Optimal. Leaf size=320 \[ -\frac {\sqrt {c} d^{3/2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {d} \sqrt {c \sin (a+b x)}}{\sqrt {c} \sqrt {d \cos (a+b x)}}\right )}{4 \sqrt {2} b}+\frac {\sqrt {c} d^{3/2} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt {d} \sqrt {c \sin (a+b x)}}{\sqrt {c} \sqrt {d \cos (a+b x)}}\right )}{4 \sqrt {2} b}+\frac {\sqrt {c} d^{3/2} \log \left (\sqrt {c}-\frac {\sqrt {2} \sqrt {d} \sqrt {c \sin (a+b x)}}{\sqrt {d \cos (a+b x)}}+\sqrt {c} \tan (a+b x)\right )}{8 \sqrt {2} b}-\frac {\sqrt {c} d^{3/2} \log \left (\sqrt {c}+\frac {\sqrt {2} \sqrt {d} \sqrt {c \sin (a+b x)}}{\sqrt {d \cos (a+b x)}}+\sqrt {c} \tan (a+b x)\right )}{8 \sqrt {2} b}+\frac {d \sqrt {d \cos (a+b x)} (c \sin (a+b x))^{3/2}}{2 b c} \]
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Rubi [A]
time = 0.18, antiderivative size = 320, normalized size of antiderivative = 1.00, number of steps
used = 11, number of rules used = 8, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {2649, 2654,
303, 1176, 631, 210, 1179, 642} \begin {gather*} -\frac {\sqrt {c} d^{3/2} \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt {d} \sqrt {c \sin (a+b x)}}{\sqrt {c} \sqrt {d \cos (a+b x)}}\right )}{4 \sqrt {2} b}+\frac {\sqrt {c} d^{3/2} \text {ArcTan}\left (\frac {\sqrt {2} \sqrt {d} \sqrt {c \sin (a+b x)}}{\sqrt {c} \sqrt {d \cos (a+b x)}}+1\right )}{4 \sqrt {2} b}+\frac {\sqrt {c} d^{3/2} \log \left (-\frac {\sqrt {2} \sqrt {d} \sqrt {c \sin (a+b x)}}{\sqrt {d \cos (a+b x)}}+\sqrt {c} \tan (a+b x)+\sqrt {c}\right )}{8 \sqrt {2} b}-\frac {\sqrt {c} d^{3/2} \log \left (\frac {\sqrt {2} \sqrt {d} \sqrt {c \sin (a+b x)}}{\sqrt {d \cos (a+b x)}}+\sqrt {c} \tan (a+b x)+\sqrt {c}\right )}{8 \sqrt {2} b}+\frac {d (c \sin (a+b x))^{3/2} \sqrt {d \cos (a+b x)}}{2 b c} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 303
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 2649
Rule 2654
Rubi steps
\begin {align*} \int (d \cos (a+b x))^{3/2} \sqrt {c \sin (a+b x)} \, dx &=\frac {d \sqrt {d \cos (a+b x)} (c \sin (a+b x))^{3/2}}{2 b c}+\frac {1}{4} d^2 \int \frac {\sqrt {c \sin (a+b x)}}{\sqrt {d \cos (a+b x)}} \, dx\\ &=\frac {d \sqrt {d \cos (a+b x)} (c \sin (a+b x))^{3/2}}{2 b c}+\frac {\left (c d^3\right ) \text {Subst}\left (\int \frac {x^2}{c^2+d^2 x^4} \, dx,x,\frac {\sqrt {c \sin (a+b x)}}{\sqrt {d \cos (a+b x)}}\right )}{2 b}\\ &=\frac {d \sqrt {d \cos (a+b x)} (c \sin (a+b x))^{3/2}}{2 b c}-\frac {\left (c d^2\right ) \text {Subst}\left (\int \frac {c-d x^2}{c^2+d^2 x^4} \, dx,x,\frac {\sqrt {c \sin (a+b x)}}{\sqrt {d \cos (a+b x)}}\right )}{4 b}+\frac {\left (c d^2\right ) \text {Subst}\left (\int \frac {c+d x^2}{c^2+d^2 x^4} \, dx,x,\frac {\sqrt {c \sin (a+b x)}}{\sqrt {d \cos (a+b x)}}\right )}{4 b}\\ &=\frac {d \sqrt {d \cos (a+b x)} (c \sin (a+b x))^{3/2}}{2 b c}+\frac {(c d) \text {Subst}\left (\int \frac {1}{\frac {c}{d}-\frac {\sqrt {2} \sqrt {c} x}{\sqrt {d}}+x^2} \, dx,x,\frac {\sqrt {c \sin (a+b x)}}{\sqrt {d \cos (a+b x)}}\right )}{8 b}+\frac {(c d) \text {Subst}\left (\int \frac {1}{\frac {c}{d}+\frac {\sqrt {2} \sqrt {c} x}{\sqrt {d}}+x^2} \, dx,x,\frac {\sqrt {c \sin (a+b x)}}{\sqrt {d \cos (a+b x)}}\right )}{8 b}+\frac {\left (\sqrt {c} d^{3/2}\right ) \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt {c}}{\sqrt {d}}+2 x}{-\frac {c}{d}-\frac {\sqrt {2} \sqrt {c} x}{\sqrt {d}}-x^2} \, dx,x,\frac {\sqrt {c \sin (a+b x)}}{\sqrt {d \cos (a+b x)}}\right )}{8 \sqrt {2} b}+\frac {\left (\sqrt {c} d^{3/2}\right ) \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt {c}}{\sqrt {d}}-2 x}{-\frac {c}{d}+\frac {\sqrt {2} \sqrt {c} x}{\sqrt {d}}-x^2} \, dx,x,\frac {\sqrt {c \sin (a+b x)}}{\sqrt {d \cos (a+b x)}}\right )}{8 \sqrt {2} b}\\ &=\frac {\sqrt {c} d^{3/2} \log \left (\sqrt {c}-\frac {\sqrt {2} \sqrt {d} \sqrt {c \sin (a+b x)}}{\sqrt {d \cos (a+b x)}}+\sqrt {c} \tan (a+b x)\right )}{8 \sqrt {2} b}-\frac {\sqrt {c} d^{3/2} \log \left (\sqrt {c}+\frac {\sqrt {2} \sqrt {d} \sqrt {c \sin (a+b x)}}{\sqrt {d \cos (a+b x)}}+\sqrt {c} \tan (a+b x)\right )}{8 \sqrt {2} b}+\frac {d \sqrt {d \cos (a+b x)} (c \sin (a+b x))^{3/2}}{2 b c}+\frac {\left (\sqrt {c} d^{3/2}\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt {d} \sqrt {c \sin (a+b x)}}{\sqrt {c} \sqrt {d \cos (a+b x)}}\right )}{4 \sqrt {2} b}-\frac {\left (\sqrt {c} d^{3/2}\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt {d} \sqrt {c \sin (a+b x)}}{\sqrt {c} \sqrt {d \cos (a+b x)}}\right )}{4 \sqrt {2} b}\\ &=-\frac {\sqrt {c} d^{3/2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {d} \sqrt {c \sin (a+b x)}}{\sqrt {c} \sqrt {d \cos (a+b x)}}\right )}{4 \sqrt {2} b}+\frac {\sqrt {c} d^{3/2} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt {d} \sqrt {c \sin (a+b x)}}{\sqrt {c} \sqrt {d \cos (a+b x)}}\right )}{4 \sqrt {2} b}+\frac {\sqrt {c} d^{3/2} \log \left (\sqrt {c}-\frac {\sqrt {2} \sqrt {d} \sqrt {c \sin (a+b x)}}{\sqrt {d \cos (a+b x)}}+\sqrt {c} \tan (a+b x)\right )}{8 \sqrt {2} b}-\frac {\sqrt {c} d^{3/2} \log \left (\sqrt {c}+\frac {\sqrt {2} \sqrt {d} \sqrt {c \sin (a+b x)}}{\sqrt {d \cos (a+b x)}}+\sqrt {c} \tan (a+b x)\right )}{8 \sqrt {2} b}+\frac {d \sqrt {d \cos (a+b x)} (c \sin (a+b x))^{3/2}}{2 b c}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 0.08, size = 70, normalized size = 0.22 \begin {gather*} \frac {2 d^2 \cos ^2(a+b x)^{3/4} \, _2F_1\left (-\frac {1}{4},\frac {3}{4};\frac {7}{4};\sin ^2(a+b x)\right ) \sqrt {c \sin (a+b x)} \tan (a+b x)}{3 b \sqrt {d \cos (a+b x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 4 vs. order
3.
time = 0.10, size = 514, normalized size = 1.61
method | result | size |
default | \(\frac {\left (-i \sqrt {\frac {1-\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {-1+\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {-1+\cos \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \EllipticPi \left (\sqrt {\frac {1-\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}, \frac {1}{2}-\frac {i}{2}, \frac {\sqrt {2}}{2}\right )+i \sqrt {\frac {1-\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {-1+\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {-1+\cos \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \EllipticPi \left (\sqrt {\frac {1-\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}, \frac {1}{2}+\frac {i}{2}, \frac {\sqrt {2}}{2}\right )+\sqrt {\frac {1-\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {-1+\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {-1+\cos \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \EllipticPi \left (\sqrt {\frac {1-\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}, \frac {1}{2}-\frac {i}{2}, \frac {\sqrt {2}}{2}\right )+\sqrt {\frac {1-\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {-1+\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {-1+\cos \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \EllipticPi \left (\sqrt {\frac {1-\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}, \frac {1}{2}+\frac {i}{2}, \frac {\sqrt {2}}{2}\right )+2 \left (\cos ^{2}\left (b x +a \right )\right ) \sqrt {2}-2 \cos \left (b x +a \right ) \sqrt {2}\right ) \left (d \cos \left (b x +a \right )\right )^{\frac {3}{2}} \sqrt {c \sin \left (b x +a \right )}\, \sin \left (b x +a \right ) \sqrt {2}}{8 b \left (-1+\cos \left (b x +a \right )\right ) \cos \left (b x +a \right )^{2}}\) | \(514\) |
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. 2205 vs.
\(2 (238) = 476\).
time = 28.54, size = 2205, normalized size = 6.89 \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 \sqrt {c \sin {\left (a + b x \right )}} \left (d \cos {\left (a + b x \right )}\right )^{\frac {3}{2}}\, 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 [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\left (d\,\cos \left (a+b\,x\right )\right )}^{3/2}\,\sqrt {c\,\sin \left (a+b\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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