Optimal. Leaf size=650 \[ -\frac{\sqrt [4]{b} (a+b) \left (\sqrt{a}+\sqrt{b} \tan ^2(c+d x)\right ) \sqrt{\frac{a+b \tan ^4(c+d x)}{\left (\sqrt{a}+\sqrt{b} \tan ^2(c+d x)\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \tan (c+d x)}{\sqrt [4]{a}}\right ),\frac{1}{2}\right )}{2 \sqrt [4]{a} d \left (\sqrt{a}-\sqrt{b}\right ) \sqrt{a+b \tan ^4(c+d x)}}+\frac{\sqrt [4]{b} \left (\sqrt{a}-\sqrt{b}\right ) \left (\sqrt{a}+\sqrt{b} \tan ^2(c+d x)\right ) \sqrt{\frac{a+b \tan ^4(c+d x)}{\left (\sqrt{a}+\sqrt{b} \tan ^2(c+d x)\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \tan (c+d x)}{\sqrt [4]{a}}\right ),\frac{1}{2}\right )}{2 \sqrt [4]{a} d \sqrt{a+b \tan ^4(c+d x)}}+\frac{\sqrt{a+b} \tan ^{-1}\left (\frac{\sqrt{a+b} \tan (c+d x)}{\sqrt{a+b \tan ^4(c+d x)}}\right )}{2 d}+\frac{\sqrt{b} \tan (c+d x) \sqrt{a+b \tan ^4(c+d x)}}{d \left (\sqrt{a}+\sqrt{b} \tan ^2(c+d x)\right )}-\frac{\sqrt [4]{a} \sqrt [4]{b} \left (\sqrt{a}+\sqrt{b} \tan ^2(c+d x)\right ) \sqrt{\frac{a+b \tan ^4(c+d x)}{\left (\sqrt{a}+\sqrt{b} \tan ^2(c+d x)\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \tan (c+d x)}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{d \sqrt{a+b \tan ^4(c+d x)}}+\frac{\left (\sqrt{a}+\sqrt{b}\right ) (a+b) \left (\sqrt{a}+\sqrt{b} \tan ^2(c+d x)\right ) \sqrt{\frac{a+b \tan ^4(c+d x)}{\left (\sqrt{a}+\sqrt{b} \tan ^2(c+d x)\right )^2}} \Pi \left (-\frac{\left (\sqrt{a}-\sqrt{b}\right )^2}{4 \sqrt{a} \sqrt{b}};2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \tan (c+d x)}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{4 \sqrt [4]{a} \sqrt [4]{b} d \left (\sqrt{a}-\sqrt{b}\right ) \sqrt{a+b \tan ^4(c+d x)}} \]
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Rubi [A] time = 0.559916, antiderivative size = 650, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.438, Rules used = {3661, 1209, 1198, 220, 1196, 1217, 1707} \[ \frac{\sqrt{a+b} \tan ^{-1}\left (\frac{\sqrt{a+b} \tan (c+d x)}{\sqrt{a+b \tan ^4(c+d x)}}\right )}{2 d}+\frac{\sqrt{b} \tan (c+d x) \sqrt{a+b \tan ^4(c+d x)}}{d \left (\sqrt{a}+\sqrt{b} \tan ^2(c+d x)\right )}-\frac{\sqrt [4]{b} (a+b) \left (\sqrt{a}+\sqrt{b} \tan ^2(c+d x)\right ) \sqrt{\frac{a+b \tan ^4(c+d x)}{\left (\sqrt{a}+\sqrt{b} \tan ^2(c+d x)\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \tan (c+d x)}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{2 \sqrt [4]{a} d \left (\sqrt{a}-\sqrt{b}\right ) \sqrt{a+b \tan ^4(c+d x)}}+\frac{\sqrt [4]{b} \left (\sqrt{a}-\sqrt{b}\right ) \left (\sqrt{a}+\sqrt{b} \tan ^2(c+d x)\right ) \sqrt{\frac{a+b \tan ^4(c+d x)}{\left (\sqrt{a}+\sqrt{b} \tan ^2(c+d x)\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \tan (c+d x)}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{2 \sqrt [4]{a} d \sqrt{a+b \tan ^4(c+d x)}}-\frac{\sqrt [4]{a} \sqrt [4]{b} \left (\sqrt{a}+\sqrt{b} \tan ^2(c+d x)\right ) \sqrt{\frac{a+b \tan ^4(c+d x)}{\left (\sqrt{a}+\sqrt{b} \tan ^2(c+d x)\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \tan (c+d x)}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{d \sqrt{a+b \tan ^4(c+d x)}}+\frac{\left (\sqrt{a}+\sqrt{b}\right ) (a+b) \left (\sqrt{a}+\sqrt{b} \tan ^2(c+d x)\right ) \sqrt{\frac{a+b \tan ^4(c+d x)}{\left (\sqrt{a}+\sqrt{b} \tan ^2(c+d x)\right )^2}} \Pi \left (-\frac{\left (\sqrt{a}-\sqrt{b}\right )^2}{4 \sqrt{a} \sqrt{b}};2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \tan (c+d x)}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{4 \sqrt [4]{a} \sqrt [4]{b} d \left (\sqrt{a}-\sqrt{b}\right ) \sqrt{a+b \tan ^4(c+d x)}} \]
Antiderivative was successfully verified.
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Rule 3661
Rule 1209
Rule 1198
Rule 220
Rule 1196
Rule 1217
Rule 1707
Rubi steps
\begin{align*} \int \sqrt{a+b \tan ^4(c+d x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\sqrt{a+b x^4}}{1+x^2} \, dx,x,\tan (c+d x)\right )}{d}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{b-b x^2}{\sqrt{a+b x^4}} \, dx,x,\tan (c+d x)\right )}{d}+\frac{(a+b) \operatorname{Subst}\left (\int \frac{1}{\left (1+x^2\right ) \sqrt{a+b x^4}} \, dx,x,\tan (c+d x)\right )}{d}\\ &=-\frac{\left (\sqrt{a} \sqrt{b}\right ) \operatorname{Subst}\left (\int \frac{1-\frac{\sqrt{b} x^2}{\sqrt{a}}}{\sqrt{a+b x^4}} \, dx,x,\tan (c+d x)\right )}{d}+\frac{\left (\left (\sqrt{a}-\sqrt{b}\right ) \sqrt{b}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x^4}} \, dx,x,\tan (c+d x)\right )}{d}+\frac{\left (\sqrt{a} (a+b)\right ) \operatorname{Subst}\left (\int \frac{1+\frac{\sqrt{b} x^2}{\sqrt{a}}}{\left (1+x^2\right ) \sqrt{a+b x^4}} \, dx,x,\tan (c+d x)\right )}{\left (\sqrt{a}-\sqrt{b}\right ) d}-\frac{\left (\sqrt{b} (a+b)\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x^4}} \, dx,x,\tan (c+d x)\right )}{\left (\sqrt{a}-\sqrt{b}\right ) d}\\ &=\frac{\sqrt{a+b} \tan ^{-1}\left (\frac{\sqrt{a+b} \tan (c+d x)}{\sqrt{a+b \tan ^4(c+d x)}}\right )}{2 d}+\frac{\sqrt{b} \tan (c+d x) \sqrt{a+b \tan ^4(c+d x)}}{d \left (\sqrt{a}+\sqrt{b} \tan ^2(c+d x)\right )}-\frac{\sqrt [4]{a} \sqrt [4]{b} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \tan (c+d x)}{\sqrt [4]{a}}\right )|\frac{1}{2}\right ) \left (\sqrt{a}+\sqrt{b} \tan ^2(c+d x)\right ) \sqrt{\frac{a+b \tan ^4(c+d x)}{\left (\sqrt{a}+\sqrt{b} \tan ^2(c+d x)\right )^2}}}{d \sqrt{a+b \tan ^4(c+d x)}}+\frac{\left (\sqrt{a}-\sqrt{b}\right ) \sqrt [4]{b} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \tan (c+d x)}{\sqrt [4]{a}}\right )|\frac{1}{2}\right ) \left (\sqrt{a}+\sqrt{b} \tan ^2(c+d x)\right ) \sqrt{\frac{a+b \tan ^4(c+d x)}{\left (\sqrt{a}+\sqrt{b} \tan ^2(c+d x)\right )^2}}}{2 \sqrt [4]{a} d \sqrt{a+b \tan ^4(c+d x)}}-\frac{\sqrt [4]{b} (a+b) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \tan (c+d x)}{\sqrt [4]{a}}\right )|\frac{1}{2}\right ) \left (\sqrt{a}+\sqrt{b} \tan ^2(c+d x)\right ) \sqrt{\frac{a+b \tan ^4(c+d x)}{\left (\sqrt{a}+\sqrt{b} \tan ^2(c+d x)\right )^2}}}{2 \sqrt [4]{a} \left (\sqrt{a}-\sqrt{b}\right ) d \sqrt{a+b \tan ^4(c+d x)}}+\frac{\left (\sqrt{a}+\sqrt{b}\right ) (a+b) \Pi \left (-\frac{\left (\sqrt{a}-\sqrt{b}\right )^2}{4 \sqrt{a} \sqrt{b}};2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \tan (c+d x)}{\sqrt [4]{a}}\right )|\frac{1}{2}\right ) \left (\sqrt{a}+\sqrt{b} \tan ^2(c+d x)\right ) \sqrt{\frac{a+b \tan ^4(c+d x)}{\left (\sqrt{a}+\sqrt{b} \tan ^2(c+d x)\right )^2}}}{4 \sqrt [4]{a} \left (\sqrt{a}-\sqrt{b}\right ) \sqrt [4]{b} d \sqrt{a+b \tan ^4(c+d x)}}\\ \end{align*}
Mathematica [C] time = 0.725444, size = 219, normalized size = 0.34 \[ \frac{\sqrt{\frac{b \tan ^4(c+d x)}{a}+1} \left (\left (\sqrt{a}-i \sqrt{b}\right ) \left (\left (\sqrt{b}-i \sqrt{a}\right ) \Pi \left (-\frac{i \sqrt{a}}{\sqrt{b}};\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} \tan (c+d x)\right )\right |-1\right )-\sqrt{b} \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} \tan (c+d x)\right ),-1\right )\right )+\sqrt{a} \sqrt{b} E\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} \tan (c+d x)\right )\right |-1\right )\right )}{d \sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} \sqrt{a+b \tan ^4(c+d x)}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.12, size = 531, normalized size = 0.8 \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 \sqrt{b \tan \left (d x + c\right )^{4} + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\sqrt{b \tan \left (d x + c\right )^{4} + a}, x\right ) \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 \sqrt{a + b \tan ^{4}{\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 \sqrt{b \tan \left (d x + c\right )^{4} + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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