Optimal. Leaf size=155 \[ \frac {-2 a c+b^2+c (b-2 c) \tan ^2(d+e x)-b c}{e (a-b+c) \left (b^2-4 a c\right ) \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}-\frac {\tanh ^{-1}\left (\frac {2 a+(b-2 c) \tan ^2(d+e x)-b}{2 \sqrt {a-b+c} \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )}{2 e (a-b+c)^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.22, antiderivative size = 155, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {3700, 1247, 740, 12, 724, 206} \[ \frac {-2 a c+b^2+c (b-2 c) \tan ^2(d+e x)-b c}{e (a-b+c) \left (b^2-4 a c\right ) \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}-\frac {\tanh ^{-1}\left (\frac {2 a+(b-2 c) \tan ^2(d+e x)-b}{2 \sqrt {a-b+c} \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )}{2 e (a-b+c)^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 206
Rule 724
Rule 740
Rule 1247
Rule 3700
Rubi steps
\begin {align*} \int \frac {\tan (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}{\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 {1}{(1+x) \left (a+b x+c x^2\right )^{3/2}} \, dx,x,\tan ^2(d+e x)\right )}{2 e}\\ &=\frac {b^2-2 a c-b c+(b-2 c) 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 {-\frac {b^2}{2}+2 a c}{(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 {b^2-2 a c-b c+(b-2 c) 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 {b^2-2 a c-b c+(b-2 c) 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 {b^2-2 a c-b c+(b-2 c) 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*}
________________________________________________________________________________________
Mathematica [A] time = 2.98, size = 156, normalized size = 1.01 \[ -\frac {\frac {2 \left (2 a c-b^2-c (b-2 c) \tan ^2(d+e x)+b c\right )}{(a-b+c) \left (b^2-4 a c\right ) \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}+\frac {\tanh ^{-1}\left (\frac {2 a+(b-2 c) \tan ^2(d+e x)-b}{2 \sqrt {a-b+c} \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )}{(a-b+c)^{3/2}}}{2 e} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 1.06, size = 1099, normalized size = 7.09 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.38, size = 417, normalized size = 2.69 \[ -\frac {2 c \sqrt {\left (\tan ^{2}\left (e x +d \right )-\frac {-b +\sqrt {-4 c a +b^{2}}}{2 c}\right )^{2} c +\sqrt {-4 c a +b^{2}}\, \left (\tan ^{2}\left (e x +d \right )-\frac {-b +\sqrt {-4 c a +b^{2}}}{2 c}\right )}}{e \left (\sqrt {-4 c a +b^{2}}-b +2 c \right ) \left (-4 c a +b^{2}\right ) \left (\tan ^{2}\left (e x +d \right )-\frac {\sqrt {-4 c a +b^{2}}}{2 c}+\frac {b}{2 c}\right )}+\frac {2 c \ln \left (\frac {2 a -2 b +2 c +\left (b -2 c \right ) \left (1+\tan ^{2}\left (e x +d \right )\right )+2 \sqrt {a -b +c}\, \sqrt {\left (1+\tan ^{2}\left (e x +d \right )\right )^{2} c +\left (b -2 c \right ) \left (1+\tan ^{2}\left (e x +d \right )\right )+a -b +c}}{1+\tan ^{2}\left (e x +d \right )}\right )}{e \left (\sqrt {-4 c a +b^{2}}-b +2 c \right ) \left (b -2 c +\sqrt {-4 c a +b^{2}}\right ) \sqrt {a -b +c}}+\frac {2 c \sqrt {\left (\tan ^{2}\left (e x +d \right )+\frac {b +\sqrt {-4 c a +b^{2}}}{2 c}\right )^{2} c -\sqrt {-4 c a +b^{2}}\, \left (\tan ^{2}\left (e x +d \right )+\frac {b +\sqrt {-4 c a +b^{2}}}{2 c}\right )}}{e \left (b -2 c +\sqrt {-4 c a +b^{2}}\right ) \left (-4 c a +b^{2}\right ) \left (\tan ^{2}\left (e x +d \right )+\frac {\sqrt {-4 c a +b^{2}}}{2 c}+\frac {b}{2 c}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\tan \left (e x + d\right )}{{\left (c \tan \left (e x + d\right )^{4} + b \tan \left (e x + d\right )^{2} + a\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\mathrm {tan}\left (d+e\,x\right )}{{\left (c\,{\mathrm {tan}\left (d+e\,x\right )}^4+b\,{\mathrm {tan}\left (d+e\,x\right )}^2+a\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\tan {\left (d + e x \right )}}{\left (a + b \tan ^{2}{\left (d + e x \right )} + c \tan ^{4}{\left (d + e x \right )}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________