Optimal. Leaf size=451 \[ -\frac{2 \left (3 a^2+13 b^2\right ) \cos ^5(c+d x) \sqrt{a+b \sin (c+d x)}}{429 d}-\frac{2 \cos ^3(c+d x) \sqrt{a+b \sin (c+d x)} \left (-7 a b \left (a^2+63 b^2\right ) \sin (c+d x)-33 a^2 b^2+8 a^4-39 b^4\right )}{9009 b^2 d}+\frac{4 \cos (c+d x) \sqrt{a+b \sin (c+d x)} \left (-24 a b \left (-5 a^2 b^2+a^4-60 b^4\right ) \sin (c+d x)-165 a^4 b^2+450 a^2 b^4+32 a^6+195 b^6\right )}{45045 b^4 d}-\frac{8 \left (-197 a^6 b^2+615 a^4 b^4-255 a^2 b^6+32 a^8-195 b^8\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{45045 b^5 d \sqrt{a+b \sin (c+d x)}}+\frac{8 a \left (-189 a^4 b^2+570 a^2 b^4+32 a^6+1635 b^6\right ) \sqrt{a+b \sin (c+d x)} E\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{45045 b^5 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}-\frac{2 \cos ^5(c+d x) (a+b \sin (c+d x))^{5/2}}{15 d}-\frac{2 a \cos ^5(c+d x) (a+b \sin (c+d x))^{3/2}}{39 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 1.07487, antiderivative size = 451, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.241, Rules used = {2862, 2865, 2752, 2663, 2661, 2655, 2653} \[ -\frac{2 \left (3 a^2+13 b^2\right ) \cos ^5(c+d x) \sqrt{a+b \sin (c+d x)}}{429 d}-\frac{2 \cos ^3(c+d x) \sqrt{a+b \sin (c+d x)} \left (-7 a b \left (a^2+63 b^2\right ) \sin (c+d x)-33 a^2 b^2+8 a^4-39 b^4\right )}{9009 b^2 d}+\frac{4 \cos (c+d x) \sqrt{a+b \sin (c+d x)} \left (-24 a b \left (-5 a^2 b^2+a^4-60 b^4\right ) \sin (c+d x)-165 a^4 b^2+450 a^2 b^4+32 a^6+195 b^6\right )}{45045 b^4 d}-\frac{8 \left (-197 a^6 b^2+615 a^4 b^4-255 a^2 b^6+32 a^8-195 b^8\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{45045 b^5 d \sqrt{a+b \sin (c+d x)}}+\frac{8 a \left (-189 a^4 b^2+570 a^2 b^4+32 a^6+1635 b^6\right ) \sqrt{a+b \sin (c+d x)} E\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{45045 b^5 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}-\frac{2 \cos ^5(c+d x) (a+b \sin (c+d x))^{5/2}}{15 d}-\frac{2 a \cos ^5(c+d x) (a+b \sin (c+d x))^{3/2}}{39 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2862
Rule 2865
Rule 2752
Rule 2663
Rule 2661
Rule 2655
Rule 2653
Rubi steps
\begin{align*} \int \cos ^4(c+d x) \sin (c+d x) (a+b \sin (c+d x))^{5/2} \, dx &=-\frac{2 \cos ^5(c+d x) (a+b \sin (c+d x))^{5/2}}{15 d}+\frac{2}{15} \int \cos ^4(c+d x) \left (\frac{5 b}{2}+\frac{5}{2} a \sin (c+d x)\right ) (a+b \sin (c+d x))^{3/2} \, dx\\ &=-\frac{2 a \cos ^5(c+d x) (a+b \sin (c+d x))^{3/2}}{39 d}-\frac{2 \cos ^5(c+d x) (a+b \sin (c+d x))^{5/2}}{15 d}+\frac{4}{195} \int \cos ^4(c+d x) \sqrt{a+b \sin (c+d x)} \left (20 a b+\frac{5}{4} \left (3 a^2+13 b^2\right ) \sin (c+d x)\right ) \, dx\\ &=-\frac{2 \left (3 a^2+13 b^2\right ) \cos ^5(c+d x) \sqrt{a+b \sin (c+d x)}}{429 d}-\frac{2 a \cos ^5(c+d x) (a+b \sin (c+d x))^{3/2}}{39 d}-\frac{2 \cos ^5(c+d x) (a+b \sin (c+d x))^{5/2}}{15 d}+\frac{8 \int \frac{\cos ^4(c+d x) \left (\frac{5}{8} b \left (179 a^2+13 b^2\right )+\frac{15}{8} a \left (a^2+63 b^2\right ) \sin (c+d x)\right )}{\sqrt{a+b \sin (c+d x)}} \, dx}{2145}\\ &=-\frac{2 \left (3 a^2+13 b^2\right ) \cos ^5(c+d x) \sqrt{a+b \sin (c+d x)}}{429 d}-\frac{2 a \cos ^5(c+d x) (a+b \sin (c+d x))^{3/2}}{39 d}-\frac{2 \cos ^5(c+d x) (a+b \sin (c+d x))^{5/2}}{15 d}-\frac{2 \cos ^3(c+d x) \sqrt{a+b \sin (c+d x)} \left (8 a^4-33 a^2 b^2-39 b^4-7 a b \left (a^2+63 b^2\right ) \sin (c+d x)\right )}{9009 b^2 d}+\frac{32 \int \frac{\cos ^2(c+d x) \left (-\frac{15}{16} b \left (a^4-474 a^2 b^2-39 b^4\right )-\frac{15}{2} a \left (a^4-5 a^2 b^2-60 b^4\right ) \sin (c+d x)\right )}{\sqrt{a+b \sin (c+d x)}} \, dx}{45045 b^2}\\ &=-\frac{2 \left (3 a^2+13 b^2\right ) \cos ^5(c+d x) \sqrt{a+b \sin (c+d x)}}{429 d}-\frac{2 a \cos ^5(c+d x) (a+b \sin (c+d x))^{3/2}}{39 d}-\frac{2 \cos ^5(c+d x) (a+b \sin (c+d x))^{5/2}}{15 d}-\frac{2 \cos ^3(c+d x) \sqrt{a+b \sin (c+d x)} \left (8 a^4-33 a^2 b^2-39 b^4-7 a b \left (a^2+63 b^2\right ) \sin (c+d x)\right )}{9009 b^2 d}+\frac{4 \cos (c+d x) \sqrt{a+b \sin (c+d x)} \left (32 a^6-165 a^4 b^2+450 a^2 b^4+195 b^6-24 a b \left (a^4-5 a^2 b^2-60 b^4\right ) \sin (c+d x)\right )}{45045 b^4 d}+\frac{128 \int \frac{\frac{15}{32} b \left (8 a^6-45 a^4 b^2+1890 a^2 b^4+195 b^6\right )+\frac{15}{32} a \left (32 a^6-189 a^4 b^2+570 a^2 b^4+1635 b^6\right ) \sin (c+d x)}{\sqrt{a+b \sin (c+d x)}} \, dx}{675675 b^4}\\ &=-\frac{2 \left (3 a^2+13 b^2\right ) \cos ^5(c+d x) \sqrt{a+b \sin (c+d x)}}{429 d}-\frac{2 a \cos ^5(c+d x) (a+b \sin (c+d x))^{3/2}}{39 d}-\frac{2 \cos ^5(c+d x) (a+b \sin (c+d x))^{5/2}}{15 d}-\frac{2 \cos ^3(c+d x) \sqrt{a+b \sin (c+d x)} \left (8 a^4-33 a^2 b^2-39 b^4-7 a b \left (a^2+63 b^2\right ) \sin (c+d x)\right )}{9009 b^2 d}+\frac{4 \cos (c+d x) \sqrt{a+b \sin (c+d x)} \left (32 a^6-165 a^4 b^2+450 a^2 b^4+195 b^6-24 a b \left (a^4-5 a^2 b^2-60 b^4\right ) \sin (c+d x)\right )}{45045 b^4 d}+\frac{\left (4 a \left (32 a^6-189 a^4 b^2+570 a^2 b^4+1635 b^6\right )\right ) \int \sqrt{a+b \sin (c+d x)} \, dx}{45045 b^5}-\frac{\left (4 \left (32 a^8-197 a^6 b^2+615 a^4 b^4-255 a^2 b^6-195 b^8\right )\right ) \int \frac{1}{\sqrt{a+b \sin (c+d x)}} \, dx}{45045 b^5}\\ &=-\frac{2 \left (3 a^2+13 b^2\right ) \cos ^5(c+d x) \sqrt{a+b \sin (c+d x)}}{429 d}-\frac{2 a \cos ^5(c+d x) (a+b \sin (c+d x))^{3/2}}{39 d}-\frac{2 \cos ^5(c+d x) (a+b \sin (c+d x))^{5/2}}{15 d}-\frac{2 \cos ^3(c+d x) \sqrt{a+b \sin (c+d x)} \left (8 a^4-33 a^2 b^2-39 b^4-7 a b \left (a^2+63 b^2\right ) \sin (c+d x)\right )}{9009 b^2 d}+\frac{4 \cos (c+d x) \sqrt{a+b \sin (c+d x)} \left (32 a^6-165 a^4 b^2+450 a^2 b^4+195 b^6-24 a b \left (a^4-5 a^2 b^2-60 b^4\right ) \sin (c+d x)\right )}{45045 b^4 d}+\frac{\left (4 a \left (32 a^6-189 a^4 b^2+570 a^2 b^4+1635 b^6\right ) \sqrt{a+b \sin (c+d x)}\right ) \int \sqrt{\frac{a}{a+b}+\frac{b \sin (c+d x)}{a+b}} \, dx}{45045 b^5 \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}-\frac{\left (4 \left (32 a^8-197 a^6 b^2+615 a^4 b^4-255 a^2 b^6-195 b^8\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}{45045 b^5 \sqrt{a+b \sin (c+d x)}}\\ &=-\frac{2 \left (3 a^2+13 b^2\right ) \cos ^5(c+d x) \sqrt{a+b \sin (c+d x)}}{429 d}-\frac{2 a \cos ^5(c+d x) (a+b \sin (c+d x))^{3/2}}{39 d}-\frac{2 \cos ^5(c+d x) (a+b \sin (c+d x))^{5/2}}{15 d}+\frac{8 a \left (32 a^6-189 a^4 b^2+570 a^2 b^4+1635 b^6\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)}}{45045 b^5 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}-\frac{8 \left (32 a^8-197 a^6 b^2+615 a^4 b^4-255 a^2 b^6-195 b^8\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}}}{45045 b^5 d \sqrt{a+b \sin (c+d x)}}-\frac{2 \cos ^3(c+d x) \sqrt{a+b \sin (c+d x)} \left (8 a^4-33 a^2 b^2-39 b^4-7 a b \left (a^2+63 b^2\right ) \sin (c+d x)\right )}{9009 b^2 d}+\frac{4 \cos (c+d x) \sqrt{a+b \sin (c+d x)} \left (32 a^6-165 a^4 b^2+450 a^2 b^4+195 b^6-24 a b \left (a^4-5 a^2 b^2-60 b^4\right ) \sin (c+d x)\right )}{45045 b^4 d}\\ \end{align*}
Mathematica [A] time = 21.1943, size = 450, normalized size = 1. \[ \frac{b \cos (c+d x) \left (-5840 a^4 b^3 \sin (c+d x)-80 a^4 b^3 \sin (3 (c+d x))+186768 a^2 b^5 \sin (c+d x)-101688 a^2 b^5 \sin (3 (c+d x))-46536 a^2 b^5 \sin (5 (c+d x))+8 \left (-18192 a^3 b^4+32 a^5 b^2-18741 a b^6\right ) \cos (2 (c+d x))-224 \left (161 a^3 b^4-54 a b^6\right ) \cos (4 (c+d x))-23936 a^5 b^2-36512 a^3 b^4+1024 a^6 b \sin (c+d x)+4096 a^7+20328 a b^6 \cos (6 (c+d x))+67584 a b^6+8151 b^7 \sin (c+d x)-22269 b^7 \sin (3 (c+d x))-2457 b^7 \sin (5 (c+d x))+3003 b^7 \sin (7 (c+d x))\right )+256 \left (-197 a^6 b^2+615 a^4 b^4-255 a^2 b^6+32 a^8-195 b^8\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left (\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right )-256 a \left (-189 a^5 b^2-189 a^4 b^3+570 a^3 b^4+570 a^2 b^5+32 a^6 b+32 a^7+1635 a b^6+1635 b^7\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} E\left (\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right )}{1441440 b^5 d \sqrt{a+b \sin (c+d x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 1.499, size = 1801, normalized size = 4. \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \sin \left (d x + c\right ) + a\right )}^{\frac{5}{2}} \cos \left (d x + c\right )^{4} \sin \left (d x + c\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-{\left (2 \, a b \cos \left (d x + c\right )^{6} - 2 \, a b \cos \left (d x + c\right )^{4} +{\left (b^{2} \cos \left (d x + c\right )^{6} -{\left (a^{2} + b^{2}\right )} \cos \left (d x + c\right )^{4}\right )} \sin \left (d x + c\right )\right )} \sqrt{b \sin \left (d x + c\right ) + a}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \sin \left (d x + c\right ) + a\right )}^{\frac{5}{2}} \cos \left (d x + c\right )^{4} \sin \left (d x + c\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]