Optimal. Leaf size=161 \[ \frac{\cos ^5(c+d x) \left (1-\frac{a+b \tan (c+d x)}{a-\sqrt{-b^2}}\right )^{5/2} \left (1-\frac{a+b \tan (c+d x)}{a+\sqrt{-b^2}}\right )^{5/2} (a+b \tan (c+d x))^{n+1} F_1\left (n+1;\frac{5}{2},\frac{5}{2};n+2;\frac{a+b \tan (c+d x)}{a-\sqrt{-b^2}},\frac{a+b \tan (c+d x)}{a+\sqrt{-b^2}}\right )}{b d (n+1)} \]
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Rubi [A] time = 0.134122, antiderivative size = 161, normalized size of antiderivative = 1., 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 = {3512, 760, 133} \[ \frac{\cos ^5(c+d x) \left (1-\frac{a+b \tan (c+d x)}{a-\sqrt{-b^2}}\right )^{5/2} \left (1-\frac{a+b \tan (c+d x)}{a+\sqrt{-b^2}}\right )^{5/2} (a+b \tan (c+d x))^{n+1} F_1\left (n+1;\frac{5}{2},\frac{5}{2};n+2;\frac{a+b \tan (c+d x)}{a-\sqrt{-b^2}},\frac{a+b \tan (c+d x)}{a+\sqrt{-b^2}}\right )}{b d (n+1)} \]
Antiderivative was successfully verified.
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Rule 3512
Rule 760
Rule 133
Rubi steps
\begin{align*} \int \cos ^3(c+d x) (a+b \tan (c+d x))^n \, dx &=\frac{\left (\cos (c+d x) \sqrt{\sec ^2(c+d x)}\right ) \operatorname{Subst}\left (\int \frac{(a+x)^n}{\left (1+\frac{x^2}{b^2}\right )^{5/2}} \, dx,x,b \tan (c+d x)\right )}{b d}\\ &=\frac{\left (\cos ^5(c+d x) \left (1-\frac{a+b \tan (c+d x)}{a-\frac{b^2}{\sqrt{-b^2}}}\right )^{5/2} \left (1-\frac{a+b \tan (c+d x)}{a+\frac{b^2}{\sqrt{-b^2}}}\right )^{5/2}\right ) \operatorname{Subst}\left (\int \frac{x^n}{\left (1-\frac{x}{a-\sqrt{-b^2}}\right )^{5/2} \left (1-\frac{x}{a+\sqrt{-b^2}}\right )^{5/2}} \, dx,x,a+b \tan (c+d x)\right )}{b d}\\ &=\frac{F_1\left (1+n;\frac{5}{2},\frac{5}{2};2+n;\frac{a+b \tan (c+d x)}{a-\sqrt{-b^2}},\frac{a+b \tan (c+d x)}{a+\sqrt{-b^2}}\right ) \cos ^5(c+d x) (a+b \tan (c+d x))^{1+n} \left (1-\frac{a+b \tan (c+d x)}{a-\sqrt{-b^2}}\right )^{5/2} \left (1-\frac{a+b \tan (c+d x)}{a+\sqrt{-b^2}}\right )^{5/2}}{b d (1+n)}\\ \end{align*}
Mathematica [C] time = 6.1758, size = 341, normalized size = 2.12 \[ \frac{2 (n+2) \left (a^2+b^2\right )^2 \cos ^7(c+d x) (\tan (c+d x)-i) (\tan (c+d x)+i) (a+b \tan (c+d x))^{n+1} F_1\left (n+1;\frac{5}{2},\frac{5}{2};n+2;\frac{a+b \tan (c+d x)}{a-i b},\frac{a+b \tan (c+d x)}{a+i b}\right )}{b d (n+1) (a-i b) (a+i b) \left (2 (n+2) \left (a^2+b^2\right ) F_1\left (n+1;\frac{5}{2},\frac{5}{2};n+2;\frac{a+b \tan (c+d x)}{a-i b},\frac{a+b \tan (c+d x)}{a+i b}\right )+5 (a+b \tan (c+d x)) \left ((a-i b) F_1\left (n+2;\frac{5}{2},\frac{7}{2};n+3;\frac{a+b \tan (c+d x)}{a-i b},\frac{a+b \tan (c+d x)}{a+i b}\right )+(a+i b) F_1\left (n+2;\frac{7}{2},\frac{5}{2};n+3;\frac{a+b \tan (c+d x)}{a-i b},\frac{a+b \tan (c+d x)}{a+i b}\right )\right )\right )} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.613, size = 0, normalized size = 0. \begin{align*} \int \left ( \cos \left ( dx+c \right ) \right ) ^{3} \left ( a+b\tan \left ( dx+c \right ) \right ) ^{n}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \tan \left (d x + c\right ) + a\right )}^{n} \cos \left (d x + c\right )^{3}\,{d x} \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 \tan \left (d x + c\right ) + a\right )}^{n} \cos \left (d x + c\right )^{3}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [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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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \tan \left (d x + c\right ) + a\right )}^{n} \cos \left (d x + c\right )^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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