Optimal. Leaf size=123 \[ \frac {\left (a^2+8 a b+4 b^2\right ) \sec (e+f x)}{4 f}-\frac {\left (3 a^2+24 a b+8 b^2\right ) \tanh ^{-1}(\cos (e+f x))}{8 f}-\frac {a^2 \csc ^4(e+f x) \sec (e+f x)}{4 f}-\frac {a (a+8 b) \cot (e+f x) \csc (e+f x)}{8 f}+\frac {b^2 \sec ^3(e+f x)}{3 f} \]
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Rubi [A] time = 0.13, antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {3664, 463, 455, 1153, 207} \[ \frac {\left (a^2+8 a b+4 b^2\right ) \sec (e+f x)}{4 f}-\frac {\left (3 a^2+24 a b+8 b^2\right ) \tanh ^{-1}(\cos (e+f x))}{8 f}-\frac {a^2 \csc ^4(e+f x) \sec (e+f x)}{4 f}-\frac {a (a+8 b) \cot (e+f x) \csc (e+f x)}{8 f}+\frac {b^2 \sec ^3(e+f x)}{3 f} \]
Antiderivative was successfully verified.
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Rule 207
Rule 455
Rule 463
Rule 1153
Rule 3664
Rubi steps
\begin {align*} \int \csc ^5(e+f x) \left (a+b \tan ^2(e+f x)\right )^2 \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x^4 \left (a-b+b x^2\right )^2}{\left (-1+x^2\right )^3} \, dx,x,\sec (e+f x)\right )}{f}\\ &=-\frac {a^2 \csc ^4(e+f x) \sec (e+f x)}{4 f}+\frac {\operatorname {Subst}\left (\int \frac {x^4 \left (a^2+8 a b-4 b^2+4 b^2 x^2\right )}{\left (-1+x^2\right )^2} \, dx,x,\sec (e+f x)\right )}{4 f}\\ &=-\frac {a (a+8 b) \cot (e+f x) \csc (e+f x)}{8 f}-\frac {a^2 \csc ^4(e+f x) \sec (e+f x)}{4 f}-\frac {\operatorname {Subst}\left (\int \frac {-a (a+8 b)-2 a (a+8 b) x^2-8 b^2 x^4}{-1+x^2} \, dx,x,\sec (e+f x)\right )}{8 f}\\ &=-\frac {a (a+8 b) \cot (e+f x) \csc (e+f x)}{8 f}-\frac {a^2 \csc ^4(e+f x) \sec (e+f x)}{4 f}-\frac {\operatorname {Subst}\left (\int \left (-2 \left (a^2+8 a b+4 b^2\right )-8 b^2 x^2+\frac {-3 a^2-24 a b-8 b^2}{-1+x^2}\right ) \, dx,x,\sec (e+f x)\right )}{8 f}\\ &=-\frac {a (a+8 b) \cot (e+f x) \csc (e+f x)}{8 f}+\frac {\left (a^2+8 a b+4 b^2\right ) \sec (e+f x)}{4 f}-\frac {a^2 \csc ^4(e+f x) \sec (e+f x)}{4 f}+\frac {b^2 \sec ^3(e+f x)}{3 f}+\frac {\left (3 a^2+24 a b+8 b^2\right ) \operatorname {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sec (e+f x)\right )}{8 f}\\ &=-\frac {\left (3 a^2+24 a b+8 b^2\right ) \tanh ^{-1}(\cos (e+f x))}{8 f}-\frac {a (a+8 b) \cot (e+f x) \csc (e+f x)}{8 f}+\frac {\left (a^2+8 a b+4 b^2\right ) \sec (e+f x)}{4 f}-\frac {a^2 \csc ^4(e+f x) \sec (e+f x)}{4 f}+\frac {b^2 \sec ^3(e+f x)}{3 f}\\ \end {align*}
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Mathematica [B] time = 6.19, size = 447, normalized size = 3.63 \[ \frac {\left (3 a^2+24 a b+8 b^2\right ) \log \left (\sin \left (\frac {1}{2} (e+f x)\right )\right )}{8 f}+\frac {\left (-3 a^2-24 a b-8 b^2\right ) \log \left (\cos \left (\frac {1}{2} (e+f x)\right )\right )}{8 f}+\frac {\left (-3 a^2-8 a b\right ) \csc ^2\left (\frac {1}{2} (e+f x)\right )}{32 f}+\frac {\left (3 a^2+8 a b\right ) \sec ^2\left (\frac {1}{2} (e+f x)\right )}{32 f}-\frac {a^2 \csc ^4\left (\frac {1}{2} (e+f x)\right )}{64 f}+\frac {a^2 \sec ^4\left (\frac {1}{2} (e+f x)\right )}{64 f}+\frac {-12 a b \sin \left (\frac {1}{2} (e+f x)\right )-7 b^2 \sin \left (\frac {1}{2} (e+f x)\right )}{6 f \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )}+\frac {12 a b \sin \left (\frac {1}{2} (e+f x)\right )+7 b^2 \sin \left (\frac {1}{2} (e+f x)\right )}{6 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )}+\frac {b^2 \sin \left (\frac {1}{2} (e+f x)\right )}{6 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^3}+\frac {b^2}{12 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^2}+\frac {b^2}{12 f \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )^2}-\frac {b^2 \sin \left (\frac {1}{2} (e+f x)\right )}{6 f \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.54, size = 284, normalized size = 2.31 \[ \frac {6 \, {\left (3 \, a^{2} + 24 \, a b + 8 \, b^{2}\right )} \cos \left (f x + e\right )^{6} - 10 \, {\left (3 \, a^{2} + 24 \, a b + 8 \, b^{2}\right )} \cos \left (f x + e\right )^{4} + 16 \, {\left (6 \, a b + b^{2}\right )} \cos \left (f x + e\right )^{2} + 16 \, b^{2} - 3 \, {\left ({\left (3 \, a^{2} + 24 \, a b + 8 \, b^{2}\right )} \cos \left (f x + e\right )^{7} - 2 \, {\left (3 \, a^{2} + 24 \, a b + 8 \, b^{2}\right )} \cos \left (f x + e\right )^{5} + {\left (3 \, a^{2} + 24 \, a b + 8 \, b^{2}\right )} \cos \left (f x + e\right )^{3}\right )} \log \left (\frac {1}{2} \, \cos \left (f x + e\right ) + \frac {1}{2}\right ) + 3 \, {\left ({\left (3 \, a^{2} + 24 \, a b + 8 \, b^{2}\right )} \cos \left (f x + e\right )^{7} - 2 \, {\left (3 \, a^{2} + 24 \, a b + 8 \, b^{2}\right )} \cos \left (f x + e\right )^{5} + {\left (3 \, a^{2} + 24 \, a b + 8 \, b^{2}\right )} \cos \left (f x + e\right )^{3}\right )} \log \left (-\frac {1}{2} \, \cos \left (f x + e\right ) + \frac {1}{2}\right )}{48 \, {\left (f \cos \left (f x + e\right )^{7} - 2 \, f \cos \left (f x + e\right )^{5} + f \cos \left (f x + e\right )^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.64, size = 183, normalized size = 1.49 \[ -\frac {a^{2} \cot \left (f x +e \right ) \left (\csc ^{3}\left (f x +e \right )\right )}{4 f}-\frac {3 a^{2} \csc \left (f x +e \right ) \cot \left (f x +e \right )}{8 f}+\frac {3 a^{2} \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right )}{8 f}-\frac {a b}{f \sin \left (f x +e \right )^{2} \cos \left (f x +e \right )}+\frac {3 a b}{f \cos \left (f x +e \right )}+\frac {3 a b \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right )}{f}+\frac {b^{2}}{3 f \cos \left (f x +e \right )^{3}}+\frac {b^{2}}{f \cos \left (f x +e \right )}+\frac {b^{2} \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right )}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.69, size = 163, normalized size = 1.33 \[ -\frac {3 \, {\left (3 \, a^{2} + 24 \, a b + 8 \, b^{2}\right )} \log \left (\cos \left (f x + e\right ) + 1\right ) - 3 \, {\left (3 \, a^{2} + 24 \, a b + 8 \, b^{2}\right )} \log \left (\cos \left (f x + e\right ) - 1\right ) - \frac {2 \, {\left (3 \, {\left (3 \, a^{2} + 24 \, a b + 8 \, b^{2}\right )} \cos \left (f x + e\right )^{6} - 5 \, {\left (3 \, a^{2} + 24 \, a b + 8 \, b^{2}\right )} \cos \left (f x + e\right )^{4} + 8 \, {\left (6 \, a b + b^{2}\right )} \cos \left (f x + e\right )^{2} + 8 \, b^{2}\right )}}{\cos \left (f x + e\right )^{7} - 2 \, \cos \left (f x + e\right )^{5} + \cos \left (f x + e\right )^{3}}}{48 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 12.06, size = 243, normalized size = 1.98 \[ \frac {a^2\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4}{64\,f}+\frac {\ln \left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )\,\left (\frac {3\,a^2}{8}+3\,a\,b+b^2\right )}{f}-\frac {{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (\frac {5\,a^2}{4}+4\,b\,a\right )-{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8\,\left (2\,a^2+68\,a\,b+64\,b^2\right )-{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\left (\frac {21\,a^2}{4}+76\,a\,b+\frac {128\,b^2}{3}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6\,\left (\frac {23\,a^2}{4}+140\,a\,b+64\,b^2\right )+\frac {a^2}{4}}{f\,\left (-16\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{10}+48\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8-48\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6+16\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\right )}+\frac {{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (\frac {a^2}{8}+\frac {b\,a}{4}\right )}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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