Optimal. Leaf size=88 \[ -\frac {a+b \sin ^{-1}(c+d x)}{3 d e^4 (c+d x)^3}-\frac {b \sqrt {1-(c+d x)^2}}{6 d e^4 (c+d x)^2}-\frac {b \tanh ^{-1}\left (\sqrt {1-(c+d x)^2}\right )}{6 d e^4} \]
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Rubi [A] time = 0.07, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {4805, 12, 4627, 266, 51, 63, 206} \[ -\frac {a+b \sin ^{-1}(c+d x)}{3 d e^4 (c+d x)^3}-\frac {b \sqrt {1-(c+d x)^2}}{6 d e^4 (c+d x)^2}-\frac {b \tanh ^{-1}\left (\sqrt {1-(c+d x)^2}\right )}{6 d e^4} \]
Antiderivative was successfully verified.
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Rule 12
Rule 51
Rule 63
Rule 206
Rule 266
Rule 4627
Rule 4805
Rubi steps
\begin {align*} \int \frac {a+b \sin ^{-1}(c+d x)}{(c e+d e x)^4} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {a+b \sin ^{-1}(x)}{e^4 x^4} \, dx,x,c+d x\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \frac {a+b \sin ^{-1}(x)}{x^4} \, dx,x,c+d x\right )}{d e^4}\\ &=-\frac {a+b \sin ^{-1}(c+d x)}{3 d e^4 (c+d x)^3}+\frac {b \operatorname {Subst}\left (\int \frac {1}{x^3 \sqrt {1-x^2}} \, dx,x,c+d x\right )}{3 d e^4}\\ &=-\frac {a+b \sin ^{-1}(c+d x)}{3 d e^4 (c+d x)^3}+\frac {b \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x} x^2} \, dx,x,(c+d x)^2\right )}{6 d e^4}\\ &=-\frac {b \sqrt {1-(c+d x)^2}}{6 d e^4 (c+d x)^2}-\frac {a+b \sin ^{-1}(c+d x)}{3 d e^4 (c+d x)^3}+\frac {b \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x} x} \, dx,x,(c+d x)^2\right )}{12 d e^4}\\ &=-\frac {b \sqrt {1-(c+d x)^2}}{6 d e^4 (c+d x)^2}-\frac {a+b \sin ^{-1}(c+d x)}{3 d e^4 (c+d x)^3}-\frac {b \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt {1-(c+d x)^2}\right )}{6 d e^4}\\ &=-\frac {b \sqrt {1-(c+d x)^2}}{6 d e^4 (c+d x)^2}-\frac {a+b \sin ^{-1}(c+d x)}{3 d e^4 (c+d x)^3}-\frac {b \tanh ^{-1}\left (\sqrt {1-(c+d x)^2}\right )}{6 d e^4}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 86, normalized size = 0.98 \[ -\frac {2 a+b \sqrt {-c^2-2 c d x-d^2 x^2+1} (c+d x)+2 b \sin ^{-1}(c+d x)+b (c+d x)^3 \tanh ^{-1}\left (\sqrt {1-(c+d x)^2}\right )}{6 d e^4 (c+d x)^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.58, size = 209, normalized size = 2.38 \[ -\frac {4 \, b \arcsin \left (d x + c\right ) + {\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3}\right )} \log \left (\sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1} + 1\right ) - {\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3}\right )} \log \left (\sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1} - 1\right ) + 2 \, \sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1} {\left (b d x + b c\right )} + 4 \, a}{12 \, {\left (d^{4} e^{4} x^{3} + 3 \, c d^{3} e^{4} x^{2} + 3 \, c^{2} d^{2} e^{4} x + c^{3} d e^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 2.34, size = 376, normalized size = 4.27 \[ -\frac {{\left (d x + c\right )}^{3} b \arcsin \left (d x + c\right ) e^{\left (-4\right )}}{24 \, d {\left (\sqrt {-{\left (d x + c\right )}^{2} + 1} + 1\right )}^{3}} - \frac {{\left (d x + c\right )} b \arcsin \left (d x + c\right ) e^{\left (-4\right )}}{8 \, d {\left (\sqrt {-{\left (d x + c\right )}^{2} + 1} + 1\right )}} - \frac {b {\left (\sqrt {-{\left (d x + c\right )}^{2} + 1} + 1\right )} \arcsin \left (d x + c\right ) e^{\left (-4\right )}}{8 \, {\left (d x + c\right )} d} - \frac {b {\left (\sqrt {-{\left (d x + c\right )}^{2} + 1} + 1\right )}^{3} \arcsin \left (d x + c\right ) e^{\left (-4\right )}}{24 \, {\left (d x + c\right )}^{3} d} - \frac {b e^{\left (-4\right )} \log \left (\sqrt {-{\left (d x + c\right )}^{2} + 1} + 1\right )}{6 \, d} + \frac {b e^{\left (-4\right )} \log \left ({\left | d x + c \right |}\right )}{6 \, d} - \frac {{\left (d x + c\right )}^{3} a e^{\left (-4\right )}}{24 \, d {\left (\sqrt {-{\left (d x + c\right )}^{2} + 1} + 1\right )}^{3}} + \frac {{\left (d x + c\right )}^{2} b e^{\left (-4\right )}}{24 \, d {\left (\sqrt {-{\left (d x + c\right )}^{2} + 1} + 1\right )}^{2}} - \frac {{\left (d x + c\right )} a e^{\left (-4\right )}}{8 \, d {\left (\sqrt {-{\left (d x + c\right )}^{2} + 1} + 1\right )}} - \frac {a {\left (\sqrt {-{\left (d x + c\right )}^{2} + 1} + 1\right )} e^{\left (-4\right )}}{8 \, {\left (d x + c\right )} d} - \frac {b {\left (\sqrt {-{\left (d x + c\right )}^{2} + 1} + 1\right )}^{2} e^{\left (-4\right )}}{24 \, {\left (d x + c\right )}^{2} d} - \frac {a {\left (\sqrt {-{\left (d x + c\right )}^{2} + 1} + 1\right )}^{3} e^{\left (-4\right )}}{24 \, {\left (d x + c\right )}^{3} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 78, normalized size = 0.89 \[ \frac {-\frac {a}{3 e^{4} \left (d x +c \right )^{3}}+\frac {b \left (-\frac {\arcsin \left (d x +c \right )}{3 \left (d x +c \right )^{3}}-\frac {\sqrt {1-\left (d x +c \right )^{2}}}{6 \left (d x +c \right )^{2}}-\frac {\arctanh \left (\frac {1}{\sqrt {1-\left (d x +c \right )^{2}}}\right )}{6}\right )}{e^{4}}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {{\left ({\left (d^{4} e^{4} x^{3} + 3 \, c d^{3} e^{4} x^{2} + 3 \, c^{2} d^{2} e^{4} x + c^{3} d e^{4}\right )} \int \frac {e^{\left (\frac {1}{2} \, \log \left (d x + c + 1\right ) + \frac {1}{2} \, \log \left (-d x - c + 1\right )\right )}}{d^{7} e^{4} x^{7} + 7 \, c d^{6} e^{4} x^{6} + {\left (21 \, c^{2} - 1\right )} d^{5} e^{4} x^{5} + 5 \, {\left (7 \, c^{3} - c\right )} d^{4} e^{4} x^{4} + 5 \, {\left (7 \, c^{4} - 2 \, c^{2}\right )} d^{3} e^{4} x^{3} + {\left (21 \, c^{5} - 10 \, c^{3}\right )} d^{2} e^{4} x^{2} + {\left (7 \, c^{6} - 5 \, c^{4}\right )} d e^{4} x + {\left (c^{7} - c^{5}\right )} e^{4} - {\left (d^{5} e^{4} x^{5} + 5 \, c d^{4} e^{4} x^{4} + {\left (10 \, c^{2} - 1\right )} d^{3} e^{4} x^{3} + {\left (10 \, c^{3} - 3 \, c\right )} d^{2} e^{4} x^{2} + {\left (5 \, c^{4} - 3 \, c^{2}\right )} d e^{4} x + {\left (c^{5} - c^{3}\right )} e^{4}\right )} {\left (d x + c + 1\right )} {\left (d x + c - 1\right )}}\,{d x} + \arctan \left (d x + c, \sqrt {d x + c + 1} \sqrt {-d x - c + 1}\right )\right )} b}{3 \, {\left (d^{4} e^{4} x^{3} + 3 \, c d^{3} e^{4} x^{2} + 3 \, c^{2} d^{2} e^{4} x + c^{3} d e^{4}\right )}} - \frac {a}{3 \, {\left (d^{4} e^{4} x^{3} + 3 \, c d^{3} e^{4} x^{2} + 3 \, c^{2} d^{2} e^{4} x + c^{3} d e^{4}\right )}} \]
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 \[ \int \frac {a+b\,\mathrm {asin}\left (c+d\,x\right )}{{\left (c\,e+d\,e\,x\right )}^4} \,d x \]
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 \[ \frac {\int \frac {a}{c^{4} + 4 c^{3} d x + 6 c^{2} d^{2} x^{2} + 4 c d^{3} x^{3} + d^{4} x^{4}}\, dx + \int \frac {b \operatorname {asin}{\left (c + d x \right )}}{c^{4} + 4 c^{3} d x + 6 c^{2} d^{2} x^{2} + 4 c d^{3} x^{3} + d^{4} x^{4}}\, dx}{e^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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