Optimal. Leaf size=81 \[ \frac {\sec (e+f x) \cos ^2(e+f x)^{\frac {1}{2} (n p+1)} \, _2F_1\left (\frac {n p}{2},\frac {1}{2} (n p+1);\frac {1}{2} (n p+2);\sin ^2(e+f x)\right ) \left (b (c \tan (e+f x))^n\right )^p}{f n p} \]
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Rubi [A] time = 0.12, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3659, 2601, 2577} \[ \frac {\sec (e+f x) \cos ^2(e+f x)^{\frac {1}{2} (n p+1)} \, _2F_1\left (\frac {n p}{2},\frac {1}{2} (n p+1);\frac {1}{2} (n p+2);\sin ^2(e+f x)\right ) \left (b (c \tan (e+f x))^n\right )^p}{f n p} \]
Antiderivative was successfully verified.
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Rule 2577
Rule 2601
Rule 3659
Rubi steps
\begin {align*} \int \csc (e+f x) \left (b (c \tan (e+f x))^n\right )^p \, dx &=\left ((c \tan (e+f x))^{-n p} \left (b (c \tan (e+f x))^n\right )^p\right ) \int \csc (e+f x) (c \tan (e+f x))^{n p} \, dx\\ &=\left (\cos ^{n p}(e+f x) \sin ^{-n p}(e+f x) \left (b (c \tan (e+f x))^n\right )^p\right ) \int \cos ^{-n p}(e+f x) \sin ^{-1+n p}(e+f x) \, dx\\ &=\frac {\cos ^2(e+f x)^{\frac {1}{2} (1+n p)} \, _2F_1\left (\frac {n p}{2},\frac {1}{2} (1+n p);\frac {1}{2} (2+n p);\sin ^2(e+f x)\right ) \sec (e+f x) \left (b (c \tan (e+f x))^n\right )^p}{f n p}\\ \end {align*}
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Mathematica [A] time = 0.20, size = 77, normalized size = 0.95 \[ \frac {\, _2F_1\left (\frac {n p}{2},n p;\frac {n p}{2}+1;\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) \left (\cos (e+f x) \sec ^2\left (\frac {1}{2} (e+f x)\right )\right )^{n p} \left (b (c \tan (e+f x))^n\right )^p}{f n p} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (\left (c \tan \left (f x + e\right )\right )^{n} b\right )^{p} \csc \left (f x + e\right ), x\right ) \]
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 \[ \int \left (\left (c \tan \left (f x + e\right )\right )^{n} b\right )^{p} \csc \left (f x + e\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 25.66, size = 0, normalized size = 0.00 \[ \int \csc \left (f x +e \right ) \left (b \left (c \tan \left (f x +e \right )\right )^{n}\right )^{p}\, dx \]
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 \[ \int \left (\left (c \tan \left (f x + e\right )\right )^{n} b\right )^{p} \csc \left (f x + e\right )\,{d x} \]
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 \[ \int \frac {{\left (b\,{\left (c\,\mathrm {tan}\left (e+f\,x\right )\right )}^n\right )}^p}{\sin \left (e+f\,x\right )} \,d x \]
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 \[ \int \left (b \left (c \tan {\left (e + f x \right )}\right )^{n}\right )^{p} \csc {\left (e + f x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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