Optimal. Leaf size=415 \[ \frac{2 \sqrt{2} b \sqrt{\cos (c+d x)} \sqrt{e \tan (c+d x)} \Pi \left (-\frac{\sqrt{a-b}}{\sqrt{a+b}};\left .\sin ^{-1}\left (\frac{\sqrt{\sin (c+d x)}}{\sqrt{\cos (c+d x)+1}}\right )\right |-1\right )}{a d \sqrt{a-b} \sqrt{a+b} \sqrt{\sin (c+d x)}}-\frac{2 \sqrt{2} b \sqrt{\cos (c+d x)} \sqrt{e \tan (c+d x)} \Pi \left (\frac{\sqrt{a-b}}{\sqrt{a+b}};\left .\sin ^{-1}\left (\frac{\sqrt{\sin (c+d x)}}{\sqrt{\cos (c+d x)+1}}\right )\right |-1\right )}{a d \sqrt{a-b} \sqrt{a+b} \sqrt{\sin (c+d x)}}-\frac{\sqrt{e} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right )}{\sqrt{2} a d}+\frac{\sqrt{e} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}+1\right )}{\sqrt{2} a d}+\frac{\sqrt{e} \log \left (\sqrt{e} \tan (c+d x)-\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right )}{2 \sqrt{2} a d}-\frac{\sqrt{e} \log \left (\sqrt{e} \tan (c+d x)+\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right )}{2 \sqrt{2} a d} \]
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Rubi [A] time = 0.704931, antiderivative size = 415, normalized size of antiderivative = 1., number of steps used = 21, number of rules used = 16, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.64, Rules used = {3890, 3476, 329, 297, 1162, 617, 204, 1165, 628, 2733, 2730, 2906, 2905, 490, 1213, 537} \[ \frac{2 \sqrt{2} b \sqrt{\cos (c+d x)} \sqrt{e \tan (c+d x)} \Pi \left (-\frac{\sqrt{a-b}}{\sqrt{a+b}};\left .\sin ^{-1}\left (\frac{\sqrt{\sin (c+d x)}}{\sqrt{\cos (c+d x)+1}}\right )\right |-1\right )}{a d \sqrt{a-b} \sqrt{a+b} \sqrt{\sin (c+d x)}}-\frac{2 \sqrt{2} b \sqrt{\cos (c+d x)} \sqrt{e \tan (c+d x)} \Pi \left (\frac{\sqrt{a-b}}{\sqrt{a+b}};\left .\sin ^{-1}\left (\frac{\sqrt{\sin (c+d x)}}{\sqrt{\cos (c+d x)+1}}\right )\right |-1\right )}{a d \sqrt{a-b} \sqrt{a+b} \sqrt{\sin (c+d x)}}-\frac{\sqrt{e} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right )}{\sqrt{2} a d}+\frac{\sqrt{e} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}+1\right )}{\sqrt{2} a d}+\frac{\sqrt{e} \log \left (\sqrt{e} \tan (c+d x)-\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right )}{2 \sqrt{2} a d}-\frac{\sqrt{e} \log \left (\sqrt{e} \tan (c+d x)+\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right )}{2 \sqrt{2} a d} \]
Antiderivative was successfully verified.
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Rule 3890
Rule 3476
Rule 329
Rule 297
Rule 1162
Rule 617
Rule 204
Rule 1165
Rule 628
Rule 2733
Rule 2730
Rule 2906
Rule 2905
Rule 490
Rule 1213
Rule 537
Rubi steps
\begin{align*} \int \frac{\sqrt{e \tan (c+d x)}}{a+b \sec (c+d x)} \, dx &=\frac{\int \sqrt{e \tan (c+d x)} \, dx}{a}-\frac{b \int \frac{\sqrt{e \tan (c+d x)}}{b+a \cos (c+d x)} \, dx}{a}\\ &=\frac{e \operatorname{Subst}\left (\int \frac{\sqrt{x}}{e^2+x^2} \, dx,x,e \tan (c+d x)\right )}{a d}-\frac{\left (b \sqrt{e \cot (c+d x)} \sqrt{e \tan (c+d x)}\right ) \int \frac{1}{(b+a \cos (c+d x)) \sqrt{e \cot (c+d x)}} \, dx}{a}\\ &=\frac{(2 e) \operatorname{Subst}\left (\int \frac{x^2}{e^2+x^4} \, dx,x,\sqrt{e \tan (c+d x)}\right )}{a d}-\frac{\left (b \sqrt{-\cos (c+d x)} \sqrt{e \tan (c+d x)}\right ) \int \frac{\sqrt{\sin (c+d x)}}{\sqrt{-\cos (c+d x)} (b+a \cos (c+d x))} \, dx}{a \sqrt{\sin (c+d x)}}\\ &=-\frac{e \operatorname{Subst}\left (\int \frac{e-x^2}{e^2+x^4} \, dx,x,\sqrt{e \tan (c+d x)}\right )}{a d}+\frac{e \operatorname{Subst}\left (\int \frac{e+x^2}{e^2+x^4} \, dx,x,\sqrt{e \tan (c+d x)}\right )}{a d}-\frac{\left (b \sqrt{\cos (c+d x)} \sqrt{e \tan (c+d x)}\right ) \int \frac{\sqrt{\sin (c+d x)}}{\sqrt{\cos (c+d x)} (b+a \cos (c+d x))} \, dx}{a \sqrt{\sin (c+d x)}}\\ &=\frac{\sqrt{e} \operatorname{Subst}\left (\int \frac{\sqrt{2} \sqrt{e}+2 x}{-e-\sqrt{2} \sqrt{e} x-x^2} \, dx,x,\sqrt{e \tan (c+d x)}\right )}{2 \sqrt{2} a d}+\frac{\sqrt{e} \operatorname{Subst}\left (\int \frac{\sqrt{2} \sqrt{e}-2 x}{-e+\sqrt{2} \sqrt{e} x-x^2} \, dx,x,\sqrt{e \tan (c+d x)}\right )}{2 \sqrt{2} a d}+\frac{e \operatorname{Subst}\left (\int \frac{1}{e-\sqrt{2} \sqrt{e} x+x^2} \, dx,x,\sqrt{e \tan (c+d x)}\right )}{2 a d}+\frac{e \operatorname{Subst}\left (\int \frac{1}{e+\sqrt{2} \sqrt{e} x+x^2} \, dx,x,\sqrt{e \tan (c+d x)}\right )}{2 a d}-\frac{\left (4 \sqrt{2} b \sqrt{\cos (c+d x)} \sqrt{e \tan (c+d x)}\right ) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{1-x^4} \left (a+b+(-a+b) x^4\right )} \, dx,x,\frac{\sqrt{\sin (c+d x)}}{\sqrt{1+\cos (c+d x)}}\right )}{a d \sqrt{\sin (c+d x)}}\\ &=\frac{\sqrt{e} \log \left (\sqrt{e}+\sqrt{e} \tan (c+d x)-\sqrt{2} \sqrt{e \tan (c+d x)}\right )}{2 \sqrt{2} a d}-\frac{\sqrt{e} \log \left (\sqrt{e}+\sqrt{e} \tan (c+d x)+\sqrt{2} \sqrt{e \tan (c+d x)}\right )}{2 \sqrt{2} a d}+\frac{\sqrt{e} \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right )}{\sqrt{2} a d}-\frac{\sqrt{e} \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right )}{\sqrt{2} a d}-\frac{\left (2 \sqrt{2} b \sqrt{\cos (c+d x)} \sqrt{e \tan (c+d x)}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (\sqrt{a+b}-\sqrt{a-b} x^2\right ) \sqrt{1-x^4}} \, dx,x,\frac{\sqrt{\sin (c+d x)}}{\sqrt{1+\cos (c+d x)}}\right )}{a \sqrt{a-b} d \sqrt{\sin (c+d x)}}+\frac{\left (2 \sqrt{2} b \sqrt{\cos (c+d x)} \sqrt{e \tan (c+d x)}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (\sqrt{a+b}+\sqrt{a-b} x^2\right ) \sqrt{1-x^4}} \, dx,x,\frac{\sqrt{\sin (c+d x)}}{\sqrt{1+\cos (c+d x)}}\right )}{a \sqrt{a-b} d \sqrt{\sin (c+d x)}}\\ &=-\frac{\sqrt{e} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right )}{\sqrt{2} a d}+\frac{\sqrt{e} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right )}{\sqrt{2} a d}+\frac{\sqrt{e} \log \left (\sqrt{e}+\sqrt{e} \tan (c+d x)-\sqrt{2} \sqrt{e \tan (c+d x)}\right )}{2 \sqrt{2} a d}-\frac{\sqrt{e} \log \left (\sqrt{e}+\sqrt{e} \tan (c+d x)+\sqrt{2} \sqrt{e \tan (c+d x)}\right )}{2 \sqrt{2} a d}-\frac{\left (2 \sqrt{2} b \sqrt{\cos (c+d x)} \sqrt{e \tan (c+d x)}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x^2} \sqrt{1+x^2} \left (\sqrt{a+b}-\sqrt{a-b} x^2\right )} \, dx,x,\frac{\sqrt{\sin (c+d x)}}{\sqrt{1+\cos (c+d x)}}\right )}{a \sqrt{a-b} d \sqrt{\sin (c+d x)}}+\frac{\left (2 \sqrt{2} b \sqrt{\cos (c+d x)} \sqrt{e \tan (c+d x)}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x^2} \sqrt{1+x^2} \left (\sqrt{a+b}+\sqrt{a-b} x^2\right )} \, dx,x,\frac{\sqrt{\sin (c+d x)}}{\sqrt{1+\cos (c+d x)}}\right )}{a \sqrt{a-b} d \sqrt{\sin (c+d x)}}\\ &=-\frac{\sqrt{e} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right )}{\sqrt{2} a d}+\frac{\sqrt{e} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right )}{\sqrt{2} a d}+\frac{\sqrt{e} \log \left (\sqrt{e}+\sqrt{e} \tan (c+d x)-\sqrt{2} \sqrt{e \tan (c+d x)}\right )}{2 \sqrt{2} a d}-\frac{\sqrt{e} \log \left (\sqrt{e}+\sqrt{e} \tan (c+d x)+\sqrt{2} \sqrt{e \tan (c+d x)}\right )}{2 \sqrt{2} a d}+\frac{2 \sqrt{2} b \sqrt{\cos (c+d x)} \Pi \left (-\frac{\sqrt{a-b}}{\sqrt{a+b}};\left .\sin ^{-1}\left (\frac{\sqrt{\sin (c+d x)}}{\sqrt{1+\cos (c+d x)}}\right )\right |-1\right ) \sqrt{e \tan (c+d x)}}{a \sqrt{a-b} \sqrt{a+b} d \sqrt{\sin (c+d x)}}-\frac{2 \sqrt{2} b \sqrt{\cos (c+d x)} \Pi \left (\frac{\sqrt{a-b}}{\sqrt{a+b}};\left .\sin ^{-1}\left (\frac{\sqrt{\sin (c+d x)}}{\sqrt{1+\cos (c+d x)}}\right )\right |-1\right ) \sqrt{e \tan (c+d x)}}{a \sqrt{a-b} \sqrt{a+b} d \sqrt{\sin (c+d x)}}\\ \end{align*}
Mathematica [C] time = 5.27994, size = 232, normalized size = 0.56 \[ -\frac{4 \sqrt{\tan \left (\frac{1}{2} (c+d x)\right )} \csc (c+d x) \sqrt{e \tan (c+d x)} (a \cos (c+d x)+b) \left (\frac{b \left (\Pi \left (-\frac{\sqrt{a-b}}{\sqrt{a+b}};\left .-\sin ^{-1}\left (\sqrt{\tan \left (\frac{1}{2} (c+d x)\right )}\right )\right |-1\right )-\Pi \left (\frac{\sqrt{a-b}}{\sqrt{a+b}};\left .-\sin ^{-1}\left (\sqrt{\tan \left (\frac{1}{2} (c+d x)\right )}\right )\right |-1\right )\right )}{\sqrt{a-b} \sqrt{a+b}}+i \Pi \left (-i;\left .-\sin ^{-1}\left (\sqrt{\tan \left (\frac{1}{2} (c+d x)\right )}\right )\right |-1\right )-i \Pi \left (i;\left .-\sin ^{-1}\left (\sqrt{\tan \left (\frac{1}{2} (c+d x)\right )}\right )\right |-1\right )\right )}{a d \sqrt{\cos (c+d x) \sec ^2\left (\frac{1}{2} (c+d x)\right )} (a+b \sec (c+d x))} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.254, size = 874, normalized size = 2.1 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{e \tan \left (d x + c\right )}}{b \sec \left (d x + c\right ) + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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{\sqrt{e \tan{\left (c + d x \right )}}}{a + b \sec{\left (c + d x \right )}}\, 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{\sqrt{e \tan \left (d x + c\right )}}{b \sec \left (d x + c\right ) + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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