Optimal. Leaf size=74 \[ -\frac {\tanh ^{-1}\left (\frac {a-b \tan ^2(x)}{\sqrt {a+b} \sqrt {a+b \tan ^4(x)}}\right )}{2 (a+b)^{3/2}}+\frac {a+b \tan ^2(x)}{2 a (a+b) \sqrt {a+b \tan ^4(x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.07, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {3751, 1262,
755, 12, 739, 212} \begin {gather*} \frac {a+b \tan ^2(x)}{2 a (a+b) \sqrt {a+b \tan ^4(x)}}-\frac {\tanh ^{-1}\left (\frac {a-b \tan ^2(x)}{\sqrt {a+b} \sqrt {a+b \tan ^4(x)}}\right )}{2 (a+b)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 212
Rule 739
Rule 755
Rule 1262
Rule 3751
Rubi steps
\begin {align*} \int \frac {\tan (x)}{\left (a+b \tan ^4(x)\right )^{3/2}} \, dx &=\text {Subst}\left (\int \frac {x}{\left (1+x^2\right ) \left (a+b x^4\right )^{3/2}} \, dx,x,\tan (x)\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \frac {1}{(1+x) \left (a+b x^2\right )^{3/2}} \, dx,x,\tan ^2(x)\right )\\ &=\frac {a+b \tan ^2(x)}{2 a (a+b) \sqrt {a+b \tan ^4(x)}}+\frac {\text {Subst}\left (\int \frac {a}{(1+x) \sqrt {a+b x^2}} \, dx,x,\tan ^2(x)\right )}{2 a (a+b)}\\ &=\frac {a+b \tan ^2(x)}{2 a (a+b) \sqrt {a+b \tan ^4(x)}}+\frac {\text {Subst}\left (\int \frac {1}{(1+x) \sqrt {a+b x^2}} \, dx,x,\tan ^2(x)\right )}{2 (a+b)}\\ &=\frac {a+b \tan ^2(x)}{2 a (a+b) \sqrt {a+b \tan ^4(x)}}-\frac {\text {Subst}\left (\int \frac {1}{a+b-x^2} \, dx,x,\frac {a-b \tan ^2(x)}{\sqrt {a+b \tan ^4(x)}}\right )}{2 (a+b)}\\ &=-\frac {\tanh ^{-1}\left (\frac {a-b \tan ^2(x)}{\sqrt {a+b} \sqrt {a+b \tan ^4(x)}}\right )}{2 (a+b)^{3/2}}+\frac {a+b \tan ^2(x)}{2 a (a+b) \sqrt {a+b \tan ^4(x)}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.20, size = 73, normalized size = 0.99 \begin {gather*} \frac {1}{2} \left (-\frac {\tanh ^{-1}\left (\frac {a-b \tan ^2(x)}{\sqrt {a+b} \sqrt {a+b \tan ^4(x)}}\right )}{(a+b)^{3/2}}+\frac {a+b \tan ^2(x)}{a (a+b) \sqrt {a+b \tan ^4(x)}}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(247\) vs.
\(2(62)=124\).
time = 0.09, size = 248, normalized size = 3.35
method | result | size |
derivativedivides | \(-\frac {\sqrt {b \left (\tan ^{2}\left (x \right )+\frac {\sqrt {-a b}}{b}\right )^{2}-2 \sqrt {-a b}\, \left (\tan ^{2}\left (x \right )+\frac {\sqrt {-a b}}{b}\right )}}{4 \left (\sqrt {-a b}-b \right ) a \left (\tan ^{2}\left (x \right )+\frac {\sqrt {-a b}}{b}\right )}+\frac {\sqrt {b \left (\tan ^{2}\left (x \right )-\frac {\sqrt {-a b}}{b}\right )^{2}+2 \sqrt {-a b}\, \left (\tan ^{2}\left (x \right )-\frac {\sqrt {-a b}}{b}\right )}}{4 \left (\sqrt {-a b}+b \right ) a \left (\tan ^{2}\left (x \right )-\frac {\sqrt {-a b}}{b}\right )}+\frac {b \ln \left (\frac {2 a +2 b -2 b \left (1+\tan ^{2}\left (x \right )\right )+2 \sqrt {a +b}\, \sqrt {b \left (1+\tan ^{2}\left (x \right )\right )^{2}-2 b \left (1+\tan ^{2}\left (x \right )\right )+a +b}}{1+\tan ^{2}\left (x \right )}\right )}{2 \left (\sqrt {-a b}+b \right ) \left (\sqrt {-a b}-b \right ) \sqrt {a +b}}\) | \(248\) |
default | \(-\frac {\sqrt {b \left (\tan ^{2}\left (x \right )+\frac {\sqrt {-a b}}{b}\right )^{2}-2 \sqrt {-a b}\, \left (\tan ^{2}\left (x \right )+\frac {\sqrt {-a b}}{b}\right )}}{4 \left (\sqrt {-a b}-b \right ) a \left (\tan ^{2}\left (x \right )+\frac {\sqrt {-a b}}{b}\right )}+\frac {\sqrt {b \left (\tan ^{2}\left (x \right )-\frac {\sqrt {-a b}}{b}\right )^{2}+2 \sqrt {-a b}\, \left (\tan ^{2}\left (x \right )-\frac {\sqrt {-a b}}{b}\right )}}{4 \left (\sqrt {-a b}+b \right ) a \left (\tan ^{2}\left (x \right )-\frac {\sqrt {-a b}}{b}\right )}+\frac {b \ln \left (\frac {2 a +2 b -2 b \left (1+\tan ^{2}\left (x \right )\right )+2 \sqrt {a +b}\, \sqrt {b \left (1+\tan ^{2}\left (x \right )\right )^{2}-2 b \left (1+\tan ^{2}\left (x \right )\right )+a +b}}{1+\tan ^{2}\left (x \right )}\right )}{2 \left (\sqrt {-a b}+b \right ) \left (\sqrt {-a b}-b \right ) \sqrt {a +b}}\) | \(248\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 150 vs.
\(2 (64) = 128\).
time = 2.68, size = 319, normalized size = 4.31 \begin {gather*} \left [\frac {{\left (a b \tan \left (x\right )^{4} + a^{2}\right )} \sqrt {a + b} \log \left (\frac {{\left (a b + 2 \, b^{2}\right )} \tan \left (x\right )^{4} - 2 \, a b \tan \left (x\right )^{2} + 2 \, \sqrt {b \tan \left (x\right )^{4} + a} {\left (b \tan \left (x\right )^{2} - a\right )} \sqrt {a + b} + 2 \, a^{2} + a b}{\tan \left (x\right )^{4} + 2 \, \tan \left (x\right )^{2} + 1}\right ) + 2 \, \sqrt {b \tan \left (x\right )^{4} + a} {\left ({\left (a b + b^{2}\right )} \tan \left (x\right )^{2} + a^{2} + a b\right )}}{4 \, {\left ({\left (a^{3} b + 2 \, a^{2} b^{2} + a b^{3}\right )} \tan \left (x\right )^{4} + a^{4} + 2 \, a^{3} b + a^{2} b^{2}\right )}}, -\frac {{\left (a b \tan \left (x\right )^{4} + a^{2}\right )} \sqrt {-a - b} \arctan \left (\frac {\sqrt {b \tan \left (x\right )^{4} + a} {\left (b \tan \left (x\right )^{2} - a\right )} \sqrt {-a - b}}{{\left (a b + b^{2}\right )} \tan \left (x\right )^{4} + a^{2} + a b}\right ) - \sqrt {b \tan \left (x\right )^{4} + a} {\left ({\left (a b + b^{2}\right )} \tan \left (x\right )^{2} + a^{2} + a b\right )}}{2 \, {\left ({\left (a^{3} b + 2 \, a^{2} b^{2} + a b^{3}\right )} \tan \left (x\right )^{4} + a^{4} + 2 \, a^{3} b + a^{2} b^{2}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\tan {\left (x \right )}}{\left (a + b \tan ^{4}{\left (x \right )}\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.44, size = 119, normalized size = 1.61 \begin {gather*} \frac {\frac {{\left (a b + b^{2}\right )} \tan \left (x\right )^{2}}{a^{3} + 2 \, a^{2} b + a b^{2}} + \frac {a^{2} + a b}{a^{3} + 2 \, a^{2} b + a b^{2}}}{2 \, \sqrt {b \tan \left (x\right )^{4} + a}} - \frac {\arctan \left (\frac {\sqrt {b} \tan \left (x\right )^{2} - \sqrt {b \tan \left (x\right )^{4} + a} + \sqrt {b}}{\sqrt {-a - b}}\right )}{{\left (a + b\right )} \sqrt {-a - b}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\mathrm {tan}\left (x\right )}{{\left (b\,{\mathrm {tan}\left (x\right )}^4+a\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________