Optimal. Leaf size=76 \[ -\frac {\sqrt {b} (a-b) \tan ^{-1}\left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a}}\right )}{a^{5/2} f}-\frac {(a-b) \cot (e+f x)}{a^2 f}-\frac {\cot ^3(e+f x)}{3 a f} \]
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Rubi [A] time = 0.09, antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3663, 453, 325, 205} \[ -\frac {\sqrt {b} (a-b) \tan ^{-1}\left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a}}\right )}{a^{5/2} f}-\frac {(a-b) \cot (e+f x)}{a^2 f}-\frac {\cot ^3(e+f x)}{3 a f} \]
Antiderivative was successfully verified.
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Rule 205
Rule 325
Rule 453
Rule 3663
Rubi steps
\begin {align*} \int \frac {\csc ^4(e+f x)}{a+b \tan ^2(e+f x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1+x^2}{x^4 \left (a+b x^2\right )} \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac {\cot ^3(e+f x)}{3 a f}+\frac {(a-b) \operatorname {Subst}\left (\int \frac {1}{x^2 \left (a+b x^2\right )} \, dx,x,\tan (e+f x)\right )}{a f}\\ &=-\frac {(a-b) \cot (e+f x)}{a^2 f}-\frac {\cot ^3(e+f x)}{3 a f}-\frac {((a-b) b) \operatorname {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\tan (e+f x)\right )}{a^2 f}\\ &=-\frac {(a-b) \sqrt {b} \tan ^{-1}\left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a}}\right )}{a^{5/2} f}-\frac {(a-b) \cot (e+f x)}{a^2 f}-\frac {\cot ^3(e+f x)}{3 a f}\\ \end {align*}
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Mathematica [A] time = 0.31, size = 73, normalized size = 0.96 \[ \frac {3 \sqrt {b} (b-a) \tan ^{-1}\left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a}}\right )-\sqrt {a} \cot (e+f x) \left (a \csc ^2(e+f x)+2 a-3 b\right )}{3 a^{5/2} f} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.47, size = 373, normalized size = 4.91 \[ \left [-\frac {4 \, {\left (2 \, a - 3 \, b\right )} \cos \left (f x + e\right )^{3} + 3 \, {\left ({\left (a - b\right )} \cos \left (f x + e\right )^{2} - a + b\right )} \sqrt {-\frac {b}{a}} \log \left (\frac {{\left (a^{2} + 6 \, a b + b^{2}\right )} \cos \left (f x + e\right )^{4} - 2 \, {\left (3 \, a b + b^{2}\right )} \cos \left (f x + e\right )^{2} - 4 \, {\left ({\left (a^{2} + a b\right )} \cos \left (f x + e\right )^{3} - a b \cos \left (f x + e\right )\right )} \sqrt {-\frac {b}{a}} \sin \left (f x + e\right ) + b^{2}}{{\left (a^{2} - 2 \, a b + b^{2}\right )} \cos \left (f x + e\right )^{4} + 2 \, {\left (a b - b^{2}\right )} \cos \left (f x + e\right )^{2} + b^{2}}\right ) \sin \left (f x + e\right ) - 12 \, {\left (a - b\right )} \cos \left (f x + e\right )}{12 \, {\left (a^{2} f \cos \left (f x + e\right )^{2} - a^{2} f\right )} \sin \left (f x + e\right )}, -\frac {2 \, {\left (2 \, a - 3 \, b\right )} \cos \left (f x + e\right )^{3} - 3 \, {\left ({\left (a - b\right )} \cos \left (f x + e\right )^{2} - a + b\right )} \sqrt {\frac {b}{a}} \arctan \left (\frac {{\left ({\left (a + b\right )} \cos \left (f x + e\right )^{2} - b\right )} \sqrt {\frac {b}{a}}}{2 \, b \cos \left (f x + e\right ) \sin \left (f x + e\right )}\right ) \sin \left (f x + e\right ) - 6 \, {\left (a - b\right )} \cos \left (f x + e\right )}{6 \, {\left (a^{2} f \cos \left (f x + e\right )^{2} - a^{2} f\right )} \sin \left (f x + e\right )}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 2.99, size = 97, normalized size = 1.28 \[ -\frac {\frac {3 \, {\left (\pi \left \lfloor \frac {f x + e}{\pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\relax (b) + \arctan \left (\frac {b \tan \left (f x + e\right )}{\sqrt {a b}}\right )\right )} {\left (a b - b^{2}\right )}}{\sqrt {a b} a^{2}} + \frac {3 \, a \tan \left (f x + e\right )^{2} - 3 \, b \tan \left (f x + e\right )^{2} + a}{a^{2} \tan \left (f x + e\right )^{3}}}{3 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.58, size = 107, normalized size = 1.41 \[ -\frac {b \arctan \left (\frac {\tan \left (f x +e \right ) b}{\sqrt {a b}}\right )}{f a \sqrt {a b}}+\frac {b^{2} \arctan \left (\frac {\tan \left (f x +e \right ) b}{\sqrt {a b}}\right )}{f \,a^{2} \sqrt {a b}}-\frac {1}{3 f a \tan \left (f x +e \right )^{3}}-\frac {1}{f a \tan \left (f x +e \right )}+\frac {b}{f \,a^{2} \tan \left (f x +e \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.88, size = 68, normalized size = 0.89 \[ -\frac {\frac {3 \, {\left (a b - b^{2}\right )} \arctan \left (\frac {b \tan \left (f x + e\right )}{\sqrt {a b}}\right )}{\sqrt {a b} a^{2}} + \frac {3 \, {\left (a - b\right )} \tan \left (f x + e\right )^{2} + a}{a^{2} \tan \left (f x + e\right )^{3}}}{3 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 11.05, size = 67, normalized size = 0.88 \[ -\frac {\frac {1}{3\,a}+\frac {{\mathrm {tan}\left (e+f\,x\right )}^2\,\left (a-b\right )}{a^2}}{f\,{\mathrm {tan}\left (e+f\,x\right )}^3}-\frac {\sqrt {b}\,\mathrm {atan}\left (\frac {\sqrt {b}\,\mathrm {tan}\left (e+f\,x\right )}{\sqrt {a}}\right )\,\left (a-b\right )}{a^{5/2}\,f} \]
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 \[ \int \frac {\csc ^{4}{\left (e + f x \right )}}{a + b \tan ^{2}{\left (e + f x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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