Optimal. Leaf size=191 \[ d^2 x \left (a+b \csc ^{-1}(c x)\right )+\frac {2}{3} d e x^3 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{5} e^2 x^5 \left (a+b \csc ^{-1}(c x)\right )+\frac {b e^2 x^4 \sqrt {c^2 x^2-1}}{20 c \sqrt {c^2 x^2}}+\frac {b x \left (120 c^4 d^2+40 c^2 d e+9 e^2\right ) \tanh ^{-1}\left (\frac {c x}{\sqrt {c^2 x^2-1}}\right )}{120 c^4 \sqrt {c^2 x^2}}+\frac {b e x^2 \sqrt {c^2 x^2-1} \left (40 c^2 d+9 e\right )}{120 c^3 \sqrt {c^2 x^2}} \]
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Rubi [A] time = 0.11, antiderivative size = 191, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 7, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {194, 5229, 12, 1159, 388, 217, 206} \[ d^2 x \left (a+b \csc ^{-1}(c x)\right )+\frac {2}{3} d e x^3 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{5} e^2 x^5 \left (a+b \csc ^{-1}(c x)\right )+\frac {b x \left (120 c^4 d^2+40 c^2 d e+9 e^2\right ) \tanh ^{-1}\left (\frac {c x}{\sqrt {c^2 x^2-1}}\right )}{120 c^4 \sqrt {c^2 x^2}}+\frac {b e x^2 \sqrt {c^2 x^2-1} \left (40 c^2 d+9 e\right )}{120 c^3 \sqrt {c^2 x^2}}+\frac {b e^2 x^4 \sqrt {c^2 x^2-1}}{20 c \sqrt {c^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 194
Rule 206
Rule 217
Rule 388
Rule 1159
Rule 5229
Rubi steps
\begin {align*} \int \left (d+e x^2\right )^2 \left (a+b \csc ^{-1}(c x)\right ) \, dx &=d^2 x \left (a+b \csc ^{-1}(c x)\right )+\frac {2}{3} d e x^3 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{5} e^2 x^5 \left (a+b \csc ^{-1}(c x)\right )+\frac {(b c x) \int \frac {15 d^2+10 d e x^2+3 e^2 x^4}{15 \sqrt {-1+c^2 x^2}} \, dx}{\sqrt {c^2 x^2}}\\ &=d^2 x \left (a+b \csc ^{-1}(c x)\right )+\frac {2}{3} d e x^3 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{5} e^2 x^5 \left (a+b \csc ^{-1}(c x)\right )+\frac {(b c x) \int \frac {15 d^2+10 d e x^2+3 e^2 x^4}{\sqrt {-1+c^2 x^2}} \, dx}{15 \sqrt {c^2 x^2}}\\ &=\frac {b e^2 x^4 \sqrt {-1+c^2 x^2}}{20 c \sqrt {c^2 x^2}}+d^2 x \left (a+b \csc ^{-1}(c x)\right )+\frac {2}{3} d e x^3 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{5} e^2 x^5 \left (a+b \csc ^{-1}(c x)\right )+\frac {(b x) \int \frac {60 c^2 d^2+e \left (40 c^2 d+9 e\right ) x^2}{\sqrt {-1+c^2 x^2}} \, dx}{60 c \sqrt {c^2 x^2}}\\ &=\frac {b e \left (40 c^2 d+9 e\right ) x^2 \sqrt {-1+c^2 x^2}}{120 c^3 \sqrt {c^2 x^2}}+\frac {b e^2 x^4 \sqrt {-1+c^2 x^2}}{20 c \sqrt {c^2 x^2}}+d^2 x \left (a+b \csc ^{-1}(c x)\right )+\frac {2}{3} d e x^3 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{5} e^2 x^5 \left (a+b \csc ^{-1}(c x)\right )-\frac {\left (b \left (-120 c^4 d^2-e \left (40 c^2 d+9 e\right )\right ) x\right ) \int \frac {1}{\sqrt {-1+c^2 x^2}} \, dx}{120 c^3 \sqrt {c^2 x^2}}\\ &=\frac {b e \left (40 c^2 d+9 e\right ) x^2 \sqrt {-1+c^2 x^2}}{120 c^3 \sqrt {c^2 x^2}}+\frac {b e^2 x^4 \sqrt {-1+c^2 x^2}}{20 c \sqrt {c^2 x^2}}+d^2 x \left (a+b \csc ^{-1}(c x)\right )+\frac {2}{3} d e x^3 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{5} e^2 x^5 \left (a+b \csc ^{-1}(c x)\right )-\frac {\left (b \left (-120 c^4 d^2-e \left (40 c^2 d+9 e\right )\right ) x\right ) \operatorname {Subst}\left (\int \frac {1}{1-c^2 x^2} \, dx,x,\frac {x}{\sqrt {-1+c^2 x^2}}\right )}{120 c^3 \sqrt {c^2 x^2}}\\ &=\frac {b e \left (40 c^2 d+9 e\right ) x^2 \sqrt {-1+c^2 x^2}}{120 c^3 \sqrt {c^2 x^2}}+\frac {b e^2 x^4 \sqrt {-1+c^2 x^2}}{20 c \sqrt {c^2 x^2}}+d^2 x \left (a+b \csc ^{-1}(c x)\right )+\frac {2}{3} d e x^3 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{5} e^2 x^5 \left (a+b \csc ^{-1}(c x)\right )+\frac {b \left (120 c^4 d^2+40 c^2 d e+9 e^2\right ) x \tanh ^{-1}\left (\frac {c x}{\sqrt {-1+c^2 x^2}}\right )}{120 c^4 \sqrt {c^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.26, size = 151, normalized size = 0.79 \[ \frac {c^2 x \left (8 a c^3 \left (15 d^2+10 d e x^2+3 e^2 x^4\right )+b e x \sqrt {1-\frac {1}{c^2 x^2}} \left (c^2 \left (40 d+6 e x^2\right )+9 e\right )\right )+8 b c^5 x \csc ^{-1}(c x) \left (15 d^2+10 d e x^2+3 e^2 x^4\right )+b \left (120 c^4 d^2+40 c^2 d e+9 e^2\right ) \log \left (x \left (\sqrt {1-\frac {1}{c^2 x^2}}+1\right )\right )}{120 c^5} \]
Antiderivative was successfully verified.
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fricas [A] time = 2.60, size = 237, normalized size = 1.24 \[ \frac {24 \, a c^{5} e^{2} x^{5} + 80 \, a c^{5} d e x^{3} + 120 \, a c^{5} d^{2} x + 8 \, {\left (3 \, b c^{5} e^{2} x^{5} + 10 \, b c^{5} d e x^{3} + 15 \, b c^{5} d^{2} x - 15 \, b c^{5} d^{2} - 10 \, b c^{5} d e - 3 \, b c^{5} e^{2}\right )} \operatorname {arccsc}\left (c x\right ) - 16 \, {\left (15 \, b c^{5} d^{2} + 10 \, b c^{5} d e + 3 \, b c^{5} e^{2}\right )} \arctan \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (120 \, b c^{4} d^{2} + 40 \, b c^{2} d e + 9 \, b e^{2}\right )} \log \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) + {\left (6 \, b c^{3} e^{2} x^{3} + {\left (40 \, b c^{3} d e + 9 \, b c e^{2}\right )} x\right )} \sqrt {c^{2} x^{2} - 1}}{120 \, c^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 3.20, size = 1027, normalized size = 5.38 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 371, normalized size = 1.94 \[ \frac {a \,e^{2} x^{5}}{5}+\frac {2 a \,x^{3} d e}{3}+a x \,d^{2}+\frac {b \,\mathrm {arccsc}\left (c x \right ) e^{2} x^{5}}{5}+\frac {2 b \,\mathrm {arccsc}\left (c x \right ) x^{3} d e}{3}+b \,\mathrm {arccsc}\left (c x \right ) x \,d^{2}+\frac {b \sqrt {c^{2} x^{2}-1}\, d^{2} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{c^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}+\frac {b \,x^{4} e^{2}}{20 c \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {b \,x^{2} e^{2}}{40 c^{3} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {b e d \,x^{2}}{3 c \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}-\frac {b e d}{3 c^{3} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {b \sqrt {c^{2} x^{2}-1}\, e d \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{3 c^{4} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}-\frac {3 b \,e^{2}}{40 c^{5} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {3 b \sqrt {c^{2} x^{2}-1}\, e^{2} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{40 c^{6} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 296, normalized size = 1.55 \[ \frac {1}{5} \, a e^{2} x^{5} + \frac {2}{3} \, a d e x^{3} + \frac {1}{6} \, {\left (4 \, x^{3} \operatorname {arccsc}\left (c x\right ) + \frac {\frac {2 \, \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{c^{2} {\left (\frac {1}{c^{2} x^{2}} - 1\right )} + c^{2}} + \frac {\log \left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}{c^{2}} - \frac {\log \left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} - 1\right )}{c^{2}}}{c}\right )} b d e + \frac {1}{80} \, {\left (16 \, x^{5} \operatorname {arccsc}\left (c x\right ) - \frac {\frac {2 \, {\left (3 \, {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} - 5 \, \sqrt {-\frac {1}{c^{2} x^{2}} + 1}\right )}}{c^{4} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{2} + 2 \, c^{4} {\left (\frac {1}{c^{2} x^{2}} - 1\right )} + c^{4}} - \frac {3 \, \log \left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}{c^{4}} + \frac {3 \, \log \left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} - 1\right )}{c^{4}}}{c}\right )} b e^{2} + a d^{2} x + \frac {{\left (2 \, c x \operatorname {arccsc}\left (c x\right ) + \log \left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right ) - \log \left (-\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )\right )} b d^{2}}{2 \, c} \]
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 \[ \int {\left (e\,x^2+d\right )}^2\,\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 10.12, size = 355, normalized size = 1.86 \[ a d^{2} x + \frac {2 a d e x^{3}}{3} + \frac {a e^{2} x^{5}}{5} + b d^{2} x \operatorname {acsc}{\left (c x \right )} + \frac {2 b d e x^{3} \operatorname {acsc}{\left (c x \right )}}{3} + \frac {b e^{2} x^{5} \operatorname {acsc}{\left (c x \right )}}{5} + \frac {b d^{2} \left (\begin {cases} \operatorname {acosh}{\left (c x \right )} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\- i \operatorname {asin}{\left (c x \right )} & \text {otherwise} \end {cases}\right )}{c} + \frac {2 b d e \left (\begin {cases} \frac {x \sqrt {c^{2} x^{2} - 1}}{2 c} + \frac {\operatorname {acosh}{\left (c x \right )}}{2 c^{2}} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\- \frac {i c x^{3}}{2 \sqrt {- c^{2} x^{2} + 1}} + \frac {i x}{2 c \sqrt {- c^{2} x^{2} + 1}} - \frac {i \operatorname {asin}{\left (c x \right )}}{2 c^{2}} & \text {otherwise} \end {cases}\right )}{3 c} + \frac {b e^{2} \left (\begin {cases} \frac {c x^{5}}{4 \sqrt {c^{2} x^{2} - 1}} + \frac {x^{3}}{8 c \sqrt {c^{2} x^{2} - 1}} - \frac {3 x}{8 c^{3} \sqrt {c^{2} x^{2} - 1}} + \frac {3 \operatorname {acosh}{\left (c x \right )}}{8 c^{4}} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\- \frac {i c x^{5}}{4 \sqrt {- c^{2} x^{2} + 1}} - \frac {i x^{3}}{8 c \sqrt {- c^{2} x^{2} + 1}} + \frac {3 i x}{8 c^{3} \sqrt {- c^{2} x^{2} + 1}} - \frac {3 i \operatorname {asin}{\left (c x \right )}}{8 c^{4}} & \text {otherwise} \end {cases}\right )}{5 c} \]
Verification of antiderivative is not currently implemented for this CAS.
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