Optimal. Leaf size=79 \[ -\frac {3 (a+4 b) \tanh ^{-1}(\cos (e+f x))}{8 f}-\frac {(5 a+4 b) \cot (e+f x) \csc (e+f x)}{8 f}-\frac {a \cot ^3(e+f x) \csc (e+f x)}{4 f}+\frac {b \sec (e+f x)}{f} \]
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Rubi [A] time = 0.07, antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {3664, 455, 1157, 388, 207} \[ -\frac {3 (a+4 b) \tanh ^{-1}(\cos (e+f x))}{8 f}-\frac {(5 a+4 b) \cot (e+f x) \csc (e+f x)}{8 f}-\frac {a \cot ^3(e+f x) \csc (e+f x)}{4 f}+\frac {b \sec (e+f x)}{f} \]
Antiderivative was successfully verified.
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Rule 207
Rule 388
Rule 455
Rule 1157
Rule 3664
Rubi steps
\begin {align*} \int \csc ^5(e+f x) \left (a+b \tan ^2(e+f x)\right ) \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x^4 \left (a-b+b x^2\right )}{\left (-1+x^2\right )^3} \, dx,x,\sec (e+f x)\right )}{f}\\ &=-\frac {a \cot ^3(e+f x) \csc (e+f x)}{4 f}-\frac {\operatorname {Subst}\left (\int \frac {-a-4 a x^2-4 b x^4}{\left (-1+x^2\right )^2} \, dx,x,\sec (e+f x)\right )}{4 f}\\ &=-\frac {(5 a+4 b) \cot (e+f x) \csc (e+f x)}{8 f}-\frac {a \cot ^3(e+f x) \csc (e+f x)}{4 f}-\frac {\operatorname {Subst}\left (\int \frac {-3 a-4 b-8 b x^2}{-1+x^2} \, dx,x,\sec (e+f x)\right )}{8 f}\\ &=-\frac {(5 a+4 b) \cot (e+f x) \csc (e+f x)}{8 f}-\frac {a \cot ^3(e+f x) \csc (e+f x)}{4 f}+\frac {b \sec (e+f x)}{f}+\frac {(3 (a+4 b)) \operatorname {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sec (e+f x)\right )}{8 f}\\ &=-\frac {3 (a+4 b) \tanh ^{-1}(\cos (e+f x))}{8 f}-\frac {(5 a+4 b) \cot (e+f x) \csc (e+f x)}{8 f}-\frac {a \cot ^3(e+f x) \csc (e+f x)}{4 f}+\frac {b \sec (e+f x)}{f}\\ \end {align*}
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Mathematica [B] time = 6.06, size = 276, normalized size = 3.49 \[ -\frac {a \csc ^4\left (\frac {1}{2} (e+f x)\right )}{64 f}-\frac {3 a \csc ^2\left (\frac {1}{2} (e+f x)\right )}{32 f}+\frac {a \sec ^4\left (\frac {1}{2} (e+f x)\right )}{64 f}+\frac {3 a \sec ^2\left (\frac {1}{2} (e+f x)\right )}{32 f}+\frac {3 a \log \left (\sin \left (\frac {1}{2} (e+f x)\right )\right )}{8 f}-\frac {3 a \log \left (\cos \left (\frac {1}{2} (e+f x)\right )\right )}{8 f}-\frac {b \csc ^2\left (\frac {1}{2} (e+f x)\right )}{8 f}+\frac {b \sec ^2\left (\frac {1}{2} (e+f x)\right )}{8 f}+\frac {3 b \log \left (\sin \left (\frac {1}{2} (e+f x)\right )\right )}{2 f}-\frac {3 b \log \left (\cos \left (\frac {1}{2} (e+f x)\right )\right )}{2 f}+\frac {b \sin \left (\frac {1}{2} (e+f x)\right )}{f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )}-\frac {b \sin \left (\frac {1}{2} (e+f x)\right )}{f \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.70, size = 178, normalized size = 2.25 \[ \frac {6 \, {\left (a + 4 \, b\right )} \cos \left (f x + e\right )^{4} - 10 \, {\left (a + 4 \, b\right )} \cos \left (f x + e\right )^{2} - 3 \, {\left ({\left (a + 4 \, b\right )} \cos \left (f x + e\right )^{5} - 2 \, {\left (a + 4 \, b\right )} \cos \left (f x + e\right )^{3} + {\left (a + 4 \, b\right )} \cos \left (f x + e\right )\right )} \log \left (\frac {1}{2} \, \cos \left (f x + e\right ) + \frac {1}{2}\right ) + 3 \, {\left ({\left (a + 4 \, b\right )} \cos \left (f x + e\right )^{5} - 2 \, {\left (a + 4 \, b\right )} \cos \left (f x + e\right )^{3} + {\left (a + 4 \, b\right )} \cos \left (f x + e\right )\right )} \log \left (-\frac {1}{2} \, \cos \left (f x + e\right ) + \frac {1}{2}\right ) + 16 \, b}{16 \, {\left (f \cos \left (f x + e\right )^{5} - 2 \, f \cos \left (f x + e\right )^{3} + f \cos \left (f x + e\right )\right )}} \]
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 = 0.53, size = 120, normalized size = 1.52 \[ -\frac {a \cot \left (f x +e \right ) \left (\csc ^{3}\left (f x +e \right )\right )}{4 f}-\frac {3 a \cot \left (f x +e \right ) \csc \left (f x +e \right )}{8 f}+\frac {3 a \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right )}{8 f}-\frac {b}{2 f \sin \left (f x +e \right )^{2} \cos \left (f x +e \right )}+\frac {3 b}{2 f \cos \left (f x +e \right )}+\frac {3 b \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right )}{2 f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.58, size = 101, normalized size = 1.28 \[ -\frac {3 \, {\left (a + 4 \, b\right )} \log \left (\cos \left (f x + e\right ) + 1\right ) - 3 \, {\left (a + 4 \, b\right )} \log \left (\cos \left (f x + e\right ) - 1\right ) - \frac {2 \, {\left (3 \, {\left (a + 4 \, b\right )} \cos \left (f x + e\right )^{4} - 5 \, {\left (a + 4 \, b\right )} \cos \left (f x + e\right )^{2} + 8 \, b\right )}}{\cos \left (f x + e\right )^{5} - 2 \, \cos \left (f x + e\right )^{3} + \cos \left (f x + e\right )}}{16 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 11.97, size = 138, normalized size = 1.75 \[ \frac {{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (\frac {a}{8}+\frac {b}{8}\right )}{f}-\frac {\left (-2\,a-34\,b\right )\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4+\left (\frac {7\,a}{4}+2\,b\right )\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+\frac {a}{4}}{f\,\left (16\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4-16\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6\right )}+\frac {\ln \left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )\,\left (\frac {3\,a}{8}+\frac {3\,b}{2}\right )}{f}+\frac {a\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4}{64\,f} \]
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 \[ \int \left (a + b \tan ^{2}{\left (e + f x \right )}\right ) \csc ^{5}{\left (e + f x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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