Optimal. Leaf size=287 \[ -\frac{2 b \tan ^{-1}\left (\frac{\sqrt [3]{a} \tan \left (\frac{1}{2} (c+d x)\right )+\sqrt [3]{b}}{\sqrt{a^{2/3}-b^{2/3}}}\right )}{3 a^{5/3} d \sqrt{a^{2/3}-b^{2/3}}}-\frac{2 b \tan ^{-1}\left (\frac{\sqrt [3]{a} \tan \left (\frac{1}{2} (c+d x)\right )+(-1)^{2/3} \sqrt [3]{b}}{\sqrt{a^{2/3}+\sqrt [3]{-1} b^{2/3}}}\right )}{3 a^{5/3} d \sqrt{a^{2/3}+\sqrt [3]{-1} b^{2/3}}}+\frac{2 b \tan ^{-1}\left (\frac{\sqrt [3]{-1} \left ((-1)^{2/3} \sqrt [3]{a} \tan \left (\frac{1}{2} (c+d x)\right )+\sqrt [3]{b}\right )}{\sqrt{a^{2/3}-(-1)^{2/3} b^{2/3}}}\right )}{3 a^{5/3} d \sqrt{a^{2/3}-(-1)^{2/3} b^{2/3}}}-\frac{\tanh ^{-1}(\cos (c+d x))}{2 a d}-\frac{\cot (c+d x) \csc (c+d x)}{2 a d} \]
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Rubi [A] time = 0.403905, antiderivative size = 287, 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, 3768, 3770, 3213, 2660, 618, 204} \[ -\frac{2 b \tan ^{-1}\left (\frac{\sqrt [3]{a} \tan \left (\frac{1}{2} (c+d x)\right )+\sqrt [3]{b}}{\sqrt{a^{2/3}-b^{2/3}}}\right )}{3 a^{5/3} d \sqrt{a^{2/3}-b^{2/3}}}-\frac{2 b \tan ^{-1}\left (\frac{\sqrt [3]{a} \tan \left (\frac{1}{2} (c+d x)\right )+(-1)^{2/3} \sqrt [3]{b}}{\sqrt{a^{2/3}+\sqrt [3]{-1} b^{2/3}}}\right )}{3 a^{5/3} d \sqrt{a^{2/3}+\sqrt [3]{-1} b^{2/3}}}+\frac{2 b \tan ^{-1}\left (\frac{\sqrt [3]{-1} \left ((-1)^{2/3} \sqrt [3]{a} \tan \left (\frac{1}{2} (c+d x)\right )+\sqrt [3]{b}\right )}{\sqrt{a^{2/3}-(-1)^{2/3} b^{2/3}}}\right )}{3 a^{5/3} d \sqrt{a^{2/3}-(-1)^{2/3} b^{2/3}}}-\frac{\tanh ^{-1}(\cos (c+d x))}{2 a d}-\frac{\cot (c+d x) \csc (c+d x)}{2 a d} \]
Antiderivative was successfully verified.
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Rule 3220
Rule 3768
Rule 3770
Rule 3213
Rule 2660
Rule 618
Rule 204
Rubi steps
\begin{align*} \int \frac{\csc ^3(c+d x)}{a+b \sin ^3(c+d x)} \, dx &=\int \left (\frac{\csc ^3(c+d x)}{a}-\frac{b}{a \left (a+b \sin ^3(c+d x)\right )}\right ) \, dx\\ &=\frac{\int \csc ^3(c+d x) \, dx}{a}-\frac{b \int \frac{1}{a+b \sin ^3(c+d x)} \, dx}{a}\\ &=-\frac{\cot (c+d x) \csc (c+d x)}{2 a d}+\frac{\int \csc (c+d x) \, dx}{2 a}-\frac{b \int \left (-\frac{1}{3 a^{2/3} \left (-\sqrt [3]{a}-\sqrt [3]{b} \sin (c+d x)\right )}-\frac{1}{3 a^{2/3} \left (-\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b} \sin (c+d x)\right )}-\frac{1}{3 a^{2/3} \left (-\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} \sin (c+d x)\right )}\right ) \, dx}{a}\\ &=-\frac{\tanh ^{-1}(\cos (c+d x))}{2 a d}-\frac{\cot (c+d x) \csc (c+d x)}{2 a d}+\frac{b \int \frac{1}{-\sqrt [3]{a}-\sqrt [3]{b} \sin (c+d x)} \, dx}{3 a^{5/3}}+\frac{b \int \frac{1}{-\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b} \sin (c+d x)} \, dx}{3 a^{5/3}}+\frac{b \int \frac{1}{-\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} \sin (c+d x)} \, dx}{3 a^{5/3}}\\ &=-\frac{\tanh ^{-1}(\cos (c+d x))}{2 a d}-\frac{\cot (c+d x) \csc (c+d x)}{2 a d}+\frac{(2 b) \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^{5/3} d}+\frac{(2 b) \operatorname{Subst}\left (\int \frac{1}{-\sqrt [3]{a}+2 \sqrt [3]{-1} \sqrt [3]{b} x-\sqrt [3]{a} x^2} \, dx,x,\tan \left (\frac{1}{2} (c+d x)\right )\right )}{3 a^{5/3} d}+\frac{(2 b) \operatorname{Subst}\left (\int \frac{1}{-\sqrt [3]{a}-2 (-1)^{2/3} \sqrt [3]{b} x-\sqrt [3]{a} x^2} \, dx,x,\tan \left (\frac{1}{2} (c+d x)\right )\right )}{3 a^{5/3} d}\\ &=-\frac{\tanh ^{-1}(\cos (c+d x))}{2 a d}-\frac{\cot (c+d x) \csc (c+d x)}{2 a d}-\frac{(4 b) \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^{5/3} d}-\frac{(4 b) \operatorname{Subst}\left (\int \frac{1}{-4 \left (a^{2/3}+\sqrt [3]{-1} b^{2/3}\right )-x^2} \, dx,x,-2 (-1)^{2/3} \sqrt [3]{b}-2 \sqrt [3]{a} \tan \left (\frac{1}{2} (c+d x)\right )\right )}{3 a^{5/3} d}-\frac{(4 b) \operatorname{Subst}\left (\int \frac{1}{-4 \left (a^{2/3}-(-1)^{2/3} b^{2/3}\right )-x^2} \, dx,x,2 \sqrt [3]{-1} \sqrt [3]{b}-2 \sqrt [3]{a} \tan \left (\frac{1}{2} (c+d x)\right )\right )}{3 a^{5/3} d}\\ &=\frac{2 b \tan ^{-1}\left (\frac{\sqrt [3]{-1} \sqrt [3]{b}-\sqrt [3]{a} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a^{2/3}-(-1)^{2/3} b^{2/3}}}\right )}{3 a^{5/3} \sqrt{a^{2/3}-(-1)^{2/3} b^{2/3}} d}-\frac{2 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^{5/3} \sqrt{a^{2/3}-b^{2/3}} d}-\frac{2 b \tan ^{-1}\left (\frac{(-1)^{2/3} \sqrt [3]{b}+\sqrt [3]{a} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a^{2/3}+\sqrt [3]{-1} b^{2/3}}}\right )}{3 a^{5/3} \sqrt{a^{2/3}+\sqrt [3]{-1} b^{2/3}} d}-\frac{\tanh ^{-1}(\cos (c+d x))}{2 a d}-\frac{\cot (c+d x) \csc (c+d x)}{2 a d}\\ \end{align*}
Mathematica [C] time = 0.396276, size = 181, normalized size = 0.63 \[ \frac{-3 \left (\csc ^2\left (\frac{1}{2} (c+d x)\right )-\sec ^2\left (\frac{1}{2} (c+d x)\right )-4 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )+4 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )\right )+16 i b \text{RootSum}\left [-8 i \text{$\#$1}^3 a+\text{$\#$1}^6 b-3 \text{$\#$1}^4 b+3 \text{$\#$1}^2 b-b\& ,\frac{2 \text{$\#$1} \tan ^{-1}\left (\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right )-i \text{$\#$1} \log \left (\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right )}{\text{$\#$1}^4 b-2 \text{$\#$1}^2 b-4 i \text{$\#$1} a+b}\& \right ]}{24 a d} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.213, size = 144, normalized size = 0.5 \begin{align*}{\frac{1}{8\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}}-{\frac{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}}^{4}+2\,{{\it \_R}}^{2}+1}{{{\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}{8\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-2}}+{\frac{1}{2\,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 )^{3}}{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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