Optimal. Leaf size=320 \[ \frac {c^{3/2} \sqrt {d} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {c} \sqrt {d \cos (a+b x)}}{\sqrt {d} \sqrt {c \sin (a+b x)}}\right )}{4 \sqrt {2} b}-\frac {c^{3/2} \sqrt {d} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d \cos (a+b x)}}{\sqrt {d} \sqrt {c \sin (a+b x)}}+1\right )}{4 \sqrt {2} b}-\frac {c^{3/2} \sqrt {d} \log \left (-\frac {\sqrt {2} \sqrt {c} \sqrt {d \cos (a+b x)}}{\sqrt {c \sin (a+b x)}}+\sqrt {d} \cot (a+b x)+\sqrt {d}\right )}{8 \sqrt {2} b}+\frac {c^{3/2} \sqrt {d} \log \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d \cos (a+b x)}}{\sqrt {c \sin (a+b x)}}+\sqrt {d} \cot (a+b x)+\sqrt {d}\right )}{8 \sqrt {2} b}-\frac {c \sqrt {c \sin (a+b x)} (d \cos (a+b x))^{3/2}}{2 b d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.28, 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 = {2568, 2575, 297, 1162, 617, 204, 1165, 628} \[ \frac {c^{3/2} \sqrt {d} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {c} \sqrt {d \cos (a+b x)}}{\sqrt {d} \sqrt {c \sin (a+b x)}}\right )}{4 \sqrt {2} b}-\frac {c^{3/2} \sqrt {d} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d \cos (a+b x)}}{\sqrt {d} \sqrt {c \sin (a+b x)}}+1\right )}{4 \sqrt {2} b}-\frac {c^{3/2} \sqrt {d} \log \left (-\frac {\sqrt {2} \sqrt {c} \sqrt {d \cos (a+b x)}}{\sqrt {c \sin (a+b x)}}+\sqrt {d} \cot (a+b x)+\sqrt {d}\right )}{8 \sqrt {2} b}+\frac {c^{3/2} \sqrt {d} \log \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d \cos (a+b x)}}{\sqrt {c \sin (a+b x)}}+\sqrt {d} \cot (a+b x)+\sqrt {d}\right )}{8 \sqrt {2} b}-\frac {c \sqrt {c \sin (a+b x)} (d \cos (a+b x))^{3/2}}{2 b d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 204
Rule 297
Rule 617
Rule 628
Rule 1162
Rule 1165
Rule 2568
Rule 2575
Rubi steps
\begin {align*} \int \sqrt {d \cos (a+b x)} (c \sin (a+b x))^{3/2} \, dx &=-\frac {c (d \cos (a+b x))^{3/2} \sqrt {c \sin (a+b x)}}{2 b d}+\frac {1}{4} c^2 \int \frac {\sqrt {d \cos (a+b x)}}{\sqrt {c \sin (a+b x)}} \, dx\\ &=-\frac {c (d \cos (a+b x))^{3/2} \sqrt {c \sin (a+b x)}}{2 b d}-\frac {\left (c^3 d\right ) \operatorname {Subst}\left (\int \frac {x^2}{d^2+c^2 x^4} \, dx,x,\frac {\sqrt {d \cos (a+b x)}}{\sqrt {c \sin (a+b x)}}\right )}{2 b}\\ &=-\frac {c (d \cos (a+b x))^{3/2} \sqrt {c \sin (a+b x)}}{2 b d}+\frac {\left (c^2 d\right ) \operatorname {Subst}\left (\int \frac {d-c x^2}{d^2+c^2 x^4} \, dx,x,\frac {\sqrt {d \cos (a+b x)}}{\sqrt {c \sin (a+b x)}}\right )}{4 b}-\frac {\left (c^2 d\right ) \operatorname {Subst}\left (\int \frac {d+c x^2}{d^2+c^2 x^4} \, dx,x,\frac {\sqrt {d \cos (a+b x)}}{\sqrt {c \sin (a+b x)}}\right )}{4 b}\\ &=-\frac {c (d \cos (a+b x))^{3/2} \sqrt {c \sin (a+b x)}}{2 b d}-\frac {\left (c^{3/2} \sqrt {d}\right ) \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt {d}}{\sqrt {c}}+2 x}{-\frac {d}{c}-\frac {\sqrt {2} \sqrt {d} x}{\sqrt {c}}-x^2} \, dx,x,\frac {\sqrt {d \cos (a+b x)}}{\sqrt {c \sin (a+b x)}}\right )}{8 \sqrt {2} b}-\frac {\left (c^{3/2} \sqrt {d}\right ) \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt {d}}{\sqrt {c}}-2 x}{-\frac {d}{c}+\frac {\sqrt {2} \sqrt {d} x}{\sqrt {c}}-x^2} \, dx,x,\frac {\sqrt {d \cos (a+b x)}}{\sqrt {c \sin (a+b x)}}\right )}{8 \sqrt {2} b}-\frac {(c d) \operatorname {Subst}\left (\int \frac {1}{\frac {d}{c}-\frac {\sqrt {2} \sqrt {d} x}{\sqrt {c}}+x^2} \, dx,x,\frac {\sqrt {d \cos (a+b x)}}{\sqrt {c \sin (a+b x)}}\right )}{8 b}-\frac {(c d) \operatorname {Subst}\left (\int \frac {1}{\frac {d}{c}+\frac {\sqrt {2} \sqrt {d} x}{\sqrt {c}}+x^2} \, dx,x,\frac {\sqrt {d \cos (a+b x)}}{\sqrt {c \sin (a+b x)}}\right )}{8 b}\\ &=-\frac {c^{3/2} \sqrt {d} \log \left (\sqrt {d}+\sqrt {d} \cot (a+b x)-\frac {\sqrt {2} \sqrt {c} \sqrt {d \cos (a+b x)}}{\sqrt {c \sin (a+b x)}}\right )}{8 \sqrt {2} b}+\frac {c^{3/2} \sqrt {d} \log \left (\sqrt {d}+\sqrt {d} \cot (a+b x)+\frac {\sqrt {2} \sqrt {c} \sqrt {d \cos (a+b x)}}{\sqrt {c \sin (a+b x)}}\right )}{8 \sqrt {2} b}-\frac {c (d \cos (a+b x))^{3/2} \sqrt {c \sin (a+b x)}}{2 b d}-\frac {\left (c^{3/2} \sqrt {d}\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt {c} \sqrt {d \cos (a+b x)}}{\sqrt {d} \sqrt {c \sin (a+b x)}}\right )}{4 \sqrt {2} b}+\frac {\left (c^{3/2} \sqrt {d}\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt {c} \sqrt {d \cos (a+b x)}}{\sqrt {d} \sqrt {c \sin (a+b x)}}\right )}{4 \sqrt {2} b}\\ &=\frac {c^{3/2} \sqrt {d} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {c} \sqrt {d \cos (a+b x)}}{\sqrt {d} \sqrt {c \sin (a+b x)}}\right )}{4 \sqrt {2} b}-\frac {c^{3/2} \sqrt {d} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt {c} \sqrt {d \cos (a+b x)}}{\sqrt {d} \sqrt {c \sin (a+b x)}}\right )}{4 \sqrt {2} b}-\frac {c^{3/2} \sqrt {d} \log \left (\sqrt {d}+\sqrt {d} \cot (a+b x)-\frac {\sqrt {2} \sqrt {c} \sqrt {d \cos (a+b x)}}{\sqrt {c \sin (a+b x)}}\right )}{8 \sqrt {2} b}+\frac {c^{3/2} \sqrt {d} \log \left (\sqrt {d}+\sqrt {d} \cot (a+b x)+\frac {\sqrt {2} \sqrt {c} \sqrt {d \cos (a+b x)}}{\sqrt {c \sin (a+b x)}}\right )}{8 \sqrt {2} b}-\frac {c (d \cos (a+b x))^{3/2} \sqrt {c \sin (a+b x)}}{2 b d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.06, size = 67, normalized size = 0.21 \[ \frac {2 \sqrt [4]{\cos ^2(a+b x)} \tan (a+b x) (c \sin (a+b x))^{3/2} \sqrt {d \cos (a+b x)} \, _2F_1\left (\frac {1}{4},\frac {5}{4};\frac {9}{4};\sin ^2(a+b x)\right )}{5 b} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 27.20, size = 1868, normalized size = 5.84 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {d \cos \left (b x + a\right )} \left (c \sin \left (b x + a\right )\right )^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 0.11, size = 656, normalized size = 2.05 \[ \frac {\left (c \sin \left (b x +a \right )\right )^{\frac {3}{2}} \sqrt {d \cos \left (b x +a \right )}\, \left (i \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 )}{\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 \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 )}{\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 )-\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 )}{\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 \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 )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {-1+\cos \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \EllipticF \left (\sqrt {\frac {1-\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}, \frac {\sqrt {2}}{2}\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 )}{\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 ^{3}\left (b x +a \right )\right ) \sqrt {2}+2 \left (\cos ^{2}\left (b x +a \right )\right ) \sqrt {2}\right ) \sqrt {2}}{8 b \sin \left (b x +a \right ) \cos \left (b x +a \right ) \left (-1+\cos \left (b x +a \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {d \cos \left (b x + a\right )} \left (c \sin \left (b x + a\right )\right )^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \sqrt {d\,\cos \left (a+b\,x\right )}\,{\left (c\,\sin \left (a+b\,x\right )\right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________