Optimal. Leaf size=90 \[ -\frac {x}{(a-b)^2}+\frac {(a+b) \text {ArcTan}\left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a}}\right )}{2 \sqrt {a} (a-b)^2 \sqrt {b} f}+\frac {\tan (e+f x)}{2 (a-b) f \left (a+b \tan ^2(e+f x)\right )} \]
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Rubi [A]
time = 0.07, antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {3751, 482, 536,
209, 211} \begin {gather*} \frac {(a+b) \text {ArcTan}\left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a}}\right )}{2 \sqrt {a} \sqrt {b} f (a-b)^2}+\frac {\tan (e+f x)}{2 f (a-b) \left (a+b \tan ^2(e+f x)\right )}-\frac {x}{(a-b)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 211
Rule 482
Rule 536
Rule 3751
Rubi steps
\begin {align*} \int \frac {\tan ^2(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^2} \, dx &=\frac {\text {Subst}\left (\int \frac {x^2}{\left (1+x^2\right ) \left (a+b x^2\right )^2} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {\tan (e+f x)}{2 (a-b) f \left (a+b \tan ^2(e+f x)\right )}-\frac {\text {Subst}\left (\int \frac {1-x^2}{\left (1+x^2\right ) \left (a+b x^2\right )} \, dx,x,\tan (e+f x)\right )}{2 (a-b) f}\\ &=\frac {\tan (e+f x)}{2 (a-b) f \left (a+b \tan ^2(e+f x)\right )}-\frac {\text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\tan (e+f x)\right )}{(a-b)^2 f}+\frac {(a+b) \text {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\tan (e+f x)\right )}{2 (a-b)^2 f}\\ &=-\frac {x}{(a-b)^2}+\frac {(a+b) \tan ^{-1}\left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a}}\right )}{2 \sqrt {a} (a-b)^2 \sqrt {b} f}+\frac {\tan (e+f x)}{2 (a-b) f \left (a+b \tan ^2(e+f x)\right )}\\ \end {align*}
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Mathematica [A]
time = 0.39, size = 87, normalized size = 0.97 \begin {gather*} \frac {-2 (e+f x)+\frac {(a+b) \text {ArcTan}\left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b}}+\frac {(a-b) \sin (2 (e+f x))}{a+b+(a-b) \cos (2 (e+f x))}}{2 (a-b)^2 f} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.21, size = 83, normalized size = 0.92
method | result | size |
derivativedivides | \(\frac {-\frac {\arctan \left (\tan \left (f x +e \right )\right )}{\left (a -b \right )^{2}}+\frac {\frac {\left (\frac {a}{2}-\frac {b}{2}\right ) \tan \left (f x +e \right )}{a +b \left (\tan ^{2}\left (f x +e \right )\right )}+\frac {\left (a +b \right ) \arctan \left (\frac {b \tan \left (f x +e \right )}{\sqrt {a b}}\right )}{2 \sqrt {a b}}}{\left (a -b \right )^{2}}}{f}\) | \(83\) |
default | \(\frac {-\frac {\arctan \left (\tan \left (f x +e \right )\right )}{\left (a -b \right )^{2}}+\frac {\frac {\left (\frac {a}{2}-\frac {b}{2}\right ) \tan \left (f x +e \right )}{a +b \left (\tan ^{2}\left (f x +e \right )\right )}+\frac {\left (a +b \right ) \arctan \left (\frac {b \tan \left (f x +e \right )}{\sqrt {a b}}\right )}{2 \sqrt {a b}}}{\left (a -b \right )^{2}}}{f}\) | \(83\) |
risch | \(-\frac {x}{a^{2}-2 a b +b^{2}}+\frac {i \left (a \,{\mathrm e}^{2 i \left (f x +e \right )}+b \,{\mathrm e}^{2 i \left (f x +e \right )}+a -b \right )}{f \left (a -b \right )^{2} \left (a \,{\mathrm e}^{4 i \left (f x +e \right )}-b \,{\mathrm e}^{4 i \left (f x +e \right )}+2 a \,{\mathrm e}^{2 i \left (f x +e \right )}+2 b \,{\mathrm e}^{2 i \left (f x +e \right )}+a -b \right )}+\frac {\ln \left ({\mathrm e}^{2 i \left (f x +e \right )}-\frac {2 i a b -\sqrt {-a b}\, a -\sqrt {-a b}\, b}{\left (a -b \right ) \sqrt {-a b}}\right ) a}{4 \sqrt {-a b}\, \left (a -b \right )^{2} f}+\frac {\ln \left ({\mathrm e}^{2 i \left (f x +e \right )}-\frac {2 i a b -\sqrt {-a b}\, a -\sqrt {-a b}\, b}{\left (a -b \right ) \sqrt {-a b}}\right ) b}{4 \sqrt {-a b}\, \left (a -b \right )^{2} f}-\frac {\ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+\frac {2 i a b +\sqrt {-a b}\, a +\sqrt {-a b}\, b}{\left (a -b \right ) \sqrt {-a b}}\right ) a}{4 \sqrt {-a b}\, \left (a -b \right )^{2} f}-\frac {\ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+\frac {2 i a b +\sqrt {-a b}\, a +\sqrt {-a b}\, b}{\left (a -b \right ) \sqrt {-a b}}\right ) b}{4 \sqrt {-a b}\, \left (a -b \right )^{2} f}\) | \(382\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 101, normalized size = 1.12 \begin {gather*} \frac {\frac {{\left (a + b\right )} \arctan \left (\frac {b \tan \left (f x + e\right )}{\sqrt {a b}}\right )}{{\left (a^{2} - 2 \, a b + b^{2}\right )} \sqrt {a b}} - \frac {2 \, {\left (f x + e\right )}}{a^{2} - 2 \, a b + b^{2}} + \frac {\tan \left (f x + e\right )}{{\left (a b - b^{2}\right )} \tan \left (f x + e\right )^{2} + a^{2} - a b}}{2 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 176 vs.
\(2 (81) = 162\).
time = 2.54, size = 409, normalized size = 4.54 \begin {gather*} \left [-\frac {8 \, a b^{2} f x \tan \left (f x + e\right )^{2} + 8 \, a^{2} b f x + {\left ({\left (a b + b^{2}\right )} \tan \left (f x + e\right )^{2} + a^{2} + a b\right )} \sqrt {-a b} \log \left (\frac {b^{2} \tan \left (f x + e\right )^{4} - 6 \, a b \tan \left (f x + e\right )^{2} + a^{2} - 4 \, {\left (b \tan \left (f x + e\right )^{3} - a \tan \left (f x + e\right )\right )} \sqrt {-a b}}{b^{2} \tan \left (f x + e\right )^{4} + 2 \, a b \tan \left (f x + e\right )^{2} + a^{2}}\right ) - 4 \, {\left (a^{2} b - a b^{2}\right )} \tan \left (f x + e\right )}{8 \, {\left ({\left (a^{3} b^{2} - 2 \, a^{2} b^{3} + a b^{4}\right )} f \tan \left (f x + e\right )^{2} + {\left (a^{4} b - 2 \, a^{3} b^{2} + a^{2} b^{3}\right )} f\right )}}, -\frac {4 \, a b^{2} f x \tan \left (f x + e\right )^{2} + 4 \, a^{2} b f x - {\left ({\left (a b + b^{2}\right )} \tan \left (f x + e\right )^{2} + a^{2} + a b\right )} \sqrt {a b} \arctan \left (\frac {{\left (b \tan \left (f x + e\right )^{2} - a\right )} \sqrt {a b}}{2 \, a b \tan \left (f x + e\right )}\right ) - 2 \, {\left (a^{2} b - a b^{2}\right )} \tan \left (f x + e\right )}{4 \, {\left ({\left (a^{3} b^{2} - 2 \, a^{2} b^{3} + a b^{4}\right )} f \tan \left (f x + e\right )^{2} + {\left (a^{4} b - 2 \, a^{3} b^{2} + a^{2} b^{3}\right )} f\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 2113 vs.
\(2 (73) = 146\).
time = 15.32, size = 2113, normalized size = 23.48 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.85, size = 112, normalized size = 1.24 \begin {gather*} \frac {\frac {{\left (\pi \left \lfloor \frac {f x + e}{\pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (b\right ) + \arctan \left (\frac {b \tan \left (f x + e\right )}{\sqrt {a b}}\right )\right )} {\left (a + b\right )}}{{\left (a^{2} - 2 \, a b + b^{2}\right )} \sqrt {a b}} - \frac {2 \, {\left (f x + e\right )}}{a^{2} - 2 \, a b + b^{2}} + \frac {\tan \left (f x + e\right )}{{\left (b \tan \left (f x + e\right )^{2} + a\right )} {\left (a - b\right )}}}{2 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 13.04, size = 2136, normalized size = 23.73 \begin {gather*} \frac {\mathrm {tan}\left (e+f\,x\right )}{2\,f\,\left (b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a\right )\,\left (a-b\right )}-\frac {2\,\mathrm {atan}\left (\frac {\frac {\frac {-\frac {\mathrm {tan}\left (e+f\,x\right )\,\left (16\,a^5\,b^2-48\,a^4\,b^3+32\,a^3\,b^4+32\,a^2\,b^5-48\,a\,b^6+16\,b^7\right )}{2\,\left (a^2-2\,a\,b+b^2\right )\,\left (2\,a^2-4\,a\,b+2\,b^2\right )}+\frac {\left (2\,a^4\,b^2-8\,a^3\,b^3+12\,a^2\,b^4-8\,a\,b^5+2\,b^6\right )\,1{}\mathrm {i}}{a^3-3\,a^2\,b+3\,a\,b^2-b^3}}{2\,a^2-4\,a\,b+2\,b^2}+\frac {\mathrm {tan}\left (e+f\,x\right )\,\left (a^2\,b+2\,a\,b^2+5\,b^3\right )}{2\,\left (a^2-2\,a\,b+b^2\right )}}{2\,a^2-4\,a\,b+2\,b^2}-\frac {\frac {\frac {\mathrm {tan}\left (e+f\,x\right )\,\left (16\,a^5\,b^2-48\,a^4\,b^3+32\,a^3\,b^4+32\,a^2\,b^5-48\,a\,b^6+16\,b^7\right )}{2\,\left (a^2-2\,a\,b+b^2\right )\,\left (2\,a^2-4\,a\,b+2\,b^2\right )}+\frac {\left (2\,a^4\,b^2-8\,a^3\,b^3+12\,a^2\,b^4-8\,a\,b^5+2\,b^6\right )\,1{}\mathrm {i}}{a^3-3\,a^2\,b+3\,a\,b^2-b^3}}{2\,a^2-4\,a\,b+2\,b^2}-\frac {\mathrm {tan}\left (e+f\,x\right )\,\left (a^2\,b+2\,a\,b^2+5\,b^3\right )}{2\,\left (a^2-2\,a\,b+b^2\right )}}{2\,a^2-4\,a\,b+2\,b^2}}{\frac {\frac {\left (-\frac {\mathrm {tan}\left (e+f\,x\right )\,\left (16\,a^5\,b^2-48\,a^4\,b^3+32\,a^3\,b^4+32\,a^2\,b^5-48\,a\,b^6+16\,b^7\right )}{2\,\left (a^2-2\,a\,b+b^2\right )\,\left (2\,a^2-4\,a\,b+2\,b^2\right )}+\frac {\left (2\,a^4\,b^2-8\,a^3\,b^3+12\,a^2\,b^4-8\,a\,b^5+2\,b^6\right )\,1{}\mathrm {i}}{a^3-3\,a^2\,b+3\,a\,b^2-b^3}\right )\,1{}\mathrm {i}}{2\,a^2-4\,a\,b+2\,b^2}+\frac {\mathrm {tan}\left (e+f\,x\right )\,\left (a^2\,b+2\,a\,b^2+5\,b^3\right )\,1{}\mathrm {i}}{2\,\left (a^2-2\,a\,b+b^2\right )}}{2\,a^2-4\,a\,b+2\,b^2}+\frac {\frac {\left (\frac {\mathrm {tan}\left (e+f\,x\right )\,\left (16\,a^5\,b^2-48\,a^4\,b^3+32\,a^3\,b^4+32\,a^2\,b^5-48\,a\,b^6+16\,b^7\right )}{2\,\left (a^2-2\,a\,b+b^2\right )\,\left (2\,a^2-4\,a\,b+2\,b^2\right )}+\frac {\left (2\,a^4\,b^2-8\,a^3\,b^3+12\,a^2\,b^4-8\,a\,b^5+2\,b^6\right )\,1{}\mathrm {i}}{a^3-3\,a^2\,b+3\,a\,b^2-b^3}\right )\,1{}\mathrm {i}}{2\,a^2-4\,a\,b+2\,b^2}-\frac {\mathrm {tan}\left (e+f\,x\right )\,\left (a^2\,b+2\,a\,b^2+5\,b^3\right )\,1{}\mathrm {i}}{2\,\left (a^2-2\,a\,b+b^2\right )}}{2\,a^2-4\,a\,b+2\,b^2}+\frac {\frac {b^2}{2}+\frac {a\,b}{2}}{a^3-3\,a^2\,b+3\,a\,b^2-b^3}}\right )}{f\,\left (2\,a^2-4\,a\,b+2\,b^2\right )}-\frac {\mathrm {atan}\left (\frac {\frac {\sqrt {-a\,b}\,\left (a+b\right )\,\left (\frac {\mathrm {tan}\left (e+f\,x\right )\,\left (a^2\,b+2\,a\,b^2+5\,b^3\right )}{2\,\left (a^2-2\,a\,b+b^2\right )}-\frac {\left (\frac {2\,a^4\,b^2-8\,a^3\,b^3+12\,a^2\,b^4-8\,a\,b^5+2\,b^6}{a^3-3\,a^2\,b+3\,a\,b^2-b^3}-\frac {\mathrm {tan}\left (e+f\,x\right )\,\sqrt {-a\,b}\,\left (a+b\right )\,\left (16\,a^5\,b^2-48\,a^4\,b^3+32\,a^3\,b^4+32\,a^2\,b^5-48\,a\,b^6+16\,b^7\right )}{8\,\left (a^2-2\,a\,b+b^2\right )\,\left (a^3\,b-2\,a^2\,b^2+a\,b^3\right )}\right )\,\sqrt {-a\,b}\,\left (a+b\right )}{4\,\left (a^3\,b-2\,a^2\,b^2+a\,b^3\right )}\right )\,1{}\mathrm {i}}{4\,\left (a^3\,b-2\,a^2\,b^2+a\,b^3\right )}+\frac {\sqrt {-a\,b}\,\left (a+b\right )\,\left (\frac {\mathrm {tan}\left (e+f\,x\right )\,\left (a^2\,b+2\,a\,b^2+5\,b^3\right )}{2\,\left (a^2-2\,a\,b+b^2\right )}+\frac {\left (\frac {2\,a^4\,b^2-8\,a^3\,b^3+12\,a^2\,b^4-8\,a\,b^5+2\,b^6}{a^3-3\,a^2\,b+3\,a\,b^2-b^3}+\frac {\mathrm {tan}\left (e+f\,x\right )\,\sqrt {-a\,b}\,\left (a+b\right )\,\left (16\,a^5\,b^2-48\,a^4\,b^3+32\,a^3\,b^4+32\,a^2\,b^5-48\,a\,b^6+16\,b^7\right )}{8\,\left (a^2-2\,a\,b+b^2\right )\,\left (a^3\,b-2\,a^2\,b^2+a\,b^3\right )}\right )\,\sqrt {-a\,b}\,\left (a+b\right )}{4\,\left (a^3\,b-2\,a^2\,b^2+a\,b^3\right )}\right )\,1{}\mathrm {i}}{4\,\left (a^3\,b-2\,a^2\,b^2+a\,b^3\right )}}{\frac {\frac {b^2}{2}+\frac {a\,b}{2}}{a^3-3\,a^2\,b+3\,a\,b^2-b^3}-\frac {\sqrt {-a\,b}\,\left (a+b\right )\,\left (\frac {\mathrm {tan}\left (e+f\,x\right )\,\left (a^2\,b+2\,a\,b^2+5\,b^3\right )}{2\,\left (a^2-2\,a\,b+b^2\right )}-\frac {\left (\frac {2\,a^4\,b^2-8\,a^3\,b^3+12\,a^2\,b^4-8\,a\,b^5+2\,b^6}{a^3-3\,a^2\,b+3\,a\,b^2-b^3}-\frac {\mathrm {tan}\left (e+f\,x\right )\,\sqrt {-a\,b}\,\left (a+b\right )\,\left (16\,a^5\,b^2-48\,a^4\,b^3+32\,a^3\,b^4+32\,a^2\,b^5-48\,a\,b^6+16\,b^7\right )}{8\,\left (a^2-2\,a\,b+b^2\right )\,\left (a^3\,b-2\,a^2\,b^2+a\,b^3\right )}\right )\,\sqrt {-a\,b}\,\left (a+b\right )}{4\,\left (a^3\,b-2\,a^2\,b^2+a\,b^3\right )}\right )}{4\,\left (a^3\,b-2\,a^2\,b^2+a\,b^3\right )}+\frac {\sqrt {-a\,b}\,\left (a+b\right )\,\left (\frac {\mathrm {tan}\left (e+f\,x\right )\,\left (a^2\,b+2\,a\,b^2+5\,b^3\right )}{2\,\left (a^2-2\,a\,b+b^2\right )}+\frac {\left (\frac {2\,a^4\,b^2-8\,a^3\,b^3+12\,a^2\,b^4-8\,a\,b^5+2\,b^6}{a^3-3\,a^2\,b+3\,a\,b^2-b^3}+\frac {\mathrm {tan}\left (e+f\,x\right )\,\sqrt {-a\,b}\,\left (a+b\right )\,\left (16\,a^5\,b^2-48\,a^4\,b^3+32\,a^3\,b^4+32\,a^2\,b^5-48\,a\,b^6+16\,b^7\right )}{8\,\left (a^2-2\,a\,b+b^2\right )\,\left (a^3\,b-2\,a^2\,b^2+a\,b^3\right )}\right )\,\sqrt {-a\,b}\,\left (a+b\right )}{4\,\left (a^3\,b-2\,a^2\,b^2+a\,b^3\right )}\right )}{4\,\left (a^3\,b-2\,a^2\,b^2+a\,b^3\right )}}\right )\,\sqrt {-a\,b}\,\left (a+b\right )\,1{}\mathrm {i}}{2\,f\,\left (a^3\,b-2\,a^2\,b^2+a\,b^3\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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