Optimal. Leaf size=358 \[ \frac {\left (d+e x^2\right )^4 \left (a+b \cosh ^{-1}(c x)\right )}{8 e}+\frac {b x \left (1-c^2 x^2\right ) \left (d+e x^2\right )^3}{64 c \sqrt {c x-1} \sqrt {c x+1}}+\frac {7 b x \left (1-c^2 x^2\right ) \left (2 c^2 d+e\right ) \left (d+e x^2\right )^2}{384 c^3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {5 b x \left (1-c^2 x^2\right ) \left (2 c^2 d+e\right ) \left (40 c^4 d^2+40 c^2 d e+21 e^2\right )}{3072 c^7 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b x \left (1-c^2 x^2\right ) \left (104 c^4 d^2+104 c^2 d e+35 e^2\right ) \left (d+e x^2\right )}{1536 c^5 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b \sqrt {c^2 x^2-1} \left (128 c^8 d^4+256 c^6 d^3 e+288 c^4 d^2 e^2+160 c^2 d e^3+35 e^4\right ) \tanh ^{-1}\left (\frac {c x}{\sqrt {c^2 x^2-1}}\right )}{1024 c^8 e \sqrt {c x-1} \sqrt {c x+1}} \]
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Rubi [A] time = 0.36, antiderivative size = 358, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {5788, 902, 416, 528, 388, 217, 206} \[ \frac {\left (d+e x^2\right )^4 \left (a+b \cosh ^{-1}(c x)\right )}{8 e}+\frac {b x \left (1-c^2 x^2\right ) \left (104 c^4 d^2+104 c^2 d e+35 e^2\right ) \left (d+e x^2\right )}{1536 c^5 \sqrt {c x-1} \sqrt {c x+1}}+\frac {5 b x \left (1-c^2 x^2\right ) \left (2 c^2 d+e\right ) \left (40 c^4 d^2+40 c^2 d e+21 e^2\right )}{3072 c^7 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b \sqrt {c^2 x^2-1} \left (288 c^4 d^2 e^2+256 c^6 d^3 e+128 c^8 d^4+160 c^2 d e^3+35 e^4\right ) \tanh ^{-1}\left (\frac {c x}{\sqrt {c^2 x^2-1}}\right )}{1024 c^8 e \sqrt {c x-1} \sqrt {c x+1}}+\frac {b x \left (1-c^2 x^2\right ) \left (d+e x^2\right )^3}{64 c \sqrt {c x-1} \sqrt {c x+1}}+\frac {7 b x \left (1-c^2 x^2\right ) \left (2 c^2 d+e\right ) \left (d+e x^2\right )^2}{384 c^3 \sqrt {c x-1} \sqrt {c x+1}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 217
Rule 388
Rule 416
Rule 528
Rule 902
Rule 5788
Rubi steps
\begin {align*} \int x \left (d+e x^2\right )^3 \left (a+b \cosh ^{-1}(c x)\right ) \, dx &=\frac {\left (d+e x^2\right )^4 \left (a+b \cosh ^{-1}(c x)\right )}{8 e}-\frac {(b c) \int \frac {\left (d+e x^2\right )^4}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{8 e}\\ &=\frac {\left (d+e x^2\right )^4 \left (a+b \cosh ^{-1}(c x)\right )}{8 e}-\frac {\left (b c \sqrt {-1+c^2 x^2}\right ) \int \frac {\left (d+e x^2\right )^4}{\sqrt {-1+c^2 x^2}} \, dx}{8 e \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {b x \left (1-c^2 x^2\right ) \left (d+e x^2\right )^3}{64 c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (d+e x^2\right )^4 \left (a+b \cosh ^{-1}(c x)\right )}{8 e}-\frac {\left (b \sqrt {-1+c^2 x^2}\right ) \int \frac {\left (d+e x^2\right )^2 \left (d \left (8 c^2 d+e\right )+7 e \left (2 c^2 d+e\right ) x^2\right )}{\sqrt {-1+c^2 x^2}} \, dx}{64 c e \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {7 b \left (2 c^2 d+e\right ) x \left (1-c^2 x^2\right ) \left (d+e x^2\right )^2}{384 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b x \left (1-c^2 x^2\right ) \left (d+e x^2\right )^3}{64 c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (d+e x^2\right )^4 \left (a+b \cosh ^{-1}(c x)\right )}{8 e}-\frac {\left (b \sqrt {-1+c^2 x^2}\right ) \int \frac {\left (d+e x^2\right ) \left (d \left (48 c^4 d^2+20 c^2 d e+7 e^2\right )+e \left (104 c^4 d^2+104 c^2 d e+35 e^2\right ) x^2\right )}{\sqrt {-1+c^2 x^2}} \, dx}{384 c^3 e \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {b \left (104 c^4 d^2+104 c^2 d e+35 e^2\right ) x \left (1-c^2 x^2\right ) \left (d+e x^2\right )}{1536 c^5 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {7 b \left (2 c^2 d+e\right ) x \left (1-c^2 x^2\right ) \left (d+e x^2\right )^2}{384 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b x \left (1-c^2 x^2\right ) \left (d+e x^2\right )^3}{64 c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (d+e x^2\right )^4 \left (a+b \cosh ^{-1}(c x)\right )}{8 e}-\frac {\left (b \sqrt {-1+c^2 x^2}\right ) \int \frac {d \left (192 c^6 d^3+184 c^4 d^2 e+132 c^2 d e^2+35 e^3\right )+5 e \left (2 c^2 d+e\right ) \left (40 c^4 d^2+40 c^2 d e+21 e^2\right ) x^2}{\sqrt {-1+c^2 x^2}} \, dx}{1536 c^5 e \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {5 b \left (2 c^2 d+e\right ) \left (40 c^4 d^2+40 c^2 d e+21 e^2\right ) x \left (1-c^2 x^2\right )}{3072 c^7 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b \left (104 c^4 d^2+104 c^2 d e+35 e^2\right ) x \left (1-c^2 x^2\right ) \left (d+e x^2\right )}{1536 c^5 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {7 b \left (2 c^2 d+e\right ) x \left (1-c^2 x^2\right ) \left (d+e x^2\right )^2}{384 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b x \left (1-c^2 x^2\right ) \left (d+e x^2\right )^3}{64 c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (d+e x^2\right )^4 \left (a+b \cosh ^{-1}(c x)\right )}{8 e}--\frac {\left (b \left (-5 e \left (2 c^2 d+e\right ) \left (40 c^4 d^2+40 c^2 d e+21 e^2\right )-2 c^2 d \left (192 c^6 d^3+184 c^4 d^2 e+132 c^2 d e^2+35 e^3\right )\right ) \sqrt {-1+c^2 x^2}\right ) \int \frac {1}{\sqrt {-1+c^2 x^2}} \, dx}{3072 c^7 e \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {5 b \left (2 c^2 d+e\right ) \left (40 c^4 d^2+40 c^2 d e+21 e^2\right ) x \left (1-c^2 x^2\right )}{3072 c^7 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b \left (104 c^4 d^2+104 c^2 d e+35 e^2\right ) x \left (1-c^2 x^2\right ) \left (d+e x^2\right )}{1536 c^5 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {7 b \left (2 c^2 d+e\right ) x \left (1-c^2 x^2\right ) \left (d+e x^2\right )^2}{384 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b x \left (1-c^2 x^2\right ) \left (d+e x^2\right )^3}{64 c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (d+e x^2\right )^4 \left (a+b \cosh ^{-1}(c x)\right )}{8 e}--\frac {\left (b \left (-5 e \left (2 c^2 d+e\right ) \left (40 c^4 d^2+40 c^2 d e+21 e^2\right )-2 c^2 d \left (192 c^6 d^3+184 c^4 d^2 e+132 c^2 d e^2+35 e^3\right )\right ) \sqrt {-1+c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{1-c^2 x^2} \, dx,x,\frac {x}{\sqrt {-1+c^2 x^2}}\right )}{3072 c^7 e \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {5 b \left (2 c^2 d+e\right ) \left (40 c^4 d^2+40 c^2 d e+21 e^2\right ) x \left (1-c^2 x^2\right )}{3072 c^7 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b \left (104 c^4 d^2+104 c^2 d e+35 e^2\right ) x \left (1-c^2 x^2\right ) \left (d+e x^2\right )}{1536 c^5 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {7 b \left (2 c^2 d+e\right ) x \left (1-c^2 x^2\right ) \left (d+e x^2\right )^2}{384 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b x \left (1-c^2 x^2\right ) \left (d+e x^2\right )^3}{64 c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (d+e x^2\right )^4 \left (a+b \cosh ^{-1}(c x)\right )}{8 e}-\frac {b \left (128 c^8 d^4+256 c^6 d^3 e+288 c^4 d^2 e^2+160 c^2 d e^3+35 e^4\right ) \sqrt {-1+c^2 x^2} \tanh ^{-1}\left (\frac {c x}{\sqrt {-1+c^2 x^2}}\right )}{1024 c^8 e \sqrt {-1+c x} \sqrt {1+c x}}\\ \end {align*}
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Mathematica [A] time = 0.41, size = 256, normalized size = 0.72 \[ \frac {c x \left (384 a c^7 x \left (4 d^3+6 d^2 e x^2+4 d e^2 x^4+e^3 x^6\right )-b \sqrt {c x-1} \sqrt {c x+1} \left (16 c^6 \left (48 d^3+36 d^2 e x^2+16 d e^2 x^4+3 e^3 x^6\right )+8 c^4 e \left (108 d^2+40 d e x^2+7 e^2 x^4\right )+10 c^2 e^2 \left (48 d+7 e x^2\right )+105 e^3\right )\right )+384 b c^8 x^2 \cosh ^{-1}(c x) \left (4 d^3+6 d^2 e x^2+4 d e^2 x^4+e^3 x^6\right )-6 b \left (256 c^6 d^3+288 c^4 d^2 e+160 c^2 d e^2+35 e^3\right ) \tanh ^{-1}\left (\sqrt {\frac {c x-1}{c x+1}}\right )}{3072 c^8} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.54, size = 286, normalized size = 0.80 \[ \frac {384 \, a c^{8} e^{3} x^{8} + 1536 \, a c^{8} d e^{2} x^{6} + 2304 \, a c^{8} d^{2} e x^{4} + 1536 \, a c^{8} d^{3} x^{2} + 3 \, {\left (128 \, b c^{8} e^{3} x^{8} + 512 \, b c^{8} d e^{2} x^{6} + 768 \, b c^{8} d^{2} e x^{4} + 512 \, b c^{8} d^{3} x^{2} - 256 \, b c^{6} d^{3} - 288 \, b c^{4} d^{2} e - 160 \, b c^{2} d e^{2} - 35 \, b e^{3}\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (48 \, b c^{7} e^{3} x^{7} + 8 \, {\left (32 \, b c^{7} d e^{2} + 7 \, b c^{5} e^{3}\right )} x^{5} + 2 \, {\left (288 \, b c^{7} d^{2} e + 160 \, b c^{5} d e^{2} + 35 \, b c^{3} e^{3}\right )} x^{3} + 3 \, {\left (256 \, b c^{7} d^{3} + 288 \, b c^{5} d^{2} e + 160 \, b c^{3} d e^{2} + 35 \, b c e^{3}\right )} x\right )} \sqrt {c^{2} x^{2} - 1}}{3072 \, c^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 553, normalized size = 1.54 \[ \frac {a \,e^{3} x^{8}}{8}+\frac {a d \,e^{2} x^{6}}{2}+\frac {3 a \,d^{2} e \,x^{4}}{4}+\frac {d^{3} a \,x^{2}}{2}+\frac {b \,\mathrm {arccosh}\left (c x \right ) e^{3} x^{8}}{8}+\frac {b \,\mathrm {arccosh}\left (c x \right ) d \,e^{2} x^{6}}{2}+\frac {3 b \,\mathrm {arccosh}\left (c x \right ) d^{2} e \,x^{4}}{4}+\frac {d^{3} b \,\mathrm {arccosh}\left (c x \right ) x^{2}}{2}-\frac {b \sqrt {c x -1}\, \sqrt {c x +1}\, e^{3} x^{7}}{64 c}-\frac {b \sqrt {c x -1}\, \sqrt {c x +1}\, x^{5} d \,e^{2}}{12 c}-\frac {3 b \sqrt {c x -1}\, \sqrt {c x +1}\, x^{3} d^{2} e}{16 c}-\frac {b \,d^{3} x \sqrt {c x -1}\, \sqrt {c x +1}}{4 c}-\frac {d^{3} b \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{4 c^{2} \sqrt {c^{2} x^{2}-1}}-\frac {7 b \sqrt {c x -1}\, \sqrt {c x +1}\, e^{3} x^{5}}{384 c^{3}}-\frac {5 b \sqrt {c x -1}\, \sqrt {c x +1}\, x^{3} d \,e^{2}}{48 c^{3}}-\frac {9 b \sqrt {c x -1}\, \sqrt {c x +1}\, x \,d^{2} e}{32 c^{3}}-\frac {9 b \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right ) d^{2} e}{32 c^{4} \sqrt {c^{2} x^{2}-1}}-\frac {35 b \sqrt {c x -1}\, \sqrt {c x +1}\, e^{3} x^{3}}{1536 c^{5}}-\frac {5 b \sqrt {c x -1}\, \sqrt {c x +1}\, x d \,e^{2}}{32 c^{5}}-\frac {5 b \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right ) d \,e^{2}}{32 c^{6} \sqrt {c^{2} x^{2}-1}}-\frac {35 b \sqrt {c x -1}\, \sqrt {c x +1}\, e^{3} x}{1024 c^{7}}-\frac {35 b \sqrt {c x -1}\, \sqrt {c x +1}\, e^{3} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{1024 c^{8} \sqrt {c^{2} x^{2}-1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.46, size = 409, normalized size = 1.14 \[ \frac {1}{8} \, a e^{3} x^{8} + \frac {1}{2} \, a d e^{2} x^{6} + \frac {3}{4} \, a d^{2} e x^{4} + \frac {1}{2} \, a d^{3} x^{2} + \frac {1}{4} \, {\left (2 \, x^{2} \operatorname {arcosh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} - 1} x}{c^{2}} + \frac {\log \left (2 \, c^{2} x + 2 \, \sqrt {c^{2} x^{2} - 1} c\right )}{c^{3}}\right )}\right )} b d^{3} + \frac {3}{32} \, {\left (8 \, x^{4} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {2 \, \sqrt {c^{2} x^{2} - 1} x^{3}}{c^{2}} + \frac {3 \, \sqrt {c^{2} x^{2} - 1} x}{c^{4}} + \frac {3 \, \log \left (2 \, c^{2} x + 2 \, \sqrt {c^{2} x^{2} - 1} c\right )}{c^{5}}\right )} c\right )} b d^{2} e + \frac {1}{96} \, {\left (48 \, x^{6} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {8 \, \sqrt {c^{2} x^{2} - 1} x^{5}}{c^{2}} + \frac {10 \, \sqrt {c^{2} x^{2} - 1} x^{3}}{c^{4}} + \frac {15 \, \sqrt {c^{2} x^{2} - 1} x}{c^{6}} + \frac {15 \, \log \left (2 \, c^{2} x + 2 \, \sqrt {c^{2} x^{2} - 1} c\right )}{c^{7}}\right )} c\right )} b d e^{2} + \frac {1}{3072} \, {\left (384 \, x^{8} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {48 \, \sqrt {c^{2} x^{2} - 1} x^{7}}{c^{2}} + \frac {56 \, \sqrt {c^{2} x^{2} - 1} x^{5}}{c^{4}} + \frac {70 \, \sqrt {c^{2} x^{2} - 1} x^{3}}{c^{6}} + \frac {105 \, \sqrt {c^{2} x^{2} - 1} x}{c^{8}} + \frac {105 \, \log \left (2 \, c^{2} x + 2 \, \sqrt {c^{2} x^{2} - 1} c\right )}{c^{9}}\right )} c\right )} b e^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int x\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (e\,x^2+d\right )}^3 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 11.72, size = 490, normalized size = 1.37 \[ \begin {cases} \frac {a d^{3} x^{2}}{2} + \frac {3 a d^{2} e x^{4}}{4} + \frac {a d e^{2} x^{6}}{2} + \frac {a e^{3} x^{8}}{8} + \frac {b d^{3} x^{2} \operatorname {acosh}{\left (c x \right )}}{2} + \frac {3 b d^{2} e x^{4} \operatorname {acosh}{\left (c x \right )}}{4} + \frac {b d e^{2} x^{6} \operatorname {acosh}{\left (c x \right )}}{2} + \frac {b e^{3} x^{8} \operatorname {acosh}{\left (c x \right )}}{8} - \frac {b d^{3} x \sqrt {c^{2} x^{2} - 1}}{4 c} - \frac {3 b d^{2} e x^{3} \sqrt {c^{2} x^{2} - 1}}{16 c} - \frac {b d e^{2} x^{5} \sqrt {c^{2} x^{2} - 1}}{12 c} - \frac {b e^{3} x^{7} \sqrt {c^{2} x^{2} - 1}}{64 c} - \frac {b d^{3} \operatorname {acosh}{\left (c x \right )}}{4 c^{2}} - \frac {9 b d^{2} e x \sqrt {c^{2} x^{2} - 1}}{32 c^{3}} - \frac {5 b d e^{2} x^{3} \sqrt {c^{2} x^{2} - 1}}{48 c^{3}} - \frac {7 b e^{3} x^{5} \sqrt {c^{2} x^{2} - 1}}{384 c^{3}} - \frac {9 b d^{2} e \operatorname {acosh}{\left (c x \right )}}{32 c^{4}} - \frac {5 b d e^{2} x \sqrt {c^{2} x^{2} - 1}}{32 c^{5}} - \frac {35 b e^{3} x^{3} \sqrt {c^{2} x^{2} - 1}}{1536 c^{5}} - \frac {5 b d e^{2} \operatorname {acosh}{\left (c x \right )}}{32 c^{6}} - \frac {35 b e^{3} x \sqrt {c^{2} x^{2} - 1}}{1024 c^{7}} - \frac {35 b e^{3} \operatorname {acosh}{\left (c x \right )}}{1024 c^{8}} & \text {for}\: c \neq 0 \\\left (a + \frac {i \pi b}{2}\right ) \left (\frac {d^{3} x^{2}}{2} + \frac {3 d^{2} e x^{4}}{4} + \frac {d e^{2} x^{6}}{2} + \frac {e^{3} x^{8}}{8}\right ) & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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