Optimal. Leaf size=186 \[ \frac{1}{2} c \sin (2 e) \text{CosIntegral}(2 f x) \sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}+c \cos (e) \text{CosIntegral}(f x) \sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}-c \sin (e) \text{Si}(f x) \sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}+\frac{1}{2} c \cos (2 e) \text{Si}(2 f x) \sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c} \]
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Rubi [A] time = 0.661755, antiderivative size = 186, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 7, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.212, Rules used = {4604, 6741, 12, 6742, 3303, 3299, 3302} \[ \frac{1}{2} c \sin (2 e) \text{CosIntegral}(2 f x) \sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}+c \cos (e) \text{CosIntegral}(f x) \sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}-c \sin (e) \text{Si}(f x) \sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}+\frac{1}{2} c \cos (2 e) \text{Si}(2 f x) \sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c} \]
Antiderivative was successfully verified.
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Rule 4604
Rule 6741
Rule 12
Rule 6742
Rule 3303
Rule 3299
Rule 3302
Rubi steps
\begin{align*} \int \frac{\sqrt{a-a \sin (e+f x)} (c+c \sin (e+f x))^{3/2}}{x} \, dx &=\left (\sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}\right ) \int \frac{\cos (e+f x) (c+c \sin (e+f x))}{x} \, dx\\ &=\left (\sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}\right ) \int \frac{c \cos (e+f x) (1+\sin (e+f x))}{x} \, dx\\ &=\left (c \sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}\right ) \int \frac{\cos (e+f x) (1+\sin (e+f x))}{x} \, dx\\ &=\left (c \sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}\right ) \int \left (\frac{\cos (e+f x)}{x}+\frac{\sin (2 e+2 f x)}{2 x}\right ) \, dx\\ &=\frac{1}{2} \left (c \sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}\right ) \int \frac{\sin (2 e+2 f x)}{x} \, dx+\left (c \sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}\right ) \int \frac{\cos (e+f x)}{x} \, dx\\ &=\left (c \cos (e) \sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}\right ) \int \frac{\cos (f x)}{x} \, dx+\frac{1}{2} \left (c \cos (2 e) \sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}\right ) \int \frac{\sin (2 f x)}{x} \, dx-\left (c \sec (e+f x) \sin (e) \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}\right ) \int \frac{\sin (f x)}{x} \, dx+\frac{1}{2} \left (c \sec (e+f x) \sin (2 e) \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}\right ) \int \frac{\cos (2 f x)}{x} \, dx\\ &=c \cos (e) \text{Ci}(f x) \sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}+\frac{1}{2} c \text{Ci}(2 f x) \sec (e+f x) \sin (2 e) \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}-c \sec (e+f x) \sin (e) \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)} \text{Si}(f x)+\frac{1}{2} c \cos (2 e) \sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)} \text{Si}(2 f x)\\ \end{align*}
Mathematica [C] time = 1.24243, size = 150, normalized size = 0.81 \[ \frac{c e^{-i (e-f x)} \sqrt{-i c e^{-i (e+f x)} \left (e^{i (e+f x)}+i\right )^2} \left (2 e^{i e} \text{ExpIntegralEi}(-i f x)+2 e^{3 i e} \text{ExpIntegralEi}(i f x)+i \left (\text{ExpIntegralEi}(-2 i f x)-e^{4 i e} \text{ExpIntegralEi}(2 i f x)\right )\right ) \sqrt{a-a \sin (e+f x)}}{2 \sqrt{2} \left (1+e^{2 i (e+f x)}\right )} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.078, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{x} \left ( c+c\sin \left ( fx+e \right ) \right ) ^{{\frac{3}{2}}}\sqrt{a-a\sin \left ( fx+e \right ) }}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{-a \sin \left (f x + e\right ) + a}{\left (c \sin \left (f x + e\right ) + c\right )}^{\frac{3}{2}}}{x}\,{d x} \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: UnboundLocalError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{-a \sin \left (f x + e\right ) + a}{\left (c \sin \left (f x + e\right ) + c\right )}^{\frac{3}{2}}}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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