Optimal. Leaf size=79 \[ -\frac {1}{\left (a^2+c\right ) x}+\frac {b \left (a^2-c\right ) \tan ^{-1}\left (\frac {a+b x}{\sqrt {c}}\right )}{\sqrt {c} \left (a^2+c\right )^2}-\frac {2 a b \log (x)}{\left (a^2+c\right )^2}+\frac {a b \log \left (c+(a+b x)^2\right )}{\left (a^2+c\right )^2} \]
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Rubi [A]
time = 0.06, antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {378, 724, 815,
649, 209, 266} \begin {gather*} \frac {b \left (a^2-c\right ) \text {ArcTan}\left (\frac {a+b x}{\sqrt {c}}\right )}{\sqrt {c} \left (a^2+c\right )^2}-\frac {2 a b \log (x)}{\left (a^2+c\right )^2}+\frac {a b \log \left ((a+b x)^2+c\right )}{\left (a^2+c\right )^2}-\frac {1}{x \left (a^2+c\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 266
Rule 378
Rule 649
Rule 724
Rule 815
Rubi steps
\begin {align*} \int \frac {1}{x^2 \left (c+(a+b x)^2\right )} \, dx &=b \text {Subst}\left (\int \frac {1}{(-a+x)^2 \left (c+x^2\right )} \, dx,x,a+b x\right )\\ &=-\frac {1}{\left (a^2+c\right ) x}+\frac {b \text {Subst}\left (\int \frac {-a-x}{(-a+x) \left (c+x^2\right )} \, dx,x,a+b x\right )}{a^2+c}\\ &=-\frac {1}{\left (a^2+c\right ) x}+\frac {b \text {Subst}\left (\int \left (\frac {2 a}{\left (a^2+c\right ) (a-x)}+\frac {a^2-c+2 a x}{\left (a^2+c\right ) \left (c+x^2\right )}\right ) \, dx,x,a+b x\right )}{a^2+c}\\ &=-\frac {1}{\left (a^2+c\right ) x}-\frac {2 a b \log (x)}{\left (a^2+c\right )^2}+\frac {b \text {Subst}\left (\int \frac {a^2-c+2 a x}{c+x^2} \, dx,x,a+b x\right )}{\left (a^2+c\right )^2}\\ &=-\frac {1}{\left (a^2+c\right ) x}-\frac {2 a b \log (x)}{\left (a^2+c\right )^2}+\frac {(2 a b) \text {Subst}\left (\int \frac {x}{c+x^2} \, dx,x,a+b x\right )}{\left (a^2+c\right )^2}+\frac {\left (b \left (a^2-c\right )\right ) \text {Subst}\left (\int \frac {1}{c+x^2} \, dx,x,a+b x\right )}{\left (a^2+c\right )^2}\\ &=-\frac {1}{\left (a^2+c\right ) x}+\frac {b \left (a^2-c\right ) \tan ^{-1}\left (\frac {a+b x}{\sqrt {c}}\right )}{\sqrt {c} \left (a^2+c\right )^2}-\frac {2 a b \log (x)}{\left (a^2+c\right )^2}+\frac {a b \log \left (c+(a+b x)^2\right )}{\left (a^2+c\right )^2}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 81, normalized size = 1.03 \begin {gather*} \frac {b \left (a^2-c\right ) x \tan ^{-1}\left (\frac {a+b x}{\sqrt {c}}\right )-\sqrt {c} \left (a^2+c+2 a b x \log (x)-a b x \log \left (a^2+c+2 a b x+b^2 x^2\right )\right )}{\sqrt {c} \left (a^2+c\right )^2 x} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.29, size = 96, normalized size = 1.22
method | result | size |
default | \(\frac {b^{2} \left (\frac {a \ln \left (b^{2} x^{2}+2 a b x +a^{2}+c \right )}{b}+\frac {\left (a^{2}-c \right ) \arctan \left (\frac {2 b^{2} x +2 a b}{2 b \sqrt {c}}\right )}{b \sqrt {c}}\right )}{\left (a^{2}+c \right )^{2}}-\frac {1}{\left (a^{2}+c \right ) x}-\frac {2 a b \ln \left (x \right )}{\left (a^{2}+c \right )^{2}}\) | \(96\) |
risch | \(-\frac {1}{\left (a^{2}+c \right ) x}-\frac {2 a b \ln \left (x \right )}{a^{4}+2 a^{2} c +c^{2}}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\left (c \,a^{4}+2 a^{2} c^{2}+c^{3}\right ) \textit {\_Z}^{2}-4 a b c \textit {\_Z} +b^{2}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (-a^{6} b +a^{4} b c +5 a^{2} b \,c^{2}+3 b \,c^{3}\right ) \textit {\_R}^{2}+\left (-4 b^{2} a^{3}-4 a \,b^{2} c \right ) \textit {\_R} +2 b^{3}\right ) x +\left (-a^{7}-3 a^{5} c -3 a^{3} c^{2}-a \,c^{3}\right ) \textit {\_R}^{2}+\left (-3 b \,a^{4}-2 a^{2} b c +b \,c^{2}\right ) \textit {\_R} +4 a \,b^{2}\right )\right )}{2}\) | \(191\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 123, normalized size = 1.56 \begin {gather*} \frac {a b \log \left (b^{2} x^{2} + 2 \, a b x + a^{2} + c\right )}{a^{4} + 2 \, a^{2} c + c^{2}} - \frac {2 \, a b \log \left (x\right )}{a^{4} + 2 \, a^{2} c + c^{2}} + \frac {{\left (a^{2} b^{2} - b^{2} c\right )} \arctan \left (\frac {b^{2} x + a b}{b \sqrt {c}}\right )}{{\left (a^{4} + 2 \, a^{2} c + c^{2}\right )} b \sqrt {c}} - \frac {1}{{\left (a^{2} + c\right )} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.42, size = 229, normalized size = 2.90 \begin {gather*} \left [\frac {2 \, a b c x \log \left (b^{2} x^{2} + 2 \, a b x + a^{2} + c\right ) - 4 \, a b c x \log \left (x\right ) + {\left (a^{2} b - b c\right )} \sqrt {-c} x \log \left (\frac {b^{2} x^{2} + 2 \, a b x + a^{2} + 2 \, {\left (b x + a\right )} \sqrt {-c} - c}{b^{2} x^{2} + 2 \, a b x + a^{2} + c}\right ) - 2 \, a^{2} c - 2 \, c^{2}}{2 \, {\left (a^{4} c + 2 \, a^{2} c^{2} + c^{3}\right )} x}, \frac {a b c x \log \left (b^{2} x^{2} + 2 \, a b x + a^{2} + c\right ) - 2 \, a b c x \log \left (x\right ) + {\left (a^{2} b - b c\right )} \sqrt {c} x \arctan \left (\frac {b x + a}{\sqrt {c}}\right ) - a^{2} c - c^{2}}{{\left (a^{4} c + 2 \, a^{2} c^{2} + c^{3}\right )} x}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1620 vs.
\(2 (73) = 146\).
time = 7.26, size = 1620, normalized size = 20.51 \begin {gather*} - \frac {2 a b \log {\left (x + \frac {- \frac {16 a^{13} b^{2} c}{\left (a^{2} + c\right )^{4}} + \frac {48 a^{11} b^{2} c^{2}}{\left (a^{2} + c\right )^{4}} + \frac {352 a^{9} b^{2} c^{3}}{\left (a^{2} + c\right )^{4}} - \frac {20 a^{9} b^{2} c}{\left (a^{2} + c\right )^{2}} + \frac {608 a^{7} b^{2} c^{4}}{\left (a^{2} + c\right )^{4}} - \frac {64 a^{7} b^{2} c^{2}}{\left (a^{2} + c\right )^{2}} + \frac {432 a^{5} b^{2} c^{5}}{\left (a^{2} + c\right )^{4}} - \frac {72 a^{5} b^{2} c^{3}}{\left (a^{2} + c\right )^{2}} + 36 a^{5} b^{2} c + \frac {112 a^{3} b^{2} c^{6}}{\left (a^{2} + c\right )^{4}} - \frac {32 a^{3} b^{2} c^{4}}{\left (a^{2} + c\right )^{2}} - 88 a^{3} b^{2} c^{2} - \frac {4 a b^{2} c^{5}}{\left (a^{2} + c\right )^{2}} + 4 a b^{2} c^{3}}{a^{6} b^{3} + 33 a^{4} b^{3} c - 33 a^{2} b^{3} c^{2} - b^{3} c^{3}} \right )}}{\left (a^{2} + c\right )^{2}} + \left (\frac {a b}{\left (a^{2} + c\right )^{2}} - \frac {b \sqrt {- c} \left (a^{2} - c\right )}{2 c \left (a^{4} + 2 a^{2} c + c^{2}\right )}\right ) \log {\left (x + \frac {- 4 a^{11} c \left (\frac {a b}{\left (a^{2} + c\right )^{2}} - \frac {b \sqrt {- c} \left (a^{2} - c\right )}{2 c \left (a^{4} + 2 a^{2} c + c^{2}\right )}\right )^{2} + 12 a^{9} c^{2} \left (\frac {a b}{\left (a^{2} + c\right )^{2}} - \frac {b \sqrt {- c} \left (a^{2} - c\right )}{2 c \left (a^{4} + 2 a^{2} c + c^{2}\right )}\right )^{2} + 10 a^{8} b c \left (\frac {a b}{\left (a^{2} + c\right )^{2}} - \frac {b \sqrt {- c} \left (a^{2} - c\right )}{2 c \left (a^{4} + 2 a^{2} c + c^{2}\right )}\right ) + 88 a^{7} c^{3} \left (\frac {a b}{\left (a^{2} + c\right )^{2}} - \frac {b \sqrt {- c} \left (a^{2} - c\right )}{2 c \left (a^{4} + 2 a^{2} c + c^{2}\right )}\right )^{2} + 32 a^{6} b c^{2} \left (\frac {a b}{\left (a^{2} + c\right )^{2}} - \frac {b \sqrt {- c} \left (a^{2} - c\right )}{2 c \left (a^{4} + 2 a^{2} c + c^{2}\right )}\right ) + 36 a^{5} b^{2} c + 152 a^{5} c^{4} \left (\frac {a b}{\left (a^{2} + c\right )^{2}} - \frac {b \sqrt {- c} \left (a^{2} - c\right )}{2 c \left (a^{4} + 2 a^{2} c + c^{2}\right )}\right )^{2} + 36 a^{4} b c^{3} \left (\frac {a b}{\left (a^{2} + c\right )^{2}} - \frac {b \sqrt {- c} \left (a^{2} - c\right )}{2 c \left (a^{4} + 2 a^{2} c + c^{2}\right )}\right ) - 88 a^{3} b^{2} c^{2} + 108 a^{3} c^{5} \left (\frac {a b}{\left (a^{2} + c\right )^{2}} - \frac {b \sqrt {- c} \left (a^{2} - c\right )}{2 c \left (a^{4} + 2 a^{2} c + c^{2}\right )}\right )^{2} + 16 a^{2} b c^{4} \left (\frac {a b}{\left (a^{2} + c\right )^{2}} - \frac {b \sqrt {- c} \left (a^{2} - c\right )}{2 c \left (a^{4} + 2 a^{2} c + c^{2}\right )}\right ) + 4 a b^{2} c^{3} + 28 a c^{6} \left (\frac {a b}{\left (a^{2} + c\right )^{2}} - \frac {b \sqrt {- c} \left (a^{2} - c\right )}{2 c \left (a^{4} + 2 a^{2} c + c^{2}\right )}\right )^{2} + 2 b c^{5} \left (\frac {a b}{\left (a^{2} + c\right )^{2}} - \frac {b \sqrt {- c} \left (a^{2} - c\right )}{2 c \left (a^{4} + 2 a^{2} c + c^{2}\right )}\right )}{a^{6} b^{3} + 33 a^{4} b^{3} c - 33 a^{2} b^{3} c^{2} - b^{3} c^{3}} \right )} + \left (\frac {a b}{\left (a^{2} + c\right )^{2}} + \frac {b \sqrt {- c} \left (a^{2} - c\right )}{2 c \left (a^{4} + 2 a^{2} c + c^{2}\right )}\right ) \log {\left (x + \frac {- 4 a^{11} c \left (\frac {a b}{\left (a^{2} + c\right )^{2}} + \frac {b \sqrt {- c} \left (a^{2} - c\right )}{2 c \left (a^{4} + 2 a^{2} c + c^{2}\right )}\right )^{2} + 12 a^{9} c^{2} \left (\frac {a b}{\left (a^{2} + c\right )^{2}} + \frac {b \sqrt {- c} \left (a^{2} - c\right )}{2 c \left (a^{4} + 2 a^{2} c + c^{2}\right )}\right )^{2} + 10 a^{8} b c \left (\frac {a b}{\left (a^{2} + c\right )^{2}} + \frac {b \sqrt {- c} \left (a^{2} - c\right )}{2 c \left (a^{4} + 2 a^{2} c + c^{2}\right )}\right ) + 88 a^{7} c^{3} \left (\frac {a b}{\left (a^{2} + c\right )^{2}} + \frac {b \sqrt {- c} \left (a^{2} - c\right )}{2 c \left (a^{4} + 2 a^{2} c + c^{2}\right )}\right )^{2} + 32 a^{6} b c^{2} \left (\frac {a b}{\left (a^{2} + c\right )^{2}} + \frac {b \sqrt {- c} \left (a^{2} - c\right )}{2 c \left (a^{4} + 2 a^{2} c + c^{2}\right )}\right ) + 36 a^{5} b^{2} c + 152 a^{5} c^{4} \left (\frac {a b}{\left (a^{2} + c\right )^{2}} + \frac {b \sqrt {- c} \left (a^{2} - c\right )}{2 c \left (a^{4} + 2 a^{2} c + c^{2}\right )}\right )^{2} + 36 a^{4} b c^{3} \left (\frac {a b}{\left (a^{2} + c\right )^{2}} + \frac {b \sqrt {- c} \left (a^{2} - c\right )}{2 c \left (a^{4} + 2 a^{2} c + c^{2}\right )}\right ) - 88 a^{3} b^{2} c^{2} + 108 a^{3} c^{5} \left (\frac {a b}{\left (a^{2} + c\right )^{2}} + \frac {b \sqrt {- c} \left (a^{2} - c\right )}{2 c \left (a^{4} + 2 a^{2} c + c^{2}\right )}\right )^{2} + 16 a^{2} b c^{4} \left (\frac {a b}{\left (a^{2} + c\right )^{2}} + \frac {b \sqrt {- c} \left (a^{2} - c\right )}{2 c \left (a^{4} + 2 a^{2} c + c^{2}\right )}\right ) + 4 a b^{2} c^{3} + 28 a c^{6} \left (\frac {a b}{\left (a^{2} + c\right )^{2}} + \frac {b \sqrt {- c} \left (a^{2} - c\right )}{2 c \left (a^{4} + 2 a^{2} c + c^{2}\right )}\right )^{2} + 2 b c^{5} \left (\frac {a b}{\left (a^{2} + c\right )^{2}} + \frac {b \sqrt {- c} \left (a^{2} - c\right )}{2 c \left (a^{4} + 2 a^{2} c + c^{2}\right )}\right )}{a^{6} b^{3} + 33 a^{4} b^{3} c - 33 a^{2} b^{3} c^{2} - b^{3} c^{3}} \right )} - \frac {1}{x \left (a^{2} + c\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.09, size = 117, normalized size = 1.48 \begin {gather*} \frac {a b \log \left (b^{2} x^{2} + 2 \, a b x + a^{2} + c\right )}{a^{4} + 2 \, a^{2} c + c^{2}} - \frac {2 \, a b \log \left ({\left | x \right |}\right )}{a^{4} + 2 \, a^{2} c + c^{2}} + \frac {{\left (a^{2} b^{2} - b^{2} c\right )} \arctan \left (\frac {b x + a}{\sqrt {c}}\right )}{{\left (a^{4} + 2 \, a^{2} c + c^{2}\right )} b \sqrt {c}} - \frac {1}{{\left (a^{2} + c\right )} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.58, size = 425, normalized size = 5.38 \begin {gather*} \frac {\ln \left ({\left (-c\right )}^{13/2}-35\,a^2\,{\left (-c\right )}^{11/2}+34\,a^4\,{\left (-c\right )}^{9/2}+34\,a^6\,{\left (-c\right )}^{7/2}-35\,a^8\,{\left (-c\right )}^{5/2}+a^{10}\,{\left (-c\right )}^{3/2}+a\,c^6-a^{11}\,c+35\,a^3\,c^5+34\,a^5\,c^4-34\,a^7\,c^3-35\,a^9\,c^2+b\,c^6\,x-a^{10}\,b\,c\,x+35\,a^2\,b\,c^5\,x+34\,a^4\,b\,c^4\,x-34\,a^6\,b\,c^3\,x-35\,a^8\,b\,c^2\,x\right )\,\left (b\,{\left (-c\right )}^{3/2}+2\,a\,b\,c+a^2\,b\,\sqrt {-c}\right )}{2\,\left (a^4\,c+2\,a^2\,c^2+c^3\right )}-\frac {1}{x\,\left (a^2+c\right )}-\frac {\ln \left ({\left (-c\right )}^{13/2}-35\,a^2\,{\left (-c\right )}^{11/2}+34\,a^4\,{\left (-c\right )}^{9/2}+34\,a^6\,{\left (-c\right )}^{7/2}-35\,a^8\,{\left (-c\right )}^{5/2}+a^{10}\,{\left (-c\right )}^{3/2}-a\,c^6+a^{11}\,c-35\,a^3\,c^5-34\,a^5\,c^4+34\,a^7\,c^3+35\,a^9\,c^2-b\,c^6\,x+a^{10}\,b\,c\,x-35\,a^2\,b\,c^5\,x-34\,a^4\,b\,c^4\,x+34\,a^6\,b\,c^3\,x+35\,a^8\,b\,c^2\,x\right )\,\left (b\,{\left (-c\right )}^{3/2}-2\,a\,b\,c+a^2\,b\,\sqrt {-c}\right )}{2\,\left (a^4\,c+2\,a^2\,c^2+c^3\right )}-\frac {2\,a\,b\,\ln \left (x\right )}{{\left (a^2+c\right )}^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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