Optimal. Leaf size=204 \[ -\frac {2 i a b \text {Li}_2\left (-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt {x}\right )}}{(a+i b)^2}\right )}{d^2 \left (a^2+b^2\right )^2}+\frac {2 b \left (2 a d \sqrt {x}+b\right ) \log \left (1+\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt {x}\right )}}{(a+i b)^2}\right )}{d^2 \left (a^2+b^2\right )^2}-\frac {2 b \sqrt {x}}{d \left (a^2+b^2\right ) \left (a+b \tan \left (c+d \sqrt {x}\right )\right )}+\frac {\left (2 a d \sqrt {x}+b\right )^2}{2 a d^2 (a+i b) \left (a^2+b^2\right )}-\frac {x}{a^2+b^2} \]
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Rubi [A] time = 0.26, antiderivative size = 204, normalized size of antiderivative = 1.00, 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 2190
Rule 2279
Rule 2391
Rule 3732
Rule 3733
Rule 3739
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*}
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Mathematica [B] time = 6.59, size = 772, normalized size = 3.78 \[ -\frac {2 \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 )^2 \left (\frac {a \left (i \text {Li}_2\left (e^{2 i \left (c+\tan ^{-1}\left (\frac {a}{b}\right )+d \sqrt {x}\right )}\right )+i \left (2 \tan ^{-1}\left (\frac {a}{b}\right )-\pi \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 )}{d^2 (a-i b) (a+i b) \sqrt {\frac {a^2+b^2}{b^2}} \left (a+b \tan \left (c+d \sqrt {x}\right )\right )^2}-\frac {4 b c \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 )^2 \left (a \log \left (a \cos \left (c+d \sqrt {x}\right )+b \sin \left (c+d \sqrt {x}\right )\right )-b \left (c+d \sqrt {x}\right )\right )}{d^2 (a-i b) (a+i b) \left (a^2+b^2\right ) \left (a+b \tan \left (c+d \sqrt {x}\right )\right )^2}+\frac {2 b^2 \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 )^2 \left (a \log \left (a \cos \left (c+d \sqrt {x}\right )+b \sin \left (c+d \sqrt {x}\right )\right )-b \left (c+d \sqrt {x}\right )\right )}{a d^2 (a-i b) (a+i b) \left (a^2+b^2\right ) \left (a+b \tan \left (c+d \sqrt {x}\right )\right )^2}+\frac {2 \sec ^2\left (c+d \sqrt {x}\right ) \left (b^2 \left (c+d \sqrt {x}\right ) \sin \left (c+d \sqrt {x}\right )-b^2 c \sin \left (c+d \sqrt {x}\right )\right ) \left (a \cos \left (c+d \sqrt {x}\right )+b \sin \left (c+d \sqrt {x}\right )\right )}{a d^2 (a-i b) (a+i b) \left (a+b \tan \left (c+d \sqrt {x}\right )\right )^2}+\frac {\left (d \sqrt {x}-c\right ) \left (c+d \sqrt {x}\right ) \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 )^2}{d^2 (a-i b) (a+i b) \left (a+b \tan \left (c+d \sqrt {x}\right )\right )^2} \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 0.70, size = 834, normalized size = 4.09 \[ -\frac {2 \, b^{3} d \sqrt {x} - {\left (a^{3} - a b^{2}\right )} d^{2} x + {\left (a^{3} - a b^{2}\right )} d^{2} - {\left (i \, a b^{2} \tan \left (d \sqrt {x} + c\right ) + i \, a^{2} b\right )} {\rm Li}_2\left (\frac {2 \, {\left (i \, a b - b^{2}\right )} \tan \left (d \sqrt {x} + c\right )^{2} - 2 \, a^{2} - 2 i \, a b - {\left (-2 i \, a^{2} + 4 \, a b + 2 i \, b^{2}\right )} \tan \left (d \sqrt {x} + c\right )}{{\left (a^{2} + b^{2}\right )} \tan \left (d \sqrt {x} + c\right )^{2} + a^{2} + b^{2}} + 1\right ) - {\left (-i \, a b^{2} \tan \left (d \sqrt {x} + c\right ) - i \, a^{2} b\right )} {\rm Li}_2\left (\frac {2 \, {\left (-i \, a b - b^{2}\right )} \tan \left (d \sqrt {x} + c\right )^{2} - 2 \, a^{2} + 2 i \, a b - {\left (2 i \, a^{2} + 4 \, a b - 2 i \, b^{2}\right )} \tan \left (d \sqrt {x} + c\right )}{{\left (a^{2} + b^{2}\right )} \tan \left (d \sqrt {x} + c\right )^{2} + a^{2} + b^{2}} + 1\right ) - 2 \, {\left (a^{2} b d \sqrt {x} + a^{2} b c + {\left (a b^{2} d \sqrt {x} + a b^{2} c\right )} \tan \left (d \sqrt {x} + c\right )\right )} \log \left (-\frac {2 \, {\left (i \, a b - b^{2}\right )} \tan \left (d \sqrt {x} + c\right )^{2} - 2 \, a^{2} - 2 i \, a b - {\left (-2 i \, a^{2} + 4 \, a b + 2 i \, b^{2}\right )} \tan \left (d \sqrt {x} + c\right )}{{\left (a^{2} + b^{2}\right )} \tan \left (d \sqrt {x} + c\right )^{2} + a^{2} + b^{2}}\right ) - 2 \, {\left (a^{2} b d \sqrt {x} + a^{2} b c + {\left (a b^{2} d \sqrt {x} + a b^{2} c\right )} \tan \left (d \sqrt {x} + c\right )\right )} \log \left (-\frac {2 \, {\left (-i \, a b - b^{2}\right )} \tan \left (d \sqrt {x} + c\right )^{2} - 2 \, a^{2} + 2 i \, a b - {\left (2 i \, a^{2} + 4 \, a b - 2 i \, b^{2}\right )} \tan \left (d \sqrt {x} + c\right )}{{\left (a^{2} + b^{2}\right )} \tan \left (d \sqrt {x} + c\right )^{2} + a^{2} + b^{2}}\right ) + {\left (2 \, a^{2} b c - a b^{2} + {\left (2 \, a b^{2} c - b^{3}\right )} \tan \left (d \sqrt {x} + c\right )\right )} \log \left (\frac {{\left (i \, a b + b^{2}\right )} \tan \left (d \sqrt {x} + c\right )^{2} - a^{2} + i \, a b + {\left (i \, a^{2} + i \, b^{2}\right )} \tan \left (d \sqrt {x} + c\right )}{\tan \left (d \sqrt {x} + c\right )^{2} + 1}\right ) + {\left (2 \, a^{2} b c - a b^{2} + {\left (2 \, a b^{2} c - b^{3}\right )} \tan \left (d \sqrt {x} + c\right )\right )} \log \left (\frac {{\left (i \, a b - b^{2}\right )} \tan \left (d \sqrt {x} + c\right )^{2} + a^{2} + i \, a b + {\left (i \, a^{2} + i \, b^{2}\right )} \tan \left (d \sqrt {x} + c\right )}{\tan \left (d \sqrt {x} + c\right )^{2} + 1}\right ) - {\left (2 \, a b^{2} d \sqrt {x} + {\left (a^{2} b - b^{3}\right )} d^{2} x - {\left (a^{2} b - b^{3}\right )} d^{2}\right )} \tan \left (d \sqrt {x} + c\right )}{{\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} d^{2} \tan \left (d \sqrt {x} + c\right ) + {\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b \tan \left (d \sqrt {x} + c\right ) + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.27, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (a +b \tan \left (c +d \sqrt {x}\right )\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.44, size = 1003, normalized size = 4.92 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{{\left (a+b\,\mathrm {tan}\left (c+d\,\sqrt {x}\right )\right )}^2} \,d x \]
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 \[ \int \frac {1}{\left (a + b \tan {\left (c + d \sqrt {x} \right )}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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