Optimal. Leaf size=528 \[ \frac {8 a \left (8 a^2-21 b^2\right ) \sin (c+d x) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{1287 b^4 d}-\frac {2 \left (80 a^2-221 b^2\right ) \sin ^2(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{2145 b^3 d}-\frac {8 \left (160 a^4-375 a^2 b^2+117 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{45045 b^5 d}+\frac {16 a \left (32 a^4-47 a^2 b^2-27 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{45045 b^5 d}-\frac {16 a \left (32 a^6-111 a^4 b^2+102 a^2 b^4-471 b^6\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{45045 b^6 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {8 \left (64 a^6-174 a^4 b^2+81 a^2 b^4-195 b^6\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{45045 b^5 d}+\frac {8 \left (64 a^8-238 a^6 b^2+255 a^4 b^4-276 a^2 b^6+195 b^8\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} F\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{45045 b^6 d \sqrt {a+b \sin (c+d x)}}+\frac {4 a \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{39 b^2 d}-\frac {2 \sin ^4(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{15 b d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 1.21, antiderivative size = 528, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 9, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.290, Rules used = {2895, 3049, 3023, 2753, 2752, 2663, 2661, 2655, 2653} \[ -\frac {2 \left (80 a^2-221 b^2\right ) \sin ^2(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{2145 b^3 d}+\frac {8 a \left (8 a^2-21 b^2\right ) \sin (c+d x) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{1287 b^4 d}-\frac {8 \left (-375 a^2 b^2+160 a^4+117 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{45045 b^5 d}+\frac {16 a \left (-47 a^2 b^2+32 a^4-27 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{45045 b^5 d}+\frac {8 \left (-174 a^4 b^2+81 a^2 b^4+64 a^6-195 b^6\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{45045 b^5 d}+\frac {8 \left (-238 a^6 b^2+255 a^4 b^4-276 a^2 b^6+64 a^8+195 b^8\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} F\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{45045 b^6 d \sqrt {a+b \sin (c+d x)}}-\frac {16 a \left (-111 a^4 b^2+102 a^2 b^4+32 a^6-471 b^6\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{45045 b^6 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {4 a \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{39 b^2 d}-\frac {2 \sin ^4(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{15 b d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2653
Rule 2655
Rule 2661
Rule 2663
Rule 2752
Rule 2753
Rule 2895
Rule 3023
Rule 3049
Rubi steps
\begin {align*} \int \cos ^4(c+d x) \sin ^2(c+d x) (a+b \sin (c+d x))^{3/2} \, dx &=\frac {4 a \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{39 b^2 d}-\frac {2 \cos (c+d x) \sin ^4(c+d x) (a+b \sin (c+d x))^{5/2}}{15 b d}-\frac {4 \int \sin ^2(c+d x) (a+b \sin (c+d x))^{3/2} \left (\frac {15}{4} \left (4 a^2-13 b^2\right )+\frac {3}{2} a b \sin (c+d x)-\frac {1}{4} \left (80 a^2-221 b^2\right ) \sin ^2(c+d x)\right ) \, dx}{195 b^2}\\ &=-\frac {2 \left (80 a^2-221 b^2\right ) \cos (c+d x) \sin ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{2145 b^3 d}+\frac {4 a \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{39 b^2 d}-\frac {2 \cos (c+d x) \sin ^4(c+d x) (a+b \sin (c+d x))^{5/2}}{15 b d}-\frac {8 \int \sin (c+d x) (a+b \sin (c+d x))^{3/2} \left (-\frac {1}{2} a \left (80 a^2-221 b^2\right )-\frac {3}{2} b \left (5 a^2+13 b^2\right ) \sin (c+d x)+\frac {15}{2} a \left (8 a^2-21 b^2\right ) \sin ^2(c+d x)\right ) \, dx}{2145 b^3}\\ &=\frac {8 a \left (8 a^2-21 b^2\right ) \cos (c+d x) \sin (c+d x) (a+b \sin (c+d x))^{5/2}}{1287 b^4 d}-\frac {2 \left (80 a^2-221 b^2\right ) \cos (c+d x) \sin ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{2145 b^3 d}+\frac {4 a \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{39 b^2 d}-\frac {2 \cos (c+d x) \sin ^4(c+d x) (a+b \sin (c+d x))^{5/2}}{15 b d}-\frac {16 \int (a+b \sin (c+d x))^{3/2} \left (\frac {15}{2} a^2 \left (8 a^2-21 b^2\right )+6 a b \left (5 a^2-9 b^2\right ) \sin (c+d x)-\frac {3}{4} \left (160 a^4-375 a^2 b^2+117 b^4\right ) \sin ^2(c+d x)\right ) \, dx}{19305 b^4}\\ &=-\frac {8 \left (160 a^4-375 a^2 b^2+117 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{45045 b^5 d}+\frac {8 a \left (8 a^2-21 b^2\right ) \cos (c+d x) \sin (c+d x) (a+b \sin (c+d x))^{5/2}}{1287 b^4 d}-\frac {2 \left (80 a^2-221 b^2\right ) \cos (c+d x) \sin ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{2145 b^3 d}+\frac {4 a \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{39 b^2 d}-\frac {2 \cos (c+d x) \sin ^4(c+d x) (a+b \sin (c+d x))^{5/2}}{15 b d}-\frac {32 \int (a+b \sin (c+d x))^{3/2} \left (-\frac {45}{8} b \left (16 a^4-27 a^2 b^2+39 b^4\right )+\frac {15}{4} a \left (32 a^4-47 a^2 b^2-27 b^4\right ) \sin (c+d x)\right ) \, dx}{135135 b^5}\\ &=\frac {16 a \left (32 a^4-47 a^2 b^2-27 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{45045 b^5 d}-\frac {8 \left (160 a^4-375 a^2 b^2+117 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{45045 b^5 d}+\frac {8 a \left (8 a^2-21 b^2\right ) \cos (c+d x) \sin (c+d x) (a+b \sin (c+d x))^{5/2}}{1287 b^4 d}-\frac {2 \left (80 a^2-221 b^2\right ) \cos (c+d x) \sin ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{2145 b^3 d}+\frac {4 a \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{39 b^2 d}-\frac {2 \cos (c+d x) \sin ^4(c+d x) (a+b \sin (c+d x))^{5/2}}{15 b d}-\frac {64 \int \sqrt {a+b \sin (c+d x)} \left (-\frac {45}{16} a b \left (16 a^4-41 a^2 b^2+249 b^4\right )+\frac {45}{16} \left (64 a^6-174 a^4 b^2+81 a^2 b^4-195 b^6\right ) \sin (c+d x)\right ) \, dx}{675675 b^5}\\ &=\frac {8 \left (64 a^6-174 a^4 b^2+81 a^2 b^4-195 b^6\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{45045 b^5 d}+\frac {16 a \left (32 a^4-47 a^2 b^2-27 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{45045 b^5 d}-\frac {8 \left (160 a^4-375 a^2 b^2+117 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{45045 b^5 d}+\frac {8 a \left (8 a^2-21 b^2\right ) \cos (c+d x) \sin (c+d x) (a+b \sin (c+d x))^{5/2}}{1287 b^4 d}-\frac {2 \left (80 a^2-221 b^2\right ) \cos (c+d x) \sin ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{2145 b^3 d}+\frac {4 a \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{39 b^2 d}-\frac {2 \cos (c+d x) \sin ^4(c+d x) (a+b \sin (c+d x))^{5/2}}{15 b d}-\frac {128 \int \frac {\frac {45}{32} b \left (16 a^6-51 a^4 b^2-666 a^2 b^4-195 b^6\right )+\frac {45}{16} a \left (32 a^6-111 a^4 b^2+102 a^2 b^4-471 b^6\right ) \sin (c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx}{2027025 b^5}\\ &=\frac {8 \left (64 a^6-174 a^4 b^2+81 a^2 b^4-195 b^6\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{45045 b^5 d}+\frac {16 a \left (32 a^4-47 a^2 b^2-27 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{45045 b^5 d}-\frac {8 \left (160 a^4-375 a^2 b^2+117 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{45045 b^5 d}+\frac {8 a \left (8 a^2-21 b^2\right ) \cos (c+d x) \sin (c+d x) (a+b \sin (c+d x))^{5/2}}{1287 b^4 d}-\frac {2 \left (80 a^2-221 b^2\right ) \cos (c+d x) \sin ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{2145 b^3 d}+\frac {4 a \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{39 b^2 d}-\frac {2 \cos (c+d x) \sin ^4(c+d x) (a+b \sin (c+d x))^{5/2}}{15 b d}-\frac {\left (8 a \left (32 a^6-111 a^4 b^2+102 a^2 b^4-471 b^6\right )\right ) \int \sqrt {a+b \sin (c+d x)} \, dx}{45045 b^6}+\frac {\left (4 \left (64 a^8-238 a^6 b^2+255 a^4 b^4-276 a^2 b^6+195 b^8\right )\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}} \, dx}{45045 b^6}\\ &=\frac {8 \left (64 a^6-174 a^4 b^2+81 a^2 b^4-195 b^6\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{45045 b^5 d}+\frac {16 a \left (32 a^4-47 a^2 b^2-27 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{45045 b^5 d}-\frac {8 \left (160 a^4-375 a^2 b^2+117 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{45045 b^5 d}+\frac {8 a \left (8 a^2-21 b^2\right ) \cos (c+d x) \sin (c+d x) (a+b \sin (c+d x))^{5/2}}{1287 b^4 d}-\frac {2 \left (80 a^2-221 b^2\right ) \cos (c+d x) \sin ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{2145 b^3 d}+\frac {4 a \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{39 b^2 d}-\frac {2 \cos (c+d x) \sin ^4(c+d x) (a+b \sin (c+d x))^{5/2}}{15 b d}-\frac {\left (8 a \left (32 a^6-111 a^4 b^2+102 a^2 b^4-471 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^6 \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {\left (4 \left (64 a^8-238 a^6 b^2+255 a^4 b^4-276 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^6 \sqrt {a+b \sin (c+d x)}}\\ &=\frac {8 \left (64 a^6-174 a^4 b^2+81 a^2 b^4-195 b^6\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{45045 b^5 d}+\frac {16 a \left (32 a^4-47 a^2 b^2-27 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{45045 b^5 d}-\frac {8 \left (160 a^4-375 a^2 b^2+117 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{45045 b^5 d}+\frac {8 a \left (8 a^2-21 b^2\right ) \cos (c+d x) \sin (c+d x) (a+b \sin (c+d x))^{5/2}}{1287 b^4 d}-\frac {2 \left (80 a^2-221 b^2\right ) \cos (c+d x) \sin ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{2145 b^3 d}+\frac {4 a \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{39 b^2 d}-\frac {2 \cos (c+d x) \sin ^4(c+d x) (a+b \sin (c+d x))^{5/2}}{15 b d}-\frac {16 a \left (32 a^6-111 a^4 b^2+102 a^2 b^4-471 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^6 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {8 \left (64 a^8-238 a^6 b^2+255 a^4 b^4-276 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^6 d \sqrt {a+b \sin (c+d x)}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 15.66, size = 382, normalized size = 0.72 \[ \frac {\sqrt {a+b \sin (c+d x)} \left (512 \left (32 a^7-111 a^5 b^2+102 a^3 b^4-471 a b^6\right ) E\left (\frac {1}{4} (-2 c-2 d x+\pi )|\frac {2 b}{a+b}\right )-2 b \cos (c+d x) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \left (4096 a^6-3072 a^5 b \sin (c+d x)-12416 a^4 b^2+8432 a^3 b^3 \sin (c+d x)+560 a^3 b^3 \sin (3 (c+d x))+8100 a^2 b^4+42 \left (6 a^2 b^4-13 b^6\right ) \cos (4 (c+d x))+\left (-1280 a^4 b^2+3168 a^2 b^4+21723 b^6\right ) \cos (2 (c+d x))-41424 a b^5 \sin (c+d x)+13776 a b^5 \sin (3 (c+d x))+7392 a b^5 \sin (5 (c+d x))-3003 b^6 \cos (6 (c+d x))+6786 b^6\right )-256 \left (64 a^7-64 a^6 b-174 a^5 b^2+174 a^4 b^3+81 a^3 b^4-81 a^2 b^5-195 a b^6+195 b^7\right ) F\left (\frac {1}{4} (-2 c-2 d x+\pi )|\frac {2 b}{a+b}\right )\right )}{1441440 b^6 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 1.35, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (a \cos \left (d x + c\right )^{6} - a \cos \left (d x + c\right )^{4} + {\left (b \cos \left (d x + c\right )^{6} - b \cos \left (d x + c\right )^{4}\right )} \sin \left (d x + c\right )\right )} \sqrt {b \sin \left (d x + c\right ) + a}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \cos \left (d x + c\right )^{4} \sin \left (d x + c\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 2.03, size = 1801, normalized size = 3.41 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \cos \left (d x + c\right )^{4} \sin \left (d x + c\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int {\cos \left (c+d\,x\right )}^4\,{\sin \left (c+d\,x\right )}^2\,{\left (a+b\,\sin \left (c+d\,x\right )\right )}^{3/2} \,d x \]
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]
________________________________________________________________________________________