Optimal. Leaf size=164 \[ \frac{\tan ^{-1}\left (\frac{\sqrt [4]{b^2+c^2} \sin \left (-\tan ^{-1}(b,c)+d+e x\right )}{\sqrt{2} \sqrt{\sqrt{b^2+c^2} \cos \left (-\tan ^{-1}(b,c)+d+e x\right )-\sqrt{b^2+c^2}}}\right )}{2 \sqrt{2} e \left (b^2+c^2\right )^{3/4}}+\frac{c \cos (d+e x)-b \sin (d+e x)}{2 e \sqrt{b^2+c^2} \left (-\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^{3/2}} \]
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Rubi [A] time = 0.125628, antiderivative size = 164, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {3116, 3115, 2649, 204} \[ \frac{\tan ^{-1}\left (\frac{\sqrt [4]{b^2+c^2} \sin \left (-\tan ^{-1}(b,c)+d+e x\right )}{\sqrt{2} \sqrt{\sqrt{b^2+c^2} \cos \left (-\tan ^{-1}(b,c)+d+e x\right )-\sqrt{b^2+c^2}}}\right )}{2 \sqrt{2} e \left (b^2+c^2\right )^{3/4}}+\frac{c \cos (d+e x)-b \sin (d+e x)}{2 e \sqrt{b^2+c^2} \left (-\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 3116
Rule 3115
Rule 2649
Rule 204
Rubi steps
\begin{align*} \int \frac{1}{\left (-\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^{3/2}} \, dx &=\frac{c \cos (d+e x)-b \sin (d+e x)}{2 \sqrt{b^2+c^2} e \left (-\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^{3/2}}-\frac{\int \frac{1}{\sqrt{-\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)}} \, dx}{4 \sqrt{b^2+c^2}}\\ &=\frac{c \cos (d+e x)-b \sin (d+e x)}{2 \sqrt{b^2+c^2} e \left (-\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^{3/2}}-\frac{\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}{4 \sqrt{b^2+c^2}}\\ &=\frac{c \cos (d+e x)-b \sin (d+e x)}{2 \sqrt{b^2+c^2} e \left (-\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^{3/2}}+\frac{\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 )}{2 \sqrt{b^2+c^2} e}\\ &=\frac{\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 )}{2 \sqrt{2} \left (b^2+c^2\right )^{3/4} e}+\frac{c \cos (d+e x)-b \sin (d+e x)}{2 \sqrt{b^2+c^2} e \left (-\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^{3/2}}\\ \end{align*}
Mathematica [F] time = 180.001, size = 0, normalized size = 0. \[ \text{\$Aborted} \]
Verification is Not applicable to the result.
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Maple [B] time = 1.934, size = 363, normalized size = 2.2 \begin{align*}{\frac{1}{4\,\cos \left ( ex+d-\arctan \left ( -b,c \right ) \right ) e} \left ( -\sin \left ( ex+d-\arctan \left ( -b,c \right ) \right ) \arctan \left ({\frac{\sqrt{2}}{2}\sqrt{-\sqrt{{b}^{2}+{c}^{2}}\sin \left ( ex+d-\arctan \left ( -b,c \right ) \right ) -\sqrt{{b}^{2}+{c}^{2}}}{\frac{1}{\sqrt [4]{{b}^{2}+{c}^{2}}}}} \right ) \sqrt{2} \left ({b}^{2}+{c}^{2} \right ) +\sqrt{2}\arctan \left ({\frac{\sqrt{2}}{2}\sqrt{-\sqrt{{b}^{2}+{c}^{2}}\sin \left ( ex+d-\arctan \left ( -b,c \right ) \right ) -\sqrt{{b}^{2}+{c}^{2}}}{\frac{1}{\sqrt [4]{{b}^{2}+{c}^{2}}}}} \right ){b}^{2}+\sqrt{2}\arctan \left ({\frac{\sqrt{2}}{2}\sqrt{-\sqrt{{b}^{2}+{c}^{2}}\sin \left ( ex+d-\arctan \left ( -b,c \right ) \right ) -\sqrt{{b}^{2}+{c}^{2}}}{\frac{1}{\sqrt [4]{{b}^{2}+{c}^{2}}}}} \right ){c}^{2}+2\,\sqrt{-\sqrt{{b}^{2}+{c}^{2}}\sin \left ( ex+d-\arctan \left ( -b,c \right ) \right ) -\sqrt{{b}^{2}+{c}^{2}}} \left ({b}^{2}+{c}^{2} \right ) ^{3/4} \right ) \sqrt{-\sqrt{{b}^{2}+{c}^{2}} \left ( 1+\sin \left ( ex+d-\arctan \left ( -b,c \right ) \right ) \right ) } \left ({b}^{2}+{c}^{2} \right ) ^{-{\frac{7}{4}}}{\frac{1}{\sqrt{{({b}^{2}\sin \left ( ex+d-\arctan \left ( -b,c \right ) \right ) +{c}^{2}\sin \left ( ex+d-\arctan \left ( -b,c \right ) \right ) -{b}^{2}-{c}^{2}){\frac{1}{\sqrt{{b}^{2}+{c}^{2}}}}}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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