Optimal. Leaf size=273 \[ -\frac{2 a \tan ^{-1}\left (\frac{\sqrt [3]{a} \tan \left (\frac{1}{2} (c+d x)\right )+\sqrt [3]{b}}{\sqrt{a^{2/3}-b^{2/3}}}\right )}{3 b^{5/3} d \sqrt{a^{2/3}-b^{2/3}}}+\frac{2 a \tanh ^{-1}\left (\frac{\sqrt [3]{b}-\sqrt [3]{-1} \sqrt [3]{a} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{b^{2/3}-(-1)^{2/3} a^{2/3}}}\right )}{3 b^{5/3} d \sqrt{b^{2/3}-(-1)^{2/3} a^{2/3}}}+\frac{2 a \tanh ^{-1}\left (\frac{(-1)^{2/3} \sqrt [3]{a} \tan \left (\frac{1}{2} (c+d x)\right )+\sqrt [3]{b}}{\sqrt{\sqrt [3]{-1} a^{2/3}+b^{2/3}}}\right )}{3 b^{5/3} d \sqrt{\sqrt [3]{-1} a^{2/3}+b^{2/3}}}-\frac{\sin (c+d x) \cos (c+d x)}{2 b d}+\frac{x}{2 b} \]
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Rubi [A] time = 0.566971, antiderivative size = 273, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 7, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.304, Rules used = {3220, 2635, 8, 2660, 618, 204, 206} \[ -\frac{2 a \tan ^{-1}\left (\frac{\sqrt [3]{a} \tan \left (\frac{1}{2} (c+d x)\right )+\sqrt [3]{b}}{\sqrt{a^{2/3}-b^{2/3}}}\right )}{3 b^{5/3} d \sqrt{a^{2/3}-b^{2/3}}}+\frac{2 a \tanh ^{-1}\left (\frac{\sqrt [3]{b}-\sqrt [3]{-1} \sqrt [3]{a} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{b^{2/3}-(-1)^{2/3} a^{2/3}}}\right )}{3 b^{5/3} d \sqrt{b^{2/3}-(-1)^{2/3} a^{2/3}}}+\frac{2 a \tanh ^{-1}\left (\frac{(-1)^{2/3} \sqrt [3]{a} \tan \left (\frac{1}{2} (c+d x)\right )+\sqrt [3]{b}}{\sqrt{\sqrt [3]{-1} a^{2/3}+b^{2/3}}}\right )}{3 b^{5/3} d \sqrt{\sqrt [3]{-1} a^{2/3}+b^{2/3}}}-\frac{\sin (c+d x) \cos (c+d x)}{2 b d}+\frac{x}{2 b} \]
Antiderivative was successfully verified.
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Rule 3220
Rule 2635
Rule 8
Rule 2660
Rule 618
Rule 204
Rule 206
Rubi steps
\begin{align*} \int \frac{\sin ^5(c+d x)}{a+b \sin ^3(c+d x)} \, dx &=\int \left (\frac{\sin ^2(c+d x)}{b}-\frac{a \sin ^2(c+d x)}{b \left (a+b \sin ^3(c+d x)\right )}\right ) \, dx\\ &=\frac{\int \sin ^2(c+d x) \, dx}{b}-\frac{a \int \frac{\sin ^2(c+d x)}{a+b \sin ^3(c+d x)} \, dx}{b}\\ &=-\frac{\cos (c+d x) \sin (c+d x)}{2 b d}+\frac{\int 1 \, dx}{2 b}-\frac{a \int \left (\frac{1}{3 b^{2/3} \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}+\frac{1}{3 b^{2/3} \left (-\sqrt [3]{-1} \sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}+\frac{1}{3 b^{2/3} \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}\right ) \, dx}{b}\\ &=\frac{x}{2 b}-\frac{\cos (c+d x) \sin (c+d x)}{2 b d}-\frac{a \int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)} \, dx}{3 b^{5/3}}-\frac{a \int \frac{1}{-\sqrt [3]{-1} \sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)} \, dx}{3 b^{5/3}}-\frac{a \int \frac{1}{(-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)} \, dx}{3 b^{5/3}}\\ &=\frac{x}{2 b}-\frac{\cos (c+d x) \sin (c+d x)}{2 b d}-\frac{(2 a) \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{a}+2 \sqrt [3]{b} x+\sqrt [3]{a} x^2} \, dx,x,\tan \left (\frac{1}{2} (c+d x)\right )\right )}{3 b^{5/3} d}-\frac{(2 a) \operatorname{Subst}\left (\int \frac{1}{-\sqrt [3]{-1} \sqrt [3]{a}+2 \sqrt [3]{b} x-\sqrt [3]{-1} \sqrt [3]{a} x^2} \, dx,x,\tan \left (\frac{1}{2} (c+d x)\right )\right )}{3 b^{5/3} d}-\frac{(2 a) \operatorname{Subst}\left (\int \frac{1}{(-1)^{2/3} \sqrt [3]{a}+2 \sqrt [3]{b} x+(-1)^{2/3} \sqrt [3]{a} x^2} \, dx,x,\tan \left (\frac{1}{2} (c+d x)\right )\right )}{3 b^{5/3} d}\\ &=\frac{x}{2 b}-\frac{\cos (c+d x) \sin (c+d x)}{2 b d}+\frac{(4 a) \operatorname{Subst}\left (\int \frac{1}{-4 \left (a^{2/3}-b^{2/3}\right )-x^2} \, dx,x,2 \sqrt [3]{b}+2 \sqrt [3]{a} \tan \left (\frac{1}{2} (c+d x)\right )\right )}{3 b^{5/3} d}+\frac{(4 a) \operatorname{Subst}\left (\int \frac{1}{-4 \left ((-1)^{2/3} a^{2/3}-b^{2/3}\right )-x^2} \, dx,x,2 \sqrt [3]{b}-2 \sqrt [3]{-1} \sqrt [3]{a} \tan \left (\frac{1}{2} (c+d x)\right )\right )}{3 b^{5/3} d}+\frac{(4 a) \operatorname{Subst}\left (\int \frac{1}{4 \left (\sqrt [3]{-1} a^{2/3}+b^{2/3}\right )-x^2} \, dx,x,2 \sqrt [3]{b}+2 (-1)^{2/3} \sqrt [3]{a} \tan \left (\frac{1}{2} (c+d x)\right )\right )}{3 b^{5/3} d}\\ &=\frac{x}{2 b}-\frac{2 a \tan ^{-1}\left (\frac{\sqrt [3]{b}+\sqrt [3]{a} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a^{2/3}-b^{2/3}}}\right )}{3 \sqrt{a^{2/3}-b^{2/3}} b^{5/3} d}+\frac{2 a \tanh ^{-1}\left (\frac{\sqrt [3]{b}-\sqrt [3]{-1} \sqrt [3]{a} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{-(-1)^{2/3} a^{2/3}+b^{2/3}}}\right )}{3 \sqrt{-(-1)^{2/3} a^{2/3}+b^{2/3}} b^{5/3} d}+\frac{2 a \tanh ^{-1}\left (\frac{\sqrt [3]{b}+(-1)^{2/3} \sqrt [3]{a} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{\sqrt [3]{-1} a^{2/3}+b^{2/3}}}\right )}{3 \sqrt{\sqrt [3]{-1} a^{2/3}+b^{2/3}} b^{5/3} d}-\frac{\cos (c+d x) \sin (c+d x)}{2 b d}\\ \end{align*}
Mathematica [C] time = 0.277522, size = 255, normalized size = 0.93 \[ \frac{-2 i a \text{RootSum}\left [8 \text{$\#$1}^3 a+i \text{$\#$1}^6 b-3 i \text{$\#$1}^4 b+3 i \text{$\#$1}^2 b-i b\& ,\frac{-i \text{$\#$1}^4 \log \left (\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right )+2 i \text{$\#$1}^2 \log \left (\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right )-i \log \left (\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right )+2 \text{$\#$1}^4 \tan ^{-1}\left (\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right )-4 \text{$\#$1}^2 \tan ^{-1}\left (\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right )+2 \tan ^{-1}\left (\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right )}{-4 i \text{$\#$1}^2 a+\text{$\#$1}^5 b-2 \text{$\#$1}^3 b+\text{$\#$1} b}\& \right ]+6 (c+d x)-3 \sin (2 (c+d x))}{12 b d} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.161, size = 163, normalized size = 0.6 \begin{align*} -{\frac{4\,a}{3\,bd}\sum _{{\it \_R}={\it RootOf} \left ( a{{\it \_Z}}^{6}+3\,a{{\it \_Z}}^{4}+8\,b{{\it \_Z}}^{3}+3\,a{{\it \_Z}}^{2}+a \right ) }{\frac{{{\it \_R}}^{2}}{{{\it \_R}}^{5}a+2\,{{\it \_R}}^{3}a+4\,{{\it \_R}}^{2}b+{\it \_R}\,a}\ln \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -{\it \_R} \right ) }}+{\frac{1}{bd} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3} \left ( 1+ \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \right ) ^{-2}}-{\frac{1}{bd}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \left ( 1+ \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \right ) ^{-2}}+{\frac{1}{bd}\arctan \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) } \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(-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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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 \frac{\sin \left (d x + c\right )^{5}}{b \sin \left (d x + c\right )^{3} + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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