Optimal. Leaf size=539 \[ \frac {x^2}{2 a}+\frac {2 i b x^{3/2} \log \left (1-\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d}-\frac {2 i b x^{3/2} \log \left (1-\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d}+\frac {6 b x \text {PolyLog}\left (2,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^2}-\frac {6 b x \text {PolyLog}\left (2,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^2}+\frac {12 i b \sqrt {x} \text {PolyLog}\left (3,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^3}-\frac {12 i b \sqrt {x} \text {PolyLog}\left (3,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^3}-\frac {12 b \text {PolyLog}\left (4,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^4}+\frac {12 b \text {PolyLog}\left (4,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^4} \]
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Rubi [A]
time = 0.67, antiderivative size = 539, normalized size of antiderivative = 1.00, number of steps
used = 15, number of rules used = 9, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4290, 4276,
3404, 2296, 2221, 2611, 6744, 2320, 6724} \begin {gather*} -\frac {12 b \text {Li}_4\left (\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {b^2-a^2}}\right )}{a d^4 \sqrt {b^2-a^2}}+\frac {12 b \text {Li}_4\left (\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {b^2-a^2}}\right )}{a d^4 \sqrt {b^2-a^2}}+\frac {12 i b \sqrt {x} \text {Li}_3\left (\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {b^2-a^2}}\right )}{a d^3 \sqrt {b^2-a^2}}-\frac {12 i b \sqrt {x} \text {Li}_3\left (\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {b^2-a^2}}\right )}{a d^3 \sqrt {b^2-a^2}}+\frac {6 b x \text {Li}_2\left (\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {b^2-a^2}}\right )}{a d^2 \sqrt {b^2-a^2}}-\frac {6 b x \text {Li}_2\left (\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {b^2-a^2}}\right )}{a d^2 \sqrt {b^2-a^2}}+\frac {2 i b x^{3/2} \log \left (1-\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {b^2-a^2}}\right )}{a d \sqrt {b^2-a^2}}-\frac {2 i b x^{3/2} \log \left (1-\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{\sqrt {b^2-a^2}+b}\right )}{a d \sqrt {b^2-a^2}}+\frac {x^2}{2 a} \end {gather*}
Antiderivative was successfully verified.
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Rule 2221
Rule 2296
Rule 2320
Rule 2611
Rule 3404
Rule 4276
Rule 4290
Rule 6724
Rule 6744
Rubi steps
\begin {align*} \int \frac {x}{a+b \csc \left (c+d \sqrt {x}\right )} \, dx &=2 \text {Subst}\left (\int \frac {x^3}{a+b \csc (c+d x)} \, dx,x,\sqrt {x}\right )\\ &=2 \text {Subst}\left (\int \left (\frac {x^3}{a}-\frac {b x^3}{a (b+a \sin (c+d x))}\right ) \, dx,x,\sqrt {x}\right )\\ &=\frac {x^2}{2 a}-\frac {(2 b) \text {Subst}\left (\int \frac {x^3}{b+a \sin (c+d x)} \, dx,x,\sqrt {x}\right )}{a}\\ &=\frac {x^2}{2 a}-\frac {(4 b) \text {Subst}\left (\int \frac {e^{i (c+d x)} x^3}{i a+2 b e^{i (c+d x)}-i a e^{2 i (c+d x)}} \, dx,x,\sqrt {x}\right )}{a}\\ &=\frac {x^2}{2 a}+\frac {(4 i b) \text {Subst}\left (\int \frac {e^{i (c+d x)} x^3}{2 b-2 \sqrt {-a^2+b^2}-2 i a e^{i (c+d x)}} \, dx,x,\sqrt {x}\right )}{\sqrt {-a^2+b^2}}-\frac {(4 i b) \text {Subst}\left (\int \frac {e^{i (c+d x)} x^3}{2 b+2 \sqrt {-a^2+b^2}-2 i a e^{i (c+d x)}} \, dx,x,\sqrt {x}\right )}{\sqrt {-a^2+b^2}}\\ &=\frac {x^2}{2 a}+\frac {2 i b x^{3/2} \log \left (1-\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d}-\frac {2 i b x^{3/2} \log \left (1-\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d}-\frac {(6 i b) \text {Subst}\left (\int x^2 \log \left (1-\frac {2 i a e^{i (c+d x)}}{2 b-2 \sqrt {-a^2+b^2}}\right ) \, dx,x,\sqrt {x}\right )}{a \sqrt {-a^2+b^2} d}+\frac {(6 i b) \text {Subst}\left (\int x^2 \log \left (1-\frac {2 i a e^{i (c+d x)}}{2 b+2 \sqrt {-a^2+b^2}}\right ) \, dx,x,\sqrt {x}\right )}{a \sqrt {-a^2+b^2} d}\\ &=\frac {x^2}{2 a}+\frac {2 i b x^{3/2} \log \left (1-\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d}-\frac {2 i b x^{3/2} \log \left (1-\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d}+\frac {6 b x \text {Li}_2\left (\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^2}-\frac {6 b x \text {Li}_2\left (\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^2}-\frac {(12 b) \text {Subst}\left (\int x \text {Li}_2\left (\frac {2 i a e^{i (c+d x)}}{2 b-2 \sqrt {-a^2+b^2}}\right ) \, dx,x,\sqrt {x}\right )}{a \sqrt {-a^2+b^2} d^2}+\frac {(12 b) \text {Subst}\left (\int x \text {Li}_2\left (\frac {2 i a e^{i (c+d x)}}{2 b+2 \sqrt {-a^2+b^2}}\right ) \, dx,x,\sqrt {x}\right )}{a \sqrt {-a^2+b^2} d^2}\\ &=\frac {x^2}{2 a}+\frac {2 i b x^{3/2} \log \left (1-\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d}-\frac {2 i b x^{3/2} \log \left (1-\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d}+\frac {6 b x \text {Li}_2\left (\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^2}-\frac {6 b x \text {Li}_2\left (\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^2}+\frac {12 i b \sqrt {x} \text {Li}_3\left (\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^3}-\frac {12 i b \sqrt {x} \text {Li}_3\left (\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^3}-\frac {(12 i b) \text {Subst}\left (\int \text {Li}_3\left (\frac {2 i a e^{i (c+d x)}}{2 b-2 \sqrt {-a^2+b^2}}\right ) \, dx,x,\sqrt {x}\right )}{a \sqrt {-a^2+b^2} d^3}+\frac {(12 i b) \text {Subst}\left (\int \text {Li}_3\left (\frac {2 i a e^{i (c+d x)}}{2 b+2 \sqrt {-a^2+b^2}}\right ) \, dx,x,\sqrt {x}\right )}{a \sqrt {-a^2+b^2} d^3}\\ &=\frac {x^2}{2 a}+\frac {2 i b x^{3/2} \log \left (1-\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d}-\frac {2 i b x^{3/2} \log \left (1-\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d}+\frac {6 b x \text {Li}_2\left (\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^2}-\frac {6 b x \text {Li}_2\left (\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^2}+\frac {12 i b \sqrt {x} \text {Li}_3\left (\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^3}-\frac {12 i b \sqrt {x} \text {Li}_3\left (\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^3}-\frac {(12 b) \text {Subst}\left (\int \frac {\text {Li}_3\left (\frac {i a x}{b-\sqrt {-a^2+b^2}}\right )}{x} \, dx,x,e^{i \left (c+d \sqrt {x}\right )}\right )}{a \sqrt {-a^2+b^2} d^4}+\frac {(12 b) \text {Subst}\left (\int \frac {\text {Li}_3\left (\frac {i a x}{b+\sqrt {-a^2+b^2}}\right )}{x} \, dx,x,e^{i \left (c+d \sqrt {x}\right )}\right )}{a \sqrt {-a^2+b^2} d^4}\\ &=\frac {x^2}{2 a}+\frac {2 i b x^{3/2} \log \left (1-\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d}-\frac {2 i b x^{3/2} \log \left (1-\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d}+\frac {6 b x \text {Li}_2\left (\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^2}-\frac {6 b x \text {Li}_2\left (\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^2}+\frac {12 i b \sqrt {x} \text {Li}_3\left (\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^3}-\frac {12 i b \sqrt {x} \text {Li}_3\left (\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^3}-\frac {12 b \text {Li}_4\left (\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^4}+\frac {12 b \text {Li}_4\left (\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^4}\\ \end {align*}
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Mathematica [A]
time = 1.71, size = 659, normalized size = 1.22 \begin {gather*} \frac {d^4 \sqrt {\left (a^2-b^2\right ) e^{2 i c}} x^2-4 b d^3 e^{i c} x^{3/2} \log \left (1+\frac {a e^{i \left (2 c+d \sqrt {x}\right )}}{i b e^{i c}-\sqrt {\left (a^2-b^2\right ) e^{2 i c}}}\right )+4 b d^3 e^{i c} x^{3/2} \log \left (1+\frac {a e^{i \left (2 c+d \sqrt {x}\right )}}{i b e^{i c}+\sqrt {\left (a^2-b^2\right ) e^{2 i c}}}\right )+12 i b d^2 e^{i c} x \text {PolyLog}\left (2,\frac {i a e^{i \left (2 c+d \sqrt {x}\right )}}{b e^{i c}+i \sqrt {\left (a^2-b^2\right ) e^{2 i c}}}\right )-12 i b d^2 e^{i c} x \text {PolyLog}\left (2,-\frac {a e^{i \left (2 c+d \sqrt {x}\right )}}{i b e^{i c}+\sqrt {\left (a^2-b^2\right ) e^{2 i c}}}\right )-24 b d e^{i c} \sqrt {x} \text {PolyLog}\left (3,\frac {i a e^{i \left (2 c+d \sqrt {x}\right )}}{b e^{i c}+i \sqrt {\left (a^2-b^2\right ) e^{2 i c}}}\right )+24 b d e^{i c} \sqrt {x} \text {PolyLog}\left (3,-\frac {a e^{i \left (2 c+d \sqrt {x}\right )}}{i b e^{i c}+\sqrt {\left (a^2-b^2\right ) e^{2 i c}}}\right )-24 i b e^{i c} \text {PolyLog}\left (4,\frac {i a e^{i \left (2 c+d \sqrt {x}\right )}}{b e^{i c}+i \sqrt {\left (a^2-b^2\right ) e^{2 i c}}}\right )+24 i b e^{i c} \text {PolyLog}\left (4,-\frac {a e^{i \left (2 c+d \sqrt {x}\right )}}{i b e^{i c}+\sqrt {\left (a^2-b^2\right ) e^{2 i c}}}\right )}{2 a d^4 \sqrt {\left (a^2-b^2\right ) e^{2 i c}}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.11, size = 0, normalized size = 0.00 \[\int \frac {x}{a +b \csc \left (c +d \sqrt {x}\right )}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x}{a + b \csc {\left (c + d \sqrt {x} \right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x}{a+\frac {b}{\sin \left (c+d\,\sqrt {x}\right )}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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