Optimal. Leaf size=765 \[ \text{result too large to display} \]
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Rubi [A] time = 4.17316, antiderivative size = 765, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {12, 6719, 6728, 364, 959, 510} \[ \frac{a b 2^{m+1} \tan ^3\left (\frac{1}{2} (c+d x)\right ) \left (\frac{\tan \left (\frac{1}{2} (c+d x)\right )}{\tan ^2\left (\frac{1}{2} (c+d x)\right )+1}\right )^m \left (\tan ^2\left (\frac{1}{2} (c+d x)\right )+1\right )^m F_1\left (\frac{m+3}{2};m+1,1;\frac{m+5}{2};-\tan ^2\left (\frac{1}{2} (c+d x)\right ),\frac{a^2 \tan ^2\left (\frac{1}{2} (c+d x)\right )}{\left (b-\sqrt{a^2+b^2}\right )^2}\right )}{d (m+3) \sqrt{a^2+b^2} \left (b-\sqrt{a^2+b^2}\right )^2}-\frac{a b 2^{m+1} \tan ^3\left (\frac{1}{2} (c+d x)\right ) \left (\frac{\tan \left (\frac{1}{2} (c+d x)\right )}{\tan ^2\left (\frac{1}{2} (c+d x)\right )+1}\right )^m \left (\tan ^2\left (\frac{1}{2} (c+d x)\right )+1\right )^m F_1\left (\frac{m+3}{2};m+1,1;\frac{m+5}{2};-\tan ^2\left (\frac{1}{2} (c+d x)\right ),\frac{a^2 \tan ^2\left (\frac{1}{2} (c+d x)\right )}{\left (b+\sqrt{a^2+b^2}\right )^2}\right )}{d (m+3) \sqrt{a^2+b^2} \left (\sqrt{a^2+b^2}+b\right )^2}+\frac{b 2^{m+1} \tan ^2\left (\frac{1}{2} (c+d x)\right ) \left (\frac{\tan \left (\frac{1}{2} (c+d x)\right )}{\tan ^2\left (\frac{1}{2} (c+d x)\right )+1}\right )^m \left (\tan ^2\left (\frac{1}{2} (c+d x)\right )+1\right )^m F_1\left (\frac{m+2}{2};m+1,1;\frac{m+4}{2};-\tan ^2\left (\frac{1}{2} (c+d x)\right ),\frac{a^2 \tan ^2\left (\frac{1}{2} (c+d x)\right )}{\left (b-\sqrt{a^2+b^2}\right )^2}\right )}{d (m+2) \sqrt{a^2+b^2} \left (b-\sqrt{a^2+b^2}\right )}-\frac{b 2^{m+1} \tan ^2\left (\frac{1}{2} (c+d x)\right ) \left (\frac{\tan \left (\frac{1}{2} (c+d x)\right )}{\tan ^2\left (\frac{1}{2} (c+d x)\right )+1}\right )^m \left (\tan ^2\left (\frac{1}{2} (c+d x)\right )+1\right )^m F_1\left (\frac{m+2}{2};m+1,1;\frac{m+4}{2};-\tan ^2\left (\frac{1}{2} (c+d x)\right ),\frac{a^2 \tan ^2\left (\frac{1}{2} (c+d x)\right )}{\left (b+\sqrt{a^2+b^2}\right )^2}\right )}{d (m+2) \sqrt{a^2+b^2} \left (\sqrt{a^2+b^2}+b\right )}+\frac{2^{m+1} \tan \left (\frac{1}{2} (c+d x)\right ) \left (\frac{\tan \left (\frac{1}{2} (c+d x)\right )}{\tan ^2\left (\frac{1}{2} (c+d x)\right )+1}\right )^m \left (\tan ^2\left (\frac{1}{2} (c+d x)\right )+1\right )^m \, _2F_1\left (\frac{m+1}{2},m+1;\frac{m+3}{2};-\tan ^2\left (\frac{1}{2} (c+d x)\right )\right )}{a d (m+1)} \]
Antiderivative was successfully verified.
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Rule 12
Rule 6719
Rule 6728
Rule 364
Rule 959
Rule 510
Rubi steps
\begin{align*} \int \frac{\sin ^m(c+d x)}{a+b \tan (c+d x)} \, dx &=\frac{2 \operatorname{Subst}\left (\int \frac{2^m \left (1-x^2\right ) \left (\frac{x}{1+x^2}\right )^{1+m}}{x \left (a+2 b x-a x^2\right )} \, dx,x,\tan \left (\frac{1}{2} (c+d x)\right )\right )}{d}\\ &=\frac{2^{1+m} \operatorname{Subst}\left (\int \frac{\left (1-x^2\right ) \left (\frac{x}{1+x^2}\right )^{1+m}}{x \left (a+2 b x-a x^2\right )} \, dx,x,\tan \left (\frac{1}{2} (c+d x)\right )\right )}{d}\\ &=\frac{\left (2^{1+m} \tan ^{-m}\left (\frac{1}{2} (c+d x)\right ) \left (\frac{\tan \left (\frac{1}{2} (c+d x)\right )}{1+\tan ^2\left (\frac{1}{2} (c+d x)\right )}\right )^m \left (1+\tan ^2\left (\frac{1}{2} (c+d x)\right )\right )^m\right ) \operatorname{Subst}\left (\int \frac{x^m \left (1-x^2\right ) \left (1+x^2\right )^{-1-m}}{a+2 b x-a x^2} \, dx,x,\tan \left (\frac{1}{2} (c+d x)\right )\right )}{d}\\ &=\frac{\left (2^{1+m} \tan ^{-m}\left (\frac{1}{2} (c+d x)\right ) \left (\frac{\tan \left (\frac{1}{2} (c+d x)\right )}{1+\tan ^2\left (\frac{1}{2} (c+d x)\right )}\right )^m \left (1+\tan ^2\left (\frac{1}{2} (c+d x)\right )\right )^m\right ) \operatorname{Subst}\left (\int \left (\frac{x^m \left (1+x^2\right )^{-1-m}}{a}-\frac{2 b x^{1+m} \left (1+x^2\right )^{-1-m}}{a \left (a+2 b x-a x^2\right )}\right ) \, dx,x,\tan \left (\frac{1}{2} (c+d x)\right )\right )}{d}\\ &=\frac{\left (2^{1+m} \tan ^{-m}\left (\frac{1}{2} (c+d x)\right ) \left (\frac{\tan \left (\frac{1}{2} (c+d x)\right )}{1+\tan ^2\left (\frac{1}{2} (c+d x)\right )}\right )^m \left (1+\tan ^2\left (\frac{1}{2} (c+d x)\right )\right )^m\right ) \operatorname{Subst}\left (\int x^m \left (1+x^2\right )^{-1-m} \, dx,x,\tan \left (\frac{1}{2} (c+d x)\right )\right )}{a d}-\frac{\left (2^{2+m} b \tan ^{-m}\left (\frac{1}{2} (c+d x)\right ) \left (\frac{\tan \left (\frac{1}{2} (c+d x)\right )}{1+\tan ^2\left (\frac{1}{2} (c+d x)\right )}\right )^m \left (1+\tan ^2\left (\frac{1}{2} (c+d x)\right )\right )^m\right ) \operatorname{Subst}\left (\int \frac{x^{1+m} \left (1+x^2\right )^{-1-m}}{a+2 b x-a x^2} \, dx,x,\tan \left (\frac{1}{2} (c+d x)\right )\right )}{a d}\\ &=\frac{2^{1+m} \, _2F_1\left (\frac{1+m}{2},1+m;\frac{3+m}{2};-\tan ^2\left (\frac{1}{2} (c+d x)\right )\right ) \tan \left (\frac{1}{2} (c+d x)\right ) \left (\frac{\tan \left (\frac{1}{2} (c+d x)\right )}{1+\tan ^2\left (\frac{1}{2} (c+d x)\right )}\right )^m \left (1+\tan ^2\left (\frac{1}{2} (c+d x)\right )\right )^m}{a d (1+m)}-\frac{\left (2^{2+m} b \tan ^{-m}\left (\frac{1}{2} (c+d x)\right ) \left (\frac{\tan \left (\frac{1}{2} (c+d x)\right )}{1+\tan ^2\left (\frac{1}{2} (c+d x)\right )}\right )^m \left (1+\tan ^2\left (\frac{1}{2} (c+d x)\right )\right )^m\right ) \operatorname{Subst}\left (\int \left (-\frac{a x^{1+m} \left (1+x^2\right )^{-1-m}}{\sqrt{a^2+b^2} \left (2 b-2 \sqrt{a^2+b^2}-2 a x\right )}+\frac{a x^{1+m} \left (1+x^2\right )^{-1-m}}{\sqrt{a^2+b^2} \left (2 b+2 \sqrt{a^2+b^2}-2 a x\right )}\right ) \, dx,x,\tan \left (\frac{1}{2} (c+d x)\right )\right )}{a d}\\ &=\frac{2^{1+m} \, _2F_1\left (\frac{1+m}{2},1+m;\frac{3+m}{2};-\tan ^2\left (\frac{1}{2} (c+d x)\right )\right ) \tan \left (\frac{1}{2} (c+d x)\right ) \left (\frac{\tan \left (\frac{1}{2} (c+d x)\right )}{1+\tan ^2\left (\frac{1}{2} (c+d x)\right )}\right )^m \left (1+\tan ^2\left (\frac{1}{2} (c+d x)\right )\right )^m}{a d (1+m)}+\frac{\left (2^{2+m} b \tan ^{-m}\left (\frac{1}{2} (c+d x)\right ) \left (\frac{\tan \left (\frac{1}{2} (c+d x)\right )}{1+\tan ^2\left (\frac{1}{2} (c+d x)\right )}\right )^m \left (1+\tan ^2\left (\frac{1}{2} (c+d x)\right )\right )^m\right ) \operatorname{Subst}\left (\int \frac{x^{1+m} \left (1+x^2\right )^{-1-m}}{2 b-2 \sqrt{a^2+b^2}-2 a x} \, dx,x,\tan \left (\frac{1}{2} (c+d x)\right )\right )}{\sqrt{a^2+b^2} d}-\frac{\left (2^{2+m} b \tan ^{-m}\left (\frac{1}{2} (c+d x)\right ) \left (\frac{\tan \left (\frac{1}{2} (c+d x)\right )}{1+\tan ^2\left (\frac{1}{2} (c+d x)\right )}\right )^m \left (1+\tan ^2\left (\frac{1}{2} (c+d x)\right )\right )^m\right ) \operatorname{Subst}\left (\int \frac{x^{1+m} \left (1+x^2\right )^{-1-m}}{2 b+2 \sqrt{a^2+b^2}-2 a x} \, dx,x,\tan \left (\frac{1}{2} (c+d x)\right )\right )}{\sqrt{a^2+b^2} d}\\ &=\frac{2^{1+m} \, _2F_1\left (\frac{1+m}{2},1+m;\frac{3+m}{2};-\tan ^2\left (\frac{1}{2} (c+d x)\right )\right ) \tan \left (\frac{1}{2} (c+d x)\right ) \left (\frac{\tan \left (\frac{1}{2} (c+d x)\right )}{1+\tan ^2\left (\frac{1}{2} (c+d x)\right )}\right )^m \left (1+\tan ^2\left (\frac{1}{2} (c+d x)\right )\right )^m}{a d (1+m)}+\frac{\left (2^{3+m} a b \tan ^{-m}\left (\frac{1}{2} (c+d x)\right ) \left (\frac{\tan \left (\frac{1}{2} (c+d x)\right )}{1+\tan ^2\left (\frac{1}{2} (c+d x)\right )}\right )^m \left (1+\tan ^2\left (\frac{1}{2} (c+d x)\right )\right )^m\right ) \operatorname{Subst}\left (\int \frac{x^{2+m} \left (1+x^2\right )^{-1-m}}{\left (2 b-2 \sqrt{a^2+b^2}\right )^2-4 a^2 x^2} \, dx,x,\tan \left (\frac{1}{2} (c+d x)\right )\right )}{\sqrt{a^2+b^2} d}-\frac{\left (2^{3+m} a b \tan ^{-m}\left (\frac{1}{2} (c+d x)\right ) \left (\frac{\tan \left (\frac{1}{2} (c+d x)\right )}{1+\tan ^2\left (\frac{1}{2} (c+d x)\right )}\right )^m \left (1+\tan ^2\left (\frac{1}{2} (c+d x)\right )\right )^m\right ) \operatorname{Subst}\left (\int \frac{x^{2+m} \left (1+x^2\right )^{-1-m}}{\left (2 b+2 \sqrt{a^2+b^2}\right )^2-4 a^2 x^2} \, dx,x,\tan \left (\frac{1}{2} (c+d x)\right )\right )}{\sqrt{a^2+b^2} d}+\frac{\left (2^{3+m} b \left (b-\sqrt{a^2+b^2}\right ) \tan ^{-m}\left (\frac{1}{2} (c+d x)\right ) \left (\frac{\tan \left (\frac{1}{2} (c+d x)\right )}{1+\tan ^2\left (\frac{1}{2} (c+d x)\right )}\right )^m \left (1+\tan ^2\left (\frac{1}{2} (c+d x)\right )\right )^m\right ) \operatorname{Subst}\left (\int \frac{x^{1+m} \left (1+x^2\right )^{-1-m}}{\left (2 b-2 \sqrt{a^2+b^2}\right )^2-4 a^2 x^2} \, dx,x,\tan \left (\frac{1}{2} (c+d x)\right )\right )}{\sqrt{a^2+b^2} d}-\frac{\left (2^{3+m} b \left (b+\sqrt{a^2+b^2}\right ) \tan ^{-m}\left (\frac{1}{2} (c+d x)\right ) \left (\frac{\tan \left (\frac{1}{2} (c+d x)\right )}{1+\tan ^2\left (\frac{1}{2} (c+d x)\right )}\right )^m \left (1+\tan ^2\left (\frac{1}{2} (c+d x)\right )\right )^m\right ) \operatorname{Subst}\left (\int \frac{x^{1+m} \left (1+x^2\right )^{-1-m}}{\left (2 b+2 \sqrt{a^2+b^2}\right )^2-4 a^2 x^2} \, dx,x,\tan \left (\frac{1}{2} (c+d x)\right )\right )}{\sqrt{a^2+b^2} d}\\ &=\frac{2^{1+m} \, _2F_1\left (\frac{1+m}{2},1+m;\frac{3+m}{2};-\tan ^2\left (\frac{1}{2} (c+d x)\right )\right ) \tan \left (\frac{1}{2} (c+d x)\right ) \left (\frac{\tan \left (\frac{1}{2} (c+d x)\right )}{1+\tan ^2\left (\frac{1}{2} (c+d x)\right )}\right )^m \left (1+\tan ^2\left (\frac{1}{2} (c+d x)\right )\right )^m}{a d (1+m)}+\frac{2^{1+m} b F_1\left (\frac{2+m}{2};1+m,1;\frac{4+m}{2};-\tan ^2\left (\frac{1}{2} (c+d x)\right ),\frac{a^2 \tan ^2\left (\frac{1}{2} (c+d x)\right )}{\left (b-\sqrt{a^2+b^2}\right )^2}\right ) \tan ^2\left (\frac{1}{2} (c+d x)\right ) \left (\frac{\tan \left (\frac{1}{2} (c+d x)\right )}{1+\tan ^2\left (\frac{1}{2} (c+d x)\right )}\right )^m \left (1+\tan ^2\left (\frac{1}{2} (c+d x)\right )\right )^m}{\sqrt{a^2+b^2} \left (b-\sqrt{a^2+b^2}\right ) d (2+m)}-\frac{2^{1+m} b F_1\left (\frac{2+m}{2};1+m,1;\frac{4+m}{2};-\tan ^2\left (\frac{1}{2} (c+d x)\right ),\frac{a^2 \tan ^2\left (\frac{1}{2} (c+d x)\right )}{\left (b+\sqrt{a^2+b^2}\right )^2}\right ) \tan ^2\left (\frac{1}{2} (c+d x)\right ) \left (\frac{\tan \left (\frac{1}{2} (c+d x)\right )}{1+\tan ^2\left (\frac{1}{2} (c+d x)\right )}\right )^m \left (1+\tan ^2\left (\frac{1}{2} (c+d x)\right )\right )^m}{\sqrt{a^2+b^2} \left (b+\sqrt{a^2+b^2}\right ) d (2+m)}+\frac{2^{1+m} a b F_1\left (\frac{3+m}{2};1+m,1;\frac{5+m}{2};-\tan ^2\left (\frac{1}{2} (c+d x)\right ),\frac{a^2 \tan ^2\left (\frac{1}{2} (c+d x)\right )}{\left (b-\sqrt{a^2+b^2}\right )^2}\right ) \tan ^3\left (\frac{1}{2} (c+d x)\right ) \left (\frac{\tan \left (\frac{1}{2} (c+d x)\right )}{1+\tan ^2\left (\frac{1}{2} (c+d x)\right )}\right )^m \left (1+\tan ^2\left (\frac{1}{2} (c+d x)\right )\right )^m}{\sqrt{a^2+b^2} \left (b-\sqrt{a^2+b^2}\right )^2 d (3+m)}-\frac{2^{1+m} a b F_1\left (\frac{3+m}{2};1+m,1;\frac{5+m}{2};-\tan ^2\left (\frac{1}{2} (c+d x)\right ),\frac{a^2 \tan ^2\left (\frac{1}{2} (c+d x)\right )}{\left (b+\sqrt{a^2+b^2}\right )^2}\right ) \tan ^3\left (\frac{1}{2} (c+d x)\right ) \left (\frac{\tan \left (\frac{1}{2} (c+d x)\right )}{1+\tan ^2\left (\frac{1}{2} (c+d x)\right )}\right )^m \left (1+\tan ^2\left (\frac{1}{2} (c+d x)\right )\right )^m}{\sqrt{a^2+b^2} \left (b+\sqrt{a^2+b^2}\right )^2 d (3+m)}\\ \end{align*}
Mathematica [F] time = 12.7916, size = 0, normalized size = 0. \[ \int \frac{\sin ^m(c+d x)}{a+b \tan (c+d x)} \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.164, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{m}}{a+b\tan \left ( dx+c \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 \frac{\sin \left (d x + c\right )^{m}}{b \tan \left (d x + c\right ) + a}\,{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 (\frac{\sin \left (d x + c\right )^{m}}{b \tan \left (d x + c\right ) + a}, 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 \frac{\sin ^{m}{\left (c + d x \right )}}{a + b \tan{\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 \frac{\sin \left (d x + c\right )^{m}}{b \tan \left (d x + c\right ) + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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