Optimal. Leaf size=232 \[ \frac{2 c^3 \sqrt{c \cot (a+b x)}}{b}+\frac{c^{7/2} \log \left (\sqrt{c} \cot (a+b x)-\sqrt{2} \sqrt{c \cot (a+b x)}+\sqrt{c}\right )}{2 \sqrt{2} b}-\frac{c^{7/2} \log \left (\sqrt{c} \cot (a+b x)+\sqrt{2} \sqrt{c \cot (a+b x)}+\sqrt{c}\right )}{2 \sqrt{2} b}+\frac{c^{7/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{c \cot (a+b x)}}{\sqrt{c}}\right )}{\sqrt{2} b}-\frac{c^{7/2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c \cot (a+b x)}}{\sqrt{c}}+1\right )}{\sqrt{2} b}-\frac{2 c (c \cot (a+b x))^{5/2}}{5 b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.191742, antiderivative size = 232, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 9, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.75, Rules used = {3473, 3476, 329, 211, 1165, 628, 1162, 617, 204} \[ \frac{2 c^3 \sqrt{c \cot (a+b x)}}{b}+\frac{c^{7/2} \log \left (\sqrt{c} \cot (a+b x)-\sqrt{2} \sqrt{c \cot (a+b x)}+\sqrt{c}\right )}{2 \sqrt{2} b}-\frac{c^{7/2} \log \left (\sqrt{c} \cot (a+b x)+\sqrt{2} \sqrt{c \cot (a+b x)}+\sqrt{c}\right )}{2 \sqrt{2} b}+\frac{c^{7/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{c \cot (a+b x)}}{\sqrt{c}}\right )}{\sqrt{2} b}-\frac{c^{7/2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c \cot (a+b x)}}{\sqrt{c}}+1\right )}{\sqrt{2} b}-\frac{2 c (c \cot (a+b x))^{5/2}}{5 b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3473
Rule 3476
Rule 329
Rule 211
Rule 1165
Rule 628
Rule 1162
Rule 617
Rule 204
Rubi steps
\begin{align*} \int (c \cot (a+b x))^{7/2} \, dx &=-\frac{2 c (c \cot (a+b x))^{5/2}}{5 b}-c^2 \int (c \cot (a+b x))^{3/2} \, dx\\ &=\frac{2 c^3 \sqrt{c \cot (a+b x)}}{b}-\frac{2 c (c \cot (a+b x))^{5/2}}{5 b}+c^4 \int \frac{1}{\sqrt{c \cot (a+b x)}} \, dx\\ &=\frac{2 c^3 \sqrt{c \cot (a+b x)}}{b}-\frac{2 c (c \cot (a+b x))^{5/2}}{5 b}-\frac{c^5 \operatorname{Subst}\left (\int \frac{1}{\sqrt{x} \left (c^2+x^2\right )} \, dx,x,c \cot (a+b x)\right )}{b}\\ &=\frac{2 c^3 \sqrt{c \cot (a+b x)}}{b}-\frac{2 c (c \cot (a+b x))^{5/2}}{5 b}-\frac{\left (2 c^5\right ) \operatorname{Subst}\left (\int \frac{1}{c^2+x^4} \, dx,x,\sqrt{c \cot (a+b x)}\right )}{b}\\ &=\frac{2 c^3 \sqrt{c \cot (a+b x)}}{b}-\frac{2 c (c \cot (a+b x))^{5/2}}{5 b}-\frac{c^4 \operatorname{Subst}\left (\int \frac{c-x^2}{c^2+x^4} \, dx,x,\sqrt{c \cot (a+b x)}\right )}{b}-\frac{c^4 \operatorname{Subst}\left (\int \frac{c+x^2}{c^2+x^4} \, dx,x,\sqrt{c \cot (a+b x)}\right )}{b}\\ &=\frac{2 c^3 \sqrt{c \cot (a+b x)}}{b}-\frac{2 c (c \cot (a+b x))^{5/2}}{5 b}+\frac{c^{7/2} \operatorname{Subst}\left (\int \frac{\sqrt{2} \sqrt{c}+2 x}{-c-\sqrt{2} \sqrt{c} x-x^2} \, dx,x,\sqrt{c \cot (a+b x)}\right )}{2 \sqrt{2} b}+\frac{c^{7/2} \operatorname{Subst}\left (\int \frac{\sqrt{2} \sqrt{c}-2 x}{-c+\sqrt{2} \sqrt{c} x-x^2} \, dx,x,\sqrt{c \cot (a+b x)}\right )}{2 \sqrt{2} b}-\frac{c^4 \operatorname{Subst}\left (\int \frac{1}{c-\sqrt{2} \sqrt{c} x+x^2} \, dx,x,\sqrt{c \cot (a+b x)}\right )}{2 b}-\frac{c^4 \operatorname{Subst}\left (\int \frac{1}{c+\sqrt{2} \sqrt{c} x+x^2} \, dx,x,\sqrt{c \cot (a+b x)}\right )}{2 b}\\ &=\frac{2 c^3 \sqrt{c \cot (a+b x)}}{b}-\frac{2 c (c \cot (a+b x))^{5/2}}{5 b}+\frac{c^{7/2} \log \left (\sqrt{c}+\sqrt{c} \cot (a+b x)-\sqrt{2} \sqrt{c \cot (a+b x)}\right )}{2 \sqrt{2} b}-\frac{c^{7/2} \log \left (\sqrt{c}+\sqrt{c} \cot (a+b x)+\sqrt{2} \sqrt{c \cot (a+b x)}\right )}{2 \sqrt{2} b}-\frac{c^{7/2} \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt{c \cot (a+b x)}}{\sqrt{c}}\right )}{\sqrt{2} b}+\frac{c^{7/2} \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt{c \cot (a+b x)}}{\sqrt{c}}\right )}{\sqrt{2} b}\\ &=\frac{c^{7/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{c \cot (a+b x)}}{\sqrt{c}}\right )}{\sqrt{2} b}-\frac{c^{7/2} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt{c \cot (a+b x)}}{\sqrt{c}}\right )}{\sqrt{2} b}+\frac{2 c^3 \sqrt{c \cot (a+b x)}}{b}-\frac{2 c (c \cot (a+b x))^{5/2}}{5 b}+\frac{c^{7/2} \log \left (\sqrt{c}+\sqrt{c} \cot (a+b x)-\sqrt{2} \sqrt{c \cot (a+b x)}\right )}{2 \sqrt{2} b}-\frac{c^{7/2} \log \left (\sqrt{c}+\sqrt{c} \cot (a+b x)+\sqrt{2} \sqrt{c \cot (a+b x)}\right )}{2 \sqrt{2} b}\\ \end{align*}
Mathematica [A] time = 0.507568, size = 175, normalized size = 0.75 \[ \frac{c^3 \sqrt{c \cot (a+b x)} \left (-8 \cot ^{\frac{5}{2}}(a+b x)+40 \sqrt{\cot (a+b x)}+5 \sqrt{2} \log \left (\cot (a+b x)-\sqrt{2} \sqrt{\cot (a+b x)}+1\right )-5 \sqrt{2} \log \left (\cot (a+b x)+\sqrt{2} \sqrt{\cot (a+b x)}+1\right )+10 \sqrt{2} \tan ^{-1}\left (1-\sqrt{2} \sqrt{\cot (a+b x)}\right )-10 \sqrt{2} \tan ^{-1}\left (\sqrt{2} \sqrt{\cot (a+b x)}+1\right )\right )}{20 b \sqrt{\cot (a+b x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.057, size = 200, normalized size = 0.9 \begin{align*} -{\frac{2\,c}{5\,b} \left ( c\cot \left ( bx+a \right ) \right ) ^{{\frac{5}{2}}}}+2\,{\frac{{c}^{3}\sqrt{c\cot \left ( bx+a \right ) }}{b}}-{\frac{{c}^{3}\sqrt{2}}{2\,b}\sqrt [4]{{c}^{2}}\arctan \left ({\sqrt{2}\sqrt{c\cot \left ( bx+a \right ) }{\frac{1}{\sqrt [4]{{c}^{2}}}}}+1 \right ) }+{\frac{{c}^{3}\sqrt{2}}{2\,b}\sqrt [4]{{c}^{2}}\arctan \left ( -{\sqrt{2}\sqrt{c\cot \left ( bx+a \right ) }{\frac{1}{\sqrt [4]{{c}^{2}}}}}+1 \right ) }-{\frac{{c}^{3}\sqrt{2}}{4\,b}\sqrt [4]{{c}^{2}}\ln \left ({ \left ( c\cot \left ( bx+a \right ) +\sqrt [4]{{c}^{2}}\sqrt{c\cot \left ( bx+a \right ) }\sqrt{2}+\sqrt{{c}^{2}} \right ) \left ( c\cot \left ( bx+a \right ) -\sqrt [4]{{c}^{2}}\sqrt{c\cot \left ( bx+a \right ) }\sqrt{2}+\sqrt{{c}^{2}} \right ) ^{-1}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (c \cot{\left (a + b x \right )}\right )^{\frac{7}{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (c \cot \left (b x + a\right )\right )^{\frac{7}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]