Optimal. Leaf size=143 \[ -\frac {3 (a-b)^2 \tanh ^{-1}\left (\frac {\sqrt {a} \sec (e+f x)}{\sqrt {a+b \sec ^2(e+f x)-b}}\right )}{8 a^{5/2} f}-\frac {(5 a-3 b) \cot (e+f x) \csc (e+f x) \sqrt {a+b \sec ^2(e+f x)-b}}{8 a^2 f}-\frac {\cot ^3(e+f x) \csc (e+f x) \sqrt {a+b \sec ^2(e+f x)-b}}{4 a f} \]
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Rubi [A] time = 0.16, antiderivative size = 143, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {3664, 470, 527, 12, 377, 207} \[ -\frac {3 (a-b)^2 \tanh ^{-1}\left (\frac {\sqrt {a} \sec (e+f x)}{\sqrt {a+b \sec ^2(e+f x)-b}}\right )}{8 a^{5/2} f}-\frac {(5 a-3 b) \cot (e+f x) \csc (e+f x) \sqrt {a+b \sec ^2(e+f x)-b}}{8 a^2 f}-\frac {\cot ^3(e+f x) \csc (e+f x) \sqrt {a+b \sec ^2(e+f x)-b}}{4 a f} \]
Antiderivative was successfully verified.
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Rule 12
Rule 207
Rule 377
Rule 470
Rule 527
Rule 3664
Rubi steps
\begin {align*} \int \frac {\csc ^5(e+f x)}{\sqrt {a+b \tan ^2(e+f x)}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x^4}{\left (-1+x^2\right )^3 \sqrt {a-b+b x^2}} \, dx,x,\sec (e+f x)\right )}{f}\\ &=-\frac {\cot ^3(e+f x) \csc (e+f x) \sqrt {a-b+b \sec ^2(e+f x)}}{4 a f}-\frac {\operatorname {Subst}\left (\int \frac {-a+b-2 (2 a-b) x^2}{\left (-1+x^2\right )^2 \sqrt {a-b+b x^2}} \, dx,x,\sec (e+f x)\right )}{4 a f}\\ &=-\frac {(5 a-3 b) \cot (e+f x) \csc (e+f x) \sqrt {a-b+b \sec ^2(e+f x)}}{8 a^2 f}-\frac {\cot ^3(e+f x) \csc (e+f x) \sqrt {a-b+b \sec ^2(e+f x)}}{4 a f}-\frac {\operatorname {Subst}\left (\int -\frac {3 (a-b)^2}{\left (-1+x^2\right ) \sqrt {a-b+b x^2}} \, dx,x,\sec (e+f x)\right )}{8 a^2 f}\\ &=-\frac {(5 a-3 b) \cot (e+f x) \csc (e+f x) \sqrt {a-b+b \sec ^2(e+f x)}}{8 a^2 f}-\frac {\cot ^3(e+f x) \csc (e+f x) \sqrt {a-b+b \sec ^2(e+f x)}}{4 a f}+\frac {\left (3 (a-b)^2\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-1+x^2\right ) \sqrt {a-b+b x^2}} \, dx,x,\sec (e+f x)\right )}{8 a^2 f}\\ &=-\frac {(5 a-3 b) \cot (e+f x) \csc (e+f x) \sqrt {a-b+b \sec ^2(e+f x)}}{8 a^2 f}-\frac {\cot ^3(e+f x) \csc (e+f x) \sqrt {a-b+b \sec ^2(e+f x)}}{4 a f}+\frac {\left (3 (a-b)^2\right ) \operatorname {Subst}\left (\int \frac {1}{-1+a x^2} \, dx,x,\frac {\sec (e+f x)}{\sqrt {a-b+b \sec ^2(e+f x)}}\right )}{8 a^2 f}\\ &=-\frac {3 (a-b)^2 \tanh ^{-1}\left (\frac {\sqrt {a} \sec (e+f x)}{\sqrt {a-b+b \sec ^2(e+f x)}}\right )}{8 a^{5/2} f}-\frac {(5 a-3 b) \cot (e+f x) \csc (e+f x) \sqrt {a-b+b \sec ^2(e+f x)}}{8 a^2 f}-\frac {\cot ^3(e+f x) \csc (e+f x) \sqrt {a-b+b \sec ^2(e+f x)}}{4 a f}\\ \end {align*}
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Mathematica [A] time = 4.76, size = 278, normalized size = 1.94 \[ \frac {\sqrt {\sec ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)} \left (-\sqrt {2} \sqrt {a} \cot (e+f x) \csc (e+f x) \left (2 a \csc ^2(e+f x)+3 a-3 b\right )-\frac {3 (a-b)^2 \cos (e+f x) \sec ^2\left (\frac {1}{2} (e+f x)\right ) \left (\tanh ^{-1}\left (\frac {a-(a-2 b) \tan ^2\left (\frac {1}{2} (e+f x)\right )}{\sqrt {a} \sqrt {a \left (\tan ^2\left (\frac {1}{2} (e+f x)\right )-1\right )^2+4 b \tan ^2\left (\frac {1}{2} (e+f x)\right )}}\right )+\tanh ^{-1}\left (\frac {a \left (\tan ^2\left (\frac {1}{2} (e+f x)\right )-1\right )+2 b}{\sqrt {a} \sqrt {a \left (\tan ^2\left (\frac {1}{2} (e+f x)\right )-1\right )^2+4 b \tan ^2\left (\frac {1}{2} (e+f x)\right )}}\right )\right )}{\sqrt {\sec ^4\left (\frac {1}{2} (e+f x)\right ) ((a-b) \cos (2 (e+f x))+a+b)}}\right )}{16 a^{5/2} f} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.65, size = 437, normalized size = 3.06 \[ \left [\frac {3 \, {\left ({\left (a^{2} - 2 \, a b + b^{2}\right )} \cos \left (f x + e\right )^{4} - 2 \, {\left (a^{2} - 2 \, a b + b^{2}\right )} \cos \left (f x + e\right )^{2} + a^{2} - 2 \, a b + b^{2}\right )} \sqrt {a} \log \left (-\frac {2 \, {\left ({\left (a - b\right )} \cos \left (f x + e\right )^{2} - 2 \, \sqrt {a} \sqrt {\frac {{\left (a - b\right )} \cos \left (f x + e\right )^{2} + b}{\cos \left (f x + e\right )^{2}}} \cos \left (f x + e\right ) + a + b\right )}}{\cos \left (f x + e\right )^{2} - 1}\right ) + 2 \, {\left (3 \, {\left (a^{2} - a b\right )} \cos \left (f x + e\right )^{3} - {\left (5 \, a^{2} - 3 \, a b\right )} \cos \left (f x + e\right )\right )} \sqrt {\frac {{\left (a - b\right )} \cos \left (f x + e\right )^{2} + b}{\cos \left (f x + e\right )^{2}}}}{16 \, {\left (a^{3} f \cos \left (f x + e\right )^{4} - 2 \, a^{3} f \cos \left (f x + e\right )^{2} + a^{3} f\right )}}, \frac {3 \, {\left ({\left (a^{2} - 2 \, a b + b^{2}\right )} \cos \left (f x + e\right )^{4} - 2 \, {\left (a^{2} - 2 \, a b + b^{2}\right )} \cos \left (f x + e\right )^{2} + a^{2} - 2 \, a b + b^{2}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {-a} \sqrt {\frac {{\left (a - b\right )} \cos \left (f x + e\right )^{2} + b}{\cos \left (f x + e\right )^{2}}} \cos \left (f x + e\right )}{a}\right ) + {\left (3 \, {\left (a^{2} - a b\right )} \cos \left (f x + e\right )^{3} - {\left (5 \, a^{2} - 3 \, a b\right )} \cos \left (f x + e\right )\right )} \sqrt {\frac {{\left (a - b\right )} \cos \left (f x + e\right )^{2} + b}{\cos \left (f x + e\right )^{2}}}}{8 \, {\left (a^{3} f \cos \left (f x + e\right )^{4} - 2 \, a^{3} f \cos \left (f x + e\right )^{2} + a^{3} f\right )}}\right ] \]
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 [B] time = 1.42, size = 6334, normalized size = 44.29 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [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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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{{\sin \left (e+f\,x\right )}^5\,\sqrt {b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a}} \,d x \]
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 \[ \int \frac {\csc ^{5}{\left (e + f x \right )}}{\sqrt {a + b \tan ^{2}{\left (e + f x \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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