Integrand size = 16, antiderivative size = 650 \[ \int \sqrt {a+b \tan ^4(c+d x)} \, dx=\frac {\sqrt {a+b} \arctan \left (\frac {\sqrt {a+b} \tan (c+d x)}{\sqrt {a+b \tan ^4(c+d x)}}\right )}{2 d}+\frac {\sqrt {b} \tan (c+d x) \sqrt {a+b \tan ^4(c+d x)}}{d \left (\sqrt {a}+\sqrt {b} \tan ^2(c+d x)\right )}-\frac {\sqrt [4]{a} \sqrt [4]{b} E\left (2 \arctan \left (\frac {\sqrt [4]{b} \tan (c+d x)}{\sqrt [4]{a}}\right )|\frac {1}{2}\right ) \left (\sqrt {a}+\sqrt {b} \tan ^2(c+d x)\right ) \sqrt {\frac {a+b \tan ^4(c+d x)}{\left (\sqrt {a}+\sqrt {b} \tan ^2(c+d x)\right )^2}}}{d \sqrt {a+b \tan ^4(c+d x)}}+\frac {\left (\sqrt {a}-\sqrt {b}\right ) \sqrt [4]{b} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \tan (c+d x)}{\sqrt [4]{a}}\right ),\frac {1}{2}\right ) \left (\sqrt {a}+\sqrt {b} \tan ^2(c+d x)\right ) \sqrt {\frac {a+b \tan ^4(c+d x)}{\left (\sqrt {a}+\sqrt {b} \tan ^2(c+d x)\right )^2}}}{2 \sqrt [4]{a} d \sqrt {a+b \tan ^4(c+d x)}}-\frac {\sqrt [4]{b} (a+b) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \tan (c+d x)}{\sqrt [4]{a}}\right ),\frac {1}{2}\right ) \left (\sqrt {a}+\sqrt {b} \tan ^2(c+d x)\right ) \sqrt {\frac {a+b \tan ^4(c+d x)}{\left (\sqrt {a}+\sqrt {b} \tan ^2(c+d x)\right )^2}}}{2 \sqrt [4]{a} \left (\sqrt {a}-\sqrt {b}\right ) d \sqrt {a+b \tan ^4(c+d x)}}+\frac {\left (\sqrt {a}+\sqrt {b}\right ) (a+b) \operatorname {EllipticPi}\left (-\frac {\left (\sqrt {a}-\sqrt {b}\right )^2}{4 \sqrt {a} \sqrt {b}},2 \arctan \left (\frac {\sqrt [4]{b} \tan (c+d x)}{\sqrt [4]{a}}\right ),\frac {1}{2}\right ) \left (\sqrt {a}+\sqrt {b} \tan ^2(c+d x)\right ) \sqrt {\frac {a+b \tan ^4(c+d x)}{\left (\sqrt {a}+\sqrt {b} \tan ^2(c+d x)\right )^2}}}{4 \sqrt [4]{a} \left (\sqrt {a}-\sqrt {b}\right ) \sqrt [4]{b} d \sqrt {a+b \tan ^4(c+d x)}} \]
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Time = 0.60 (sec) , antiderivative size = 650, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {3742, 1223, 1212, 226, 1210, 1231, 1721} \[ \int \sqrt {a+b \tan ^4(c+d x)} \, dx=\frac {\sqrt {a+b} \arctan \left (\frac {\sqrt {a+b} \tan (c+d x)}{\sqrt {a+b \tan ^4(c+d x)}}\right )}{2 d}-\frac {\sqrt [4]{b} (a+b) \left (\sqrt {a}+\sqrt {b} \tan ^2(c+d x)\right ) \sqrt {\frac {a+b \tan ^4(c+d x)}{\left (\sqrt {a}+\sqrt {b} \tan ^2(c+d x)\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \tan (c+d x)}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{a} d \left (\sqrt {a}-\sqrt {b}\right ) \sqrt {a+b \tan ^4(c+d x)}}+\frac {\sqrt [4]{b} \left (\sqrt {a}-\sqrt {b}\right ) \left (\sqrt {a}+\sqrt {b} \tan ^2(c+d x)\right ) \sqrt {\frac {a+b \tan ^4(c+d x)}{\left (\sqrt {a}+\sqrt {b} \tan ^2(c+d x)\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \tan (c+d x)}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{a} d \sqrt {a+b \tan ^4(c+d x)}}-\frac {\sqrt [4]{a} \sqrt [4]{b} \left (\sqrt {a}+\sqrt {b} \tan ^2(c+d x)\right ) \sqrt {\frac {a+b \tan ^4(c+d x)}{\left (\sqrt {a}+\sqrt {b} \tan ^2(c+d x)\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} \tan (c+d x)}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{d \sqrt {a+b \tan ^4(c+d x)}}+\frac {\left (\sqrt {a}+\sqrt {b}\right ) (a+b) \left (\sqrt {a}+\sqrt {b} \tan ^2(c+d x)\right ) \sqrt {\frac {a+b \tan ^4(c+d x)}{\left (\sqrt {a}+\sqrt {b} \tan ^2(c+d x)\right )^2}} \operatorname {EllipticPi}\left (-\frac {\left (\sqrt {a}-\sqrt {b}\right )^2}{4 \sqrt {a} \sqrt {b}},2 \arctan \left (\frac {\sqrt [4]{b} \tan (c+d x)}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{a} \sqrt [4]{b} d \left (\sqrt {a}-\sqrt {b}\right ) \sqrt {a+b \tan ^4(c+d x)}}+\frac {\sqrt {b} \tan (c+d x) \sqrt {a+b \tan ^4(c+d x)}}{d \left (\sqrt {a}+\sqrt {b} \tan ^2(c+d x)\right )} \]
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Rule 226
Rule 1210
Rule 1212
Rule 1223
Rule 1231
Rule 1721
Rule 3742
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\sqrt {a+b x^4}}{1+x^2} \, dx,x,\tan (c+d x)\right )}{d} \\ & = -\frac {\text {Subst}\left (\int \frac {b-b x^2}{\sqrt {a+b x^4}} \, dx,x,\tan (c+d x)\right )}{d}+\frac {(a+b) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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} \arctan \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 \arctan \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} \operatorname {EllipticF}\left (2 \arctan \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) \operatorname {EllipticF}\left (2 \arctan \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) \operatorname {EllipticPi}\left (-\frac {\left (\sqrt {a}-\sqrt {b}\right )^2}{4 \sqrt {a} \sqrt {b}},2 \arctan \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*}
Result contains complex when optimal does not.
Time = 11.05 (sec) , antiderivative size = 219, normalized size of antiderivative = 0.34 \[ \int \sqrt {a+b \tan ^4(c+d x)} \, dx=\frac {\left (\sqrt {a} \sqrt {b} E\left (\left .i \text {arcsinh}\left (\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}} \tan (c+d x)\right )\right |-1\right )+\left (\sqrt {a}-i \sqrt {b}\right ) \left (-\sqrt {b} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}} \tan (c+d x)\right ),-1\right )+\left (-i \sqrt {a}+\sqrt {b}\right ) \operatorname {EllipticPi}\left (-\frac {i \sqrt {a}}{\sqrt {b}},i \text {arcsinh}\left (\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}} \tan (c+d x)\right ),-1\right )\right )\right ) \sqrt {1+\frac {b \tan ^4(c+d x)}{a}}}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}} d \sqrt {a+b \tan ^4(c+d x)}} \]
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Result contains complex when optimal does not.
Time = 1.03 (sec) , antiderivative size = 531, normalized size of antiderivative = 0.82
method | result | size |
derivativedivides | \(\frac {-\frac {b \sqrt {1-\frac {i \sqrt {b}\, \tan \left (d x +c \right )^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, \tan \left (d x +c \right )^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (\tan \left (d x +c \right ) \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {a +\tan \left (d x +c \right )^{4} b}}+\frac {i \sqrt {b}\, \sqrt {a}\, \sqrt {1-\frac {i \sqrt {b}\, \tan \left (d x +c \right )^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, \tan \left (d x +c \right )^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (\tan \left (d x +c \right ) \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {a +\tan \left (d x +c \right )^{4} b}}-\frac {i \sqrt {b}\, \sqrt {a}\, \sqrt {1-\frac {i \sqrt {b}\, \tan \left (d x +c \right )^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, \tan \left (d x +c \right )^{2}}{\sqrt {a}}}\, \operatorname {EllipticE}\left (\tan \left (d x +c \right ) \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {a +\tan \left (d x +c \right )^{4} b}}+\frac {a \sqrt {1-\frac {i \sqrt {b}\, \tan \left (d x +c \right )^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, \tan \left (d x +c \right )^{2}}{\sqrt {a}}}\, \operatorname {EllipticPi}\left (\tan \left (d x +c \right ) \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, \frac {i \sqrt {a}}{\sqrt {b}}, \frac {\sqrt {-\frac {i \sqrt {b}}{\sqrt {a}}}}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}}\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {a +\tan \left (d x +c \right )^{4} b}}+\frac {b \sqrt {1-\frac {i \sqrt {b}\, \tan \left (d x +c \right )^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, \tan \left (d x +c \right )^{2}}{\sqrt {a}}}\, \operatorname {EllipticPi}\left (\tan \left (d x +c \right ) \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, \frac {i \sqrt {a}}{\sqrt {b}}, \frac {\sqrt {-\frac {i \sqrt {b}}{\sqrt {a}}}}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}}\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {a +\tan \left (d x +c \right )^{4} b}}}{d}\) | \(531\) |
default | \(\frac {-\frac {b \sqrt {1-\frac {i \sqrt {b}\, \tan \left (d x +c \right )^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, \tan \left (d x +c \right )^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (\tan \left (d x +c \right ) \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {a +\tan \left (d x +c \right )^{4} b}}+\frac {i \sqrt {b}\, \sqrt {a}\, \sqrt {1-\frac {i \sqrt {b}\, \tan \left (d x +c \right )^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, \tan \left (d x +c \right )^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (\tan \left (d x +c \right ) \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {a +\tan \left (d x +c \right )^{4} b}}-\frac {i \sqrt {b}\, \sqrt {a}\, \sqrt {1-\frac {i \sqrt {b}\, \tan \left (d x +c \right )^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, \tan \left (d x +c \right )^{2}}{\sqrt {a}}}\, \operatorname {EllipticE}\left (\tan \left (d x +c \right ) \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {a +\tan \left (d x +c \right )^{4} b}}+\frac {a \sqrt {1-\frac {i \sqrt {b}\, \tan \left (d x +c \right )^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, \tan \left (d x +c \right )^{2}}{\sqrt {a}}}\, \operatorname {EllipticPi}\left (\tan \left (d x +c \right ) \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, \frac {i \sqrt {a}}{\sqrt {b}}, \frac {\sqrt {-\frac {i \sqrt {b}}{\sqrt {a}}}}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}}\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {a +\tan \left (d x +c \right )^{4} b}}+\frac {b \sqrt {1-\frac {i \sqrt {b}\, \tan \left (d x +c \right )^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, \tan \left (d x +c \right )^{2}}{\sqrt {a}}}\, \operatorname {EllipticPi}\left (\tan \left (d x +c \right ) \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, \frac {i \sqrt {a}}{\sqrt {b}}, \frac {\sqrt {-\frac {i \sqrt {b}}{\sqrt {a}}}}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}}\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {a +\tan \left (d x +c \right )^{4} b}}}{d}\) | \(531\) |
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Timed out. \[ \int \sqrt {a+b \tan ^4(c+d x)} \, dx=\text {Timed out} \]
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\[ \int \sqrt {a+b \tan ^4(c+d x)} \, dx=\int \sqrt {a + b \tan ^{4}{\left (c + d x \right )}}\, dx \]
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\[ \int \sqrt {a+b \tan ^4(c+d x)} \, dx=\int { \sqrt {b \tan \left (d x + c\right )^{4} + a} \,d x } \]
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\[ \int \sqrt {a+b \tan ^4(c+d x)} \, dx=\int { \sqrt {b \tan \left (d x + c\right )^{4} + a} \,d x } \]
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Timed out. \[ \int \sqrt {a+b \tan ^4(c+d x)} \, dx=\int \sqrt {b\,{\mathrm {tan}\left (c+d\,x\right )}^4+a} \,d x \]
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