Optimal. Leaf size=204 \[ -\frac{2 i a b \text{PolyLog}\left (2,-\frac{\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt{x}\right )}}{(a+i b)^2}\right )}{d^2 \left (a^2+b^2\right )^2}+\frac{2 b \left (2 a d \sqrt{x}+b\right ) \log \left (1+\frac{\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt{x}\right )}}{(a+i b)^2}\right )}{d^2 \left (a^2+b^2\right )^2}-\frac{2 b \sqrt{x}}{d \left (a^2+b^2\right ) \left (a+b \tan \left (c+d \sqrt{x}\right )\right )}+\frac{\left (2 a d \sqrt{x}+b\right )^2}{2 a d^2 (a+i b) \left (a^2+b^2\right )}-\frac{x}{a^2+b^2} \]
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Rubi [A] time = 0.255395, antiderivative size = 204, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {3739, 3733, 3732, 2190, 2279, 2391} \[ -\frac{2 i a b \text{Li}_2\left (-\frac{\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt{x}\right )}}{(a+i b)^2}\right )}{d^2 \left (a^2+b^2\right )^2}+\frac{2 b \left (2 a d \sqrt{x}+b\right ) \log \left (1+\frac{\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt{x}\right )}}{(a+i b)^2}\right )}{d^2 \left (a^2+b^2\right )^2}-\frac{2 b \sqrt{x}}{d \left (a^2+b^2\right ) \left (a+b \tan \left (c+d \sqrt{x}\right )\right )}+\frac{\left (2 a d \sqrt{x}+b\right )^2}{2 a d^2 (a+i b) \left (a^2+b^2\right )}-\frac{x}{a^2+b^2} \]
Antiderivative was successfully verified.
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Rule 3739
Rule 3733
Rule 3732
Rule 2190
Rule 2279
Rule 2391
Rubi steps
\begin{align*} \int \frac{1}{\left (a+b \tan \left (c+d \sqrt{x}\right )\right )^2} \, dx &=2 \operatorname{Subst}\left (\int \frac{x}{(a+b \tan (c+d x))^2} \, dx,x,\sqrt{x}\right )\\ &=-\frac{x}{a^2+b^2}-\frac{2 b \sqrt{x}}{\left (a^2+b^2\right ) d \left (a+b \tan \left (c+d \sqrt{x}\right )\right )}+\frac{2 \operatorname{Subst}\left (\int \frac{b+2 a d x}{a+b \tan (c+d x)} \, dx,x,\sqrt{x}\right )}{\left (a^2+b^2\right ) d}\\ &=\frac{\left (b+2 a d \sqrt{x}\right )^2}{2 a (a+i b) \left (a^2+b^2\right ) d^2}-\frac{x}{a^2+b^2}-\frac{2 b \sqrt{x}}{\left (a^2+b^2\right ) d \left (a+b \tan \left (c+d \sqrt{x}\right )\right )}+\frac{(4 i b) \operatorname{Subst}\left (\int \frac{e^{2 i (c+d x)} (b+2 a d x)}{(a+i b)^2+\left (a^2+b^2\right ) e^{2 i (c+d x)}} \, dx,x,\sqrt{x}\right )}{\left (a^2+b^2\right ) d}\\ &=\frac{\left (b+2 a d \sqrt{x}\right )^2}{2 a (a+i b) \left (a^2+b^2\right ) d^2}-\frac{x}{a^2+b^2}+\frac{2 b \left (b+2 a d \sqrt{x}\right ) \log \left (1+\frac{\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt{x}\right )}}{(a+i b)^2}\right )}{\left (a^2+b^2\right )^2 d^2}-\frac{2 b \sqrt{x}}{\left (a^2+b^2\right ) d \left (a+b \tan \left (c+d \sqrt{x}\right )\right )}-\frac{(4 a b) \operatorname{Subst}\left (\int \log \left (1+\frac{\left (a^2+b^2\right ) e^{2 i (c+d x)}}{(a+i b)^2}\right ) \, dx,x,\sqrt{x}\right )}{\left (a^2+b^2\right )^2 d}\\ &=\frac{\left (b+2 a d \sqrt{x}\right )^2}{2 a (a+i b) \left (a^2+b^2\right ) d^2}-\frac{x}{a^2+b^2}+\frac{2 b \left (b+2 a d \sqrt{x}\right ) \log \left (1+\frac{\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt{x}\right )}}{(a+i b)^2}\right )}{\left (a^2+b^2\right )^2 d^2}-\frac{2 b \sqrt{x}}{\left (a^2+b^2\right ) d \left (a+b \tan \left (c+d \sqrt{x}\right )\right )}+\frac{(2 i a b) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{\left (a^2+b^2\right ) x}{(a+i b)^2}\right )}{x} \, dx,x,e^{2 i \left (c+d \sqrt{x}\right )}\right )}{\left (a^2+b^2\right )^2 d^2}\\ &=\frac{\left (b+2 a d \sqrt{x}\right )^2}{2 a (a+i b) \left (a^2+b^2\right ) d^2}-\frac{x}{a^2+b^2}+\frac{2 b \left (b+2 a d \sqrt{x}\right ) \log \left (1+\frac{\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt{x}\right )}}{(a+i b)^2}\right )}{\left (a^2+b^2\right )^2 d^2}-\frac{2 i a b \text{Li}_2\left (-\frac{\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt{x}\right )}}{(a+i b)^2}\right )}{\left (a^2+b^2\right )^2 d^2}-\frac{2 b \sqrt{x}}{\left (a^2+b^2\right ) d \left (a+b \tan \left (c+d \sqrt{x}\right )\right )}\\ \end{align*}
Mathematica [B] time = 6.51845, size = 517, normalized size = 2.53 \[ \frac{\sec ^2\left (c+d \sqrt{x}\right ) \left (a \cos \left (c+d \sqrt{x}\right )+b \sin \left (c+d \sqrt{x}\right )\right ) \left (-2 a b \left (a \cos \left (c+d \sqrt{x}\right )+b \sin \left (c+d \sqrt{x}\right )\right ) \left (a \left (i \text{PolyLog}\left (2,e^{2 i \left (\tan ^{-1}\left (\frac{a}{b}\right )+c+d \sqrt{x}\right )}\right )-i \left (\pi -2 \tan ^{-1}\left (\frac{a}{b}\right )\right ) \left (c+d \sqrt{x}\right )-2 \left (\tan ^{-1}\left (\frac{a}{b}\right )+c+d \sqrt{x}\right ) \log \left (1-e^{2 i \left (\tan ^{-1}\left (\frac{a}{b}\right )+c+d \sqrt{x}\right )}\right )+2 \tan ^{-1}\left (\frac{a}{b}\right ) \log \left (\sin \left (\tan ^{-1}\left (\frac{a}{b}\right )+c+d \sqrt{x}\right )\right )-\pi \log \left (1+e^{-2 i \left (c+d \sqrt{x}\right )}\right )+\pi \log \left (\cos \left (c+d \sqrt{x}\right )\right )\right )+b \sqrt{\frac{a^2}{b^2}+1} e^{i \tan ^{-1}\left (\frac{a}{b}\right )} \left (c+d \sqrt{x}\right )^2\right )+2 b^2 d \sqrt{x} \left (a^2+b^2\right ) \sin \left (c+d \sqrt{x}\right )-a \left (a^2+b^2\right ) \left (c-d \sqrt{x}\right ) \left (c+d \sqrt{x}\right ) \left (a \cos \left (c+d \sqrt{x}\right )+b \sin \left (c+d \sqrt{x}\right )\right )-2 b^2 \left (a \cos \left (c+d \sqrt{x}\right )+b \sin \left (c+d \sqrt{x}\right )\right ) \left (b \left (c+d \sqrt{x}\right )-a \log \left (a \cos \left (c+d \sqrt{x}\right )+b \sin \left (c+d \sqrt{x}\right )\right )\right )+4 a b c \left (a \cos \left (c+d \sqrt{x}\right )+b \sin \left (c+d \sqrt{x}\right )\right ) \left (b \left (c+d \sqrt{x}\right )-a \log \left (a \cos \left (c+d \sqrt{x}\right )+b \sin \left (c+d \sqrt{x}\right )\right )\right )\right )}{a d^2 \left (a^2+b^2\right )^2 \left (a+b \tan \left (c+d \sqrt{x}\right )\right )^2} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.263, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b\tan \left ( c+d\sqrt{x} \right ) \right ) ^{-2}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 2.8726, size = 1355, normalized size = 6.64 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.68268, size = 2016, normalized size = 9.88 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a + b \tan{\left (c + d \sqrt{x} \right )}\right )^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b \tan \left (d \sqrt{x} + c\right ) + a\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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