Optimal. Leaf size=247 \[ -\frac {2 b \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)}}{7 d}+\frac {4 a \left (3 a^2+29 b^2\right ) E\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {a+b \sin (c+d x)}}{105 b^2 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {4 \left (3 a^4+2 a^2 b^2-5 b^4\right ) F\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{105 b^2 d \sqrt {a+b \sin (c+d x)}}+\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (3 a^2+5 b^2+24 a b \sin (c+d x)\right )}{105 b d} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.28, antiderivative size = 247, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 7, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {2771, 2944,
2831, 2742, 2740, 2734, 2732} \begin {gather*} \frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (3 a^2+24 a b \sin (c+d x)+5 b^2\right )}{105 b d}+\frac {4 a \left (3 a^2+29 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{105 b^2 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {4 \left (3 a^4+2 a^2 b^2-5 b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} F\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{105 b^2 d \sqrt {a+b \sin (c+d x)}}-\frac {2 b \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)}}{7 d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2732
Rule 2734
Rule 2740
Rule 2742
Rule 2771
Rule 2831
Rule 2944
Rubi steps
\begin {align*} \int \cos ^2(c+d x) (a+b \sin (c+d x))^{3/2} \, dx &=-\frac {2 b \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)}}{7 d}+\frac {2}{7} \int \frac {\cos ^2(c+d x) \left (\frac {7 a^2}{2}+\frac {b^2}{2}+4 a b \sin (c+d x)\right )}{\sqrt {a+b \sin (c+d x)}} \, dx\\ &=-\frac {2 b \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)}}{7 d}+\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (3 a^2+5 b^2+24 a b \sin (c+d x)\right )}{105 b d}+\frac {8 \int \frac {\frac {1}{4} b^2 \left (27 a^2+5 b^2\right )+\frac {1}{4} a b \left (3 a^2+29 b^2\right ) \sin (c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx}{105 b^2}\\ &=-\frac {2 b \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)}}{7 d}+\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (3 a^2+5 b^2+24 a b \sin (c+d x)\right )}{105 b d}+\frac {1}{105} \left (2 a \left (29+\frac {3 a^2}{b^2}\right )\right ) \int \sqrt {a+b \sin (c+d x)} \, dx-\frac {\left (2 \left (3 a^4+2 a^2 b^2-5 b^4\right )\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}} \, dx}{105 b^2}\\ &=-\frac {2 b \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)}}{7 d}+\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (3 a^2+5 b^2+24 a b \sin (c+d x)\right )}{105 b d}+\frac {\left (2 a \left (29+\frac {3 a^2}{b^2}\right ) \sqrt {a+b \sin (c+d x)}\right ) \int \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}} \, dx}{105 \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {\left (2 \left (3 a^4+2 a^2 b^2-5 b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}\right ) \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}} \, dx}{105 b^2 \sqrt {a+b \sin (c+d x)}}\\ &=-\frac {2 b \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)}}{7 d}+\frac {4 a \left (29+\frac {3 a^2}{b^2}\right ) E\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {a+b \sin (c+d x)}}{105 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {4 \left (3 a^4+2 a^2 b^2-5 b^4\right ) F\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{105 b^2 d \sqrt {a+b \sin (c+d x)}}+\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (3 a^2+5 b^2+24 a b \sin (c+d x)\right )}{105 b d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 1.03, size = 222, normalized size = 0.90 \begin {gather*} \frac {-8 a \left (3 a^3+3 a^2 b+29 a b^2+29 b^3\right ) E\left (\frac {1}{4} (-2 c+\pi -2 d x)|\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}+8 \left (3 a^4+2 a^2 b^2-5 b^4\right ) F\left (\frac {1}{4} (-2 c+\pi -2 d x)|\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}+b \cos (c+d x) \left (12 a^3+38 a b^2-78 a b^2 \cos (2 (c+d x))+b \left (108 a^2+5 b^2\right ) \sin (c+d x)-15 b^3 \sin (3 (c+d x))\right )}{210 b^2 d \sqrt {a+b \sin (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(942\) vs.
\(2(293)=586\).
time = 2.21, size = 943, normalized size = 3.82
method | result | size |
default | \(\frac {-\frac {2 b^{5} \left (\sin ^{5}\left (d x +c \right )\right )}{7}+\frac {4 \sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}\, \sqrt {-\frac {\left (\sin \left (d x +c \right )-1\right ) b}{a +b}}\, \sqrt {-\frac {\left (1+\sin \left (d x +c \right )\right ) b}{a -b}}\, \EllipticF \left (\sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right ) a^{4} b}{35}+\frac {32 \sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}\, \sqrt {-\frac {\left (\sin \left (d x +c \right )-1\right ) b}{a +b}}\, \sqrt {-\frac {\left (1+\sin \left (d x +c \right )\right ) b}{a -b}}\, \EllipticF \left (\sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right ) a^{3} b^{2}}{35}+\frac {8 \sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}\, \sqrt {-\frac {\left (\sin \left (d x +c \right )-1\right ) b}{a +b}}\, \sqrt {-\frac {\left (1+\sin \left (d x +c \right )\right ) b}{a -b}}\, \EllipticF \left (\sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right ) a^{2} b^{3}}{105}-\frac {32 \sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}\, \sqrt {-\frac {\left (\sin \left (d x +c \right )-1\right ) b}{a +b}}\, \sqrt {-\frac {\left (1+\sin \left (d x +c \right )\right ) b}{a -b}}\, \EllipticF \left (\sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right ) a \,b^{4}}{35}-\frac {4 \sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}\, \sqrt {-\frac {\left (\sin \left (d x +c \right )-1\right ) b}{a +b}}\, \sqrt {-\frac {\left (1+\sin \left (d x +c \right )\right ) b}{a -b}}\, \EllipticF \left (\sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right ) b^{5}}{21}-\frac {4 \sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}\, \sqrt {-\frac {\left (\sin \left (d x +c \right )-1\right ) b}{a +b}}\, \sqrt {-\frac {\left (1+\sin \left (d x +c \right )\right ) b}{a -b}}\, \EllipticE \left (\sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right ) a^{5}}{35}-\frac {104 \sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}\, \sqrt {-\frac {\left (\sin \left (d x +c \right )-1\right ) b}{a +b}}\, \sqrt {-\frac {\left (1+\sin \left (d x +c \right )\right ) b}{a -b}}\, \EllipticE \left (\sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right ) a^{3} b^{2}}{105}+\frac {116 \sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}\, \sqrt {-\frac {\left (\sin \left (d x +c \right )-1\right ) b}{a +b}}\, \sqrt {-\frac {\left (1+\sin \left (d x +c \right )\right ) b}{a -b}}\, \EllipticE \left (\sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right ) a \,b^{4}}{105}-\frac {26 a \,b^{4} \left (\sin ^{4}\left (d x +c \right )\right )}{35}-\frac {18 a^{2} b^{3} \left (\sin ^{3}\left (d x +c \right )\right )}{35}+\frac {10 b^{5} \left (\sin ^{3}\left (d x +c \right )\right )}{21}-\frac {2 a^{3} b^{2} \left (\sin ^{2}\left (d x +c \right )\right )}{35}+\frac {14 a \,b^{4} \left (\sin ^{2}\left (d x +c \right )\right )}{15}+\frac {18 a^{2} b^{3} \sin \left (d x +c \right )}{35}-\frac {4 b^{5} \sin \left (d x +c \right )}{21}+\frac {2 a^{3} b^{2}}{35}-\frac {4 a \,b^{4}}{21}}{b^{3} \cos \left (d x +c \right ) \sqrt {a +b \sin \left (d x +c \right )}\, d}\) | \(943\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.14, size = 491, normalized size = 1.99 \begin {gather*} -\frac {2 \, {\left (\sqrt {2} {\left (6 \, a^{4} - 23 \, a^{2} b^{2} - 15 \, b^{4}\right )} \sqrt {i \, b} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 i \, a^{3} - 9 i \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) - 3 i \, b \sin \left (d x + c\right ) - 2 i \, a}{3 \, b}\right ) + \sqrt {2} {\left (6 \, a^{4} - 23 \, a^{2} b^{2} - 15 \, b^{4}\right )} \sqrt {-i \, b} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (-8 i \, a^{3} + 9 i \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) + 3 i \, b \sin \left (d x + c\right ) + 2 i \, a}{3 \, b}\right ) + 3 \, \sqrt {2} {\left (3 i \, a^{3} b + 29 i \, a b^{3}\right )} \sqrt {i \, b} {\rm weierstrassZeta}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 i \, a^{3} - 9 i \, a b^{2}\right )}}{27 \, b^{3}}, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 i \, a^{3} - 9 i \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) - 3 i \, b \sin \left (d x + c\right ) - 2 i \, a}{3 \, b}\right )\right ) + 3 \, \sqrt {2} {\left (-3 i \, a^{3} b - 29 i \, a b^{3}\right )} \sqrt {-i \, b} {\rm weierstrassZeta}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (-8 i \, a^{3} + 9 i \, a b^{2}\right )}}{27 \, b^{3}}, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (-8 i \, a^{3} + 9 i \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) + 3 i \, b \sin \left (d x + c\right ) + 2 i \, a}{3 \, b}\right )\right ) + 3 \, {\left (15 \, b^{4} \cos \left (d x + c\right )^{3} - 24 \, a b^{3} \cos \left (d x + c\right ) \sin \left (d x + c\right ) - {\left (3 \, a^{2} b^{2} + 5 \, b^{4}\right )} \cos \left (d x + c\right )\right )} \sqrt {b \sin \left (d x + c\right ) + a}\right )}}{315 \, b^{3} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a + b \sin {\left (c + d x \right )}\right )^{\frac {3}{2}} \cos ^{2}{\left (c + d x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\cos \left (c+d\,x\right )}^2\,{\left (a+b\,\sin \left (c+d\,x\right )\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________