Optimal. Leaf size=287 \[ -\frac {2 b \tan ^{-1}\left (\frac {\sqrt [3]{b}+\sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^{2/3}-b^{2/3}}}\right )}{3 a^{5/3} \sqrt {a^{2/3}-b^{2/3}} d}-\frac {2 b \tan ^{-1}\left (\frac {(-1)^{2/3} \sqrt [3]{b}+\sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^{2/3}+\sqrt [3]{-1} b^{2/3}}}\right )}{3 a^{5/3} \sqrt {a^{2/3}+\sqrt [3]{-1} b^{2/3}} d}+\frac {2 b \tan ^{-1}\left (\frac {\sqrt [3]{-1} \left (\sqrt [3]{b}+(-1)^{2/3} \sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )\right )}{\sqrt {a^{2/3}-(-1)^{2/3} b^{2/3}}}\right )}{3 a^{5/3} \sqrt {a^{2/3}-(-1)^{2/3} b^{2/3}} d}-\frac {\tanh ^{-1}(\cos (c+d x))}{2 a d}-\frac {\cot (c+d x) \csc (c+d x)}{2 a d} \]
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Rubi [A]
time = 0.30, antiderivative size = 287, normalized size of antiderivative = 1.00, 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 = {3299, 3853,
3855, 3292, 2739, 632, 210} \begin {gather*} -\frac {2 b \text {ArcTan}\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 \text {ArcTan}\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 \text {ArcTan}\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 632
Rule 2739
Rule 3292
Rule 3299
Rule 3853
Rule 3855
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) \text {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) \text {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) \text {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) \text {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) \text {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) \text {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*}
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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in
optimal.
time = 0.29, size = 181, normalized size = 0.63 \begin {gather*} \frac {16 i b \text {RootSum}\left [-b+3 b \text {$\#$1}^2-8 i a \text {$\#$1}^3-3 b \text {$\#$1}^4+b \text {$\#$1}^6\&,\frac {2 \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}-i \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}}{b-4 i a \text {$\#$1}-2 b \text {$\#$1}^2+b \text {$\#$1}^4}\&\right ]-3 \left (\csc ^2\left (\frac {1}{2} (c+d x)\right )+4 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-4 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-\sec ^2\left (\frac {1}{2} (c+d x)\right )\right )}{24 a d} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.96, size = 136, normalized size = 0.47
method | result | size |
derivativedivides | \(\frac {\frac {\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a}-\frac {b \left (\munderset {\textit {\_R} =\RootOf \left (a \,\textit {\_Z}^{6}+3 a \,\textit {\_Z}^{4}+8 b \,\textit {\_Z}^{3}+3 a \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {\left (\textit {\_R}^{4}+2 \textit {\_R}^{2}+1\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{5} a +2 \textit {\_R}^{3} a +4 \textit {\_R}^{2} b +\textit {\_R} a}\right )}{3 a}-\frac {1}{8 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}}{d}\) | \(136\) |
default | \(\frac {\frac {\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a}-\frac {b \left (\munderset {\textit {\_R} =\RootOf \left (a \,\textit {\_Z}^{6}+3 a \,\textit {\_Z}^{4}+8 b \,\textit {\_Z}^{3}+3 a \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {\left (\textit {\_R}^{4}+2 \textit {\_R}^{2}+1\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{5} a +2 \textit {\_R}^{3} a +4 \textit {\_R}^{2} b +\textit {\_R} a}\right )}{3 a}-\frac {1}{8 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}}{d}\) | \(136\) |
risch | \(\frac {{\mathrm e}^{3 i \left (d x +c \right )}+{\mathrm e}^{i \left (d x +c \right )}}{d a \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{2}}-8 i \left (\munderset {\textit {\_R} =\RootOf \left (\left (191102976 a^{12} d^{6}-191102976 a^{10} b^{2} d^{6}\right ) \textit {\_Z}^{6}-995328 a^{8} b^{2} d^{4} \textit {\_Z}^{4}+1728 a^{4} b^{4} d^{2} \textit {\_Z}^{2}-b^{6}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\left (-\frac {15925248 i d^{5} a^{11}}{b^{6}}+\frac {15925248 i d^{5} a^{9}}{b^{4}}\right ) \textit {\_R}^{5}+\left (-\frac {331776 i d^{4} a^{9}}{b^{5}}+\frac {331776 i d^{4} a^{7}}{b^{3}}\right ) \textit {\_R}^{4}+\left (\frac {69120 i d^{3} a^{7}}{b^{4}}+\frac {13824 i d^{3} a^{5}}{b^{2}}\right ) \textit {\_R}^{3}+\frac {1728 i d^{2} a^{5} \textit {\_R}^{2}}{b^{3}}-\frac {72 i d \,a^{3} \textit {\_R}}{b^{2}}-\frac {2 i a}{b}\right )\right )-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{2 a d}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{2 a d}\) | \(279\) |
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 [C] Result contains complex when optimal does not.
time = 14.64, size = 29431, normalized size = 102.55 \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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\csc ^{3}{\left (c + d x \right )}}{a + b \sin ^{3}{\left (c + d 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 [B]
time = 15.03, size = 1573, normalized size = 5.48 \begin {gather*} \frac {\sum _{k=1}^6\ln \left (-\frac {65536\,a\,b^9-262144\,b^{10}\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-\mathrm {root}\left (729\,a^{10}\,b^2\,z^6-729\,a^{12}\,z^6-243\,a^8\,b^2\,z^4-27\,a^4\,b^4\,z^2-b^6,z,k\right )\,a^2\,b^9\,131072-\mathrm {root}\left (729\,a^{10}\,b^2\,z^6-729\,a^{12}\,z^6-243\,a^8\,b^2\,z^4-27\,a^4\,b^4\,z^2-b^6,z,k\right )\,a^4\,b^7\,61440+{\mathrm {root}\left (729\,a^{10}\,b^2\,z^6-729\,a^{12}\,z^6-243\,a^8\,b^2\,z^4-27\,a^4\,b^4\,z^2-b^6,z,k\right )}^2\,a^5\,b^7\,860160-{\mathrm {root}\left (729\,a^{10}\,b^2\,z^6-729\,a^{12}\,z^6-243\,a^8\,b^2\,z^4-27\,a^4\,b^4\,z^2-b^6,z,k\right )}^3\,a^6\,b^7\,3244032-{\mathrm {root}\left (729\,a^{10}\,b^2\,z^6-729\,a^{12}\,z^6-243\,a^8\,b^2\,z^4-27\,a^4\,b^4\,z^2-b^6,z,k\right )}^3\,a^8\,b^5\,1105920+{\mathrm {root}\left (729\,a^{10}\,b^2\,z^6-729\,a^{12}\,z^6-243\,a^8\,b^2\,z^4-27\,a^4\,b^4\,z^2-b^6,z,k\right )}^4\,a^7\,b^7\,3538944+{\mathrm {root}\left (729\,a^{10}\,b^2\,z^6-729\,a^{12}\,z^6-243\,a^8\,b^2\,z^4-27\,a^4\,b^4\,z^2-b^6,z,k\right )}^4\,a^9\,b^5\,3870720+{\mathrm {root}\left (729\,a^{10}\,b^2\,z^6-729\,a^{12}\,z^6-243\,a^8\,b^2\,z^4-27\,a^4\,b^4\,z^2-b^6,z,k\right )}^5\,a^{10}\,b^5\,663552-{\mathrm {root}\left (729\,a^{10}\,b^2\,z^6-729\,a^{12}\,z^6-243\,a^8\,b^2\,z^4-27\,a^4\,b^4\,z^2-b^6,z,k\right )}^5\,a^{12}\,b^3\,4976640-{\mathrm {root}\left (729\,a^{10}\,b^2\,z^6-729\,a^{12}\,z^6-243\,a^8\,b^2\,z^4-27\,a^4\,b^4\,z^2-b^6,z,k\right )}^6\,a^{11}\,b^5\,7962624+{\mathrm {root}\left (729\,a^{10}\,b^2\,z^6-729\,a^{12}\,z^6-243\,a^8\,b^2\,z^4-27\,a^4\,b^4\,z^2-b^6,z,k\right )}^6\,a^{13}\,b^3\,9953280+24576\,a^2\,b^8\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+\mathrm {root}\left (729\,a^{10}\,b^2\,z^6-729\,a^{12}\,z^6-243\,a^8\,b^2\,z^4-27\,a^4\,b^4\,z^2-b^6,z,k\right )\,a^3\,b^8\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,540672-{\mathrm {root}\left (729\,a^{10}\,b^2\,z^6-729\,a^{12}\,z^6-243\,a^8\,b^2\,z^4-27\,a^4\,b^4\,z^2-b^6,z,k\right )}^2\,a^4\,b^8\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,7077888+{\mathrm {root}\left (729\,a^{10}\,b^2\,z^6-729\,a^{12}\,z^6-243\,a^8\,b^2\,z^4-27\,a^4\,b^4\,z^2-b^6,z,k\right )}^2\,a^6\,b^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,442368-{\mathrm {root}\left (729\,a^{10}\,b^2\,z^6-729\,a^{12}\,z^6-243\,a^8\,b^2\,z^4-27\,a^4\,b^4\,z^2-b^6,z,k\right )}^3\,a^5\,b^8\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,2359296+{\mathrm {root}\left (729\,a^{10}\,b^2\,z^6-729\,a^{12}\,z^6-243\,a^8\,b^2\,z^4-27\,a^4\,b^4\,z^2-b^6,z,k\right )}^3\,a^7\,b^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,7741440-{\mathrm {root}\left (729\,a^{10}\,b^2\,z^6-729\,a^{12}\,z^6-243\,a^8\,b^2\,z^4-27\,a^4\,b^4\,z^2-b^6,z,k\right )}^4\,a^8\,b^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,80953344+{\mathrm {root}\left (729\,a^{10}\,b^2\,z^6-729\,a^{12}\,z^6-243\,a^8\,b^2\,z^4-27\,a^4\,b^4\,z^2-b^6,z,k\right )}^4\,a^{10}\,b^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,1990656-{\mathrm {root}\left (729\,a^{10}\,b^2\,z^6-729\,a^{12}\,z^6-243\,a^8\,b^2\,z^4-27\,a^4\,b^4\,z^2-b^6,z,k\right )}^5\,a^9\,b^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,31850496+{\mathrm {root}\left (729\,a^{10}\,b^2\,z^6-729\,a^{12}\,z^6-243\,a^8\,b^2\,z^4-27\,a^4\,b^4\,z^2-b^6,z,k\right )}^5\,a^{11}\,b^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,26873856+{\mathrm {root}\left (729\,a^{10}\,b^2\,z^6-729\,a^{12}\,z^6-243\,a^8\,b^2\,z^4-27\,a^4\,b^4\,z^2-b^6,z,k\right )}^6\,a^{10}\,b^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,254803968-{\mathrm {root}\left (729\,a^{10}\,b^2\,z^6-729\,a^{12}\,z^6-243\,a^8\,b^2\,z^4-27\,a^4\,b^4\,z^2-b^6,z,k\right )}^6\,a^{12}\,b^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,252813312}{a^5}\right )\,\mathrm {root}\left (729\,a^{10}\,b^2\,z^6-729\,a^{12}\,z^6-243\,a^8\,b^2\,z^4-27\,a^4\,b^4\,z^2-b^6,z,k\right )}{d}-\frac {{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{8\,a\,d}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{8\,a\,d}+\frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{2\,a\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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