Optimal. Leaf size=197 \[ \frac {b^2 \tan ^{-1}\left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{11/4} \sqrt {\sqrt {a}-\sqrt {b}} d}+\frac {b^2 \tan ^{-1}\left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{11/4} \sqrt {\sqrt {a}+\sqrt {b}} d}-\frac {(a+b) \cot (c+d x)}{a^2 d}-\frac {(3 a+b) \cot ^3(c+d x)}{3 a^2 d}-\frac {3 \cot ^5(c+d x)}{5 a d}-\frac {\cot ^7(c+d x)}{7 a d} \]
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Rubi [A]
time = 0.16, antiderivative size = 197, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3296, 1301,
1180, 211} \begin {gather*} \frac {b^2 \text {ArcTan}\left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{11/4} d \sqrt {\sqrt {a}-\sqrt {b}}}+\frac {b^2 \text {ArcTan}\left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{11/4} d \sqrt {\sqrt {a}+\sqrt {b}}}-\frac {(3 a+b) \cot ^3(c+d x)}{3 a^2 d}-\frac {(a+b) \cot (c+d x)}{a^2 d}-\frac {\cot ^7(c+d x)}{7 a d}-\frac {3 \cot ^5(c+d x)}{5 a d} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 1180
Rule 1301
Rule 3296
Rubi steps
\begin {align*} \int \frac {\csc ^8(c+d x)}{a-b \sin ^4(c+d x)} \, dx &=\frac {\text {Subst}\left (\int \frac {\left (1+x^2\right )^5}{x^8 \left (a+2 a x^2+(a-b) x^4\right )} \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac {\text {Subst}\left (\int \left (\frac {1}{a x^8}+\frac {3}{a x^6}+\frac {3 a+b}{a^2 x^4}+\frac {a+b}{a^2 x^2}+\frac {b^2 \left (1+x^2\right )}{a^2 \left (a+2 a x^2+(a-b) x^4\right )}\right ) \, dx,x,\tan (c+d x)\right )}{d}\\ &=-\frac {(a+b) \cot (c+d x)}{a^2 d}-\frac {(3 a+b) \cot ^3(c+d x)}{3 a^2 d}-\frac {3 \cot ^5(c+d x)}{5 a d}-\frac {\cot ^7(c+d x)}{7 a d}+\frac {b^2 \text {Subst}\left (\int \frac {1+x^2}{a+2 a x^2+(a-b) x^4} \, dx,x,\tan (c+d x)\right )}{a^2 d}\\ &=-\frac {(a+b) \cot (c+d x)}{a^2 d}-\frac {(3 a+b) \cot ^3(c+d x)}{3 a^2 d}-\frac {3 \cot ^5(c+d x)}{5 a d}-\frac {\cot ^7(c+d x)}{7 a d}+\frac {\left (\left (\sqrt {a}+\sqrt {b}\right ) b^2\right ) \text {Subst}\left (\int \frac {1}{a+\sqrt {a} \sqrt {b}+(a-b) x^2} \, dx,x,\tan (c+d x)\right )}{2 a^{5/2} d}+\frac {\left (\left (1-\frac {\sqrt {b}}{\sqrt {a}}\right ) b^2\right ) \text {Subst}\left (\int \frac {1}{a-\sqrt {a} \sqrt {b}+(a-b) x^2} \, dx,x,\tan (c+d x)\right )}{2 a^2 d}\\ &=\frac {b^2 \tan ^{-1}\left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{11/4} \sqrt {\sqrt {a}-\sqrt {b}} d}+\frac {b^2 \tan ^{-1}\left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{11/4} \sqrt {\sqrt {a}+\sqrt {b}} d}-\frac {(a+b) \cot (c+d x)}{a^2 d}-\frac {(3 a+b) \cot ^3(c+d x)}{3 a^2 d}-\frac {3 \cot ^5(c+d x)}{5 a d}-\frac {\cot ^7(c+d x)}{7 a d}\\ \end {align*}
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Mathematica [A]
time = 6.22, size = 277, normalized size = 1.41 \begin {gather*} \frac {b^2 \tan ^{-1}\left (\frac {\left (\sqrt {a} \sqrt {b}+b\right ) \tan (c+d x)}{\sqrt {a+\sqrt {a} \sqrt {b}} \sqrt {b}}\right )}{2 a^{5/2} \sqrt {a+\sqrt {a} \sqrt {b}} d}-\frac {b^2 \tanh ^{-1}\left (\frac {\left (\sqrt {a} \sqrt {b}-b\right ) \tan (c+d x)}{\sqrt {-a+\sqrt {a} \sqrt {b}} \sqrt {b}}\right )}{2 a^{5/2} \sqrt {-a+\sqrt {a} \sqrt {b}} d}-\frac {2 (24 a \cos (c+d x)+35 b \cos (c+d x)) \csc (c+d x)}{105 a^2 d}+\frac {(-24 a \cos (c+d x)-35 b \cos (c+d x)) \csc ^3(c+d x)}{105 a^2 d}-\frac {6 \cot (c+d x) \csc ^4(c+d x)}{35 a d}-\frac {\cot (c+d x) \csc ^6(c+d x)}{7 a d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.51, size = 213, normalized size = 1.08
method | result | size |
derivativedivides | \(\frac {-\frac {1}{7 a \tan \left (d x +c \right )^{7}}-\frac {a +b}{a^{2} \tan \left (d x +c \right )}-\frac {3 a +b}{3 a^{2} \tan \left (d x +c \right )^{3}}-\frac {3}{5 a \tan \left (d x +c \right )^{5}}+\frac {b^{2} \left (a -b \right ) \left (\frac {\left (\sqrt {a b}-b \right ) \arctanh \left (\frac {\left (-a +b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}\right )}{2 \sqrt {a b}\, \left (a -b \right ) \sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}+\frac {\left (\sqrt {a b}+b \right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}\right )}{2 \sqrt {a b}\, \left (a -b \right ) \sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}\right )}{a^{2}}}{d}\) | \(213\) |
default | \(\frac {-\frac {1}{7 a \tan \left (d x +c \right )^{7}}-\frac {a +b}{a^{2} \tan \left (d x +c \right )}-\frac {3 a +b}{3 a^{2} \tan \left (d x +c \right )^{3}}-\frac {3}{5 a \tan \left (d x +c \right )^{5}}+\frac {b^{2} \left (a -b \right ) \left (\frac {\left (\sqrt {a b}-b \right ) \arctanh \left (\frac {\left (-a +b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}\right )}{2 \sqrt {a b}\, \left (a -b \right ) \sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}+\frac {\left (\sqrt {a b}+b \right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}\right )}{2 \sqrt {a b}\, \left (a -b \right ) \sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}\right )}{a^{2}}}{d}\) | \(213\) |
risch | \(\frac {4 i \left (105 b \,{\mathrm e}^{10 i \left (d x +c \right )}-455 b \,{\mathrm e}^{8 i \left (d x +c \right )}+840 a \,{\mathrm e}^{6 i \left (d x +c \right )}+770 b \,{\mathrm e}^{6 i \left (d x +c \right )}-504 a \,{\mathrm e}^{4 i \left (d x +c \right )}-630 b \,{\mathrm e}^{4 i \left (d x +c \right )}+168 a \,{\mathrm e}^{2 i \left (d x +c \right )}+245 b \,{\mathrm e}^{2 i \left (d x +c \right )}-24 a -35 b \right )}{105 d \,a^{2} \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{7}}+256 \left (\munderset {\textit {\_R} =\RootOf \left (\left (1099511627776 a^{12} d^{4}-1099511627776 a^{11} b \,d^{4}\right ) \textit {\_Z}^{4}+2097152 a^{6} b^{4} d^{2} \textit {\_Z}^{2}+b^{8}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\left (\frac {2147483648 i d^{3} a^{10}}{b^{7}}-\frac {2147483648 i d^{3} a^{9}}{b^{6}}\right ) \textit {\_R}^{3}+\left (-\frac {2097152 d^{2} a^{7}}{b^{5}}+\frac {2097152 d^{2} a^{6}}{b^{4}}\right ) \textit {\_R}^{2}+\left (\frac {2048 i d \,a^{4}}{b^{3}}+\frac {2048 i d \,a^{3}}{b^{2}}\right ) \textit {\_R} -\frac {2 a}{b}-1\right )\right )\) | \(272\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1585 vs.
\(2 (155) = 310\).
time = 0.63, size = 1585, normalized size = 8.05 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 467 vs.
\(2 (155) = 310\).
time = 0.83, size = 467, normalized size = 2.37 \begin {gather*} \frac {\frac {105 \, {\left (3 \, \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} a^{2} b^{2} - 6 \, \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} a b^{3} - \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} b^{4}\right )} {\left (\pi \left \lfloor \frac {d x + c}{\pi } + \frac {1}{2} \right \rfloor + \arctan \left (\frac {\tan \left (d x + c\right )}{\sqrt {\frac {a^{3} + \sqrt {a^{6} - {\left (a^{3} - a^{2} b\right )} a^{3}}}{a^{3} - a^{2} b}}}\right )\right )} {\left | a - b \right |}}{3 \, a^{7} - 12 \, a^{6} b + 14 \, a^{5} b^{2} - 4 \, a^{4} b^{3} - a^{3} b^{4}} + \frac {105 \, {\left (3 \, \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} a^{2} b^{2} - 6 \, \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} a b^{3} - \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} b^{4}\right )} {\left (\pi \left \lfloor \frac {d x + c}{\pi } + \frac {1}{2} \right \rfloor + \arctan \left (\frac {\tan \left (d x + c\right )}{\sqrt {\frac {a^{3} - \sqrt {a^{6} - {\left (a^{3} - a^{2} b\right )} a^{3}}}{a^{3} - a^{2} b}}}\right )\right )} {\left | a - b \right |}}{3 \, a^{7} - 12 \, a^{6} b + 14 \, a^{5} b^{2} - 4 \, a^{4} b^{3} - a^{3} b^{4}} - \frac {2 \, {\left (105 \, a \tan \left (d x + c\right )^{6} + 105 \, b \tan \left (d x + c\right )^{6} + 105 \, a \tan \left (d x + c\right )^{4} + 35 \, b \tan \left (d x + c\right )^{4} + 63 \, a \tan \left (d x + c\right )^{2} + 15 \, a\right )}}{a^{2} \tan \left (d x + c\right )^{7}}}{210 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 17.08, size = 1704, normalized size = 8.65 \begin {gather*} -\frac {\frac {1}{7\,a}+\frac {3\,{\mathrm {tan}\left (c+d\,x\right )}^2}{5\,a}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^6\,\left (a+b\right )}{a^2}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^4\,\left (3\,a+b\right )}{3\,a^2}}{d\,{\mathrm {tan}\left (c+d\,x\right )}^7}+\frac {\mathrm {atan}\left (\frac {\left (\sqrt {\frac {\sqrt {a^{11}\,b^9}+a^6\,b^4}{16\,\left (a^{11}\,b-a^{12}\right )}}\,\left (16\,a^9\,b^5-32\,a^{10}\,b^4+16\,a^{11}\,b^3+\mathrm {tan}\left (c+d\,x\right )\,\sqrt {\frac {\sqrt {a^{11}\,b^9}+a^6\,b^4}{16\,\left (a^{11}\,b-a^{12}\right )}}\,\left (64\,a^{14}\,b-128\,a^{13}\,b^2+64\,a^{12}\,b^3\right )\right )-\mathrm {tan}\left (c+d\,x\right )\,\left (4\,a^6\,b^7-4\,a^8\,b^5\right )\right )\,\sqrt {\frac {\sqrt {a^{11}\,b^9}+a^6\,b^4}{16\,\left (a^{11}\,b-a^{12}\right )}}\,1{}\mathrm {i}-\left (\sqrt {\frac {\sqrt {a^{11}\,b^9}+a^6\,b^4}{16\,\left (a^{11}\,b-a^{12}\right )}}\,\left (16\,a^9\,b^5-32\,a^{10}\,b^4+16\,a^{11}\,b^3-\mathrm {tan}\left (c+d\,x\right )\,\sqrt {\frac {\sqrt {a^{11}\,b^9}+a^6\,b^4}{16\,\left (a^{11}\,b-a^{12}\right )}}\,\left (64\,a^{14}\,b-128\,a^{13}\,b^2+64\,a^{12}\,b^3\right )\right )+\mathrm {tan}\left (c+d\,x\right )\,\left (4\,a^6\,b^7-4\,a^8\,b^5\right )\right )\,\sqrt {\frac {\sqrt {a^{11}\,b^9}+a^6\,b^4}{16\,\left (a^{11}\,b-a^{12}\right )}}\,1{}\mathrm {i}}{\left (\sqrt {\frac {\sqrt {a^{11}\,b^9}+a^6\,b^4}{16\,\left (a^{11}\,b-a^{12}\right )}}\,\left (16\,a^9\,b^5-32\,a^{10}\,b^4+16\,a^{11}\,b^3+\mathrm {tan}\left (c+d\,x\right )\,\sqrt {\frac {\sqrt {a^{11}\,b^9}+a^6\,b^4}{16\,\left (a^{11}\,b-a^{12}\right )}}\,\left (64\,a^{14}\,b-128\,a^{13}\,b^2+64\,a^{12}\,b^3\right )\right )-\mathrm {tan}\left (c+d\,x\right )\,\left (4\,a^6\,b^7-4\,a^8\,b^5\right )\right )\,\sqrt {\frac {\sqrt {a^{11}\,b^9}+a^6\,b^4}{16\,\left (a^{11}\,b-a^{12}\right )}}+\left (\sqrt {\frac {\sqrt {a^{11}\,b^9}+a^6\,b^4}{16\,\left (a^{11}\,b-a^{12}\right )}}\,\left (16\,a^9\,b^5-32\,a^{10}\,b^4+16\,a^{11}\,b^3-\mathrm {tan}\left (c+d\,x\right )\,\sqrt {\frac {\sqrt {a^{11}\,b^9}+a^6\,b^4}{16\,\left (a^{11}\,b-a^{12}\right )}}\,\left (64\,a^{14}\,b-128\,a^{13}\,b^2+64\,a^{12}\,b^3\right )\right )+\mathrm {tan}\left (c+d\,x\right )\,\left (4\,a^6\,b^7-4\,a^8\,b^5\right )\right )\,\sqrt {\frac {\sqrt {a^{11}\,b^9}+a^6\,b^4}{16\,\left (a^{11}\,b-a^{12}\right )}}-2\,a^4\,b^8+2\,a^5\,b^7}\right )\,\sqrt {\frac {\sqrt {a^{11}\,b^9}+a^6\,b^4}{16\,\left (a^{11}\,b-a^{12}\right )}}\,2{}\mathrm {i}}{d}+\frac {\mathrm {atan}\left (\frac {\left (\sqrt {-\frac {\sqrt {a^{11}\,b^9}-a^6\,b^4}{16\,\left (a^{11}\,b-a^{12}\right )}}\,\left (16\,a^9\,b^5-32\,a^{10}\,b^4+16\,a^{11}\,b^3+\mathrm {tan}\left (c+d\,x\right )\,\sqrt {-\frac {\sqrt {a^{11}\,b^9}-a^6\,b^4}{16\,\left (a^{11}\,b-a^{12}\right )}}\,\left (64\,a^{14}\,b-128\,a^{13}\,b^2+64\,a^{12}\,b^3\right )\right )-\mathrm {tan}\left (c+d\,x\right )\,\left (4\,a^6\,b^7-4\,a^8\,b^5\right )\right )\,\sqrt {-\frac {\sqrt {a^{11}\,b^9}-a^6\,b^4}{16\,\left (a^{11}\,b-a^{12}\right )}}\,1{}\mathrm {i}-\left (\sqrt {-\frac {\sqrt {a^{11}\,b^9}-a^6\,b^4}{16\,\left (a^{11}\,b-a^{12}\right )}}\,\left (16\,a^9\,b^5-32\,a^{10}\,b^4+16\,a^{11}\,b^3-\mathrm {tan}\left (c+d\,x\right )\,\sqrt {-\frac {\sqrt {a^{11}\,b^9}-a^6\,b^4}{16\,\left (a^{11}\,b-a^{12}\right )}}\,\left (64\,a^{14}\,b-128\,a^{13}\,b^2+64\,a^{12}\,b^3\right )\right )+\mathrm {tan}\left (c+d\,x\right )\,\left (4\,a^6\,b^7-4\,a^8\,b^5\right )\right )\,\sqrt {-\frac {\sqrt {a^{11}\,b^9}-a^6\,b^4}{16\,\left (a^{11}\,b-a^{12}\right )}}\,1{}\mathrm {i}}{\left (\sqrt {-\frac {\sqrt {a^{11}\,b^9}-a^6\,b^4}{16\,\left (a^{11}\,b-a^{12}\right )}}\,\left (16\,a^9\,b^5-32\,a^{10}\,b^4+16\,a^{11}\,b^3+\mathrm {tan}\left (c+d\,x\right )\,\sqrt {-\frac {\sqrt {a^{11}\,b^9}-a^6\,b^4}{16\,\left (a^{11}\,b-a^{12}\right )}}\,\left (64\,a^{14}\,b-128\,a^{13}\,b^2+64\,a^{12}\,b^3\right )\right )-\mathrm {tan}\left (c+d\,x\right )\,\left (4\,a^6\,b^7-4\,a^8\,b^5\right )\right )\,\sqrt {-\frac {\sqrt {a^{11}\,b^9}-a^6\,b^4}{16\,\left (a^{11}\,b-a^{12}\right )}}+\left (\sqrt {-\frac {\sqrt {a^{11}\,b^9}-a^6\,b^4}{16\,\left (a^{11}\,b-a^{12}\right )}}\,\left (16\,a^9\,b^5-32\,a^{10}\,b^4+16\,a^{11}\,b^3-\mathrm {tan}\left (c+d\,x\right )\,\sqrt {-\frac {\sqrt {a^{11}\,b^9}-a^6\,b^4}{16\,\left (a^{11}\,b-a^{12}\right )}}\,\left (64\,a^{14}\,b-128\,a^{13}\,b^2+64\,a^{12}\,b^3\right )\right )+\mathrm {tan}\left (c+d\,x\right )\,\left (4\,a^6\,b^7-4\,a^8\,b^5\right )\right )\,\sqrt {-\frac {\sqrt {a^{11}\,b^9}-a^6\,b^4}{16\,\left (a^{11}\,b-a^{12}\right )}}-2\,a^4\,b^8+2\,a^5\,b^7}\right )\,\sqrt {-\frac {\sqrt {a^{11}\,b^9}-a^6\,b^4}{16\,\left (a^{11}\,b-a^{12}\right )}}\,2{}\mathrm {i}}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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