Optimal. Leaf size=209 \[ -\frac {\left (-4 c (a+2 c)+b^2+4 b c\right ) \tanh ^{-1}\left (\frac {b+2 c \tan ^2(d+e x)}{2 \sqrt {c} \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )}{16 c^{3/2} e}+\frac {\left (b+2 c \tan ^2(d+e x)-4 c\right ) \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}{8 c e}+\frac {\sqrt {a-b+c} \tanh ^{-1}\left (\frac {2 a+(b-2 c) \tan ^2(d+e x)-b}{2 \sqrt {a-b+c} \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )}{2 e} \]
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Rubi [A] time = 0.35, antiderivative size = 209, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3700, 1251, 814, 843, 621, 206, 724} \[ -\frac {\left (-4 c (a+2 c)+b^2+4 b c\right ) \tanh ^{-1}\left (\frac {b+2 c \tan ^2(d+e x)}{2 \sqrt {c} \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )}{16 c^{3/2} e}+\frac {\left (b+2 c \tan ^2(d+e x)-4 c\right ) \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}{8 c e}+\frac {\sqrt {a-b+c} \tanh ^{-1}\left (\frac {2 a+(b-2 c) \tan ^2(d+e x)-b}{2 \sqrt {a-b+c} \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )}{2 e} \]
Antiderivative was successfully verified.
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Rule 206
Rule 621
Rule 724
Rule 814
Rule 843
Rule 1251
Rule 3700
Rubi steps
\begin {align*} \int \tan ^3(d+e x) \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x^3 \sqrt {a+b x^2+c x^4}}{1+x^2} \, dx,x,\tan (d+e x)\right )}{e}\\ &=\frac {\operatorname {Subst}\left (\int \frac {x \sqrt {a+b x+c x^2}}{1+x} \, dx,x,\tan ^2(d+e x)\right )}{2 e}\\ &=\frac {\left (b-4 c+2 c \tan ^2(d+e x)\right ) \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}{8 c e}-\frac {\operatorname {Subst}\left (\int \frac {\frac {1}{2} \left (b^2+4 a c-4 b c\right )+\frac {1}{2} \left (b^2+4 b c-4 c (a+2 c)\right ) x}{(1+x) \sqrt {a+b x+c x^2}} \, dx,x,\tan ^2(d+e x)\right )}{8 c e}\\ &=\frac {\left (b-4 c+2 c \tan ^2(d+e x)\right ) \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}{8 c e}-\frac {(a-b+c) \operatorname {Subst}\left (\int \frac {1}{(1+x) \sqrt {a+b x+c x^2}} \, dx,x,\tan ^2(d+e x)\right )}{2 e}-\frac {\left (b^2+4 b c-4 c (a+2 c)\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x+c x^2}} \, dx,x,\tan ^2(d+e x)\right )}{16 c e}\\ &=\frac {\left (b-4 c+2 c \tan ^2(d+e x)\right ) \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}{8 c e}+\frac {(a-b+c) \operatorname {Subst}\left (\int \frac {1}{4 a-4 b+4 c-x^2} \, dx,x,\frac {2 a-b-(-b+2 c) \tan ^2(d+e x)}{\sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )}{e}-\frac {\left (b^2+4 b c-4 c (a+2 c)\right ) \operatorname {Subst}\left (\int \frac {1}{4 c-x^2} \, dx,x,\frac {b+2 c \tan ^2(d+e x)}{\sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )}{8 c e}\\ &=\frac {\sqrt {a-b+c} \tanh ^{-1}\left (\frac {2 a-b+(b-2 c) \tan ^2(d+e x)}{2 \sqrt {a-b+c} \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )}{2 e}-\frac {\left (b^2+4 b c-4 c (a+2 c)\right ) \tanh ^{-1}\left (\frac {b+2 c \tan ^2(d+e x)}{2 \sqrt {c} \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )}{16 c^{3/2} e}+\frac {\left (b-4 c+2 c \tan ^2(d+e x)\right ) \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}{8 c e}\\ \end {align*}
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Mathematica [A] time = 1.57, size = 208, normalized size = 1.00 \[ \frac {-\left (-4 c (a+2 c)+b^2+4 b c\right ) \tanh ^{-1}\left (\frac {b+2 c \tan ^2(d+e x)}{2 \sqrt {c} \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )+8 c^{3/2} \sqrt {a-b+c} \tanh ^{-1}\left (\frac {2 a+(b-2 c) \tan ^2(d+e x)-b}{2 \sqrt {a-b+c} \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )+2 \sqrt {c} \left (b+2 c \tan ^2(d+e x)-4 c\right ) \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}{16 c^{3/2} e} \]
Antiderivative was successfully verified.
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fricas [A] time = 6.37, size = 1199, normalized size = 5.74 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.44, size = 467, normalized size = 2.23 \[ \frac {\sqrt {a +b \left (\tan ^{2}\left (e x +d \right )\right )+c \left (\tan ^{4}\left (e x +d \right )\right )}\, \left (\tan ^{2}\left (e x +d \right )\right )}{4 e}+\frac {\sqrt {a +b \left (\tan ^{2}\left (e x +d \right )\right )+c \left (\tan ^{4}\left (e x +d \right )\right )}\, b}{8 e c}+\frac {\ln \left (\frac {c \left (\tan ^{2}\left (e x +d \right )\right )+\frac {b}{2}}{\sqrt {c}}+\sqrt {a +b \left (\tan ^{2}\left (e x +d \right )\right )+c \left (\tan ^{4}\left (e x +d \right )\right )}\right ) a}{4 e \sqrt {c}}-\frac {\ln \left (\frac {c \left (\tan ^{2}\left (e x +d \right )\right )+\frac {b}{2}}{\sqrt {c}}+\sqrt {a +b \left (\tan ^{2}\left (e x +d \right )\right )+c \left (\tan ^{4}\left (e x +d \right )\right )}\right ) b^{2}}{16 e \,c^{\frac {3}{2}}}-\frac {\sqrt {\left (1+\tan ^{2}\left (e x +d \right )\right )^{2} c +\left (b -2 c \right ) \left (1+\tan ^{2}\left (e x +d \right )\right )+a -b +c}}{2 e}-\frac {\ln \left (\frac {\frac {b}{2}-c +\left (1+\tan ^{2}\left (e x +d \right )\right ) c}{\sqrt {c}}+\sqrt {\left (1+\tan ^{2}\left (e x +d \right )\right )^{2} c +\left (b -2 c \right ) \left (1+\tan ^{2}\left (e x +d \right )\right )+a -b +c}\right ) b}{4 e \sqrt {c}}+\frac {\ln \left (\frac {\frac {b}{2}-c +\left (1+\tan ^{2}\left (e x +d \right )\right ) c}{\sqrt {c}}+\sqrt {\left (1+\tan ^{2}\left (e x +d \right )\right )^{2} c +\left (b -2 c \right ) \left (1+\tan ^{2}\left (e x +d \right )\right )+a -b +c}\right ) \sqrt {c}}{2 e}+\frac {\sqrt {a -b +c}\, \ln \left (\frac {2 a -2 b +2 c +\left (b -2 c \right ) \left (1+\tan ^{2}\left (e x +d \right )\right )+2 \sqrt {a -b +c}\, \sqrt {\left (1+\tan ^{2}\left (e x +d \right )\right )^{2} c +\left (b -2 c \right ) \left (1+\tan ^{2}\left (e x +d \right )\right )+a -b +c}}{1+\tan ^{2}\left (e x +d \right )}\right )}{2 e} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {c \tan \left (e x + d\right )^{4} + b \tan \left (e x + d\right )^{2} + a} \tan \left (e x + d\right )^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int {\mathrm {tan}\left (d+e\,x\right )}^3\,\sqrt {c\,{\mathrm {tan}\left (d+e\,x\right )}^4+b\,{\mathrm {tan}\left (d+e\,x\right )}^2+a} \,d x \]
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 \[ \int \sqrt {a + b \tan ^{2}{\left (d + e x \right )} + c \tan ^{4}{\left (d + e x \right )}} \tan ^{3}{\left (d + e x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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