Optimal. Leaf size=196 \[ \frac {b \left (4 c^2 d+3 e\right ) x \sqrt {-1+c^2 x^2}}{24 c^7 \sqrt {c^2 x^2}}+\frac {b \left (8 c^2 d+9 e\right ) x \left (-1+c^2 x^2\right )^{3/2}}{72 c^7 \sqrt {c^2 x^2}}+\frac {b \left (4 c^2 d+9 e\right ) x \left (-1+c^2 x^2\right )^{5/2}}{120 c^7 \sqrt {c^2 x^2}}+\frac {b e x \left (-1+c^2 x^2\right )^{7/2}}{56 c^7 \sqrt {c^2 x^2}}+\frac {1}{6} d x^6 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{8} e x^8 \left (a+b \csc ^{-1}(c x)\right ) \]
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Rubi [A]
time = 0.11, antiderivative size = 196, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {14, 5347, 12,
457, 78} \begin {gather*} \frac {1}{6} d x^6 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{8} e x^8 \left (a+b \csc ^{-1}(c x)\right )+\frac {b x \left (c^2 x^2-1\right )^{5/2} \left (4 c^2 d+9 e\right )}{120 c^7 \sqrt {c^2 x^2}}+\frac {b x \left (c^2 x^2-1\right )^{3/2} \left (8 c^2 d+9 e\right )}{72 c^7 \sqrt {c^2 x^2}}+\frac {b x \sqrt {c^2 x^2-1} \left (4 c^2 d+3 e\right )}{24 c^7 \sqrt {c^2 x^2}}+\frac {b e x \left (c^2 x^2-1\right )^{7/2}}{56 c^7 \sqrt {c^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 78
Rule 457
Rule 5347
Rubi steps
\begin {align*} \int x^5 \left (d+e x^2\right ) \left (a+b \csc ^{-1}(c x)\right ) \, dx &=\frac {1}{6} d x^6 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{8} e x^8 \left (a+b \csc ^{-1}(c x)\right )+\frac {(b c x) \int \frac {x^5 \left (4 d+3 e x^2\right )}{24 \sqrt {-1+c^2 x^2}} \, dx}{\sqrt {c^2 x^2}}\\ &=\frac {1}{6} d x^6 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{8} e x^8 \left (a+b \csc ^{-1}(c x)\right )+\frac {(b c x) \int \frac {x^5 \left (4 d+3 e x^2\right )}{\sqrt {-1+c^2 x^2}} \, dx}{24 \sqrt {c^2 x^2}}\\ &=\frac {1}{6} d x^6 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{8} e x^8 \left (a+b \csc ^{-1}(c x)\right )+\frac {(b c x) \text {Subst}\left (\int \frac {x^2 (4 d+3 e x)}{\sqrt {-1+c^2 x}} \, dx,x,x^2\right )}{48 \sqrt {c^2 x^2}}\\ &=\frac {1}{6} d x^6 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{8} e x^8 \left (a+b \csc ^{-1}(c x)\right )+\frac {(b c x) \text {Subst}\left (\int \left (\frac {4 c^2 d+3 e}{c^6 \sqrt {-1+c^2 x}}+\frac {\left (8 c^2 d+9 e\right ) \sqrt {-1+c^2 x}}{c^6}+\frac {\left (4 c^2 d+9 e\right ) \left (-1+c^2 x\right )^{3/2}}{c^6}+\frac {3 e \left (-1+c^2 x\right )^{5/2}}{c^6}\right ) \, dx,x,x^2\right )}{48 \sqrt {c^2 x^2}}\\ &=\frac {b \left (4 c^2 d+3 e\right ) x \sqrt {-1+c^2 x^2}}{24 c^7 \sqrt {c^2 x^2}}+\frac {b \left (8 c^2 d+9 e\right ) x \left (-1+c^2 x^2\right )^{3/2}}{72 c^7 \sqrt {c^2 x^2}}+\frac {b \left (4 c^2 d+9 e\right ) x \left (-1+c^2 x^2\right )^{5/2}}{120 c^7 \sqrt {c^2 x^2}}+\frac {b e x \left (-1+c^2 x^2\right )^{7/2}}{56 c^7 \sqrt {c^2 x^2}}+\frac {1}{6} d x^6 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{8} e x^8 \left (a+b \csc ^{-1}(c x)\right )\\ \end {align*}
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Mathematica [A]
time = 0.17, size = 115, normalized size = 0.59 \begin {gather*} \frac {x \left (105 a x^5 \left (4 d+3 e x^2\right )+\frac {b \sqrt {1-\frac {1}{c^2 x^2}} \left (144 e+8 c^2 \left (28 d+9 e x^2\right )+2 c^4 \left (56 d x^2+27 e x^4\right )+c^6 \left (84 d x^4+45 e x^6\right )\right )}{c^7}+105 b x^5 \left (4 d+3 e x^2\right ) \csc ^{-1}(c x)\right )}{2520} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.42, size = 152, normalized size = 0.78
method | result | size |
derivativedivides | \(\frac {\frac {a \left (\frac {1}{6} c^{8} d \,x^{6}+\frac {1}{8} e \,c^{8} x^{8}\right )}{c^{2}}+\frac {b \left (\frac {\mathrm {arccsc}\left (c x \right ) d \,c^{8} x^{6}}{6}+\frac {\mathrm {arccsc}\left (c x \right ) e \,c^{8} x^{8}}{8}+\frac {\left (c^{2} x^{2}-1\right ) \left (45 c^{6} e \,x^{6}+84 c^{6} d \,x^{4}+54 c^{4} e \,x^{4}+112 c^{4} d \,x^{2}+72 c^{2} e \,x^{2}+224 c^{2} d +144 e \right )}{2520 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}\right )}{c^{2}}}{c^{6}}\) | \(152\) |
default | \(\frac {\frac {a \left (\frac {1}{6} c^{8} d \,x^{6}+\frac {1}{8} e \,c^{8} x^{8}\right )}{c^{2}}+\frac {b \left (\frac {\mathrm {arccsc}\left (c x \right ) d \,c^{8} x^{6}}{6}+\frac {\mathrm {arccsc}\left (c x \right ) e \,c^{8} x^{8}}{8}+\frac {\left (c^{2} x^{2}-1\right ) \left (45 c^{6} e \,x^{6}+84 c^{6} d \,x^{4}+54 c^{4} e \,x^{4}+112 c^{4} d \,x^{2}+72 c^{2} e \,x^{2}+224 c^{2} d +144 e \right )}{2520 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}\right )}{c^{2}}}{c^{6}}\) | \(152\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 185, normalized size = 0.94 \begin {gather*} \frac {1}{8} \, a x^{8} e + \frac {1}{6} \, a d x^{6} + \frac {1}{90} \, {\left (15 \, x^{6} \operatorname {arccsc}\left (c x\right ) + \frac {3 \, c^{4} x^{5} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {5}{2}} + 10 \, c^{2} x^{3} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} + 15 \, x \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{c^{5}}\right )} b d + \frac {1}{280} \, {\left (35 \, x^{8} \operatorname {arccsc}\left (c x\right ) + \frac {5 \, c^{6} x^{7} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {7}{2}} + 21 \, c^{4} x^{5} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {5}{2}} + 35 \, c^{2} x^{3} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} + 35 \, x \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{c^{7}}\right )} b e \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 130, normalized size = 0.66 \begin {gather*} \frac {315 \, a c^{8} x^{8} e + 420 \, a c^{8} d x^{6} + 105 \, {\left (3 \, b c^{8} x^{8} e + 4 \, b c^{8} d x^{6}\right )} \operatorname {arccsc}\left (c x\right ) + {\left (84 \, b c^{6} d x^{4} + 112 \, b c^{4} d x^{2} + 224 \, b c^{2} d + 9 \, {\left (5 \, b c^{6} x^{6} + 6 \, b c^{4} x^{4} + 8 \, b c^{2} x^{2} + 16 \, b\right )} e\right )} \sqrt {c^{2} x^{2} - 1}}{2520 \, c^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 5.36, size = 364, normalized size = 1.86 \begin {gather*} \frac {a d x^{6}}{6} + \frac {a e x^{8}}{8} + \frac {b d x^{6} \operatorname {acsc}{\left (c x \right )}}{6} + \frac {b e x^{8} \operatorname {acsc}{\left (c x \right )}}{8} + \frac {b d \left (\begin {cases} \frac {x^{4} \sqrt {c^{2} x^{2} - 1}}{5 c} + \frac {4 x^{2} \sqrt {c^{2} x^{2} - 1}}{15 c^{3}} + \frac {8 \sqrt {c^{2} x^{2} - 1}}{15 c^{5}} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\\frac {i x^{4} \sqrt {- c^{2} x^{2} + 1}}{5 c} + \frac {4 i x^{2} \sqrt {- c^{2} x^{2} + 1}}{15 c^{3}} + \frac {8 i \sqrt {- c^{2} x^{2} + 1}}{15 c^{5}} & \text {otherwise} \end {cases}\right )}{6 c} + \frac {b e \left (\begin {cases} \frac {x^{6} \sqrt {c^{2} x^{2} - 1}}{7 c} + \frac {6 x^{4} \sqrt {c^{2} x^{2} - 1}}{35 c^{3}} + \frac {8 x^{2} \sqrt {c^{2} x^{2} - 1}}{35 c^{5}} + \frac {16 \sqrt {c^{2} x^{2} - 1}}{35 c^{7}} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\\frac {i x^{6} \sqrt {- c^{2} x^{2} + 1}}{7 c} + \frac {6 i x^{4} \sqrt {- c^{2} x^{2} + 1}}{35 c^{3}} + \frac {8 i x^{2} \sqrt {- c^{2} x^{2} + 1}}{35 c^{5}} + \frac {16 i \sqrt {- c^{2} x^{2} + 1}}{35 c^{7}} & \text {otherwise} \end {cases}\right )}{8 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. 1244 vs.
\(2 (168) = 336\).
time = 0.54, size = 1244, normalized size = 6.35 \begin {gather*} \frac {1}{645120} \, {\left (\frac {315 \, b e x^{8} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{8} \arcsin \left (\frac {1}{c x}\right )}{c} + \frac {315 \, a e x^{8} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{8}}{c} + \frac {90 \, b e x^{7} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{7}}{c^{2}} + \frac {1680 \, b d x^{6} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{6} \arcsin \left (\frac {1}{c x}\right )}{c} + \frac {1680 \, a d x^{6} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{6}}{c} + \frac {2520 \, b e x^{6} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{6} \arcsin \left (\frac {1}{c x}\right )}{c^{3}} + \frac {2520 \, a e x^{6} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{6}}{c^{3}} + \frac {672 \, b d x^{5} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{5}}{c^{2}} + \frac {882 \, b e x^{5} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{5}}{c^{4}} + \frac {10080 \, b d x^{4} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{4} \arcsin \left (\frac {1}{c x}\right )}{c^{3}} + \frac {10080 \, a d x^{4} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{4}}{c^{3}} + \frac {8820 \, b e x^{4} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{4} \arcsin \left (\frac {1}{c x}\right )}{c^{5}} + \frac {8820 \, a e x^{4} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{4}}{c^{5}} + \frac {5600 \, b d x^{3} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{3}}{c^{4}} + \frac {4410 \, b e x^{3} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{3}}{c^{6}} + \frac {25200 \, b d x^{2} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{2} \arcsin \left (\frac {1}{c x}\right )}{c^{5}} + \frac {25200 \, a d x^{2} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{2}}{c^{5}} + \frac {17640 \, b e x^{2} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{2} \arcsin \left (\frac {1}{c x}\right )}{c^{7}} + \frac {17640 \, a e x^{2} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{2}}{c^{7}} + \frac {33600 \, b d x {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}}{c^{6}} + \frac {22050 \, b e x {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}}{c^{8}} + \frac {33600 \, b d \arcsin \left (\frac {1}{c x}\right )}{c^{7}} + \frac {33600 \, a d}{c^{7}} + \frac {22050 \, b e \arcsin \left (\frac {1}{c x}\right )}{c^{9}} + \frac {22050 \, a e}{c^{9}} - \frac {33600 \, b d}{c^{8} x {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}} - \frac {22050 \, b e}{c^{10} x {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}} + \frac {25200 \, b d \arcsin \left (\frac {1}{c x}\right )}{c^{9} x^{2} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{2}} + \frac {25200 \, a d}{c^{9} x^{2} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{2}} + \frac {17640 \, b e \arcsin \left (\frac {1}{c x}\right )}{c^{11} x^{2} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{2}} + \frac {17640 \, a e}{c^{11} x^{2} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{2}} - \frac {5600 \, b d}{c^{10} x^{3} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{3}} - \frac {4410 \, b e}{c^{12} x^{3} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{3}} + \frac {10080 \, b d \arcsin \left (\frac {1}{c x}\right )}{c^{11} x^{4} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{4}} + \frac {10080 \, a d}{c^{11} x^{4} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{4}} + \frac {8820 \, b e \arcsin \left (\frac {1}{c x}\right )}{c^{13} x^{4} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{4}} + \frac {8820 \, a e}{c^{13} x^{4} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{4}} - \frac {672 \, b d}{c^{12} x^{5} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{5}} - \frac {882 \, b e}{c^{14} x^{5} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{5}} + \frac {1680 \, b d \arcsin \left (\frac {1}{c x}\right )}{c^{13} x^{6} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{6}} + \frac {1680 \, a d}{c^{13} x^{6} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{6}} + \frac {2520 \, b e \arcsin \left (\frac {1}{c x}\right )}{c^{15} x^{6} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{6}} + \frac {2520 \, a e}{c^{15} x^{6} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{6}} - \frac {90 \, b e}{c^{16} x^{7} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{7}} + \frac {315 \, b e \arcsin \left (\frac {1}{c x}\right )}{c^{17} x^{8} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{8}} + \frac {315 \, a e}{c^{17} x^{8} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{8}}\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 x^5\,\left (e\,x^2+d\right )\,\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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