Optimal. Leaf size=119 \[ a^2 x+\frac {2 i a b \text {Li}_2\left (-e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {4 a b \sqrt {x} \log \left (1+e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d}+2 i a b x+\frac {2 b^2 \log \left (\cos \left (c+d \sqrt {x}\right )\right )}{d^2}+\frac {2 b^2 \sqrt {x} \tan \left (c+d \sqrt {x}\right )}{d}-b^2 x \]
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Rubi [A] time = 0.18, antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.562, Rules used = {3739, 3722, 3719, 2190, 2279, 2391, 3720, 3475, 30} \[ a^2 x+\frac {2 i a b \text {Li}_2\left (-e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {4 a b \sqrt {x} \log \left (1+e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d}+2 i a b x+\frac {2 b^2 \log \left (\cos \left (c+d \sqrt {x}\right )\right )}{d^2}+\frac {2 b^2 \sqrt {x} \tan \left (c+d \sqrt {x}\right )}{d}-b^2 x \]
Antiderivative was successfully verified.
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Rule 30
Rule 2190
Rule 2279
Rule 2391
Rule 3475
Rule 3719
Rule 3720
Rule 3722
Rule 3739
Rubi steps
\begin {align*} \int \left (a+b \tan \left (c+d \sqrt {x}\right )\right )^2 \, dx &=2 \operatorname {Subst}\left (\int x (a+b \tan (c+d x))^2 \, dx,x,\sqrt {x}\right )\\ &=2 \operatorname {Subst}\left (\int \left (a^2 x+2 a b x \tan (c+d x)+b^2 x \tan ^2(c+d x)\right ) \, dx,x,\sqrt {x}\right )\\ &=a^2 x+(4 a b) \operatorname {Subst}\left (\int x \tan (c+d x) \, dx,x,\sqrt {x}\right )+\left (2 b^2\right ) \operatorname {Subst}\left (\int x \tan ^2(c+d x) \, dx,x,\sqrt {x}\right )\\ &=a^2 x+2 i a b x+\frac {2 b^2 \sqrt {x} \tan \left (c+d \sqrt {x}\right )}{d}-(8 i a b) \operatorname {Subst}\left (\int \frac {e^{2 i (c+d x)} x}{1+e^{2 i (c+d x)}} \, dx,x,\sqrt {x}\right )-\left (2 b^2\right ) \operatorname {Subst}\left (\int x \, dx,x,\sqrt {x}\right )-\frac {\left (2 b^2\right ) \operatorname {Subst}\left (\int \tan (c+d x) \, dx,x,\sqrt {x}\right )}{d}\\ &=a^2 x+2 i a b x-b^2 x-\frac {4 a b \sqrt {x} \log \left (1+e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d}+\frac {2 b^2 \log \left (\cos \left (c+d \sqrt {x}\right )\right )}{d^2}+\frac {2 b^2 \sqrt {x} \tan \left (c+d \sqrt {x}\right )}{d}+\frac {(4 a b) \operatorname {Subst}\left (\int \log \left (1+e^{2 i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d}\\ &=a^2 x+2 i a b x-b^2 x-\frac {4 a b \sqrt {x} \log \left (1+e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d}+\frac {2 b^2 \log \left (\cos \left (c+d \sqrt {x}\right )\right )}{d^2}+\frac {2 b^2 \sqrt {x} \tan \left (c+d \sqrt {x}\right )}{d}-\frac {(2 i a b) \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^2}\\ &=a^2 x+2 i a b x-b^2 x-\frac {4 a b \sqrt {x} \log \left (1+e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d}+\frac {2 b^2 \log \left (\cos \left (c+d \sqrt {x}\right )\right )}{d^2}+\frac {2 i a b \text {Li}_2\left (-e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^2}+\frac {2 b^2 \sqrt {x} \tan \left (c+d \sqrt {x}\right )}{d}\\ \end {align*}
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Mathematica [B] time = 6.33, size = 308, normalized size = 2.59 \[ x \sec (c) \left (a^2 \cos (c)+2 a b \sin (c)-b^2 \cos (c)\right )-\frac {2 a b \csc (c) \sec (c) \left (d^2 x e^{-i \tan ^{-1}(\cot (c))}-\frac {\cot (c) \left (i \text {Li}_2\left (e^{2 i \left (d \sqrt {x}-\tan ^{-1}(\cot (c))\right )}\right )+i d \sqrt {x} \left (-2 \tan ^{-1}(\cot (c))-\pi \right )-2 \left (d \sqrt {x}-\tan ^{-1}(\cot (c))\right ) \log \left (1-e^{2 i \left (d \sqrt {x}-\tan ^{-1}(\cot (c))\right )}\right )-2 \tan ^{-1}(\cot (c)) \log \left (\sin \left (d \sqrt {x}-\tan ^{-1}(\cot (c))\right )\right )-\pi \log \left (1+e^{-2 i d \sqrt {x}}\right )+\pi \log \left (\cos \left (d \sqrt {x}\right )\right )\right )}{\sqrt {\cot ^2(c)+1}}\right )}{d^2 \sqrt {\csc ^2(c) \left (\sin ^2(c)+\cos ^2(c)\right )}}+\frac {2 b^2 \sec (c) \left (d \sqrt {x} \sin (c)+\cos (c) \log \left (\cos (c) \cos \left (d \sqrt {x}\right )-\sin (c) \sin \left (d \sqrt {x}\right )\right )\right )}{d^2 \left (\sin ^2(c)+\cos ^2(c)\right )}+\frac {2 b^2 \sqrt {x} \sec (c) \sin \left (d \sqrt {x}\right ) \sec \left (c+d \sqrt {x}\right )}{d} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.59, size = 196, normalized size = 1.65 \[ \frac {2 \, b^{2} d \sqrt {x} \tan \left (d \sqrt {x} + c\right ) + {\left (a^{2} - b^{2}\right )} d^{2} x - i \, a b {\rm Li}_2\left (\frac {2 \, {\left (i \, \tan \left (d \sqrt {x} + c\right ) - 1\right )}}{\tan \left (d \sqrt {x} + c\right )^{2} + 1} + 1\right ) + i \, a b {\rm Li}_2\left (\frac {2 \, {\left (-i \, \tan \left (d \sqrt {x} + c\right ) - 1\right )}}{\tan \left (d \sqrt {x} + c\right )^{2} + 1} + 1\right ) - {\left (2 \, a b d \sqrt {x} - b^{2}\right )} \log \left (-\frac {2 \, {\left (i \, \tan \left (d \sqrt {x} + c\right ) - 1\right )}}{\tan \left (d \sqrt {x} + c\right )^{2} + 1}\right ) - {\left (2 \, a b d \sqrt {x} - b^{2}\right )} \log \left (-\frac {2 \, {\left (-i \, \tan \left (d \sqrt {x} + c\right ) - 1\right )}}{\tan \left (d \sqrt {x} + c\right )^{2} + 1}\right )}{d^{2}} \]
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 \[ \int {\left (b \tan \left (d \sqrt {x} + c\right ) + a\right )}^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.23, size = 0, normalized size = 0.00 \[ \int \left (a +b \tan \left (c +d \sqrt {x}\right )\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.29, size = 498, normalized size = 4.18 \[ a^{2} x + \frac {4 \, b^{2} d \sqrt {x} + 4 \, {\left (a b \cos \left (2 \, d \sqrt {x} + 2 \, c\right ) + i \, a b \sin \left (2 \, d \sqrt {x} + 2 \, c\right ) + a b\right )} \arctan \left (\sin \left (2 \, d \sqrt {x} - 2 \, c\right ), \cos \left (2 \, d \sqrt {x} - 2 \, c\right ) + 1\right ) \arctan \left (\sin \left (d \sqrt {x}\right ), \cos \left (d \sqrt {x}\right )\right ) + {\left (-2 i \, a b \cos \left (2 \, d \sqrt {x} + 2 \, c\right ) + 2 \, a b \sin \left (2 \, d \sqrt {x} + 2 \, c\right ) - 2 i \, a b\right )} \arctan \left (\sin \left (d \sqrt {x}\right ), \cos \left (d \sqrt {x}\right )\right ) \log \left (\cos \left (2 \, d \sqrt {x} - 2 \, c\right )^{2} + \sin \left (2 \, d \sqrt {x} - 2 \, c\right )^{2} + 2 \, \cos \left (2 \, d \sqrt {x} - 2 \, c\right ) + 1\right ) - {\left ({\left (2 \, a b - i \, b^{2}\right )} d^{2} \cos \left (2 \, d \sqrt {x} + 2 \, c\right ) - {\left (-2 i \, a b - b^{2}\right )} d^{2} \sin \left (2 \, d \sqrt {x} + 2 \, c\right ) + {\left (2 \, a b - i \, b^{2}\right )} d^{2}\right )} x + {\left (2 \, b^{2} \cos \left (2 \, d \sqrt {x} + 2 \, c\right ) + 2 i \, b^{2} \sin \left (2 \, d \sqrt {x} + 2 \, c\right ) + 2 \, b^{2}\right )} \arctan \left (\sin \left (2 \, d \sqrt {x}\right ) + \sin \left (2 \, c\right ), \cos \left (2 \, d \sqrt {x}\right ) + \cos \left (2 \, c\right )\right ) - 2 \, {\left (a b \cos \left (2 \, d \sqrt {x} + 2 \, c\right ) + i \, a b \sin \left (2 \, d \sqrt {x} + 2 \, c\right ) + a b\right )} {\rm Li}_2\left (-e^{\left (2 i \, d \sqrt {x} - 2 i \, c\right )}\right ) + {\left (-i \, b^{2} \cos \left (2 \, d \sqrt {x} + 2 \, c\right ) + b^{2} \sin \left (2 \, d \sqrt {x} + 2 \, c\right ) - i \, b^{2}\right )} \log \left (\cos \left (2 \, d \sqrt {x}\right )^{2} + 2 \, \cos \left (2 \, d \sqrt {x}\right ) \cos \left (2 \, c\right ) + \cos \left (2 \, c\right )^{2} + \sin \left (2 \, d \sqrt {x}\right )^{2} + 2 \, \sin \left (2 \, d \sqrt {x}\right ) \sin \left (2 \, c\right ) + \sin \left (2 \, c\right )^{2}\right )}{-i \, d^{2} \cos \left (2 \, d \sqrt {x} + 2 \, c\right ) + d^{2} \sin \left (2 \, d \sqrt {x} + 2 \, c\right ) - i \, d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (a+b\,\mathrm {tan}\left (c+d\,\sqrt {x}\right )\right )}^2 \,d x \]
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 \[ \int \left (a + b \tan {\left (c + d \sqrt {x} \right )}\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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