Optimal. Leaf size=405 \[ \frac {8 a \left (120 a^2-179 b^2\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3465 b^4 d}-\frac {2 \left (80 a^2-117 b^2\right ) \sin ^2(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{693 b^3 d}-\frac {16 a \left (160 a^4-267 a^2 b^2+69 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{3465 b^6 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {8 \left (160 a^4-247 a^2 b^2+45 b^4\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3465 b^5 d}+\frac {8 \left (320 a^6-614 a^4 b^2+249 a^2 b^4+45 b^6\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} F\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{3465 b^6 d \sqrt {a+b \sin (c+d x)}}+\frac {20 a \sin ^3(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{99 b^2 d}-\frac {2 \sin ^4(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{11 b d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.89, antiderivative size = 405, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.258, Rules used = {2895, 3049, 3023, 2752, 2663, 2661, 2655, 2653} \[ -\frac {2 \left (80 a^2-117 b^2\right ) \sin ^2(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{693 b^3 d}+\frac {8 a \left (120 a^2-179 b^2\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3465 b^4 d}-\frac {8 \left (-247 a^2 b^2+160 a^4+45 b^4\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3465 b^5 d}+\frac {8 \left (-614 a^4 b^2+249 a^2 b^4+320 a^6+45 b^6\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} F\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{3465 b^6 d \sqrt {a+b \sin (c+d x)}}-\frac {16 a \left (-267 a^2 b^2+160 a^4+69 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{3465 b^6 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {20 a \sin ^3(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{99 b^2 d}-\frac {2 \sin ^4(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{11 b d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2653
Rule 2655
Rule 2661
Rule 2663
Rule 2752
Rule 2895
Rule 3023
Rule 3049
Rubi steps
\begin {align*} \int \frac {\cos ^4(c+d x) \sin ^2(c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx &=\frac {20 a \cos (c+d x) \sin ^3(c+d x) \sqrt {a+b \sin (c+d x)}}{99 b^2 d}-\frac {2 \cos (c+d x) \sin ^4(c+d x) \sqrt {a+b \sin (c+d x)}}{11 b d}-\frac {4 \int \frac {\sin ^2(c+d x) \left (\frac {3}{4} \left (20 a^2-33 b^2\right )-\frac {1}{2} a b \sin (c+d x)-\frac {1}{4} \left (80 a^2-117 b^2\right ) \sin ^2(c+d x)\right )}{\sqrt {a+b \sin (c+d x)}} \, dx}{99 b^2}\\ &=-\frac {2 \left (80 a^2-117 b^2\right ) \cos (c+d x) \sin ^2(c+d x) \sqrt {a+b \sin (c+d x)}}{693 b^3 d}+\frac {20 a \cos (c+d x) \sin ^3(c+d x) \sqrt {a+b \sin (c+d x)}}{99 b^2 d}-\frac {2 \cos (c+d x) \sin ^4(c+d x) \sqrt {a+b \sin (c+d x)}}{11 b d}-\frac {8 \int \frac {\sin (c+d x) \left (-\frac {1}{2} a \left (80 a^2-117 b^2\right )+\frac {1}{2} b \left (5 a^2-27 b^2\right ) \sin (c+d x)+\frac {1}{2} a \left (120 a^2-179 b^2\right ) \sin ^2(c+d x)\right )}{\sqrt {a+b \sin (c+d x)}} \, dx}{693 b^3}\\ &=\frac {8 a \left (120 a^2-179 b^2\right ) \cos (c+d x) \sin (c+d x) \sqrt {a+b \sin (c+d x)}}{3465 b^4 d}-\frac {2 \left (80 a^2-117 b^2\right ) \cos (c+d x) \sin ^2(c+d x) \sqrt {a+b \sin (c+d x)}}{693 b^3 d}+\frac {20 a \cos (c+d x) \sin ^3(c+d x) \sqrt {a+b \sin (c+d x)}}{99 b^2 d}-\frac {2 \cos (c+d x) \sin ^4(c+d x) \sqrt {a+b \sin (c+d x)}}{11 b d}-\frac {16 \int \frac {\frac {1}{2} a^2 \left (120 a^2-179 b^2\right )-2 a b \left (5 a^2-6 b^2\right ) \sin (c+d x)-\frac {3}{4} \left (160 a^4-247 a^2 b^2+45 b^4\right ) \sin ^2(c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx}{3465 b^4}\\ &=-\frac {8 \left (160 a^4-247 a^2 b^2+45 b^4\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3465 b^5 d}+\frac {8 a \left (120 a^2-179 b^2\right ) \cos (c+d x) \sin (c+d x) \sqrt {a+b \sin (c+d x)}}{3465 b^4 d}-\frac {2 \left (80 a^2-117 b^2\right ) \cos (c+d x) \sin ^2(c+d x) \sqrt {a+b \sin (c+d x)}}{693 b^3 d}+\frac {20 a \cos (c+d x) \sin ^3(c+d x) \sqrt {a+b \sin (c+d x)}}{99 b^2 d}-\frac {2 \cos (c+d x) \sin ^4(c+d x) \sqrt {a+b \sin (c+d x)}}{11 b d}-\frac {32 \int \frac {\frac {3}{8} b \left (80 a^4-111 a^2 b^2-45 b^4\right )+\frac {3}{4} a \left (160 a^4-267 a^2 b^2+69 b^4\right ) \sin (c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx}{10395 b^5}\\ &=-\frac {8 \left (160 a^4-247 a^2 b^2+45 b^4\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3465 b^5 d}+\frac {8 a \left (120 a^2-179 b^2\right ) \cos (c+d x) \sin (c+d x) \sqrt {a+b \sin (c+d x)}}{3465 b^4 d}-\frac {2 \left (80 a^2-117 b^2\right ) \cos (c+d x) \sin ^2(c+d x) \sqrt {a+b \sin (c+d x)}}{693 b^3 d}+\frac {20 a \cos (c+d x) \sin ^3(c+d x) \sqrt {a+b \sin (c+d x)}}{99 b^2 d}-\frac {2 \cos (c+d x) \sin ^4(c+d x) \sqrt {a+b \sin (c+d x)}}{11 b d}-\frac {\left (8 a \left (160 a^4-267 a^2 b^2+69 b^4\right )\right ) \int \sqrt {a+b \sin (c+d x)} \, dx}{3465 b^6}+\frac {\left (4 \left (320 a^6-614 a^4 b^2+249 a^2 b^4+45 b^6\right )\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}} \, dx}{3465 b^6}\\ &=-\frac {8 \left (160 a^4-247 a^2 b^2+45 b^4\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3465 b^5 d}+\frac {8 a \left (120 a^2-179 b^2\right ) \cos (c+d x) \sin (c+d x) \sqrt {a+b \sin (c+d x)}}{3465 b^4 d}-\frac {2 \left (80 a^2-117 b^2\right ) \cos (c+d x) \sin ^2(c+d x) \sqrt {a+b \sin (c+d x)}}{693 b^3 d}+\frac {20 a \cos (c+d x) \sin ^3(c+d x) \sqrt {a+b \sin (c+d x)}}{99 b^2 d}-\frac {2 \cos (c+d x) \sin ^4(c+d x) \sqrt {a+b \sin (c+d x)}}{11 b d}-\frac {\left (8 a \left (160 a^4-267 a^2 b^2+69 b^4\right ) \sqrt {a+b \sin (c+d x)}\right ) \int \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}} \, dx}{3465 b^6 \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {\left (4 \left (320 a^6-614 a^4 b^2+249 a^2 b^4+45 b^6\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}{3465 b^6 \sqrt {a+b \sin (c+d x)}}\\ &=-\frac {8 \left (160 a^4-247 a^2 b^2+45 b^4\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3465 b^5 d}+\frac {8 a \left (120 a^2-179 b^2\right ) \cos (c+d x) \sin (c+d x) \sqrt {a+b \sin (c+d x)}}{3465 b^4 d}-\frac {2 \left (80 a^2-117 b^2\right ) \cos (c+d x) \sin ^2(c+d x) \sqrt {a+b \sin (c+d x)}}{693 b^3 d}+\frac {20 a \cos (c+d x) \sin ^3(c+d x) \sqrt {a+b \sin (c+d x)}}{99 b^2 d}-\frac {2 \cos (c+d x) \sin ^4(c+d x) \sqrt {a+b \sin (c+d x)}}{11 b d}-\frac {16 a \left (160 a^4-267 a^2 b^2+69 b^4\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)}}{3465 b^6 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {8 \left (320 a^6-614 a^4 b^2+249 a^2 b^4+45 b^6\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}}}{3465 b^6 d \sqrt {a+b \sin (c+d x)}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 4.36, size = 326, normalized size = 0.80 \[ \frac {-64 \left (320 a^6-614 a^4 b^2+249 a^2 b^4+45 b^6\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} F\left (\frac {1}{4} (-2 c-2 d x+\pi )|\frac {2 b}{a+b}\right )+b \cos (c+d x) \left (-10240 a^5-2560 a^4 b \sin (c+d x)+16448 a^3 b^2-128 \left (5 a^3 b^2-6 a b^4\right ) \cos (2 (c+d x))+3752 a^2 b^3 \sin (c+d x)+200 a^2 b^3 \sin (3 (c+d x))+70 a b^4 \cos (4 (c+d x))-3718 a b^4+990 b^5 \sin (c+d x)-765 b^5 \sin (3 (c+d x))-315 b^5 \sin (5 (c+d x))\right )+128 a \left (160 a^5+160 a^4 b-267 a^3 b^2-267 a^2 b^3+69 a b^4+69 b^5\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} E\left (\frac {1}{4} (-2 c-2 d x+\pi )|\frac {2 b}{a+b}\right )}{27720 b^6 d \sqrt {a+b \sin (c+d x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 1.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\cos \left (d x + c\right )^{6} - \cos \left (d x + c\right )^{4}}{\sqrt {b \sin \left (d x + c\right ) + a}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cos \left (d x + c\right )^{4} \sin \left (d x + c\right )^{2}}{\sqrt {b \sin \left (d x + c\right ) + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 1.70, size = 1356, normalized size = 3.35 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cos \left (d x + c\right )^{4} \sin \left (d x + c\right )^{2}}{\sqrt {b \sin \left (d x + c\right ) + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\cos \left (c+d\,x\right )}^4\,{\sin \left (c+d\,x\right )}^2}{\sqrt {a+b\,\sin \left (c+d\,x\right )}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sin ^{2}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{\sqrt {a + b \sin {\left (c + d x \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________