Optimal. Leaf size=200 \[ \frac{6 i b x \text{PolyLog}\left (2,-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^2}-\frac{6 i b x \text{PolyLog}\left (2,e^{i \left (c+d \sqrt{x}\right )}\right )}{d^2}-\frac{12 b \sqrt{x} \text{PolyLog}\left (3,-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^3}+\frac{12 b \sqrt{x} \text{PolyLog}\left (3,e^{i \left (c+d \sqrt{x}\right )}\right )}{d^3}-\frac{12 i b \text{PolyLog}\left (4,-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^4}+\frac{12 i b \text{PolyLog}\left (4,e^{i \left (c+d \sqrt{x}\right )}\right )}{d^4}+\frac{a x^2}{2}-\frac{4 b x^{3/2} \tanh ^{-1}\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.177666, antiderivative size = 200, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 7, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.438, Rules used = {14, 4205, 4183, 2531, 6609, 2282, 6589} \[ \frac{6 i b x \text{PolyLog}\left (2,-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^2}-\frac{6 i b x \text{PolyLog}\left (2,e^{i \left (c+d \sqrt{x}\right )}\right )}{d^2}-\frac{12 b \sqrt{x} \text{PolyLog}\left (3,-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^3}+\frac{12 b \sqrt{x} \text{PolyLog}\left (3,e^{i \left (c+d \sqrt{x}\right )}\right )}{d^3}-\frac{12 i b \text{PolyLog}\left (4,-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^4}+\frac{12 i b \text{PolyLog}\left (4,e^{i \left (c+d \sqrt{x}\right )}\right )}{d^4}+\frac{a x^2}{2}-\frac{4 b x^{3/2} \tanh ^{-1}\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 14
Rule 4205
Rule 4183
Rule 2531
Rule 6609
Rule 2282
Rule 6589
Rubi steps
\begin{align*} \int x \left (a+b \csc \left (c+d \sqrt{x}\right )\right ) \, dx &=\int \left (a x+b x \csc \left (c+d \sqrt{x}\right )\right ) \, dx\\ &=\frac{a x^2}{2}+b \int x \csc \left (c+d \sqrt{x}\right ) \, dx\\ &=\frac{a x^2}{2}+(2 b) \operatorname{Subst}\left (\int x^3 \csc (c+d x) \, dx,x,\sqrt{x}\right )\\ &=\frac{a x^2}{2}-\frac{4 b x^{3/2} \tanh ^{-1}\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d}-\frac{(6 b) \operatorname{Subst}\left (\int x^2 \log \left (1-e^{i (c+d x)}\right ) \, dx,x,\sqrt{x}\right )}{d}+\frac{(6 b) \operatorname{Subst}\left (\int x^2 \log \left (1+e^{i (c+d x)}\right ) \, dx,x,\sqrt{x}\right )}{d}\\ &=\frac{a x^2}{2}-\frac{4 b x^{3/2} \tanh ^{-1}\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d}+\frac{6 i b x \text{Li}_2\left (-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^2}-\frac{6 i b x \text{Li}_2\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d^2}-\frac{(12 i b) \operatorname{Subst}\left (\int x \text{Li}_2\left (-e^{i (c+d x)}\right ) \, dx,x,\sqrt{x}\right )}{d^2}+\frac{(12 i b) \operatorname{Subst}\left (\int x \text{Li}_2\left (e^{i (c+d x)}\right ) \, dx,x,\sqrt{x}\right )}{d^2}\\ &=\frac{a x^2}{2}-\frac{4 b x^{3/2} \tanh ^{-1}\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d}+\frac{6 i b x \text{Li}_2\left (-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^2}-\frac{6 i b x \text{Li}_2\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d^2}-\frac{12 b \sqrt{x} \text{Li}_3\left (-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^3}+\frac{12 b \sqrt{x} \text{Li}_3\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d^3}+\frac{(12 b) \operatorname{Subst}\left (\int \text{Li}_3\left (-e^{i (c+d x)}\right ) \, dx,x,\sqrt{x}\right )}{d^3}-\frac{(12 b) \operatorname{Subst}\left (\int \text{Li}_3\left (e^{i (c+d x)}\right ) \, dx,x,\sqrt{x}\right )}{d^3}\\ &=\frac{a x^2}{2}-\frac{4 b x^{3/2} \tanh ^{-1}\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d}+\frac{6 i b x \text{Li}_2\left (-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^2}-\frac{6 i b x \text{Li}_2\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d^2}-\frac{12 b \sqrt{x} \text{Li}_3\left (-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^3}+\frac{12 b \sqrt{x} \text{Li}_3\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d^3}-\frac{(12 i b) \operatorname{Subst}\left (\int \frac{\text{Li}_3(-x)}{x} \, dx,x,e^{i \left (c+d \sqrt{x}\right )}\right )}{d^4}+\frac{(12 i b) \operatorname{Subst}\left (\int \frac{\text{Li}_3(x)}{x} \, dx,x,e^{i \left (c+d \sqrt{x}\right )}\right )}{d^4}\\ &=\frac{a x^2}{2}-\frac{4 b x^{3/2} \tanh ^{-1}\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d}+\frac{6 i b x \text{Li}_2\left (-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^2}-\frac{6 i b x \text{Li}_2\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d^2}-\frac{12 b \sqrt{x} \text{Li}_3\left (-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^3}+\frac{12 b \sqrt{x} \text{Li}_3\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d^3}-\frac{12 i b \text{Li}_4\left (-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^4}+\frac{12 i b \text{Li}_4\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d^4}\\ \end{align*}
Mathematica [A] time = 0.376526, size = 260, normalized size = 1.3 \[ \frac{a x^2}{2}-\frac{2 b \left (-3 i d^2 x \text{PolyLog}\left (2,-\cos \left (c+d \sqrt{x}\right )-i \sin \left (c+d \sqrt{x}\right )\right )+3 i d^2 x \text{PolyLog}\left (2,\cos \left (c+d \sqrt{x}\right )+i \sin \left (c+d \sqrt{x}\right )\right )+6 d \sqrt{x} \text{PolyLog}\left (3,-\cos \left (c+d \sqrt{x}\right )-i \sin \left (c+d \sqrt{x}\right )\right )-6 d \sqrt{x} \text{PolyLog}\left (3,\cos \left (c+d \sqrt{x}\right )+i \sin \left (c+d \sqrt{x}\right )\right )+6 i \text{PolyLog}\left (4,-\cos \left (c+d \sqrt{x}\right )-i \sin \left (c+d \sqrt{x}\right )\right )-6 i \text{PolyLog}\left (4,\cos \left (c+d \sqrt{x}\right )+i \sin \left (c+d \sqrt{x}\right )\right )+2 d^3 x^{3/2} \tanh ^{-1}\left (\cos \left (c+d \sqrt{x}\right )+i \sin \left (c+d \sqrt{x}\right )\right )\right )}{d^4} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.11, size = 0, normalized size = 0. \begin{align*} \int x \left ( a+b\csc \left ( c+d\sqrt{x} \right ) \right ) \, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.38328, size = 721, normalized size = 3.6 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (b x \csc \left (d \sqrt{x} + c\right ) + a x, x\right ) \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 x \left (a + b \csc{\left (c + d \sqrt{x} \right )}\right )\, 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 (b \csc \left (d \sqrt{x} + c\right ) + a\right )} x\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]