Optimal. Leaf size=216 \[ \frac {a \, _2F_1\left (1,\frac {1}{2} (1+n p);\frac {1}{2} (3+n p);-\tan ^2(e+f x)\right ) \tan (e+f x) \left (c (d \tan (e+f x))^p\right )^n}{\left (a^2+b^2\right ) f (1+n p)}+\frac {b^2 \, _2F_1\left (1,1+n p;2+n p;-\frac {b \tan (e+f x)}{a}\right ) \tan (e+f x) \left (c (d \tan (e+f x))^p\right )^n}{a \left (a^2+b^2\right ) f (1+n p)}-\frac {b \, _2F_1\left (1,\frac {1}{2} (2+n p);\frac {1}{2} (4+n p);-\tan ^2(e+f x)\right ) \tan ^2(e+f x) \left (c (d \tan (e+f x))^p\right )^n}{\left (a^2+b^2\right ) f (2+n p)} \]
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Rubi [A]
time = 0.21, antiderivative size = 216, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {1970, 975, 66,
822, 371} \begin {gather*} -\frac {b \tan ^2(e+f x) \, _2F_1\left (1,\frac {1}{2} (n p+2);\frac {1}{2} (n p+4);-\tan ^2(e+f x)\right ) \left (c (d \tan (e+f x))^p\right )^n}{f \left (a^2+b^2\right ) (n p+2)}+\frac {a \tan (e+f x) \, _2F_1\left (1,\frac {1}{2} (n p+1);\frac {1}{2} (n p+3);-\tan ^2(e+f x)\right ) \left (c (d \tan (e+f x))^p\right )^n}{f \left (a^2+b^2\right ) (n p+1)}+\frac {b^2 \tan (e+f x) \left (c (d \tan (e+f x))^p\right )^n \, _2F_1\left (1,n p+1;n p+2;-\frac {b \tan (e+f x)}{a}\right )}{a f \left (a^2+b^2\right ) (n p+1)} \end {gather*}
Antiderivative was successfully verified.
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Rule 66
Rule 371
Rule 822
Rule 975
Rule 1970
Rubi steps
\begin {align*} \int \frac {\left (c (d \tan (e+f x))^p\right )^n}{a+b \tan (e+f x)} \, dx &=\frac {\text {Subst}\left (\int \frac {\left (c (d x)^p\right )^n}{(a+b x) \left (1+x^2\right )} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {\left ((d \tan (e+f x))^{-n p} \left (c (d \tan (e+f x))^p\right )^n\right ) \text {Subst}\left (\int \frac {(d x)^{n p}}{(a+b x) \left (1+x^2\right )} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {\left ((d \tan (e+f x))^{-n p} \left (c (d \tan (e+f x))^p\right )^n\right ) \text {Subst}\left (\int \left (\frac {b^2 (d x)^{n p}}{\left (a^2+b^2\right ) (a+b x)}+\frac {(d x)^{n p} (a-b x)}{\left (a^2+b^2\right ) \left (1+x^2\right )}\right ) \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {\left ((d \tan (e+f x))^{-n p} \left (c (d \tan (e+f x))^p\right )^n\right ) \text {Subst}\left (\int \frac {(d x)^{n p} (a-b x)}{1+x^2} \, dx,x,\tan (e+f x)\right )}{\left (a^2+b^2\right ) f}+\frac {\left (b^2 (d \tan (e+f x))^{-n p} \left (c (d \tan (e+f x))^p\right )^n\right ) \text {Subst}\left (\int \frac {(d x)^{n p}}{a+b x} \, dx,x,\tan (e+f x)\right )}{\left (a^2+b^2\right ) f}\\ &=\frac {b^2 \, _2F_1\left (1,1+n p;2+n p;-\frac {b \tan (e+f x)}{a}\right ) \tan (e+f x) \left (c (d \tan (e+f x))^p\right )^n}{a \left (a^2+b^2\right ) f (1+n p)}+\frac {\left (a (d \tan (e+f x))^{-n p} \left (c (d \tan (e+f x))^p\right )^n\right ) \text {Subst}\left (\int \frac {(d x)^{n p}}{1+x^2} \, dx,x,\tan (e+f x)\right )}{\left (a^2+b^2\right ) f}-\frac {\left (b (d \tan (e+f x))^{-n p} \left (c (d \tan (e+f x))^p\right )^n\right ) \text {Subst}\left (\int \frac {(d x)^{1+n p}}{1+x^2} \, dx,x,\tan (e+f x)\right )}{\left (a^2+b^2\right ) d f}\\ &=\frac {a \, _2F_1\left (1,\frac {1}{2} (1+n p);\frac {1}{2} (3+n p);-\tan ^2(e+f x)\right ) \tan (e+f x) \left (c (d \tan (e+f x))^p\right )^n}{\left (a^2+b^2\right ) f (1+n p)}+\frac {b^2 \, _2F_1\left (1,1+n p;2+n p;-\frac {b \tan (e+f x)}{a}\right ) \tan (e+f x) \left (c (d \tan (e+f x))^p\right )^n}{a \left (a^2+b^2\right ) f (1+n p)}-\frac {b \, _2F_1\left (1,\frac {1}{2} (2+n p);\frac {1}{2} (4+n p);-\tan ^2(e+f x)\right ) \tan ^2(e+f x) \left (c (d \tan (e+f x))^p\right )^n}{\left (a^2+b^2\right ) f (2+n p)}\\ \end {align*}
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Mathematica [A]
time = 0.76, size = 166, normalized size = 0.77 \begin {gather*} \frac {\tan (e+f x) \left (c (d \tan (e+f x))^p\right )^n \left (a^2 (2+n p) \, _2F_1\left (1,\frac {1}{2} (1+n p);\frac {1}{2} (3+n p);-\tan ^2(e+f x)\right )+b \left (b (2+n p) \, _2F_1\left (1,1+n p;2+n p;-\frac {b \tan (e+f x)}{a}\right )-a (1+n p) \, _2F_1\left (1,1+\frac {n p}{2};2+\frac {n p}{2};-\tan ^2(e+f x)\right ) \tan (e+f x)\right )\right )}{a \left (a^2+b^2\right ) f (1+n p) (2+n p)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.33, size = 0, normalized size = 0.00 \[\int \frac {\left (c \left (d \tan \left (f x +e \right )\right )^{p}\right )^{n}}{a +b \tan \left (f x +e \right )}\, dx\]
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 \frac {\left (c \left (d \tan {\left (e + f x \right )}\right )^{p}\right )^{n}}{a + b \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 = -0.00 \begin {gather*} \int \frac {{\left (c\,{\left (d\,\mathrm {tan}\left (e+f\,x\right )\right )}^p\right )}^n}{a+b\,\mathrm {tan}\left (e+f\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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