Optimal. Leaf size=219 \[ \frac {3 a b^2 \tan (e+f x) \left (c (d \tan (e+f x))^p\right )^n}{f (1+n p)}+\frac {a \left (a^2-3 b^2\right ) \, _2F_1\left (1,\frac {1}{2} (1+n p);\frac {1}{2} (3+n p);-\tan ^2(e+f x)\right ) \tan (e+f x) \left (c (d \tan (e+f x))^p\right )^n}{f (1+n p)}+\frac {b^3 \tan ^2(e+f x) \left (c (d \tan (e+f x))^p\right )^n}{f (2+n p)}+\frac {b \left (3 a^2-b^2\right ) \, _2F_1\left (1,\frac {1}{2} (2+n p);\frac {1}{2} (4+n p);-\tan ^2(e+f x)\right ) \tan ^2(e+f x) \left (c (d \tan (e+f x))^p\right )^n}{f (2+n p)} \]
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Rubi [A]
time = 0.20, antiderivative size = 219, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {1970, 1816,
822, 371} \begin {gather*} \frac {b \left (3 a^2-b^2\right ) \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 (n p+2)}+\frac {a \left (a^2-3 b^2\right ) \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 (n p+1)}+\frac {3 a b^2 \tan (e+f x) \left (c (d \tan (e+f x))^p\right )^n}{f (n p+1)}+\frac {b^3 \tan ^2(e+f x) \left (c (d \tan (e+f x))^p\right )^n}{f (n p+2)} \end {gather*}
Antiderivative was successfully verified.
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Rule 371
Rule 822
Rule 1816
Rule 1970
Rubi steps
\begin {align*} \int \left (c (d \tan (e+f x))^p\right )^n (a+b \tan (e+f x))^3 \, dx &=\frac {\text {Subst}\left (\int \frac {\left (c (d x)^p\right )^n (a+b x)^3}{1+x^2} \, 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)^3}{1+x^2} \, 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 (3 a b^2 (d x)^{n p}+\frac {b^3 (d x)^{1+n p}}{d}+\frac {(d x)^{n p} \left (a^3-3 a b^2+b \left (3 a^2-b^2\right ) x\right )}{1+x^2}\right ) \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {3 a b^2 \tan (e+f x) \left (c (d \tan (e+f x))^p\right )^n}{f (1+n p)}+\frac {b^3 \tan ^2(e+f x) \left (c (d \tan (e+f x))^p\right )^n}{f (2+n p)}+\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} \left (a^3-3 a b^2+b \left (3 a^2-b^2\right ) x\right )}{1+x^2} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {3 a b^2 \tan (e+f x) \left (c (d \tan (e+f x))^p\right )^n}{f (1+n p)}+\frac {b^3 \tan ^2(e+f x) \left (c (d \tan (e+f x))^p\right )^n}{f (2+n p)}+\frac {\left (a \left (a^2-3 b^2\right ) (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 )}{f}+\frac {\left (b \left (3 a^2-b^2\right ) (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 )}{d f}\\ &=\frac {3 a b^2 \tan (e+f x) \left (c (d \tan (e+f x))^p\right )^n}{f (1+n p)}+\frac {a \left (a^2-3 b^2\right ) \, _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}{f (1+n p)}+\frac {b^3 \tan ^2(e+f x) \left (c (d \tan (e+f x))^p\right )^n}{f (2+n p)}+\frac {b \left (3 a^2-b^2\right ) \, _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}{f (2+n p)}\\ \end {align*}
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Mathematica [A]
time = 0.98, size = 163, normalized size = 0.74 \begin {gather*} \frac {\tan (e+f x) \left (c (d \tan (e+f x))^p\right )^n \left (a \left (a^2-3 b^2\right ) (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 (\left (3 a^2-b^2\right ) (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)+b (3 a (2+n p)+b (1+n p) \tan (e+f x))\right )\right )}{f (1+n p) (2+n p)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.23, size = 0, normalized size = 0.00 \[\int \left (c \left (d \tan \left (f x +e \right )\right )^{p}\right )^{n} \left (a +b \tan \left (f x +e \right )\right )^{3}\, dx\]
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 (c \left (d \tan {\left (e + f x \right )}\right )^{p}\right )^{n} \left (a + b \tan {\left (e + f x \right )}\right )^{3}\, 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 {\left (c\,{\left (d\,\mathrm {tan}\left (e+f\,x\right )\right )}^p\right )}^n\,{\left (a+b\,\mathrm {tan}\left (e+f\,x\right )\right )}^3 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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