Optimal. Leaf size=94 \[ \frac {\left (a^2-b^2\right ) x^2}{2 \left (a^2+b^2\right )^2}+\frac {a b \log \left (a \cos \left (c+d x^2\right )+b \sin \left (c+d x^2\right )\right )}{\left (a^2+b^2\right )^2 d}-\frac {b}{2 \left (a^2+b^2\right ) d \left (a+b \tan \left (c+d x^2\right )\right )} \]
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Rubi [A]
time = 0.09, antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3832, 3564,
3612, 3611} \begin {gather*} -\frac {b}{2 d \left (a^2+b^2\right ) \left (a+b \tan \left (c+d x^2\right )\right )}+\frac {a b \log \left (a \cos \left (c+d x^2\right )+b \sin \left (c+d x^2\right )\right )}{d \left (a^2+b^2\right )^2}+\frac {x^2 \left (a^2-b^2\right )}{2 \left (a^2+b^2\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 3564
Rule 3611
Rule 3612
Rule 3832
Rubi steps
\begin {align*} \int \frac {x}{\left (a+b \tan \left (c+d x^2\right )\right )^2} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {1}{(a+b \tan (c+d x))^2} \, dx,x,x^2\right )\\ &=-\frac {b}{2 \left (a^2+b^2\right ) d \left (a+b \tan \left (c+d x^2\right )\right )}+\frac {\text {Subst}\left (\int \frac {a-b \tan (c+d x)}{a+b \tan (c+d x)} \, dx,x,x^2\right )}{2 \left (a^2+b^2\right )}\\ &=\frac {\left (a^2-b^2\right ) x^2}{2 \left (a^2+b^2\right )^2}-\frac {b}{2 \left (a^2+b^2\right ) d \left (a+b \tan \left (c+d x^2\right )\right )}+\frac {(a b) \text {Subst}\left (\int \frac {b-a \tan (c+d x)}{a+b \tan (c+d x)} \, dx,x,x^2\right )}{\left (a^2+b^2\right )^2}\\ &=\frac {\left (a^2-b^2\right ) x^2}{2 \left (a^2+b^2\right )^2}+\frac {a b \log \left (a \cos \left (c+d x^2\right )+b \sin \left (c+d x^2\right )\right )}{\left (a^2+b^2\right )^2 d}-\frac {b}{2 \left (a^2+b^2\right ) d \left (a+b \tan \left (c+d x^2\right )\right )}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 1.39, size = 114, normalized size = 1.21 \begin {gather*} \frac {-\frac {i \log \left (i-\tan \left (c+d x^2\right )\right )}{(a+i b)^2}+\frac {i \log \left (i+\tan \left (c+d x^2\right )\right )}{(a-i b)^2}+\frac {2 b \left (2 a \log \left (a+b \tan \left (c+d x^2\right )\right )-\frac {a^2+b^2}{a+b \tan \left (c+d x^2\right )}\right )}{\left (a^2+b^2\right )^2}}{4 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.16, size = 106, normalized size = 1.13
method | result | size |
derivativedivides | \(\frac {-\frac {b}{\left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d \,x^{2}+c \right )\right )}+\frac {2 a b \ln \left (a +b \tan \left (d \,x^{2}+c \right )\right )}{\left (a^{2}+b^{2}\right )^{2}}+\frac {-a b \ln \left (1+\tan ^{2}\left (d \,x^{2}+c \right )\right )+\left (a^{2}-b^{2}\right ) \arctan \left (\tan \left (d \,x^{2}+c \right )\right )}{\left (a^{2}+b^{2}\right )^{2}}}{2 d}\) | \(106\) |
default | \(\frac {-\frac {b}{\left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d \,x^{2}+c \right )\right )}+\frac {2 a b \ln \left (a +b \tan \left (d \,x^{2}+c \right )\right )}{\left (a^{2}+b^{2}\right )^{2}}+\frac {-a b \ln \left (1+\tan ^{2}\left (d \,x^{2}+c \right )\right )+\left (a^{2}-b^{2}\right ) \arctan \left (\tan \left (d \,x^{2}+c \right )\right )}{\left (a^{2}+b^{2}\right )^{2}}}{2 d}\) | \(106\) |
norman | \(\frac {\frac {\left (a^{2}-b^{2}\right ) a \,x^{2}}{2 a^{4}+4 a^{2} b^{2}+2 b^{4}}+\frac {b \left (a^{2}-b^{2}\right ) x^{2} \tan \left (d \,x^{2}+c \right )}{2 a^{4}+4 a^{2} b^{2}+2 b^{4}}+\frac {b^{2} \tan \left (d \,x^{2}+c \right )}{2 a \left (a^{2}+b^{2}\right ) d}}{a +b \tan \left (d \,x^{2}+c \right )}+\frac {a b \ln \left (a +b \tan \left (d \,x^{2}+c \right )\right )}{d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {a b \ln \left (1+\tan ^{2}\left (d \,x^{2}+c \right )\right )}{2 d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}\) | \(191\) |
risch | \(-\frac {x^{2}}{2 \left (2 i a b -a^{2}+b^{2}\right )}-\frac {2 i a b \,x^{2}}{a^{4}+2 a^{2} b^{2}+b^{4}}-\frac {2 i a b c}{d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {i b^{2}}{\left (-i a +b \right ) d \left (i a +b \right )^{2} \left (b \,{\mathrm e}^{2 i \left (d \,x^{2}+c \right )}+i a \,{\mathrm e}^{2 i \left (d \,x^{2}+c \right )}-b +i a \right )}+\frac {a b \ln \left ({\mathrm e}^{2 i \left (d \,x^{2}+c \right )}-\frac {i b +a}{i b -a}\right )}{d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}\) | \(191\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 556 vs.
\(2 (90) = 180\).
time = 0.39, size = 556, normalized size = 5.91 \begin {gather*} \frac {{\left (a^{4} - b^{4}\right )} d x^{2} \cos \left (2 \, d x^{2} + 2 \, c\right )^{2} + {\left (a^{4} - b^{4}\right )} d x^{2} \sin \left (2 \, d x^{2} + 2 \, c\right )^{2} + {\left (a^{4} - b^{4}\right )} d x^{2} - 2 \, {\left (2 \, a b^{3} - {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} d x^{2}\right )} \cos \left (2 \, d x^{2} + 2 \, c\right ) + {\left (4 \, a^{2} b^{2} \sin \left (2 \, d x^{2} + 2 \, c\right ) + a^{3} b + a b^{3} + {\left (a^{3} b + a b^{3}\right )} \cos \left (2 \, d x^{2} + 2 \, c\right )^{2} + {\left (a^{3} b + a b^{3}\right )} \sin \left (2 \, d x^{2} + 2 \, c\right )^{2} + 2 \, {\left (a^{3} b - a b^{3}\right )} \cos \left (2 \, d x^{2} + 2 \, c\right )\right )} \log \left (\frac {{\left (a^{2} + b^{2}\right )} \cos \left (2 \, d x^{2} + 2 \, c\right )^{2} + 4 \, a b \sin \left (2 \, d x^{2} + 2 \, c\right ) + {\left (a^{2} + b^{2}\right )} \sin \left (2 \, d x^{2} + 2 \, c\right )^{2} + a^{2} + b^{2} + 2 \, {\left (a^{2} - b^{2}\right )} \cos \left (2 \, d x^{2} + 2 \, c\right )}{{\left (a^{2} + b^{2}\right )} \cos \left (2 \, c\right )^{2} + {\left (a^{2} + b^{2}\right )} \sin \left (2 \, c\right )^{2}}\right ) + 2 \, {\left (a^{2} b^{2} - b^{4} + 2 \, {\left (a^{3} b - a b^{3}\right )} d x^{2}\right )} \sin \left (2 \, d x^{2} + 2 \, c\right )}{2 \, {\left ({\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )} d \cos \left (2 \, d x^{2} + 2 \, c\right )^{2} + {\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )} d \sin \left (2 \, d x^{2} + 2 \, c\right )^{2} + 2 \, {\left (a^{6} + a^{4} b^{2} - a^{2} b^{4} - b^{6}\right )} d \cos \left (2 \, d x^{2} + 2 \, c\right ) + 4 \, {\left (a^{5} b + 2 \, a^{3} b^{3} + a b^{5}\right )} d \sin \left (2 \, d x^{2} + 2 \, c\right ) + {\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )} d\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 169, normalized size = 1.80 \begin {gather*} \frac {{\left (a^{3} - a b^{2}\right )} d x^{2} - b^{3} + {\left (a b^{2} \tan \left (d x^{2} + c\right ) + a^{2} b\right )} \log \left (\frac {b^{2} \tan \left (d x^{2} + c\right )^{2} + 2 \, a b \tan \left (d x^{2} + c\right ) + a^{2}}{\tan \left (d x^{2} + c\right )^{2} + 1}\right ) + {\left ({\left (a^{2} b - b^{3}\right )} d x^{2} + a b^{2}\right )} \tan \left (d x^{2} + c\right )}{2 \, {\left ({\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} d \tan \left (d x^{2} + c\right ) + {\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} d\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 0.70, size = 1584, normalized size = 16.85 \begin {gather*} \begin {cases} \frac {\tilde {\infty } x^{2}}{\tan ^{2}{\left (c \right )}} & \text {for}\: a = 0 \wedge b = 0 \wedge d = 0 \\\frac {x^{2}}{2 a^{2}} & \text {for}\: b = 0 \\- \frac {\left (\operatorname {atan}{\left (\tan {\left (c + d x^{2} \right )} \right )} + \pi \left \lfloor {\frac {c + d x^{2} - \frac {\pi }{2}}{\pi }}\right \rfloor \right ) \tan ^{2}{\left (c + d x^{2} \right )}}{8 b^{2} d \tan ^{2}{\left (c + d x^{2} \right )} - 16 i b^{2} d \tan {\left (c + d x^{2} \right )} - 8 b^{2} d} + \frac {2 i \left (\operatorname {atan}{\left (\tan {\left (c + d x^{2} \right )} \right )} + \pi \left \lfloor {\frac {c + d x^{2} - \frac {\pi }{2}}{\pi }}\right \rfloor \right ) \tan {\left (c + d x^{2} \right )}}{8 b^{2} d \tan ^{2}{\left (c + d x^{2} \right )} - 16 i b^{2} d \tan {\left (c + d x^{2} \right )} - 8 b^{2} d} + \frac {\operatorname {atan}{\left (\tan {\left (c + d x^{2} \right )} \right )} + \pi \left \lfloor {\frac {c + d x^{2} - \frac {\pi }{2}}{\pi }}\right \rfloor }{8 b^{2} d \tan ^{2}{\left (c + d x^{2} \right )} - 16 i b^{2} d \tan {\left (c + d x^{2} \right )} - 8 b^{2} d} - \frac {\tan {\left (c + d x^{2} \right )}}{8 b^{2} d \tan ^{2}{\left (c + d x^{2} \right )} - 16 i b^{2} d \tan {\left (c + d x^{2} \right )} - 8 b^{2} d} + \frac {2 i}{8 b^{2} d \tan ^{2}{\left (c + d x^{2} \right )} - 16 i b^{2} d \tan {\left (c + d x^{2} \right )} - 8 b^{2} d} & \text {for}\: a = - i b \\- \frac {\left (\operatorname {atan}{\left (\tan {\left (c + d x^{2} \right )} \right )} + \pi \left \lfloor {\frac {c + d x^{2} - \frac {\pi }{2}}{\pi }}\right \rfloor \right ) \tan ^{2}{\left (c + d x^{2} \right )}}{8 b^{2} d \tan ^{2}{\left (c + d x^{2} \right )} + 16 i b^{2} d \tan {\left (c + d x^{2} \right )} - 8 b^{2} d} - \frac {2 i \left (\operatorname {atan}{\left (\tan {\left (c + d x^{2} \right )} \right )} + \pi \left \lfloor {\frac {c + d x^{2} - \frac {\pi }{2}}{\pi }}\right \rfloor \right ) \tan {\left (c + d x^{2} \right )}}{8 b^{2} d \tan ^{2}{\left (c + d x^{2} \right )} + 16 i b^{2} d \tan {\left (c + d x^{2} \right )} - 8 b^{2} d} + \frac {\operatorname {atan}{\left (\tan {\left (c + d x^{2} \right )} \right )} + \pi \left \lfloor {\frac {c + d x^{2} - \frac {\pi }{2}}{\pi }}\right \rfloor }{8 b^{2} d \tan ^{2}{\left (c + d x^{2} \right )} + 16 i b^{2} d \tan {\left (c + d x^{2} \right )} - 8 b^{2} d} - \frac {\tan {\left (c + d x^{2} \right )}}{8 b^{2} d \tan ^{2}{\left (c + d x^{2} \right )} + 16 i b^{2} d \tan {\left (c + d x^{2} \right )} - 8 b^{2} d} - \frac {2 i}{8 b^{2} d \tan ^{2}{\left (c + d x^{2} \right )} + 16 i b^{2} d \tan {\left (c + d x^{2} \right )} - 8 b^{2} d} & \text {for}\: a = i b \\\frac {x^{2}}{2 \left (a + b \tan {\left (c \right )}\right )^{2}} & \text {for}\: d = 0 \\\frac {a^{3} d x^{2}}{2 a^{5} d + 2 a^{4} b d \tan {\left (c + d x^{2} \right )} + 4 a^{3} b^{2} d + 4 a^{2} b^{3} d \tan {\left (c + d x^{2} \right )} + 2 a b^{4} d + 2 b^{5} d \tan {\left (c + d x^{2} \right )}} + \frac {a^{2} b d x^{2} \tan {\left (c + d x^{2} \right )}}{2 a^{5} d + 2 a^{4} b d \tan {\left (c + d x^{2} \right )} + 4 a^{3} b^{2} d + 4 a^{2} b^{3} d \tan {\left (c + d x^{2} \right )} + 2 a b^{4} d + 2 b^{5} d \tan {\left (c + d x^{2} \right )}} + \frac {2 a^{2} b \log {\left (\frac {a}{b} + \tan {\left (c + d x^{2} \right )} \right )}}{2 a^{5} d + 2 a^{4} b d \tan {\left (c + d x^{2} \right )} + 4 a^{3} b^{2} d + 4 a^{2} b^{3} d \tan {\left (c + d x^{2} \right )} + 2 a b^{4} d + 2 b^{5} d \tan {\left (c + d x^{2} \right )}} - \frac {a^{2} b \log {\left (\tan ^{2}{\left (c + d x^{2} \right )} + 1 \right )}}{2 a^{5} d + 2 a^{4} b d \tan {\left (c + d x^{2} \right )} + 4 a^{3} b^{2} d + 4 a^{2} b^{3} d \tan {\left (c + d x^{2} \right )} + 2 a b^{4} d + 2 b^{5} d \tan {\left (c + d x^{2} \right )}} - \frac {a^{2} b}{2 a^{5} d + 2 a^{4} b d \tan {\left (c + d x^{2} \right )} + 4 a^{3} b^{2} d + 4 a^{2} b^{3} d \tan {\left (c + d x^{2} \right )} + 2 a b^{4} d + 2 b^{5} d \tan {\left (c + d x^{2} \right )}} - \frac {a b^{2} d x^{2}}{2 a^{5} d + 2 a^{4} b d \tan {\left (c + d x^{2} \right )} + 4 a^{3} b^{2} d + 4 a^{2} b^{3} d \tan {\left (c + d x^{2} \right )} + 2 a b^{4} d + 2 b^{5} d \tan {\left (c + d x^{2} \right )}} + \frac {2 a b^{2} \log {\left (\frac {a}{b} + \tan {\left (c + d x^{2} \right )} \right )} \tan {\left (c + d x^{2} \right )}}{2 a^{5} d + 2 a^{4} b d \tan {\left (c + d x^{2} \right )} + 4 a^{3} b^{2} d + 4 a^{2} b^{3} d \tan {\left (c + d x^{2} \right )} + 2 a b^{4} d + 2 b^{5} d \tan {\left (c + d x^{2} \right )}} - \frac {a b^{2} \log {\left (\tan ^{2}{\left (c + d x^{2} \right )} + 1 \right )} \tan {\left (c + d x^{2} \right )}}{2 a^{5} d + 2 a^{4} b d \tan {\left (c + d x^{2} \right )} + 4 a^{3} b^{2} d + 4 a^{2} b^{3} d \tan {\left (c + d x^{2} \right )} + 2 a b^{4} d + 2 b^{5} d \tan {\left (c + d x^{2} \right )}} - \frac {b^{3} d x^{2} \tan {\left (c + d x^{2} \right )}}{2 a^{5} d + 2 a^{4} b d \tan {\left (c + d x^{2} \right )} + 4 a^{3} b^{2} d + 4 a^{2} b^{3} d \tan {\left (c + d x^{2} \right )} + 2 a b^{4} d + 2 b^{5} d \tan {\left (c + d x^{2} \right )}} - \frac {b^{3}}{2 a^{5} d + 2 a^{4} b d \tan {\left (c + d x^{2} \right )} + 4 a^{3} b^{2} d + 4 a^{2} b^{3} d \tan {\left (c + d x^{2} \right )} + 2 a b^{4} d + 2 b^{5} d \tan {\left (c + d x^{2} \right )}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.47, size = 159, normalized size = 1.69 \begin {gather*} \frac {a b^{2} \log \left ({\left | b \tan \left (d x^{2} + c\right ) + a \right |}\right )}{a^{4} b d + 2 \, a^{2} b^{3} d + b^{5} d} - \frac {a b \log \left (\tan \left (d x^{2} + c\right )^{2} + 1\right )}{2 \, {\left (a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d\right )}} + \frac {{\left (d x^{2} + c\right )} {\left (a^{2} - b^{2}\right )}}{2 \, {\left (a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d\right )}} - \frac {a^{2} b + b^{3}}{2 \, {\left (a^{2} + b^{2}\right )}^{2} {\left (b \tan \left (d x^{2} + c\right ) + a\right )} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.60, size = 173, normalized size = 1.84 \begin {gather*} \frac {\frac {x^2\,\mathrm {tan}\left (d\,x^2+c\right )\,\left (\frac {a^2\,b}{2}-\frac {b^3}{2}\right )}{{\left (a^2+b^2\right )}^2}-\frac {x^2\,\left (\frac {a\,b^2}{2}-\frac {a^3}{2}\right )}{{\left (a^2+b^2\right )}^2}+\frac {b^2\,\mathrm {tan}\left (d\,x^2+c\right )}{2\,a\,d\,\left (a^2+b^2\right )}}{a+b\,\mathrm {tan}\left (d\,x^2+c\right )}-\frac {a\,b\,\ln \left ({\mathrm {tan}\left (d\,x^2+c\right )}^2+1\right )}{2\,\left (d\,a^4+2\,d\,a^2\,b^2+d\,b^4\right )}+\frac {a\,b\,\ln \left (a+b\,\mathrm {tan}\left (d\,x^2+c\right )\right )}{d\,{\left (a^2+b^2\right )}^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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