Optimal. Leaf size=154 \[ \frac {\tanh ^{-1}\left (\frac {2 a+(b-2 c) \tan ^2(d+e x)-b}{2 \sqrt {a-b+c} \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )}{2 e (a-b+c)^{3/2}}-\frac {c (2 a-b) \tan ^2(d+e x)+a (b-2 c)}{e (a-b+c) \left (b^2-4 a c\right ) \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}} \]
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Rubi [A] time = 0.28, antiderivative size = 154, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {3700, 1251, 822, 12, 724, 206} \[ \frac {\tanh ^{-1}\left (\frac {2 a+(b-2 c) \tan ^2(d+e x)-b}{2 \sqrt {a-b+c} \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )}{2 e (a-b+c)^{3/2}}-\frac {c (2 a-b) \tan ^2(d+e x)+a (b-2 c)}{e (a-b+c) \left (b^2-4 a c\right ) \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 206
Rule 724
Rule 822
Rule 1251
Rule 3700
Rubi steps
\begin {align*} \int \frac {\tan ^3(d+e x)}{\left (a+b \tan ^2(d+e x)+c \tan ^4(d+e x)\right )^{3/2}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x^3}{\left (1+x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}} \, dx,x,\tan (d+e x)\right )}{e}\\ &=\frac {\operatorname {Subst}\left (\int \frac {x}{(1+x) \left (a+b x+c x^2\right )^{3/2}} \, dx,x,\tan ^2(d+e x)\right )}{2 e}\\ &=-\frac {a (b-2 c)+(2 a-b) c \tan ^2(d+e x)}{(a-b+c) \left (b^2-4 a c\right ) e \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}-\frac {\operatorname {Subst}\left (\int \frac {b^2-4 a c}{2 (1+x) \sqrt {a+b x+c x^2}} \, dx,x,\tan ^2(d+e x)\right )}{(a-b+c) \left (b^2-4 a c\right ) e}\\ &=-\frac {a (b-2 c)+(2 a-b) c \tan ^2(d+e x)}{(a-b+c) \left (b^2-4 a c\right ) e \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}-\frac {\operatorname {Subst}\left (\int \frac {1}{(1+x) \sqrt {a+b x+c x^2}} \, dx,x,\tan ^2(d+e x)\right )}{2 (a-b+c) e}\\ &=-\frac {a (b-2 c)+(2 a-b) c \tan ^2(d+e x)}{(a-b+c) \left (b^2-4 a c\right ) e \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}+\frac {\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 )}{(a-b+c) e}\\ &=\frac {\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 (a-b+c)^{3/2} e}-\frac {a (b-2 c)+(2 a-b) c \tan ^2(d+e x)}{(a-b+c) \left (b^2-4 a c\right ) e \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\\ \end {align*}
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Mathematica [A] time = 2.90, size = 155, normalized size = 1.01 \[ \frac {\frac {2 c (2 a-b) \tan ^2(d+e x)+2 a (b-2 c)}{(a-b+c) \left (4 a c-b^2\right ) \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}+\frac {\tanh ^{-1}\left (\frac {2 a+(b-2 c) \tan ^2(d+e x)-b}{2 \sqrt {a-b+c} \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )}{(a-b+c)^{3/2}}}{2 e} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.15, size = 1077, normalized size = 6.99 \[ \text {result too large to display} \]
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 = 0.45, size = 508, normalized size = 3.30 \[ \frac {2 c \left (\tan ^{2}\left (e x +d \right )\right )}{e \sqrt {a +b \left (\tan ^{2}\left (e x +d \right )\right )+c \left (\tan ^{4}\left (e x +d \right )\right )}\, \left (4 c a -b^{2}\right )}+\frac {b}{e \sqrt {a +b \left (\tan ^{2}\left (e x +d \right )\right )+c \left (\tan ^{4}\left (e x +d \right )\right )}\, \left (4 c a -b^{2}\right )}+\frac {2 c \sqrt {\left (\tan ^{2}\left (e x +d \right )-\frac {-b +\sqrt {-4 c a +b^{2}}}{2 c}\right )^{2} c +\sqrt {-4 c a +b^{2}}\, \left (\tan ^{2}\left (e x +d \right )-\frac {-b +\sqrt {-4 c a +b^{2}}}{2 c}\right )}}{e \left (\sqrt {-4 c a +b^{2}}-b +2 c \right ) \left (-4 c a +b^{2}\right ) \left (\tan ^{2}\left (e x +d \right )-\frac {\sqrt {-4 c a +b^{2}}}{2 c}+\frac {b}{2 c}\right )}-\frac {2 c \ln \left (\frac {2 a -2 b +2 c +\left (b -2 c \right ) \left (1+\tan ^{2}\left (e x +d \right )\right )+2 \sqrt {a -b +c}\, \sqrt {\left (1+\tan ^{2}\left (e x +d \right )\right )^{2} c +\left (b -2 c \right ) \left (1+\tan ^{2}\left (e x +d \right )\right )+a -b +c}}{1+\tan ^{2}\left (e x +d \right )}\right )}{e \left (\sqrt {-4 c a +b^{2}}-b +2 c \right ) \left (b -2 c +\sqrt {-4 c a +b^{2}}\right ) \sqrt {a -b +c}}-\frac {2 c \sqrt {\left (\tan ^{2}\left (e x +d \right )+\frac {b +\sqrt {-4 c a +b^{2}}}{2 c}\right )^{2} c -\sqrt {-4 c a +b^{2}}\, \left (\tan ^{2}\left (e x +d \right )+\frac {b +\sqrt {-4 c a +b^{2}}}{2 c}\right )}}{e \left (b -2 c +\sqrt {-4 c a +b^{2}}\right ) \left (-4 c a +b^{2}\right ) \left (\tan ^{2}\left (e x +d \right )+\frac {\sqrt {-4 c a +b^{2}}}{2 c}+\frac {b}{2 c}\right )} \]
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 \[ \int \frac {\tan \left (e x + d\right )^{3}}{{\left (c \tan \left (e x + d\right )^{4} + b \tan \left (e x + d\right )^{2} + a\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\mathrm {tan}\left (d+e\,x\right )}^3}{{\left (c\,{\mathrm {tan}\left (d+e\,x\right )}^4+b\,{\mathrm {tan}\left (d+e\,x\right )}^2+a\right )}^{3/2}} \,d x \]
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 \[ \int \frac {\tan ^{3}{\left (d + e x \right )}}{\left (a + b \tan ^{2}{\left (d + e x \right )} + c \tan ^{4}{\left (d + e x \right )}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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