Optimal. Leaf size=26 \[ 2 a \sqrt {x}+\frac {2 b \tanh ^{-1}\left (\sin \left (c+d \sqrt {x}\right )\right )}{d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {14, 4204, 3770} \[ 2 a \sqrt {x}+\frac {2 b \tanh ^{-1}\left (\sin \left (c+d \sqrt {x}\right )\right )}{d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 14
Rule 3770
Rule 4204
Rubi steps
\begin {align*} \int \frac {a+b \sec \left (c+d \sqrt {x}\right )}{\sqrt {x}} \, dx &=\int \left (\frac {a}{\sqrt {x}}+\frac {b \sec \left (c+d \sqrt {x}\right )}{\sqrt {x}}\right ) \, dx\\ &=2 a \sqrt {x}+b \int \frac {\sec \left (c+d \sqrt {x}\right )}{\sqrt {x}} \, dx\\ &=2 a \sqrt {x}+(2 b) \operatorname {Subst}\left (\int \sec (c+d x) \, dx,x,\sqrt {x}\right )\\ &=2 a \sqrt {x}+\frac {2 b \tanh ^{-1}\left (\sin \left (c+d \sqrt {x}\right )\right )}{d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.08, size = 26, normalized size = 1.00 \[ 2 a \sqrt {x}+\frac {2 b \tanh ^{-1}\left (\sin \left (c+d \sqrt {x}\right )\right )}{d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.66, size = 41, normalized size = 1.58 \[ \frac {2 \, a d \sqrt {x} + b \log \left (\sin \left (d \sqrt {x} + c\right ) + 1\right ) - b \log \left (-\sin \left (d \sqrt {x} + c\right ) + 1\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.85, size = 50, normalized size = 1.92 \[ \frac {2 \, {\left ({\left (d \sqrt {x} + c\right )} a + b \log \left ({\left | \tan \left (\frac {1}{2} \, d \sqrt {x} + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - b \log \left ({\left | \tan \left (\frac {1}{2} \, d \sqrt {x} + \frac {1}{2} \, c\right ) - 1 \right |}\right )\right )}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.26, size = 32, normalized size = 1.23 \[ 2 a \sqrt {x}+\frac {2 b \ln \left (\sec \left (c +d \sqrt {x}\right )+\tan \left (c +d \sqrt {x}\right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.37, size = 31, normalized size = 1.19 \[ 2 \, a \sqrt {x} + \frac {2 \, b \log \left (\sec \left (d \sqrt {x} + c\right ) + \tan \left (d \sqrt {x} + c\right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 2.94, size = 71, normalized size = 2.73 \[ 2\,a\,\sqrt {x}-\frac {2\,b\,\ln \left (\frac {b\,2{}\mathrm {i}-2\,b\,{\mathrm {e}}^{d\,\sqrt {x}\,1{}\mathrm {i}}\,{\mathrm {e}}^{c\,1{}\mathrm {i}}}{\sqrt {x}}\right )}{d}+\frac {2\,b\,\ln \left (\frac {b\,2{}\mathrm {i}+2\,b\,{\mathrm {e}}^{d\,\sqrt {x}\,1{}\mathrm {i}}\,{\mathrm {e}}^{c\,1{}\mathrm {i}}}{\sqrt {x}}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 4.74, size = 56, normalized size = 2.15 \[ \begin {cases} \frac {2 a \left (c + d \sqrt {x}\right ) + 2 b \log {\left (\tan {\left (c + d \sqrt {x} \right )} + \sec {\left (c + d \sqrt {x} \right )} \right )}}{d} & \text {for}\: d \neq 0 \\- \sqrt {x} \left (- 2 a - 2 b \sec {\relax (c )}\right ) & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________