Optimal. Leaf size=219 \[ \frac{b \left (3 a^2-b^2\right ) \tan ^2(e+f x) \, _2F_1\left (1,\frac{1}{2} (n p+2);\frac{1}{2} (n p+4);-\tan ^2(e+f x)\right ) \left (c (d \tan (e+f x))^p\right )^n}{f (n p+2)}+\frac{a \left (a^2-3 b^2\right ) \tan (e+f x) \, _2F_1\left (1,\frac{1}{2} (n p+1);\frac{1}{2} (n p+3);-\tan ^2(e+f x)\right ) \left (c (d \tan (e+f x))^p\right )^n}{f (n p+1)}+\frac{3 a b^2 \tan (e+f x) \left (c (d \tan (e+f x))^p\right )^n}{f (n p+1)}+\frac{b^3 \tan ^2(e+f x) \left (c (d \tan (e+f x))^p\right )^n}{f (n p+2)} \]
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Rubi [A] time = 0.367151, antiderivative size = 219, normalized size of antiderivative = 1., 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 = {6677, 1802, 808, 364} \[ \frac{b \left (3 a^2-b^2\right ) \tan ^2(e+f x) \, _2F_1\left (1,\frac{1}{2} (n p+2);\frac{1}{2} (n p+4);-\tan ^2(e+f x)\right ) \left (c (d \tan (e+f x))^p\right )^n}{f (n p+2)}+\frac{a \left (a^2-3 b^2\right ) \tan (e+f x) \, _2F_1\left (1,\frac{1}{2} (n p+1);\frac{1}{2} (n p+3);-\tan ^2(e+f x)\right ) \left (c (d \tan (e+f x))^p\right )^n}{f (n p+1)}+\frac{3 a b^2 \tan (e+f x) \left (c (d \tan (e+f x))^p\right )^n}{f (n p+1)}+\frac{b^3 \tan ^2(e+f x) \left (c (d \tan (e+f x))^p\right )^n}{f (n p+2)} \]
Antiderivative was successfully verified.
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Rule 6677
Rule 1802
Rule 808
Rule 364
Rubi steps
\begin{align*} \int \left (c (d \tan (e+f x))^p\right )^n (a+b \tan (e+f x))^3 \, dx &=\frac{\operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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*}
Mathematica [A] time = 0.889318, size = 163, normalized size = 0.74 \[ \frac{\tan (e+f x) \left (c (d \tan (e+f x))^p\right )^n \left (a \left (a^2-3 b^2\right ) (n p+2) \, _2F_1\left (1,\frac{1}{2} (n p+1);\frac{1}{2} (n p+3);-\tan ^2(e+f x)\right )+b \left (\left (3 a^2-b^2\right ) (n p+1) \tan (e+f x) \, _2F_1\left (1,\frac{n p}{2}+1;\frac{n p}{2}+2;-\tan ^2(e+f x)\right )+b (3 a (n p+2)+b (n p+1) \tan (e+f x))\right )\right )}{f (n p+1) (n p+2)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.637, size = 0, normalized size = 0. \begin{align*} \int \left ( c \left ( d\tan \left ( fx+e \right ) \right ) ^{p} \right ) ^{n} \left ( a+b\tan \left ( fx+e \right ) \right ) ^{3}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (b^{3} \tan \left (f x + e\right )^{3} + 3 \, a b^{2} \tan \left (f x + e\right )^{2} + 3 \, a^{2} b \tan \left (f x + e\right ) + a^{3}\right )} \left (\left (d \tan \left (f x + e\right )\right )^{p} c\right )^{n}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \tan \left (f x + e\right ) + a\right )}^{3} \left (\left (d \tan \left (f x + e\right )\right )^{p} c\right )^{n}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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