Optimal. Leaf size=74 \[ -\frac {(3 a-2 b) \cot (e+f x) \sqrt {a+b \tan ^2(e+f x)}}{3 a^2 f}-\frac {\cot ^3(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{3 a f} \]
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Rubi [A]
time = 0.06, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {3744, 464, 270}
\begin {gather*} -\frac {(3 a-2 b) \cot (e+f x) \sqrt {a+b \tan ^2(e+f x)}}{3 a^2 f}-\frac {\cot ^3(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{3 a f} \end {gather*}
Antiderivative was successfully verified.
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Rule 270
Rule 464
Rule 3744
Rubi steps
\begin {align*} \int \frac {\csc ^4(e+f x)}{\sqrt {a+b \tan ^2(e+f x)}} \, dx &=\frac {\text {Subst}\left (\int \frac {1+x^2}{x^4 \sqrt {a+b x^2}} \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac {\cot ^3(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{3 a f}+\frac {(3 a-2 b) \text {Subst}\left (\int \frac {1}{x^2 \sqrt {a+b x^2}} \, dx,x,\tan (e+f x)\right )}{3 a f}\\ &=-\frac {(3 a-2 b) \cot (e+f x) \sqrt {a+b \tan ^2(e+f x)}}{3 a^2 f}-\frac {\cot ^3(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{3 a f}\\ \end {align*}
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Mathematica [A]
time = 0.45, size = 68, normalized size = 0.92 \begin {gather*} -\frac {\cot (e+f x) \left (2 a-2 b+a \csc ^2(e+f x)\right ) \sqrt {(a+b+(a-b) \cos (2 (e+f x))) \sec ^2(e+f x)}}{3 \sqrt {2} a^2 f} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.30, size = 86, normalized size = 1.16
method | result | size |
default | \(\frac {\left (2 a \left (\cos ^{2}\left (f x +e \right )\right )-2 \left (\cos ^{2}\left (f x +e \right )\right ) b -3 a +2 b \right ) \sqrt {\frac {a \left (\cos ^{2}\left (f x +e \right )\right )-\left (\cos ^{2}\left (f x +e \right )\right ) b +b}{\cos \left (f x +e \right )^{2}}}\, \cos \left (f x +e \right )}{3 f \sin \left (f x +e \right )^{3} a^{2}}\) | \(86\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 93, normalized size = 1.26 \begin {gather*} -\frac {\frac {3 \, \sqrt {b \tan \left (f x + e\right )^{2} + a}}{a \tan \left (f x + e\right )} - \frac {2 \, \sqrt {b \tan \left (f x + e\right )^{2} + a} b}{a^{2} \tan \left (f x + e\right )} + \frac {\sqrt {b \tan \left (f x + e\right )^{2} + a}}{a \tan \left (f x + e\right )^{3}}}{3 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 5.27, size = 96, normalized size = 1.30 \begin {gather*} -\frac {{\left (2 \, {\left (a - b\right )} \cos \left (f x + e\right )^{3} - {\left (3 \, a - 2 \, b\right )} \cos \left (f x + e\right )\right )} \sqrt {\frac {{\left (a - b\right )} \cos \left (f x + e\right )^{2} + b}{\cos \left (f x + e\right )^{2}}}}{3 \, {\left (a^{2} f \cos \left (f x + e\right )^{2} - a^{2} f\right )} \sin \left (f x + e\right )} \end {gather*}
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 \begin {gather*} \int \frac {\csc ^{4}{\left (e + f x \right )}}{\sqrt {a + b \tan ^{2}{\left (e + f x \right )}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 18.82, size = 145, normalized size = 1.96 \begin {gather*} -\frac {2\,\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1\right )\,\sqrt {a+\frac {b\,{\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )}^2}{{\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1\right )}^2}}\,\left (a\,1{}\mathrm {i}-b\,1{}\mathrm {i}-a\,{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}\,4{}\mathrm {i}+a\,{\mathrm {e}}^{e\,4{}\mathrm {i}+f\,x\,4{}\mathrm {i}}\,1{}\mathrm {i}+b\,{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}\,2{}\mathrm {i}-b\,{\mathrm {e}}^{e\,4{}\mathrm {i}+f\,x\,4{}\mathrm {i}}\,1{}\mathrm {i}\right )}{3\,a^2\,f\,{\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}-1\right )}^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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