Optimal. Leaf size=59 \[ -\frac {\, _2F_1\left (1,1+p;2+p;1+\frac {b (c \sec (e+f x))^n}{a}\right ) \left (a+b (c \sec (e+f x))^n\right )^{1+p}}{a f n (1+p)} \]
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Rubi [A]
time = 0.06, antiderivative size = 59, 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 = {4224, 374, 12,
272, 67} \begin {gather*} -\frac {\left (a+b (c \sec (e+f x))^n\right )^{p+1} \, _2F_1\left (1,p+1;p+2;\frac {b (c \sec (e+f x))^n}{a}+1\right )}{a f n (p+1)} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 67
Rule 272
Rule 374
Rule 4224
Rubi steps
\begin {align*} \int \left (a+b (c \sec (e+f x))^n\right )^p \tan (e+f x) \, dx &=\frac {\text {Subst}\left (\int \frac {\left (a+b (c x)^n\right )^p}{x} \, dx,x,\sec (e+f x)\right )}{f}\\ &=\frac {\text {Subst}\left (\int \frac {c \left (a+b x^n\right )^p}{x} \, dx,x,c \sec (e+f x)\right )}{c f}\\ &=\frac {\text {Subst}\left (\int \frac {\left (a+b x^n\right )^p}{x} \, dx,x,c \sec (e+f x)\right )}{f}\\ &=\frac {\text {Subst}\left (\int \frac {(a+b x)^p}{x} \, dx,x,(c \sec (e+f x))^n\right )}{f n}\\ &=-\frac {\, _2F_1\left (1,1+p;2+p;1+\frac {b (c \sec (e+f x))^n}{a}\right ) \left (a+b (c \sec (e+f x))^n\right )^{1+p}}{a f n (1+p)}\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 59, normalized size = 1.00 \begin {gather*} -\frac {\, _2F_1\left (1,1+p;2+p;1+\frac {b (c \sec (e+f x))^n}{a}\right ) \left (a+b (c \sec (e+f x))^n\right )^{1+p}}{a f n (1+p)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.04, size = 0, normalized size = 0.00 \[\int \left (a +b \left (c \sec \left (f x +e \right )\right )^{n}\right )^{p} \tan \left (f x +e \right )\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \left (a + b \left (c \sec {\left (e + f x \right )}\right )^{n}\right )^{p} \tan {\left (e + f x \right )}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \mathrm {tan}\left (e+f\,x\right )\,{\left (a+b\,{\left (\frac {c}{\cos \left (e+f\,x\right )}\right )}^n\right )}^p \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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