Optimal. Leaf size=109 \[ \frac{a \cos (c+d x) \sin ^{m+1}(c+d x) \, _2F_1\left (\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\sin ^2(c+d x)\right )}{d (m+1) \sqrt{\cos ^2(c+d x)}}+\frac{b \sin ^{m+2}(c+d x) \, _2F_1\left (1,\frac{m+2}{2};\frac{m+4}{2};\sin ^2(c+d x)\right )}{d (m+2)} \]
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Rubi [A] time = 0.14461, antiderivative size = 109, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {4401, 2643, 2564, 364} \[ \frac{a \cos (c+d x) \sin ^{m+1}(c+d x) \, _2F_1\left (\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\sin ^2(c+d x)\right )}{d (m+1) \sqrt{\cos ^2(c+d x)}}+\frac{b \sin ^{m+2}(c+d x) \, _2F_1\left (1,\frac{m+2}{2};\frac{m+4}{2};\sin ^2(c+d x)\right )}{d (m+2)} \]
Antiderivative was successfully verified.
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Rule 4401
Rule 2643
Rule 2564
Rule 364
Rubi steps
\begin{align*} \int \sin ^m(c+d x) (a+b \tan (c+d x)) \, dx &=\int \left (a \sin ^m(c+d x)+b \sec (c+d x) \sin ^{1+m}(c+d x)\right ) \, dx\\ &=a \int \sin ^m(c+d x) \, dx+b \int \sec (c+d x) \sin ^{1+m}(c+d x) \, dx\\ &=\frac{a \cos (c+d x) \, _2F_1\left (\frac{1}{2},\frac{1+m}{2};\frac{3+m}{2};\sin ^2(c+d x)\right ) \sin ^{1+m}(c+d x)}{d (1+m) \sqrt{\cos ^2(c+d x)}}+\frac{b \operatorname{Subst}\left (\int \frac{x^{1+m}}{1-x^2} \, dx,x,\sin (c+d x)\right )}{d}\\ &=\frac{a \cos (c+d x) \, _2F_1\left (\frac{1}{2},\frac{1+m}{2};\frac{3+m}{2};\sin ^2(c+d x)\right ) \sin ^{1+m}(c+d x)}{d (1+m) \sqrt{\cos ^2(c+d x)}}+\frac{b \, _2F_1\left (1,\frac{2+m}{2};\frac{4+m}{2};\sin ^2(c+d x)\right ) \sin ^{2+m}(c+d x)}{d (2+m)}\\ \end{align*}
Mathematica [A] time = 0.257188, size = 109, normalized size = 1. \[ \frac{a \sqrt{\cos ^2(c+d x)} \sec (c+d x) \sin ^{m+1}(c+d x) \, _2F_1\left (\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\sin ^2(c+d x)\right )}{d (m+1)}+\frac{b \sin ^{m+2}(c+d x) \, _2F_1\left (1,\frac{m+2}{2};\frac{m+4}{2};\sin ^2(c+d x)\right )}{d (m+2)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.488, size = 0, normalized size = 0. \begin{align*} \int \left ( \sin \left ( dx+c \right ) \right ) ^{m} \left ( a+b\tan \left ( dx+c \right ) \right ) \, 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 )} \sin \left (d x + c\right )^{m}\,{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 )} \sin \left (d x + c\right )^{m}, 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 (a + b \tan{\left (c + d x \right )}\right ) \sin ^{m}{\left (c + d x \right )}\, 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 (d x + c\right ) + a\right )} \sin \left (d x + c\right )^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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