Optimal. Leaf size=264 \[ -\frac{2 \sqrt [3]{b} \tan ^{-1}\left (\frac{\sqrt [3]{a} \tan \left (\frac{1}{2} (c+d x)\right )+\sqrt [3]{b}}{\sqrt{a^{2/3}-b^{2/3}}}\right )}{3 a d \sqrt{a^{2/3}-b^{2/3}}}+\frac{2 \sqrt [3]{b} \tanh ^{-1}\left (\frac{\sqrt [3]{b}-\sqrt [3]{-1} \sqrt [3]{a} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{b^{2/3}-(-1)^{2/3} a^{2/3}}}\right )}{3 a d \sqrt{b^{2/3}-(-1)^{2/3} a^{2/3}}}+\frac{2 \sqrt [3]{b} \tanh ^{-1}\left (\frac{(-1)^{2/3} \sqrt [3]{a} \tan \left (\frac{1}{2} (c+d x)\right )+\sqrt [3]{b}}{\sqrt{\sqrt [3]{-1} a^{2/3}+b^{2/3}}}\right )}{3 a d \sqrt{\sqrt [3]{-1} a^{2/3}+b^{2/3}}}-\frac{\tanh ^{-1}(\cos (c+d x))}{a d} \]
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Rubi [A] time = 0.364646, antiderivative size = 264, 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 = {3220, 3770, 2660, 618, 204, 206} \[ -\frac{2 \sqrt [3]{b} \tan ^{-1}\left (\frac{\sqrt [3]{a} \tan \left (\frac{1}{2} (c+d x)\right )+\sqrt [3]{b}}{\sqrt{a^{2/3}-b^{2/3}}}\right )}{3 a d \sqrt{a^{2/3}-b^{2/3}}}+\frac{2 \sqrt [3]{b} \tanh ^{-1}\left (\frac{\sqrt [3]{b}-\sqrt [3]{-1} \sqrt [3]{a} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{b^{2/3}-(-1)^{2/3} a^{2/3}}}\right )}{3 a d \sqrt{b^{2/3}-(-1)^{2/3} a^{2/3}}}+\frac{2 \sqrt [3]{b} \tanh ^{-1}\left (\frac{(-1)^{2/3} \sqrt [3]{a} \tan \left (\frac{1}{2} (c+d x)\right )+\sqrt [3]{b}}{\sqrt{\sqrt [3]{-1} a^{2/3}+b^{2/3}}}\right )}{3 a d \sqrt{\sqrt [3]{-1} a^{2/3}+b^{2/3}}}-\frac{\tanh ^{-1}(\cos (c+d x))}{a d} \]
Antiderivative was successfully verified.
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Rule 3220
Rule 3770
Rule 2660
Rule 618
Rule 204
Rule 206
Rubi steps
\begin{align*} \int \frac{\csc (c+d x)}{a+b \sin ^3(c+d x)} \, dx &=\int \left (\frac{\csc (c+d x)}{a}-\frac{b \sin ^2(c+d x)}{a \left (a+b \sin ^3(c+d x)\right )}\right ) \, dx\\ &=\frac{\int \csc (c+d x) \, dx}{a}-\frac{b \int \frac{\sin ^2(c+d x)}{a+b \sin ^3(c+d x)} \, dx}{a}\\ &=-\frac{\tanh ^{-1}(\cos (c+d x))}{a d}-\frac{b \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}\\ &=-\frac{\tanh ^{-1}(\cos (c+d x))}{a d}-\frac{\sqrt [3]{b} \int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)} \, dx}{3 a}-\frac{\sqrt [3]{b} \int \frac{1}{-\sqrt [3]{-1} \sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)} \, dx}{3 a}-\frac{\sqrt [3]{b} \int \frac{1}{(-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)} \, dx}{3 a}\\ &=-\frac{\tanh ^{-1}(\cos (c+d x))}{a d}-\frac{\left (2 \sqrt [3]{b}\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 d}-\frac{\left (2 \sqrt [3]{b}\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 d}-\frac{\left (2 \sqrt [3]{b}\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 d}\\ &=-\frac{\tanh ^{-1}(\cos (c+d x))}{a d}+\frac{\left (4 \sqrt [3]{b}\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 d}+\frac{\left (4 \sqrt [3]{b}\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 d}+\frac{\left (4 \sqrt [3]{b}\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 d}\\ &=-\frac{2 \sqrt [3]{b} \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 \sqrt{a^{2/3}-b^{2/3}} d}-\frac{\tanh ^{-1}(\cos (c+d x))}{a d}+\frac{2 \sqrt [3]{b} \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 \sqrt{-(-1)^{2/3} a^{2/3}+b^{2/3}} d}+\frac{2 \sqrt [3]{b} \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 \sqrt{\sqrt [3]{-1} a^{2/3}+b^{2/3}} d}\\ \end{align*}
Mathematica [C] time = 0.263686, size = 264, normalized size = 1. \[ -\frac{i b \text{RootSum}\left [8 \text{$\#$1}^3 a+i \text{$\#$1}^6 b-3 i \text{$\#$1}^4 b+3 i \text{$\#$1}^2 b-i b\& ,\frac{-i \text{$\#$1}^4 \log \left (\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right )+2 i \text{$\#$1}^2 \log \left (\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right )-i \log \left (\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right )+2 \text{$\#$1}^4 \tan ^{-1}\left (\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right )-4 \text{$\#$1}^2 \tan ^{-1}\left (\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right )+2 \tan ^{-1}\left (\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right )}{-4 i \text{$\#$1}^2 a+\text{$\#$1}^5 b-2 \text{$\#$1}^3 b+\text{$\#$1} b}\& \right ]-6 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )+6 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )}{6 a d} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.188, size = 98, normalized size = 0.4 \begin{align*} -{\frac{4\,b}{3\,da}\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}{da}\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 )}{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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