Optimal. Leaf size=123 \[ \frac {b d e \sqrt {1-\frac {1}{c^2 x^2}} x}{c}+\frac {b e^2 \sqrt {1-\frac {1}{c^2 x^2}} x^2}{6 c}-\frac {b d^3 \csc ^{-1}(c x)}{3 e}+\frac {(d+e x)^3 \left (a+b \csc ^{-1}(c x)\right )}{3 e}+\frac {b \left (6 c^2 d^2+e^2\right ) \tanh ^{-1}\left (\sqrt {1-\frac {1}{c^2 x^2}}\right )}{6 c^3} \]
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Rubi [A]
time = 0.19, antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 9, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.562, Rules used = {5335, 1582,
1489, 1821, 858, 222, 272, 65, 214} \begin {gather*} \frac {(d+e x)^3 \left (a+b \csc ^{-1}(c x)\right )}{3 e}+\frac {b d e x \sqrt {1-\frac {1}{c^2 x^2}}}{c}+\frac {b e^2 x^2 \sqrt {1-\frac {1}{c^2 x^2}}}{6 c}+\frac {b \left (6 c^2 d^2+e^2\right ) \tanh ^{-1}\left (\sqrt {1-\frac {1}{c^2 x^2}}\right )}{6 c^3}-\frac {b d^3 \csc ^{-1}(c x)}{3 e} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 214
Rule 222
Rule 272
Rule 858
Rule 1489
Rule 1582
Rule 1821
Rule 5335
Rubi steps
\begin {align*} \int (d+e x)^2 \left (a+b \csc ^{-1}(c x)\right ) \, dx &=\frac {(d+e x)^3 \left (a+b \csc ^{-1}(c x)\right )}{3 e}+\frac {b \int \frac {(d+e x)^3}{\sqrt {1-\frac {1}{c^2 x^2}} x^2} \, dx}{3 c e}\\ &=\frac {(d+e x)^3 \left (a+b \csc ^{-1}(c x)\right )}{3 e}+\frac {b \int \frac {\left (e+\frac {d}{x}\right )^3 x}{\sqrt {1-\frac {1}{c^2 x^2}}} \, dx}{3 c e}\\ &=\frac {(d+e x)^3 \left (a+b \csc ^{-1}(c x)\right )}{3 e}-\frac {b \text {Subst}\left (\int \frac {(e+d x)^3}{x^3 \sqrt {1-\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{3 c e}\\ &=\frac {b e^2 \sqrt {1-\frac {1}{c^2 x^2}} x^2}{6 c}+\frac {(d+e x)^3 \left (a+b \csc ^{-1}(c x)\right )}{3 e}+\frac {b \text {Subst}\left (\int \frac {-6 d e^2-e \left (6 d^2+\frac {e^2}{c^2}\right ) x-2 d^3 x^2}{x^2 \sqrt {1-\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{6 c e}\\ &=\frac {b d e \sqrt {1-\frac {1}{c^2 x^2}} x}{c}+\frac {b e^2 \sqrt {1-\frac {1}{c^2 x^2}} x^2}{6 c}+\frac {(d+e x)^3 \left (a+b \csc ^{-1}(c x)\right )}{3 e}-\frac {b \text {Subst}\left (\int \frac {e \left (6 d^2+\frac {e^2}{c^2}\right )+2 d^3 x}{x \sqrt {1-\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{6 c e}\\ &=\frac {b d e \sqrt {1-\frac {1}{c^2 x^2}} x}{c}+\frac {b e^2 \sqrt {1-\frac {1}{c^2 x^2}} x^2}{6 c}+\frac {(d+e x)^3 \left (a+b \csc ^{-1}(c x)\right )}{3 e}-\frac {\left (b d^3\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{3 c e}-\frac {\left (b \left (6 c^2 d^2+e^2\right )\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{6 c^3}\\ &=\frac {b d e \sqrt {1-\frac {1}{c^2 x^2}} x}{c}+\frac {b e^2 \sqrt {1-\frac {1}{c^2 x^2}} x^2}{6 c}-\frac {b d^3 \csc ^{-1}(c x)}{3 e}+\frac {(d+e x)^3 \left (a+b \csc ^{-1}(c x)\right )}{3 e}-\frac {\left (b \left (6 c^2 d^2+e^2\right )\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x}{c^2}}} \, dx,x,\frac {1}{x^2}\right )}{12 c^3}\\ &=\frac {b d e \sqrt {1-\frac {1}{c^2 x^2}} x}{c}+\frac {b e^2 \sqrt {1-\frac {1}{c^2 x^2}} x^2}{6 c}-\frac {b d^3 \csc ^{-1}(c x)}{3 e}+\frac {(d+e x)^3 \left (a+b \csc ^{-1}(c x)\right )}{3 e}+\frac {\left (b \left (6 c^2 d^2+e^2\right )\right ) \text {Subst}\left (\int \frac {1}{c^2-c^2 x^2} \, dx,x,\sqrt {1-\frac {1}{c^2 x^2}}\right )}{6 c}\\ &=\frac {b d e \sqrt {1-\frac {1}{c^2 x^2}} x}{c}+\frac {b e^2 \sqrt {1-\frac {1}{c^2 x^2}} x^2}{6 c}-\frac {b d^3 \csc ^{-1}(c x)}{3 e}+\frac {(d+e x)^3 \left (a+b \csc ^{-1}(c x)\right )}{3 e}+\frac {b \left (6 c^2 d^2+e^2\right ) \tanh ^{-1}\left (\sqrt {1-\frac {1}{c^2 x^2}}\right )}{6 c^3}\\ \end {align*}
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Mathematica [A]
time = 0.11, size = 122, normalized size = 0.99 \begin {gather*} \frac {c^2 x \left (b e \sqrt {1-\frac {1}{c^2 x^2}} (6 d+e x)+2 a c \left (3 d^2+3 d e x+e^2 x^2\right )\right )+2 b c^3 x \left (3 d^2+3 d e x+e^2 x^2\right ) \csc ^{-1}(c x)+b \left (6 c^2 d^2+e^2\right ) \log \left (\left (1+\sqrt {1-\frac {1}{c^2 x^2}}\right ) x\right )}{6 c^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(314\) vs.
\(2(109)=218\).
time = 0.23, size = 315, normalized size = 2.56
method | result | size |
derivativedivides | \(\frac {\frac {\left (e c x +c d \right )^{3} a}{3 c^{2} e}+\frac {b c \,\mathrm {arccsc}\left (c x \right ) d^{3}}{3 e}+b \,\mathrm {arccsc}\left (c x \right ) d^{2} c x +b c e \,\mathrm {arccsc}\left (c x \right ) d \,x^{2}+\frac {b c \,e^{2} \mathrm {arccsc}\left (c x \right ) x^{3}}{3}-\frac {b \sqrt {c^{2} x^{2}-1}\, d^{3} \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )}{3 e \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}+\frac {b \sqrt {c^{2} x^{2}-1}\, d^{2} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{\sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}+\frac {b e \left (c^{2} x^{2}-1\right ) d}{c^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}+\frac {b \,e^{2} \left (c^{2} x^{2}-1\right )}{6 c^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {b \,e^{2} \sqrt {c^{2} x^{2}-1}\, \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{6 c^{3} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}}{c}\) | \(315\) |
default | \(\frac {\frac {\left (e c x +c d \right )^{3} a}{3 c^{2} e}+\frac {b c \,\mathrm {arccsc}\left (c x \right ) d^{3}}{3 e}+b \,\mathrm {arccsc}\left (c x \right ) d^{2} c x +b c e \,\mathrm {arccsc}\left (c x \right ) d \,x^{2}+\frac {b c \,e^{2} \mathrm {arccsc}\left (c x \right ) x^{3}}{3}-\frac {b \sqrt {c^{2} x^{2}-1}\, d^{3} \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )}{3 e \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}+\frac {b \sqrt {c^{2} x^{2}-1}\, d^{2} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{\sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}+\frac {b e \left (c^{2} x^{2}-1\right ) d}{c^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}+\frac {b \,e^{2} \left (c^{2} x^{2}-1\right )}{6 c^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {b \,e^{2} \sqrt {c^{2} x^{2}-1}\, \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{6 c^{3} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}}{c}\) | \(315\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 198, normalized size = 1.61 \begin {gather*} \frac {1}{3} \, a x^{3} e^{2} + a d x^{2} e + a d^{2} x + {\left (x^{2} \operatorname {arccsc}\left (c x\right ) + \frac {x \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{c}\right )} b d e + \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} + \frac {1}{12} \, {\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 e^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.41, size = 208, normalized size = 1.69 \begin {gather*} \frac {2 \, a c^{3} x^{3} e^{2} + 6 \, a c^{3} d x^{2} e + 6 \, a c^{3} d^{2} x + 2 \, {\left (3 \, b c^{3} d^{2} x - 3 \, b c^{3} d^{2} + {\left (b c^{3} x^{3} - b c^{3}\right )} e^{2} + 3 \, {\left (b c^{3} d x^{2} - b c^{3} d\right )} e\right )} \operatorname {arccsc}\left (c x\right ) - 4 \, {\left (3 \, b c^{3} d^{2} + 3 \, b c^{3} d e + b c^{3} e^{2}\right )} \arctan \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (6 \, b c^{2} d^{2} + b e^{2}\right )} \log \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) + \sqrt {c^{2} x^{2} - 1} {\left (b c x e^{2} + 6 \, b c d e\right )}}{6 \, c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 4.90, size = 228, normalized size = 1.85 \begin {gather*} a d^{2} x + a d e x^{2} + \frac {a e^{2} x^{3}}{3} + b d^{2} x \operatorname {acsc}{\left (c x \right )} + b d e x^{2} \operatorname {acsc}{\left (c x \right )} + \frac {b e^{2} x^{3} \operatorname {acsc}{\left (c x \right )}}{3} + \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 {b d e \left (\begin {cases} \frac {\sqrt {c^{2} x^{2} - 1}}{c} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\\frac {i \sqrt {- c^{2} x^{2} + 1}}{c} & \text {otherwise} \end {cases}\right )}{c} + \frac {b e^{2} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 602 vs.
\(2 (109) = 218\).
time = 1.97, size = 602, normalized size = 4.89 \begin {gather*} \frac {1}{24} \, {\left (\frac {b e^{2} x^{3} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{3} \arcsin \left (\frac {1}{c x}\right )}{c} + \frac {a e^{2} x^{3} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{3}}{c} - \frac {24 \, b d e x^{2} {\left (\frac {1}{c^{2} x^{2}} - 1\right )} \arcsin \left (\frac {1}{c x}\right )}{c} + \frac {b e^{2} x^{2} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{2}}{c^{2}} - \frac {24 \, a d e x^{2} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}}{c} + \frac {12 \, b d^{2} x {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )} \arcsin \left (\frac {1}{c x}\right )}{c} + \frac {12 \, a d^{2} x {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}}{c} + \frac {3 \, b e^{2} x {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )} \arcsin \left (\frac {1}{c x}\right )}{c^{3}} + \frac {24 \, b d e x \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{c^{2}} + \frac {3 \, a e^{2} x {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}}{c^{3}} + \frac {24 \, b d^{2} \log \left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}{c^{2}} - \frac {24 \, b d^{2} \log \left (\frac {1}{{\left | c \right |} {\left | x \right |}}\right )}{c^{2}} + \frac {24 \, b d e \arcsin \left (\frac {1}{c x}\right )}{c^{3}} + \frac {24 \, a d e}{c^{3}} + \frac {4 \, b e^{2} \log \left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}{c^{4}} - \frac {4 \, b e^{2} \log \left (\frac {1}{{\left | c \right |} {\left | x \right |}}\right )}{c^{4}} + \frac {12 \, b d^{2} \arcsin \left (\frac {1}{c x}\right )}{c^{3} x {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}} + \frac {12 \, a d^{2}}{c^{3} x {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}} + \frac {3 \, b e^{2} \arcsin \left (\frac {1}{c x}\right )}{c^{5} x {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}} + \frac {3 \, a e^{2}}{c^{5} x {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}} - \frac {b e^{2}}{c^{6} x^{2} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{2}} + \frac {b e^{2} \arcsin \left (\frac {1}{c x}\right )}{c^{7} x^{3} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{3}} + \frac {a e^{2}}{c^{7} x^{3} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{3}}\right )} c \end {gather*}
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 \begin {gather*} \int \left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right )\,{\left (d+e\,x\right )}^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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