Optimal. Leaf size=190 \[ -\frac {a+b \sin ^{-1}\left (c+d x^2\right )}{6 x^6}-\frac {b c d^2 \sqrt {-c^2-2 c d x^2-d^2 x^4+1}}{4 \left (1-c^2\right )^2 x^2}-\frac {b d \sqrt {-c^2-2 c d x^2-d^2 x^4+1}}{12 \left (1-c^2\right ) x^4}-\frac {b \left (2 c^2+1\right ) d^3 \tanh ^{-1}\left (\frac {-c^2-c d x^2+1}{\sqrt {1-c^2} \sqrt {-c^2-2 c d x^2-d^2 x^4+1}}\right )}{12 \left (1-c^2\right )^{5/2}} \]
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Rubi [A] time = 0.22, antiderivative size = 190, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {4842, 12, 1114, 744, 806, 724, 206} \[ -\frac {a+b \sin ^{-1}\left (c+d x^2\right )}{6 x^6}-\frac {b c d^2 \sqrt {-c^2-2 c d x^2-d^2 x^4+1}}{4 \left (1-c^2\right )^2 x^2}-\frac {b d \sqrt {-c^2-2 c d x^2-d^2 x^4+1}}{12 \left (1-c^2\right ) x^4}-\frac {b \left (2 c^2+1\right ) d^3 \tanh ^{-1}\left (\frac {-c^2-c d x^2+1}{\sqrt {1-c^2} \sqrt {-c^2-2 c d x^2-d^2 x^4+1}}\right )}{12 \left (1-c^2\right )^{5/2}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 206
Rule 724
Rule 744
Rule 806
Rule 1114
Rule 4842
Rubi steps
\begin {align*} \int \frac {a+b \sin ^{-1}\left (c+d x^2\right )}{x^7} \, dx &=-\frac {a+b \sin ^{-1}\left (c+d x^2\right )}{6 x^6}+\frac {1}{6} b \int \frac {2 d}{x^5 \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 )}{6 x^6}+\frac {1}{3} (b d) \int \frac {1}{x^5 \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 )}{6 x^6}+\frac {1}{6} (b d) \operatorname {Subst}\left (\int \frac {1}{x^3 \sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx,x,x^2\right )\\ &=-\frac {b d \sqrt {1-c^2-2 c d x^2-d^2 x^4}}{12 \left (1-c^2\right ) x^4}-\frac {a+b \sin ^{-1}\left (c+d x^2\right )}{6 x^6}-\frac {(b d) \operatorname {Subst}\left (\int \frac {-3 c d-d^2 x}{x^2 \sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx,x,x^2\right )}{12 \left (1-c^2\right )}\\ &=-\frac {b d \sqrt {1-c^2-2 c d x^2-d^2 x^4}}{12 \left (1-c^2\right ) x^4}-\frac {b c d^2 \sqrt {1-c^2-2 c d x^2-d^2 x^4}}{4 \left (1-c^2\right )^2 x^2}-\frac {a+b \sin ^{-1}\left (c+d x^2\right )}{6 x^6}+\frac {\left (b \left (1+2 c^2\right ) d^3\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx,x,x^2\right )}{12 \left (1-c^2\right )^2}\\ &=-\frac {b d \sqrt {1-c^2-2 c d x^2-d^2 x^4}}{12 \left (1-c^2\right ) x^4}-\frac {b c d^2 \sqrt {1-c^2-2 c d x^2-d^2 x^4}}{4 \left (1-c^2\right )^2 x^2}-\frac {a+b \sin ^{-1}\left (c+d x^2\right )}{6 x^6}-\frac {\left (b \left (1+2 c^2\right ) d^3\right ) \operatorname {Subst}\left (\int \frac {1}{4 \left (1-c^2\right )-x^2} \, dx,x,\frac {2 \left (1-c^2-c d x^2\right )}{\sqrt {1-c^2-2 c d x^2-d^2 x^4}}\right )}{6 \left (1-c^2\right )^2}\\ &=-\frac {b d \sqrt {1-c^2-2 c d x^2-d^2 x^4}}{12 \left (1-c^2\right ) x^4}-\frac {b c d^2 \sqrt {1-c^2-2 c d x^2-d^2 x^4}}{4 \left (1-c^2\right )^2 x^2}-\frac {a+b \sin ^{-1}\left (c+d x^2\right )}{6 x^6}-\frac {b \left (1+2 c^2\right ) d^3 \tanh ^{-1}\left (\frac {1-c^2-c d x^2}{\sqrt {1-c^2} \sqrt {1-c^2-2 c d x^2-d^2 x^4}}\right )}{12 \left (1-c^2\right )^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.28, size = 176, normalized size = 0.93 \[ -\frac {a}{6 x^6}+b \left (\frac {d}{12 \left (c^2-1\right ) x^4}-\frac {c d^2}{4 \left (c^2-1\right )^2 x^2}\right ) \sqrt {-c^2-2 c d x^2-d^2 x^4+1}-\frac {b \left (2 c^2+1\right ) d^3 \tanh ^{-1}\left (\frac {-c^2-c d x^2+1}{\sqrt {1-c^2} \sqrt {-c^2-2 c d x^2-d^2 x^4+1}}\right )}{12 (c-1)^2 (c+1)^2 \sqrt {1-c^2}}-\frac {b \sin ^{-1}\left (c+d x^2\right )}{6 x^6} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.65, size = 496, normalized size = 2.61 \[ \left [-\frac {{\left (2 \, b c^{2} + b\right )} \sqrt {-c^{2} + 1} d^{3} x^{6} \log \left (\frac {{\left (2 \, c^{2} - 1\right )} d^{2} x^{4} + 2 \, c^{4} + 4 \, {\left (c^{3} - c\right )} d x^{2} + 2 \, \sqrt {-d^{2} x^{4} - 2 \, c d x^{2} - c^{2} + 1} {\left (c d x^{2} + c^{2} - 1\right )} \sqrt {-c^{2} + 1} - 4 \, c^{2} + 2}{x^{4}}\right ) + 4 \, a c^{6} - 12 \, a c^{4} + 12 \, a c^{2} + 4 \, {\left (b c^{6} - 3 \, b c^{4} + 3 \, b c^{2} - b\right )} \arcsin \left (d x^{2} + c\right ) + 2 \, {\left (3 \, {\left (b c^{3} - b c\right )} d^{2} x^{4} - {\left (b c^{4} - 2 \, b c^{2} + b\right )} d x^{2}\right )} \sqrt {-d^{2} x^{4} - 2 \, c d x^{2} - c^{2} + 1} - 4 \, a}{24 \, {\left (c^{6} - 3 \, c^{4} + 3 \, c^{2} - 1\right )} x^{6}}, \frac {{\left (2 \, b c^{2} + b\right )} \sqrt {c^{2} - 1} d^{3} x^{6} \arctan \left (\frac {\sqrt {-d^{2} x^{4} - 2 \, c d x^{2} - c^{2} + 1} {\left (c d x^{2} + c^{2} - 1\right )} \sqrt {c^{2} - 1}}{{\left (c^{2} - 1\right )} d^{2} x^{4} + c^{4} + 2 \, {\left (c^{3} - c\right )} d x^{2} - 2 \, c^{2} + 1}\right ) - 2 \, a c^{6} + 6 \, a c^{4} - 6 \, a c^{2} - 2 \, {\left (b c^{6} - 3 \, b c^{4} + 3 \, b c^{2} - b\right )} \arcsin \left (d x^{2} + c\right ) - {\left (3 \, {\left (b c^{3} - b c\right )} d^{2} x^{4} - {\left (b c^{4} - 2 \, b c^{2} + b\right )} d x^{2}\right )} \sqrt {-d^{2} x^{4} - 2 \, c d x^{2} - c^{2} + 1} + 2 \, a}{12 \, {\left (c^{6} - 3 \, c^{4} + 3 \, c^{2} - 1\right )} x^{6}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {b \arcsin \left (d x^{2} + c\right ) + a}{x^{7}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 246, normalized size = 1.29 \[ -\frac {a}{6 x^{6}}-\frac {b \arcsin \left (d \,x^{2}+c \right )}{6 x^{6}}-\frac {b d \sqrt {-d^{2} x^{4}-2 c d \,x^{2}-c^{2}+1}}{12 \left (-c^{2}+1\right ) x^{4}}-\frac {b c \,d^{2} \sqrt {-d^{2} x^{4}-2 c d \,x^{2}-c^{2}+1}}{4 \left (-c^{2}+1\right )^{2} x^{2}}-\frac {b \,d^{3} c^{2} \ln \left (\frac {-2 c^{2}+2-2 c d \,x^{2}+2 \sqrt {-c^{2}+1}\, \sqrt {-d^{2} x^{4}-2 c d \,x^{2}-c^{2}+1}}{x^{2}}\right )}{4 \left (-c^{2}+1\right )^{\frac {5}{2}}}-\frac {b \,d^{3} \ln \left (\frac {-2 c^{2}+2-2 c d \,x^{2}+2 \sqrt {-c^{2}+1}\, \sqrt {-d^{2} x^{4}-2 c d \,x^{2}-c^{2}+1}}{x^{2}}\right )}{12 \left (-c^{2}+1\right )^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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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.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {a+b\,\mathrm {asin}\left (d\,x^2+c\right )}{x^7} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {a + b \operatorname {asin}{\left (c + d x^{2} \right )}}{x^{7}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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