Optimal. Leaf size=296 \[ \frac{2 b^{4/3} \tan ^{-1}\left (\frac{\sqrt [3]{a} \tan \left (\frac{1}{2} (c+d x)\right )+\sqrt [3]{b}}{\sqrt{a^{2/3}-b^{2/3}}}\right )}{3 a^2 d \sqrt{a^{2/3}-b^{2/3}}}-\frac{2 b^{4/3} \tanh ^{-1}\left (\frac{\sqrt [3]{b}-\sqrt [3]{-1} \sqrt [3]{a} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{b^{2/3}-(-1)^{2/3} a^{2/3}}}\right )}{3 a^2 d \sqrt{b^{2/3}-(-1)^{2/3} a^{2/3}}}-\frac{2 b^{4/3} \tanh ^{-1}\left (\frac{(-1)^{2/3} \sqrt [3]{a} \tan \left (\frac{1}{2} (c+d x)\right )+\sqrt [3]{b}}{\sqrt{\sqrt [3]{-1} a^{2/3}+b^{2/3}}}\right )}{3 a^2 d \sqrt{\sqrt [3]{-1} a^{2/3}+b^{2/3}}}+\frac{b \tanh ^{-1}(\cos (c+d x))}{a^2 d}-\frac{\cot ^3(c+d x)}{3 a d}-\frac{\cot (c+d x)}{a d} \]
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Rubi [A] time = 0.393332, antiderivative size = 296, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 7, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.304, Rules used = {3220, 3770, 3767, 2660, 618, 204, 206} \[ \frac{2 b^{4/3} \tan ^{-1}\left (\frac{\sqrt [3]{a} \tan \left (\frac{1}{2} (c+d x)\right )+\sqrt [3]{b}}{\sqrt{a^{2/3}-b^{2/3}}}\right )}{3 a^2 d \sqrt{a^{2/3}-b^{2/3}}}-\frac{2 b^{4/3} \tanh ^{-1}\left (\frac{\sqrt [3]{b}-\sqrt [3]{-1} \sqrt [3]{a} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{b^{2/3}-(-1)^{2/3} a^{2/3}}}\right )}{3 a^2 d \sqrt{b^{2/3}-(-1)^{2/3} a^{2/3}}}-\frac{2 b^{4/3} \tanh ^{-1}\left (\frac{(-1)^{2/3} \sqrt [3]{a} \tan \left (\frac{1}{2} (c+d x)\right )+\sqrt [3]{b}}{\sqrt{\sqrt [3]{-1} a^{2/3}+b^{2/3}}}\right )}{3 a^2 d \sqrt{\sqrt [3]{-1} a^{2/3}+b^{2/3}}}+\frac{b \tanh ^{-1}(\cos (c+d x))}{a^2 d}-\frac{\cot ^3(c+d x)}{3 a d}-\frac{\cot (c+d x)}{a d} \]
Antiderivative was successfully verified.
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Rule 3220
Rule 3770
Rule 3767
Rule 2660
Rule 618
Rule 204
Rule 206
Rubi steps
\begin{align*} \int \frac{\csc ^4(c+d x)}{a+b \sin ^3(c+d x)} \, dx &=\int \left (-\frac{b \csc (c+d x)}{a^2}+\frac{\csc ^4(c+d x)}{a}+\frac{b^2 \sin ^2(c+d x)}{a^2 \left (a+b \sin ^3(c+d x)\right )}\right ) \, dx\\ &=\frac{\int \csc ^4(c+d x) \, dx}{a}-\frac{b \int \csc (c+d x) \, dx}{a^2}+\frac{b^2 \int \frac{\sin ^2(c+d x)}{a+b \sin ^3(c+d x)} \, dx}{a^2}\\ &=\frac{b \tanh ^{-1}(\cos (c+d x))}{a^2 d}+\frac{b^2 \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}{a^2}-\frac{\operatorname{Subst}\left (\int \left (1+x^2\right ) \, dx,x,\cot (c+d x)\right )}{a d}\\ &=\frac{b \tanh ^{-1}(\cos (c+d x))}{a^2 d}-\frac{\cot (c+d x)}{a d}-\frac{\cot ^3(c+d x)}{3 a d}+\frac{b^{4/3} \int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)} \, dx}{3 a^2}+\frac{b^{4/3} \int \frac{1}{-\sqrt [3]{-1} \sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)} \, dx}{3 a^2}+\frac{b^{4/3} \int \frac{1}{(-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)} \, dx}{3 a^2}\\ &=\frac{b \tanh ^{-1}(\cos (c+d x))}{a^2 d}-\frac{\cot (c+d x)}{a d}-\frac{\cot ^3(c+d x)}{3 a d}+\frac{\left (2 b^{4/3}\right ) \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 a^2 d}+\frac{\left (2 b^{4/3}\right ) \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 a^2 d}+\frac{\left (2 b^{4/3}\right ) \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 a^2 d}\\ &=\frac{b \tanh ^{-1}(\cos (c+d x))}{a^2 d}-\frac{\cot (c+d x)}{a d}-\frac{\cot ^3(c+d x)}{3 a d}-\frac{\left (4 b^{4/3}\right ) \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 a^2 d}-\frac{\left (4 b^{4/3}\right ) \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 a^2 d}-\frac{\left (4 b^{4/3}\right ) \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 a^2 d}\\ &=\frac{2 b^{4/3} \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 a^2 \sqrt{a^{2/3}-b^{2/3}} d}+\frac{b \tanh ^{-1}(\cos (c+d x))}{a^2 d}-\frac{2 b^{4/3} \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 a^2 \sqrt{-(-1)^{2/3} a^{2/3}+b^{2/3}} d}-\frac{2 b^{4/3} \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 a^2 \sqrt{\sqrt [3]{-1} a^{2/3}+b^{2/3}} d}-\frac{\cot (c+d x)}{a d}-\frac{\cot ^3(c+d x)}{3 a d}\\ \end{align*}
Mathematica [C] time = 2.05013, size = 333, normalized size = 1.12 \[ \frac{4 i b^2 \text{RootSum}\left [-8 i \text{$\#$1}^3 a+\text{$\#$1}^6 b-3 \text{$\#$1}^4 b+3 \text{$\#$1}^2 b-b\& ,\frac{-i \text{$\#$1}^4 \log \left (\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right )+2 i \text{$\#$1}^2 \log \left (\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right )-i \log \left (\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right )+2 \text{$\#$1}^4 \tan ^{-1}\left (\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right )-4 \text{$\#$1}^2 \tan ^{-1}\left (\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right )+2 \tan ^{-1}\left (\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right )}{-4 i \text{$\#$1}^2 a+\text{$\#$1}^5 b-2 \text{$\#$1}^3 b+\text{$\#$1} b}\& \right ]+8 a \tan \left (\frac{1}{2} (c+d x)\right )-8 a \cot \left (\frac{1}{2} (c+d x)\right )+8 a \sin ^4\left (\frac{1}{2} (c+d x)\right ) \csc ^3(c+d x)-\frac{1}{2} a \sin (c+d x) \csc ^4\left (\frac{1}{2} (c+d x)\right )-24 b \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )+24 b \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )}{24 a^2 d} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.207, size = 176, normalized size = 0.6 \begin{align*}{\frac{1}{24\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}}+{\frac{3}{8\,da}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }+{\frac{4\,{b}^{2}}{3\,{a}^{2}d}\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}{24\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-3}}-{\frac{3}{8\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-1}}-{\frac{b}{{a}^{2}d}\ln \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{\csc \left (d x + c\right )^{4}}{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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