Optimal. Leaf size=127 \[ \frac{\left (a+b \sin ^4(c+d x)\right )^{p+1} \, _2F_1\left (1,p+1;p+2;\frac{b \sin ^4(c+d x)}{a}+1\right )}{4 a d (p+1)}-\frac{\csc ^2(c+d x) \left (a+b \sin ^4(c+d x)\right )^p \left (\frac{b \sin ^4(c+d x)}{a}+1\right )^{-p} \, _2F_1\left (-\frac{1}{2},-p;\frac{1}{2};-\frac{b \sin ^4(c+d x)}{a}\right )}{2 d} \]
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Rubi [A] time = 0.101792, antiderivative size = 127, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261, Rules used = {3229, 764, 365, 364, 266, 65} \[ \frac{\left (a+b \sin ^4(c+d x)\right )^{p+1} \, _2F_1\left (1,p+1;p+2;\frac{b \sin ^4(c+d x)}{a}+1\right )}{4 a d (p+1)}-\frac{\csc ^2(c+d x) \left (a+b \sin ^4(c+d x)\right )^p \left (\frac{b \sin ^4(c+d x)}{a}+1\right )^{-p} \, _2F_1\left (-\frac{1}{2},-p;\frac{1}{2};-\frac{b \sin ^4(c+d x)}{a}\right )}{2 d} \]
Antiderivative was successfully verified.
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Rule 3229
Rule 764
Rule 365
Rule 364
Rule 266
Rule 65
Rubi steps
\begin{align*} \int \cot ^3(c+d x) \left (a+b \sin ^4(c+d x)\right )^p \, dx &=\frac{\operatorname{Subst}\left (\int \frac{(1-x) \left (a+b x^2\right )^p}{x^2} \, dx,x,\sin ^2(c+d x)\right )}{2 d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{\left (a+b x^2\right )^p}{x^2} \, dx,x,\sin ^2(c+d x)\right )}{2 d}-\frac{\operatorname{Subst}\left (\int \frac{\left (a+b x^2\right )^p}{x} \, dx,x,\sin ^2(c+d x)\right )}{2 d}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{(a+b x)^p}{x} \, dx,x,\sin ^4(c+d x)\right )}{4 d}+\frac{\left (\left (a+b \sin ^4(c+d x)\right )^p \left (1+\frac{b \sin ^4(c+d x)}{a}\right )^{-p}\right ) \operatorname{Subst}\left (\int \frac{\left (1+\frac{b x^2}{a}\right )^p}{x^2} \, dx,x,\sin ^2(c+d x)\right )}{2 d}\\ &=\frac{\, _2F_1\left (1,1+p;2+p;1+\frac{b \sin ^4(c+d x)}{a}\right ) \left (a+b \sin ^4(c+d x)\right )^{1+p}}{4 a d (1+p)}-\frac{\csc ^2(c+d x) \, _2F_1\left (-\frac{1}{2},-p;\frac{1}{2};-\frac{b \sin ^4(c+d x)}{a}\right ) \left (a+b \sin ^4(c+d x)\right )^p \left (1+\frac{b \sin ^4(c+d x)}{a}\right )^{-p}}{2 d}\\ \end{align*}
Mathematica [A] time = 0.649591, size = 119, normalized size = 0.94 \[ \frac{\left (a+b \sin ^4(c+d x)\right )^p \left (\frac{\left (a+b \sin ^4(c+d x)\right ) \, _2F_1\left (1,p+1;p+2;\frac{b \sin ^4(c+d x)}{a}+1\right )}{a (p+1)}-2 \csc ^2(c+d x) \left (\frac{b \sin ^4(c+d x)}{a}+1\right )^{-p} \, _2F_1\left (-\frac{1}{2},-p;\frac{1}{2};-\frac{b \sin ^4(c+d x)}{a}\right )\right )}{4 d} \]
Antiderivative was successfully verified.
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Maple [F] time = 1.055, size = 0, normalized size = 0. \begin{align*} \int \left ( \cot \left ( dx+c \right ) \right ) ^{3} \left ( a+b \left ( \sin \left ( dx+c \right ) \right ) ^{4} \right ) ^{p}\, 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 \sin \left (d x + c\right )^{4} + a\right )}^{p} \cot \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 \cos \left (d x + c\right )^{4} - 2 \, b \cos \left (d x + c\right )^{2} + a + b\right )}^{p} \cot \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 \sin \left (d x + c\right )^{4} + a\right )}^{p} \cot \left (d x + c\right )^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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