^2-d^2 x^4+1}}-\frac {2 b d^{3/2} \sqrt {1-\frac {d x^2}{1-c}} \sqrt {\frac {d x^2}{c+1}+1} E\left (\sin ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {1-c}}\right )|-\frac {1-c}{c+1}\right )}{3 \sqrt {1-c} \sqrt {-c^2-2 c d x^2-d^2 x^4+1}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.25, antiderivative size = 284, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {4842, 12, 1123, 1140, 493, 424, 419} \[ -\frac {a+b \sin ^{-1}\left (c+d x^2\right )}{3 x^3}-\frac {2 b d \sqrt {-c^2-2 c d x^2-d^2 x^4+1}}{3 \left (1-c^2\right ) x}+\frac {2 b d^{3/2} \sqrt {1-\frac {d x^2}{1-c}} \sqrt {\frac {d x^2}{c+1}+1} F\left (\sin ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {1-c}}\right )|-\frac {1-c}{c+1}\right )}{3 \sqrt {1-c} \sqrt {-c^2-2 c d x^2-d^2 x^4+1}}-\frac {2 b d^{3/2} \sqrt {1-\frac {d x^2}{1-c}} \sqrt {\frac {d x^2}{c+1}+1} E\left (\sin ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {1-c}}\right )|-\frac {1-c}{c+1}\right )}{3 \sqrt {1-c} \sqrt {-c^2-2 c d x^2-d^2 x^4+1}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 419
Rule 424
Rule 493
Rule 1123
Rule 1140
Rule 4842
Rubi steps
\begin {align*} \int \frac {a+b \sin ^{-1}\left (c+d x^2\right )}{x^4} \, dx &=-\frac {a+b \sin ^{-1}\left (c+d x^2\right )}{3 x^3}+\frac {1}{3} b \int \frac {2 d}{x^2 \sqrt {1-c^2-2 c d x^2-d^2 x^4}} \, dx\\ &=-\frac {a+b \sin ^{-1}\left (c+d x^2\right )}{3 x^3}+\frac {1}{3} (2 b d) \int \frac {1}{x^2 \sqrt {1-c^2-2 c d x^2-d^2 x^4}} \, dx\\ &=-\frac {2 b d \sqrt {1-c^2-2 c d x^2-d^2 x^4}}{3 \left (1-c^2\right ) x}-\frac {a+b \sin ^{-1}\left (c+d x^2\right )}{3 x^3}-\frac {(2 b d) \int \frac {d^2 x^2}{\sqrt {1-c^2-2 c d x^2-d^2 x^4}} \, dx}{3 \left (1-c^2\right )}\\ &=-\frac {2 b d \sqrt {1-c^2-2 c d x^2-d^2 x^4}}{3 \left (1-c^2\right ) x}-\frac {a+b \sin ^{-1}\left (c+d x^2\right )}{3 x^3}-\frac {\left (2 b d^3\right ) \int \frac {x^2}{\sqrt {1-c^2-2 c d x^2-d^2 x^4}} \, dx}{3 \left (1-c^2\right )}\\ &=-\frac {2 b d \sqrt {1-c^2-2 c d x^2-d^2 x^4}}{3 \left (1-c^2\right ) x}-\frac {a+b \sin ^{-1}\left (c+d x^2\right )}{3 x^3}-\frac {\left (2 b d^3 \sqrt {1-\frac {2 d^2 x^2}{-2 d-2 c d}} \sqrt {1-\frac {2 d^2 x^2}{2 d-2 c d}}\right ) \int \frac {x^2}{\sqrt {1-\frac {2 d^2 x^2}{-2 d-2 c d}} \sqrt {1-\frac {2 d^2 x^2}{2 d-2 c d}}} \, dx}{3 \left (1-c^2\right ) \sqrt {1-c^2-2 c d x^2-d^2 x^4}}\\ &=-\frac {2 b d \sqrt {1-c^2-2 c d x^2-d^2 x^4}}{3 \left (1-c^2\right ) x}-\frac {a+b \sin ^{-1}\left (c+d x^2\right )}{3 x^3}+\frac {\left (2 b (1+c) d^2 \sqrt {1-\frac {2 d^2 x^2}{-2 d-2 c d}} \sqrt {1-\frac {2 d^2 x^2}{2 d-2 c d}}\right ) \int \frac {1}{\sqrt {1-\frac {2 d^2 x^2}{-2 d-2 c d}} \sqrt {1-\frac {2 d^2 x^2}{2 d-2 c d}}} \, dx}{3 \left (1-c^2\right ) \sqrt {1-c^2-2 c d x^2-d^2 x^4}}-\frac {\left (2 b (1+c) d^2 \sqrt {1-\frac {2 d^2 x^2}{-2 d-2 c d}} \sqrt {1-\frac {2 d^2 x^2}{2 d-2 c d}}\right ) \int \frac {\sqrt {1-\frac {2 d^2 x^2}{-2 d-2 c d}}}{\sqrt {1-\frac {2 d^2 x^2}{2 d-2 c d}}} \, dx}{3 \left (1-c^2\right ) \sqrt {1-c^2-2 c d x^2-d^2 x^4}}\\ &=-\frac {2 b d \sqrt {1-c^2-2 c d x^2-d^2 x^4}}{3 \left (1-c^2\right ) x}-\frac {a+b \sin ^{-1}\left (c+d x^2\right )}{3 x^3}-\frac {2 b d^{3/2} \sqrt {1-\frac {d x^2}{1-c}} \sqrt {1+\frac {d x^2}{1+c}} E\left (\sin ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {1-c}}\right )|-\frac {1-c}{1+c}\right )}{3 \sqrt {1-c} \sqrt {1-c^2-2 c d x^2-d^2 x^4}}+\frac {2 b d^{3/2} \sqrt {1-\frac {d x^2}{1-c}} \sqrt {1+\frac {d x^2}{1+c}} F\left (\sin ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {1-c}}\right )|-\frac {1-c}{1+c}\right )}{3 \sqrt {1-c} \sqrt {1-c^2-2 c d x^2-d^2 x^4}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.42, size = 243, normalized size = 0.86 \[ -\frac {a}{3 x^3}+\frac {2 b d \sqrt {-c^2-2 c d x^2-d^2 x^4+1}}{3 \left (c^2-1\right ) x}+\frac {2 i b (1-c) d^2 \sqrt {1-\frac {d x^2}{-c-1}} \sqrt {1-\frac {d x^2}{1-c}} \left (E\left (i \sinh ^{-1}\left (\sqrt {-\frac {d}{-c-1}} x\right )|\frac {-c-1}{1-c}\right )-F\left (i \sinh ^{-1}\left (\sqrt {-\frac {d}{-c-1}} x\right )|\frac {-c-1}{1-c}\right )\right )}{3 (c-1) (c+1) \sqrt {-\frac {d}{-c-1}} \sqrt {-c^2-2 c d x^2-d^2 x^4+1}}-\frac {b \sin ^{-1}\left (c+d x^2\right )}{3 x^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 1.38, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b \arcsin \left (d x^{2} + c\right ) + a}{x^{4}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {b \arcsin \left (d x^{2} + c\right ) + a}{x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 207, normalized size = 0.73 \[ -\frac {a}{3 x^{3}}+b \left (-\frac {\arcsin \left (d \,x^{2}+c \right )}{3 x^{3}}+\frac {2 d \left (\frac {\sqrt {-d^{2} x^{4}-2 c d \,x^{2}-c^{2}+1}}{\left (c^{2}-1\right ) x}-\frac {2 d^{2} \left (-c^{2}+1\right ) \sqrt {1+\frac {d \,x^{2}}{-1+c}}\, \sqrt {1+\frac {d \,x^{2}}{1+c}}\, \left (\EllipticF \left (x \sqrt {-\frac {d}{-1+c}}, \sqrt {-1+\frac {2 c}{1+c}}\right )-\EllipticE \left (x \sqrt {-\frac {d}{-1+c}}, \sqrt {-1+\frac {2 c}{1+c}}\right )\right )}{\left (c^{2}-1\right ) \sqrt {-\frac {d}{-1+c}}\, \sqrt {-d^{2} x^{4}-2 c d \,x^{2}-c^{2}+1}\, \left (-2 d c +2 d \right )}\right )}{3}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {a+b\,\mathrm {asin}\left (d\,x^2+c\right )}{x^4} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {a + b \operatorname {asin}{\left (c + d x^{2} \right )}}{x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________