Optimal. Leaf size=94 \[ -\frac {b}{2 d \left (a^2+b^2\right ) \left (a+b \tan \left (c+d x^2\right )\right )}+\frac {a b \log \left (a \cos \left (c+d x^2\right )+b \sin \left (c+d x^2\right )\right )}{d \left (a^2+b^2\right )^2}+\frac {x^2 \left (a^2-b^2\right )}{2 \left (a^2+b^2\right )^2} \]
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Rubi [A] time = 0.13, 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 = {3747, 3483, 3531, 3530} \[ -\frac {b}{2 d \left (a^2+b^2\right ) \left (a+b \tan \left (c+d x^2\right )\right )}+\frac {a b \log \left (a \cos \left (c+d x^2\right )+b \sin \left (c+d x^2\right )\right )}{d \left (a^2+b^2\right )^2}+\frac {x^2 \left (a^2-b^2\right )}{2 \left (a^2+b^2\right )^2} \]
Antiderivative was successfully verified.
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Rule 3483
Rule 3530
Rule 3531
Rule 3747
Rubi steps
\begin {align*} \int \frac {x}{\left (a+b \tan \left (c+d x^2\right )\right )^2} \, dx &=\frac {1}{2} \operatorname {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 {\operatorname {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) \operatorname {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] time = 1.30, size = 114, normalized size = 1.21 \[ \frac {\frac {2 b \left (2 a \log \left (a+b \tan \left (c+d x^2\right )\right )-\frac {a^2+b^2}{a+b \tan \left (c+d x^2\right )}\right )}{\left (a^2+b^2\right )^2}-\frac {i \log \left (-\tan \left (c+d x^2\right )+i\right )}{(a+i b)^2}+\frac {i \log \left (\tan \left (c+d x^2\right )+i\right )}{(a-i b)^2}}{4 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.52, size = 169, normalized size = 1.80 \[ \frac {{\left (a^{3} - a b^{2}\right )} d x^{2} - b^{3} + {\left (a b^{2} \tan \left (d x^{2} + c\right ) + a^{2} b\right )} \log \left (\frac {b^{2} \tan \left (d x^{2} + c\right )^{2} + 2 \, a b \tan \left (d x^{2} + c\right ) + a^{2}}{\tan \left (d x^{2} + c\right )^{2} + 1}\right ) + {\left ({\left (a^{2} b - b^{3}\right )} d x^{2} + a b^{2}\right )} \tan \left (d x^{2} + c\right )}{2 \, {\left ({\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} d \tan \left (d x^{2} + c\right ) + {\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.42, size = 159, normalized size = 1.69 \[ \frac {a b^{2} \log \left ({\left | b \tan \left (d x^{2} + c\right ) + a \right |}\right )}{a^{4} b d + 2 \, a^{2} b^{3} d + b^{5} d} - \frac {a b \log \left (\tan \left (d x^{2} + c\right )^{2} + 1\right )}{2 \, {\left (a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d\right )}} + \frac {{\left (d x^{2} + c\right )} {\left (a^{2} - b^{2}\right )}}{2 \, {\left (a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d\right )}} - \frac {a^{2} b + b^{3}}{2 \, {\left (a^{2} + b^{2}\right )}^{2} {\left (b \tan \left (d x^{2} + c\right ) + a\right )} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.16, size = 140, normalized size = 1.49 \[ -\frac {b}{2 \left (a^{2}+b^{2}\right ) d \left (a +b \tan \left (d \,x^{2}+c \right )\right )}+\frac {a b \ln \left (a +b \tan \left (d \,x^{2}+c \right )\right )}{d \left (a^{2}+b^{2}\right )^{2}}-\frac {a b \ln \left (1+\tan ^{2}\left (d \,x^{2}+c \right )\right )}{2 d \left (a^{2}+b^{2}\right )^{2}}+\frac {\arctan \left (\tan \left (d \,x^{2}+c \right )\right ) a^{2}}{2 d \left (a^{2}+b^{2}\right )^{2}}-\frac {\arctan \left (\tan \left (d \,x^{2}+c \right )\right ) b^{2}}{2 d \left (a^{2}+b^{2}\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.41, size = 556, normalized size = 5.91 \[ \frac {{\left (a^{4} - b^{4}\right )} d x^{2} \cos \left (2 \, d x^{2} + 2 \, c\right )^{2} + {\left (a^{4} - b^{4}\right )} d x^{2} \sin \left (2 \, d x^{2} + 2 \, c\right )^{2} + {\left (a^{4} - b^{4}\right )} d x^{2} - 2 \, {\left (2 \, a b^{3} - {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} d x^{2}\right )} \cos \left (2 \, d x^{2} + 2 \, c\right ) + {\left (4 \, a^{2} b^{2} \sin \left (2 \, d x^{2} + 2 \, c\right ) + a^{3} b + a b^{3} + {\left (a^{3} b + a b^{3}\right )} \cos \left (2 \, d x^{2} + 2 \, c\right )^{2} + {\left (a^{3} b + a b^{3}\right )} \sin \left (2 \, d x^{2} + 2 \, c\right )^{2} + 2 \, {\left (a^{3} b - a b^{3}\right )} \cos \left (2 \, d x^{2} + 2 \, c\right )\right )} \log \left (\frac {{\left (a^{2} + b^{2}\right )} \cos \left (2 \, d x^{2} + 2 \, c\right )^{2} + 4 \, a b \sin \left (2 \, d x^{2} + 2 \, c\right ) + {\left (a^{2} + b^{2}\right )} \sin \left (2 \, d x^{2} + 2 \, c\right )^{2} + a^{2} + b^{2} + 2 \, {\left (a^{2} - b^{2}\right )} \cos \left (2 \, d x^{2} + 2 \, c\right )}{{\left (a^{2} + b^{2}\right )} \cos \left (2 \, c\right )^{2} + {\left (a^{2} + b^{2}\right )} \sin \left (2 \, c\right )^{2}}\right ) + 2 \, {\left (a^{2} b^{2} - b^{4} + 2 \, {\left (a^{3} b - a b^{3}\right )} d x^{2}\right )} \sin \left (2 \, d x^{2} + 2 \, c\right )}{2 \, {\left ({\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )} d \cos \left (2 \, d x^{2} + 2 \, c\right )^{2} + {\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )} d \sin \left (2 \, d x^{2} + 2 \, c\right )^{2} + 2 \, {\left (a^{6} + a^{4} b^{2} - a^{2} b^{4} - b^{6}\right )} d \cos \left (2 \, d x^{2} + 2 \, c\right ) + 4 \, {\left (a^{5} b + 2 \, a^{3} b^{3} + a b^{5}\right )} d \sin \left (2 \, d x^{2} + 2 \, c\right ) + {\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )} d\right )}} \]
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 \[ \frac {\frac {x^2\,\mathrm {tan}\left (d\,x^2+c\right )\,\left (\frac {a^2\,b}{2}-\frac {b^3}{2}\right )}{{\left (a^2+b^2\right )}^2}-\frac {x^2\,\left (\frac {a\,b^2}{2}-\frac {a^3}{2}\right )}{{\left (a^2+b^2\right )}^2}+\frac {b^2\,\mathrm {tan}\left (d\,x^2+c\right )}{2\,a\,d\,\left (a^2+b^2\right )}}{a+b\,\mathrm {tan}\left (d\,x^2+c\right )}-\frac {a\,b\,\ln \left ({\mathrm {tan}\left (d\,x^2+c\right )}^2+1\right )}{2\,\left (d\,a^4+2\,d\,a^2\,b^2+d\,b^4\right )}+\frac {a\,b\,\ln \left (a+b\,\mathrm {tan}\left (d\,x^2+c\right )\right )}{d\,{\left (a^2+b^2\right )}^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.56, size = 1584, normalized size = 16.85 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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