Optimal. Leaf size=192 \[ -\frac {\left (15 a^2+20 a b (p+1)+4 b^2 \left (p^2+3 p+2\right )\right ) \cot (e+f x) \left (a+b \tan ^2(e+f x)+b\right )^p \left (\frac {b \tan ^2(e+f x)}{a+b}+1\right )^{-p} \, _2F_1\left (-\frac {1}{2},-p;\frac {1}{2};-\frac {b \tan ^2(e+f x)}{a+b}\right )}{15 f (a+b)^2}-\frac {\cot ^5(e+f x) \left (a+b \tan ^2(e+f x)+b\right )^{p+1}}{5 f (a+b)}-\frac {(10 a+b (2 p+7)) \cot ^3(e+f x) \left (a+b \tan ^2(e+f x)+b\right )^{p+1}}{15 f (a+b)^2} \]
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Rubi [A] time = 0.17, antiderivative size = 192, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {4132, 462, 453, 365, 364} \[ -\frac {\left (15 a^2+20 a b (p+1)+4 b^2 \left (p^2+3 p+2\right )\right ) \cot (e+f x) \left (a+b \tan ^2(e+f x)+b\right )^p \left (\frac {b \tan ^2(e+f x)}{a+b}+1\right )^{-p} \, _2F_1\left (-\frac {1}{2},-p;\frac {1}{2};-\frac {b \tan ^2(e+f x)}{a+b}\right )}{15 f (a+b)^2}-\frac {\cot ^5(e+f x) \left (a+b \tan ^2(e+f x)+b\right )^{p+1}}{5 f (a+b)}-\frac {(10 a+b (2 p+7)) \cot ^3(e+f x) \left (a+b \tan ^2(e+f x)+b\right )^{p+1}}{15 f (a+b)^2} \]
Antiderivative was successfully verified.
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Rule 364
Rule 365
Rule 453
Rule 462
Rule 4132
Rubi steps
\begin {align*} \int \csc ^6(e+f x) \left (a+b \sec ^2(e+f x)\right )^p \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (1+x^2\right )^2 \left (a+b+b x^2\right )^p}{x^6} \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac {\cot ^5(e+f x) \left (a+b+b \tan ^2(e+f x)\right )^{1+p}}{5 (a+b) f}+\frac {\operatorname {Subst}\left (\int \frac {\left (a+b+b x^2\right )^p \left (10 a+b (7+2 p)+5 (a+b) x^2\right )}{x^4} \, dx,x,\tan (e+f x)\right )}{5 (a+b) f}\\ &=-\frac {(10 a+b (7+2 p)) \cot ^3(e+f x) \left (a+b+b \tan ^2(e+f x)\right )^{1+p}}{15 (a+b)^2 f}-\frac {\cot ^5(e+f x) \left (a+b+b \tan ^2(e+f x)\right )^{1+p}}{5 (a+b) f}+\frac {\left (15 a^2+20 a b (1+p)+4 b^2 \left (2+3 p+p^2\right )\right ) \operatorname {Subst}\left (\int \frac {\left (a+b+b x^2\right )^p}{x^2} \, dx,x,\tan (e+f x)\right )}{15 (a+b)^2 f}\\ &=-\frac {(10 a+b (7+2 p)) \cot ^3(e+f x) \left (a+b+b \tan ^2(e+f x)\right )^{1+p}}{15 (a+b)^2 f}-\frac {\cot ^5(e+f x) \left (a+b+b \tan ^2(e+f x)\right )^{1+p}}{5 (a+b) f}+\frac {\left (\left (15 a^2+20 a b (1+p)+4 b^2 \left (2+3 p+p^2\right )\right ) \left (a+b+b \tan ^2(e+f x)\right )^p \left (1+\frac {b \tan ^2(e+f x)}{a+b}\right )^{-p}\right ) \operatorname {Subst}\left (\int \frac {\left (1+\frac {b x^2}{a+b}\right )^p}{x^2} \, dx,x,\tan (e+f x)\right )}{15 (a+b)^2 f}\\ &=-\frac {(10 a+b (7+2 p)) \cot ^3(e+f x) \left (a+b+b \tan ^2(e+f x)\right )^{1+p}}{15 (a+b)^2 f}-\frac {\cot ^5(e+f x) \left (a+b+b \tan ^2(e+f x)\right )^{1+p}}{5 (a+b) f}-\frac {\left (15 a^2+20 a b (1+p)+4 b^2 \left (2+3 p+p^2\right )\right ) \cot (e+f x) \, _2F_1\left (-\frac {1}{2},-p;\frac {1}{2};-\frac {b \tan ^2(e+f x)}{a+b}\right ) \left (a+b+b \tan ^2(e+f x)\right )^p \left (1+\frac {b \tan ^2(e+f x)}{a+b}\right )^{-p}}{15 (a+b)^2 f}\\ \end {align*}
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Mathematica [A] time = 2.00, size = 149, normalized size = 0.78 \[ -\frac {\cot (e+f x) \left (\frac {b \tan ^2(e+f x)}{a+b}+1\right )^{-p} \left (a+b \sec ^2(e+f x)\right )^p \left (15 \, _2F_1\left (-\frac {1}{2},-p;\frac {1}{2};-\frac {b \tan ^2(e+f x)}{a+b}\right )+3 \cot ^4(e+f x) \, _2F_1\left (-\frac {5}{2},-p;-\frac {3}{2};-\frac {b \tan ^2(e+f x)}{a+b}\right )+10 \cot ^2(e+f x) \, _2F_1\left (-\frac {3}{2},-p;-\frac {1}{2};-\frac {b \tan ^2(e+f x)}{a+b}\right )\right )}{15 f} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.59, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b \sec \left (f x + e\right )^{2} + a\right )}^{p} \csc \left (f x + e\right )^{6}, x\right ) \]
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 \[ \int {\left (b \sec \left (f x + e\right )^{2} + a\right )}^{p} \csc \left (f x + e\right )^{6}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.75, size = 0, normalized size = 0.00 \[ \int \left (\csc ^{6}\left (f x +e \right )\right ) \left (a +b \left (\sec ^{2}\left (f x +e \right )\right )\right )^{p}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \sec \left (f x + e\right )^{2} + a\right )}^{p} \csc \left (f x + e\right )^{6}\,{d x} \]
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 \[ \int \frac {{\left (a+\frac {b}{{\cos \left (e+f\,x\right )}^2}\right )}^p}{{\sin \left (e+f\,x\right )}^6} \,d x \]
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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