Optimal. Leaf size=232 \[ -\frac {3 \tan ^{-1}\left (\frac {\sqrt [4]{b^2+c^2} \sin \left (-\tan ^{-1}(b,c)+d+e x\right )}{\sqrt {2} \sqrt {\sqrt {b^2+c^2} \cos \left (-\tan ^{-1}(b,c)+d+e x\right )-\sqrt {b^2+c^2}}}\right )}{16 \sqrt {2} e \left (b^2+c^2\right )^{5/4}}-\frac {3 (c \cos (d+e x)-b \sin (d+e x))}{16 e \left (b^2+c^2\right ) \left (-\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^{3/2}}+\frac {c \cos (d+e x)-b \sin (d+e x)}{4 e \sqrt {b^2+c^2} \left (-\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^{5/2}} \]
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Rubi [A] time = 0.17, antiderivative size = 232, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {3116, 3115, 2649, 204} \[ -\frac {3 \tan ^{-1}\left (\frac {\sqrt [4]{b^2+c^2} \sin \left (-\tan ^{-1}(b,c)+d+e x\right )}{\sqrt {2} \sqrt {\sqrt {b^2+c^2} \cos \left (-\tan ^{-1}(b,c)+d+e x\right )-\sqrt {b^2+c^2}}}\right )}{16 \sqrt {2} e \left (b^2+c^2\right )^{5/4}}-\frac {3 (c \cos (d+e x)-b \sin (d+e x))}{16 e \left (b^2+c^2\right ) \left (-\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^{3/2}}+\frac {c \cos (d+e x)-b \sin (d+e x)}{4 e \sqrt {b^2+c^2} \left (-\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^{5/2}} \]
Antiderivative was successfully verified.
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Rule 204
Rule 2649
Rule 3115
Rule 3116
Rubi steps
\begin {align*} \int \frac {1}{\left (-\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^{5/2}} \, dx &=\frac {c \cos (d+e x)-b \sin (d+e x)}{4 \sqrt {b^2+c^2} e \left (-\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^{5/2}}-\frac {3 \int \frac {1}{\left (-\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^{3/2}} \, dx}{8 \sqrt {b^2+c^2}}\\ &=\frac {c \cos (d+e x)-b \sin (d+e x)}{4 \sqrt {b^2+c^2} e \left (-\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^{5/2}}-\frac {3 (c \cos (d+e x)-b \sin (d+e x))}{16 \left (b^2+c^2\right ) e \left (-\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^{3/2}}+\frac {3 \int \frac {1}{\sqrt {-\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)}} \, dx}{32 \left (b^2+c^2\right )}\\ &=\frac {c \cos (d+e x)-b \sin (d+e x)}{4 \sqrt {b^2+c^2} e \left (-\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^{5/2}}-\frac {3 (c \cos (d+e x)-b \sin (d+e x))}{16 \left (b^2+c^2\right ) e \left (-\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^{3/2}}+\frac {3 \int \frac {1}{\sqrt {-\sqrt {b^2+c^2}+\sqrt {b^2+c^2} \cos \left (d+e x-\tan ^{-1}(b,c)\right )}} \, dx}{32 \left (b^2+c^2\right )}\\ &=\frac {c \cos (d+e x)-b \sin (d+e x)}{4 \sqrt {b^2+c^2} e \left (-\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^{5/2}}-\frac {3 (c \cos (d+e x)-b \sin (d+e x))}{16 \left (b^2+c^2\right ) e \left (-\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^{3/2}}-\frac {3 \operatorname {Subst}\left (\int \frac {1}{-2 \sqrt {b^2+c^2}-x^2} \, dx,x,-\frac {\sqrt {b^2+c^2} \sin \left (d+e x-\tan ^{-1}(b,c)\right )}{\sqrt {-\sqrt {b^2+c^2}+\sqrt {b^2+c^2} \cos \left (d+e x-\tan ^{-1}(b,c)\right )}}\right )}{16 \left (b^2+c^2\right ) e}\\ &=-\frac {3 \tan ^{-1}\left (\frac {\sqrt [4]{b^2+c^2} \sin \left (d+e x-\tan ^{-1}(b,c)\right )}{\sqrt {2} \sqrt {-\sqrt {b^2+c^2}+\sqrt {b^2+c^2} \cos \left (d+e x-\tan ^{-1}(b,c)\right )}}\right )}{16 \sqrt {2} \left (b^2+c^2\right )^{5/4} e}+\frac {c \cos (d+e x)-b \sin (d+e x)}{4 \sqrt {b^2+c^2} e \left (-\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^{5/2}}-\frac {3 (c \cos (d+e x)-b \sin (d+e x))}{16 \left (b^2+c^2\right ) e \left (-\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^{3/2}}\\ \end {align*}
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Mathematica [F] time = 180.06, size = 0, normalized size = 0.00 \[ \text {\$Aborted} \]
Verification is Not applicable to the result.
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fricas [B] time = 1.20, size = 655, normalized size = 2.82 \[ \frac {\frac {3 \, \sqrt {\frac {1}{2}} {\left (5 \, b^{4} c \cos \left (e x + d\right ) + {\left (5 \, b^{4} c - 10 \, b^{2} c^{3} + c^{5}\right )} \cos \left (e x + d\right )^{5} - 10 \, {\left (b^{4} c - b^{2} c^{3}\right )} \cos \left (e x + d\right )^{3} - {\left (b^{5} + {\left (b^{5} - 10 \, b^{3} c^{2} + 5 \, b c^{4}\right )} \cos \left (e x + d\right )^{4} - 2 \, {\left (b^{5} - 5 \, b^{3} c^{2}\right )} \cos \left (e x + d\right )^{2}\right )} \sin \left (e x + d\right )\right )} \arctan \left (-\frac {\sqrt {\frac {1}{2}} {\left (b \cos \left (e x + d\right ) + c \sin \left (e x + d\right ) + \sqrt {b^{2} + c^{2}}\right )} \sqrt {b \cos \left (e x + d\right ) + c \sin \left (e x + d\right ) - \sqrt {b^{2} + c^{2}}}}{{\left (b^{2} + c^{2}\right )}^{\frac {1}{4}} {\left (c \cos \left (e x + d\right ) - b \sin \left (e x + d\right )\right )}}\right )}{{\left (b^{2} + c^{2}\right )}^{\frac {1}{4}}} + {\left (3 \, {\left (b^{4} - 6 \, b^{2} c^{2} + c^{4}\right )} \cos \left (e x + d\right )^{4} - 7 \, b^{4} - 26 \, b^{2} c^{2} - 16 \, c^{4} - 6 \, {\left (2 \, b^{4} - 3 \, b^{2} c^{2} - c^{4}\right )} \cos \left (e x + d\right )^{2} + 12 \, {\left ({\left (b^{3} c - b c^{3}\right )} \cos \left (e x + d\right )^{3} - {\left (2 \, b^{3} c + b c^{3}\right )} \cos \left (e x + d\right )\right )} \sin \left (e x + d\right ) + 2 \, {\left ({\left (b^{3} - 3 \, b c^{2}\right )} \cos \left (e x + d\right )^{3} - 3 \, {\left (3 \, b^{3} + 2 \, b c^{2}\right )} \cos \left (e x + d\right ) - {\left (9 \, b^{2} c + 8 \, c^{3} - {\left (3 \, b^{2} c - c^{3}\right )} \cos \left (e x + d\right )^{2}\right )} \sin \left (e x + d\right )\right )} \sqrt {b^{2} + c^{2}}\right )} \sqrt {b \cos \left (e x + d\right ) + c \sin \left (e x + d\right ) - \sqrt {b^{2} + c^{2}}}}{16 \, {\left ({\left (5 \, b^{6} c - 5 \, b^{4} c^{3} - 9 \, b^{2} c^{5} + c^{7}\right )} e \cos \left (e x + d\right )^{5} - 10 \, {\left (b^{6} c - b^{2} c^{5}\right )} e \cos \left (e x + d\right )^{3} + 5 \, {\left (b^{6} c + b^{4} c^{3}\right )} e \cos \left (e x + d\right ) - {\left ({\left (b^{7} - 9 \, b^{5} c^{2} - 5 \, b^{3} c^{4} + 5 \, b c^{6}\right )} e \cos \left (e x + d\right )^{4} - 2 \, {\left (b^{7} - 4 \, b^{5} c^{2} - 5 \, b^{3} c^{4}\right )} e \cos \left (e x + d\right )^{2} + {\left (b^{7} + b^{5} c^{2}\right )} e\right )} \sin \left (e x + d\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.45, size = 363, normalized size = 1.56 \[ -\frac {\left (-\sin \left (e x +d -\arctan \left (-b , c\right )\right ) \sqrt {2}\, \arctan \left (\frac {\sqrt {-\sqrt {b^{2}+c^{2}}\, \sin \left (e x +d -\arctan \left (-b , c\right )\right )-\sqrt {b^{2}+c^{2}}}\, \sqrt {2}}{2 \left (b^{2}+c^{2}\right )^{\frac {1}{4}}}\right ) \left (b^{2}+c^{2}\right )+2 \sqrt {-\sqrt {b^{2}+c^{2}}\, \sin \left (e x +d -\arctan \left (-b , c\right )\right )-\sqrt {b^{2}+c^{2}}}\, \left (b^{2}+c^{2}\right )^{\frac {3}{4}}+\sqrt {2}\, \arctan \left (\frac {\sqrt {-\sqrt {b^{2}+c^{2}}\, \sin \left (e x +d -\arctan \left (-b , c\right )\right )-\sqrt {b^{2}+c^{2}}}\, \sqrt {2}}{2 \left (b^{2}+c^{2}\right )^{\frac {1}{4}}}\right ) b^{2}+\sqrt {2}\, \arctan \left (\frac {\sqrt {-\sqrt {b^{2}+c^{2}}\, \sin \left (e x +d -\arctan \left (-b , c\right )\right )-\sqrt {b^{2}+c^{2}}}\, \sqrt {2}}{2 \left (b^{2}+c^{2}\right )^{\frac {1}{4}}}\right ) c^{2}\right ) \sqrt {-\sqrt {b^{2}+c^{2}}\, \left (1+\sin \left (e x +d -\arctan \left (-b , c\right )\right )\right )}}{4 \left (b^{2}+c^{2}\right )^{\frac {5}{4}} \cos \left (e x +d -\arctan \left (-b , c\right )\right ) \sqrt {\frac {b^{2} \sin \left (e x +d -\arctan \left (-b , c\right )\right )+c^{2} \sin \left (e x +d -\arctan \left (-b , c\right )\right )-b^{2}-c^{2}}{\sqrt {b^{2}+c^{2}}}}\, e} \]
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 \[ \text {Exception raised: RuntimeError} \]
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 \[ \int \frac {1}{{\left (b\,\cos \left (d+e\,x\right )+c\,\sin \left (d+e\,x\right )-\sqrt {b^2+c^2}\right )}^{5/2}} \,d x \]
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 \[ \int \frac {1}{\left (b \cos {\left (d + e x \right )} + c \sin {\left (d + e x \right )} - \sqrt {b^{2} + c^{2}}\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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