Integrand size = 24, antiderivative size = 407 \[ \int (c+d x)^{5/2} \cos (a+b x) \sin ^3(a+b x) \, dx=\frac {15 d^2 \sqrt {c+d x} \cos (2 a+2 b x)}{128 b^3}-\frac {(c+d x)^{5/2} \cos (2 a+2 b x)}{8 b}-\frac {15 d^2 \sqrt {c+d x} \cos (4 a+4 b x)}{2048 b^3}+\frac {(c+d x)^{5/2} \cos (4 a+4 b x)}{32 b}+\frac {15 d^{5/2} \sqrt {\frac {\pi }{2}} \cos \left (4 a-\frac {4 b c}{d}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{4096 b^{7/2}}-\frac {15 d^{5/2} \sqrt {\pi } \cos \left (2 a-\frac {2 b c}{d}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {\pi }}\right )}{256 b^{7/2}}-\frac {15 d^{5/2} \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\frac {2 \sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right ) \sin \left (4 a-\frac {4 b c}{d}\right )}{4096 b^{7/2}}+\frac {15 d^{5/2} \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {\pi }}\right ) \sin \left (2 a-\frac {2 b c}{d}\right )}{256 b^{7/2}}+\frac {5 d (c+d x)^{3/2} \sin (2 a+2 b x)}{32 b^2}-\frac {5 d (c+d x)^{3/2} \sin (4 a+4 b x)}{256 b^2} \]
[Out]
Time = 0.81 (sec) , antiderivative size = 407, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {4491, 3377, 3387, 3386, 3432, 3385, 3433} \[ \int (c+d x)^{5/2} \cos (a+b x) \sin ^3(a+b x) \, dx=\frac {15 \sqrt {\frac {\pi }{2}} d^{5/2} \cos \left (4 a-\frac {4 b c}{d}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{4096 b^{7/2}}-\frac {15 \sqrt {\pi } d^{5/2} \cos \left (2 a-\frac {2 b c}{d}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {\pi }}\right )}{256 b^{7/2}}-\frac {15 \sqrt {\frac {\pi }{2}} d^{5/2} \sin \left (4 a-\frac {4 b c}{d}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{4096 b^{7/2}}+\frac {15 \sqrt {\pi } d^{5/2} \sin \left (2 a-\frac {2 b c}{d}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {\pi }}\right )}{256 b^{7/2}}+\frac {15 d^2 \sqrt {c+d x} \cos (2 a+2 b x)}{128 b^3}-\frac {15 d^2 \sqrt {c+d x} \cos (4 a+4 b x)}{2048 b^3}+\frac {5 d (c+d x)^{3/2} \sin (2 a+2 b x)}{32 b^2}-\frac {5 d (c+d x)^{3/2} \sin (4 a+4 b x)}{256 b^2}-\frac {(c+d x)^{5/2} \cos (2 a+2 b x)}{8 b}+\frac {(c+d x)^{5/2} \cos (4 a+4 b x)}{32 b} \]
[In]
[Out]
Rule 3377
Rule 3385
Rule 3386
Rule 3387
Rule 3432
Rule 3433
Rule 4491
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{4} (c+d x)^{5/2} \sin (2 a+2 b x)-\frac {1}{8} (c+d x)^{5/2} \sin (4 a+4 b x)\right ) \, dx \\ & = -\left (\frac {1}{8} \int (c+d x)^{5/2} \sin (4 a+4 b x) \, dx\right )+\frac {1}{4} \int (c+d x)^{5/2} \sin (2 a+2 b x) \, dx \\ & = -\frac {(c+d x)^{5/2} \cos (2 a+2 b x)}{8 b}+\frac {(c+d x)^{5/2} \cos (4 a+4 b x)}{32 b}-\frac {(5 d) \int (c+d x)^{3/2} \cos (4 a+4 b x) \, dx}{64 b}+\frac {(5 d) \int (c+d x)^{3/2} \cos (2 a+2 b x) \, dx}{16 b} \\ & = -\frac {(c+d x)^{5/2} \cos (2 a+2 b x)}{8 b}+\frac {(c+d x)^{5/2} \cos (4 a+4 b x)}{32 b}+\frac {5 d (c+d x)^{3/2} \sin (2 a+2 b x)}{32 b^2}-\frac {5 d (c+d x)^{3/2} \sin (4 a+4 b x)}{256 b^2}+\frac {\left (15 d^2\right ) \int \sqrt {c+d x} \sin (4 a+4 b x) \, dx}{512 b^2}-\frac {\left (15 d^2\right ) \int \sqrt {c+d x} \sin (2 a+2 b x) \, dx}{64 b^2} \\ & = \frac {15 d^2 \sqrt {c+d x} \cos (2 a+2 b x)}{128 b^3}-\frac {(c+d x)^{5/2} \cos (2 a+2 b x)}{8 b}-\frac {15 d^2 \sqrt {c+d x} \cos (4 a+4 b x)}{2048 b^3}+\frac {(c+d x)^{5/2} \cos (4 a+4 b x)}{32 b}+\frac {5 d (c+d x)^{3/2} \sin (2 a+2 b x)}{32 b^2}-\frac {5 d (c+d x)^{3/2} \sin (4 a+4 b x)}{256 b^2}+\frac {\left (15 d^3\right ) \int \frac {\cos (4 a+4 b x)}{\sqrt {c+d x}} \, dx}{4096 b^3}-\frac {\left (15 d^3\right ) \int \frac {\cos (2 a+2 b x)}{\sqrt {c+d x}} \, dx}{256 b^3} \\ & = \frac {15 d^2 \sqrt {c+d x} \cos (2 a+2 b x)}{128 b^3}-\frac {(c+d x)^{5/2} \cos (2 a+2 b x)}{8 b}-\frac {15 d^2 \sqrt {c+d x} \cos (4 a+4 b x)}{2048 b^3}+\frac {(c+d x)^{5/2} \cos (4 a+4 b x)}{32 b}+\frac {5 d (c+d x)^{3/2} \sin (2 a+2 b x)}{32 b^2}-\frac {5 d (c+d x)^{3/2} \sin (4 a+4 b x)}{256 b^2}+\frac {\left (15 d^3 \cos \left (4 a-\frac {4 b c}{d}\right )\right ) \int \frac {\cos \left (\frac {4 b c}{d}+4 b x\right )}{\sqrt {c+d x}} \, dx}{4096 b^3}-\frac {\left (15 d^3 \cos \left (2 a-\frac {2 b c}{d}\right )\right ) \int \frac {\cos \left (\frac {2 b c}{d}+2 b x\right )}{\sqrt {c+d x}} \, dx}{256 b^3}-\frac {\left (15 d^3 \sin \left (4 a-\frac {4 b c}{d}\right )\right ) \int \frac {\sin \left (\frac {4 b c}{d}+4 b x\right )}{\sqrt {c+d x}} \, dx}{4096 b^3}+\frac {\left (15 d^3 \sin \left (2 a-\frac {2 b c}{d}\right )\right ) \int \frac {\sin \left (\frac {2 b c}{d}+2 b x\right )}{\sqrt {c+d x}} \, dx}{256 b^3} \\ & = \frac {15 d^2 \sqrt {c+d x} \cos (2 a+2 b x)}{128 b^3}-\frac {(c+d x)^{5/2} \cos (2 a+2 b x)}{8 b}-\frac {15 d^2 \sqrt {c+d x} \cos (4 a+4 b x)}{2048 b^3}+\frac {(c+d x)^{5/2} \cos (4 a+4 b x)}{32 b}+\frac {5 d (c+d x)^{3/2} \sin (2 a+2 b x)}{32 b^2}-\frac {5 d (c+d x)^{3/2} \sin (4 a+4 b x)}{256 b^2}+\frac {\left (15 d^2 \cos \left (4 a-\frac {4 b c}{d}\right )\right ) \text {Subst}\left (\int \cos \left (\frac {4 b x^2}{d}\right ) \, dx,x,\sqrt {c+d x}\right )}{2048 b^3}-\frac {\left (15 d^2 \cos \left (2 a-\frac {2 b c}{d}\right )\right ) \text {Subst}\left (\int \cos \left (\frac {2 b x^2}{d}\right ) \, dx,x,\sqrt {c+d x}\right )}{128 b^3}-\frac {\left (15 d^2 \sin \left (4 a-\frac {4 b c}{d}\right )\right ) \text {Subst}\left (\int \sin \left (\frac {4 b x^2}{d}\right ) \, dx,x,\sqrt {c+d x}\right )}{2048 b^3}+\frac {\left (15 d^2 \sin \left (2 a-\frac {2 b c}{d}\right )\right ) \text {Subst}\left (\int \sin \left (\frac {2 b x^2}{d}\right ) \, dx,x,\sqrt {c+d x}\right )}{128 b^3} \\ & = \frac {15 d^2 \sqrt {c+d x} \cos (2 a+2 b x)}{128 b^3}-\frac {(c+d x)^{5/2} \cos (2 a+2 b x)}{8 b}-\frac {15 d^2 \sqrt {c+d x} \cos (4 a+4 b x)}{2048 b^3}+\frac {(c+d x)^{5/2} \cos (4 a+4 b x)}{32 b}+\frac {15 d^{5/2} \sqrt {\frac {\pi }{2}} \cos \left (4 a-\frac {4 b c}{d}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{4096 b^{7/2}}-\frac {15 d^{5/2} \sqrt {\pi } \cos \left (2 a-\frac {2 b c}{d}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {\pi }}\right )}{256 b^{7/2}}-\frac {15 d^{5/2} \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\frac {2 \sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right ) \sin \left (4 a-\frac {4 b c}{d}\right )}{4096 b^{7/2}}+\frac {15 d^{5/2} \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {\pi }}\right ) \sin \left (2 a-\frac {2 b c}{d}\right )}{256 b^{7/2}}+\frac {5 d (c+d x)^{3/2} \sin (2 a+2 b x)}{32 b^2}-\frac {5 d (c+d x)^{3/2} \sin (4 a+4 b x)}{256 b^2} \\ \end{align*}
Result contains complex when optimal does not.
Time = 6.03 (sec) , antiderivative size = 1332, normalized size of antiderivative = 3.27 \[ \int (c+d x)^{5/2} \cos (a+b x) \sin ^3(a+b x) \, dx=\frac {\left (\frac {1}{128}+\frac {i}{128}\right ) c \sqrt {d} e^{-\frac {2 i (a d+b (c+d x))}{d}} \left ((2+2 i) \sqrt {b} \sqrt {d} e^{\frac {2 i b c}{d}} \sqrt {c+d x} \left (3+4 i b x+e^{4 i (a+b x)} (-3+4 i b x)\right )+i (4 b c+3 i d) e^{\frac {2 i b (2 c+d x)}{d}} \sqrt {\pi } \text {erf}\left (\frac {(1+i) \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )+(4 i b c+3 d) e^{2 i (2 a+b x)} \sqrt {\pi } \text {erfi}\left (\frac {(1+i) \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )\right )}{b^{5/2}}-\frac {c \sqrt {d} e^{-\frac {4 i (a d+b (c+d x))}{d}} \left (-4 \sqrt {b} \sqrt {d} e^{\frac {4 i b c}{d}} \sqrt {c+d x} \left (-3 i+8 b x+e^{8 i (a+b x)} (3 i+8 b x)\right )+(-1)^{3/4} (8 b c+3 i d) e^{\frac {4 i b (2 c+d x)}{d}} \sqrt {\pi } \text {erf}\left (\frac {2 \sqrt [4]{-1} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )+\sqrt [4]{-1} (8 i b c+3 d) e^{4 i (2 a+b x)} \sqrt {\pi } \text {erfi}\left (\frac {2 \sqrt [4]{-1} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )\right )}{1024 b^{5/2}}+\frac {1}{4} c^2 \left (-\frac {e^{2 i \left (a-\frac {b c}{d}\right )} \sqrt {c+d x} \Gamma \left (\frac {3}{2},-\frac {2 i b (c+d x)}{d}\right )}{4 \sqrt {2} b \sqrt {-\frac {i b (c+d x)}{d}}}-\frac {e^{-2 i \left (a-\frac {b c}{d}\right )} \sqrt {c+d x} \Gamma \left (\frac {3}{2},\frac {2 i b (c+d x)}{d}\right )}{4 \sqrt {2} b \sqrt {\frac {i b (c+d x)}{d}}}\right )-\frac {c^2 e^{-\frac {4 i (b c+a d)}{d}} \sqrt {c+d x} \left (-\frac {e^{8 i a} \Gamma \left (\frac {3}{2},-\frac {4 i b (c+d x)}{d}\right )}{\sqrt {-\frac {i b (c+d x)}{d}}}-\frac {e^{\frac {8 i b c}{d}} \Gamma \left (\frac {3}{2},\frac {4 i b (c+d x)}{d}\right )}{\sqrt {\frac {i b (c+d x)}{d}}}\right )}{128 b}+\frac {\sqrt {d} \left ((1-i) e^{2 i \left (a-\frac {b c}{d}\right )} \left ((2+2 i) \sqrt {b} \sqrt {d} e^{\frac {2 i b (c+d x)}{d}} \sqrt {c+d x} \left (15 d-16 b^2 d x^2+4 i b (c-5 d x)\right )+\left (16 b^2 c^2-24 i b c d-15 d^2\right ) \sqrt {\pi } \text {erfi}\left (\frac {(1+i) \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )\right )+i (\cos (2 (a+b x))-i \sin (2 (a+b x))) \left (4 \sqrt {b} \sqrt {d} \sqrt {c+d x} \left (-15 i d+16 i b^2 d x^2-4 b (c-5 d x)\right )-(1+i) \left (16 b^2 c^2+24 i b c d-15 d^2\right ) \sqrt {\pi } \text {erf}\left (\frac {(1+i) \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right ) \left (\cos \left (\frac {2 b (c+d x)}{d}\right )+i \sin \left (\frac {2 b (c+d x)}{d}\right )\right )\right )\right )}{1024 b^{7/2}}-\frac {\sqrt {d} \left (e^{4 i \left (a-\frac {b c}{d}\right )} \left (4 \sqrt {b} \sqrt {d} e^{\frac {4 i b (c+d x)}{d}} \sqrt {c+d x} \left (15 d-64 b^2 d x^2+8 i b (c-5 d x)\right )+(-1)^{3/4} \left (-64 b^2 c^2+48 i b c d+15 d^2\right ) \sqrt {\pi } \text {erfi}\left (\frac {2 \sqrt [4]{-1} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )\right )+i (\cos (4 (a+b x))-i \sin (4 (a+b x))) \left (4 \sqrt {b} \sqrt {d} \sqrt {c+d x} \left (-15 i d+64 i b^2 d x^2-8 b (c-5 d x)\right )-(1+i) \left (64 b^2 c^2+48 i b c d-15 d^2\right ) \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {(1+i) \sqrt {2} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right ) \left (\cos \left (\frac {4 b (c+d x)}{d}\right )+i \sin \left (\frac {4 b (c+d x)}{d}\right )\right )\right )\right )}{16384 b^{7/2}} \]
[In]
[Out]
Time = 3.56 (sec) , antiderivative size = 470, normalized size of antiderivative = 1.15
method | result | size |
derivativedivides | \(\frac {-\frac {d \left (d x +c \right )^{\frac {5}{2}} \cos \left (\frac {2 b \left (d x +c \right )}{d}+\frac {2 a d -2 c b}{d}\right )}{8 b}+\frac {5 d \left (\frac {d \left (d x +c \right )^{\frac {3}{2}} \sin \left (\frac {2 b \left (d x +c \right )}{d}+\frac {2 a d -2 c b}{d}\right )}{4 b}-\frac {3 d \left (-\frac {d \sqrt {d x +c}\, \cos \left (\frac {2 b \left (d x +c \right )}{d}+\frac {2 a d -2 c b}{d}\right )}{4 b}+\frac {d \sqrt {\pi }\, \left (\cos \left (\frac {2 a d -2 c b}{d}\right ) \operatorname {FresnelC}\left (\frac {2 b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )-\sin \left (\frac {2 a d -2 c b}{d}\right ) \operatorname {FresnelS}\left (\frac {2 b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{8 b \sqrt {\frac {b}{d}}}\right )}{4 b}\right )}{8 b}+\frac {d \left (d x +c \right )^{\frac {5}{2}} \cos \left (\frac {4 b \left (d x +c \right )}{d}+\frac {4 a d -4 c b}{d}\right )}{32 b}-\frac {5 d \left (\frac {d \left (d x +c \right )^{\frac {3}{2}} \sin \left (\frac {4 b \left (d x +c \right )}{d}+\frac {4 a d -4 c b}{d}\right )}{8 b}-\frac {3 d \left (-\frac {d \sqrt {d x +c}\, \cos \left (\frac {4 b \left (d x +c \right )}{d}+\frac {4 a d -4 c b}{d}\right )}{8 b}+\frac {d \sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (\frac {4 a d -4 c b}{d}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {2}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )-\sin \left (\frac {4 a d -4 c b}{d}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {2}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{32 b \sqrt {\frac {b}{d}}}\right )}{8 b}\right )}{32 b}}{d}\) | \(470\) |
default | \(\frac {-\frac {d \left (d x +c \right )^{\frac {5}{2}} \cos \left (\frac {2 b \left (d x +c \right )}{d}+\frac {2 a d -2 c b}{d}\right )}{8 b}+\frac {5 d \left (\frac {d \left (d x +c \right )^{\frac {3}{2}} \sin \left (\frac {2 b \left (d x +c \right )}{d}+\frac {2 a d -2 c b}{d}\right )}{4 b}-\frac {3 d \left (-\frac {d \sqrt {d x +c}\, \cos \left (\frac {2 b \left (d x +c \right )}{d}+\frac {2 a d -2 c b}{d}\right )}{4 b}+\frac {d \sqrt {\pi }\, \left (\cos \left (\frac {2 a d -2 c b}{d}\right ) \operatorname {FresnelC}\left (\frac {2 b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )-\sin \left (\frac {2 a d -2 c b}{d}\right ) \operatorname {FresnelS}\left (\frac {2 b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{8 b \sqrt {\frac {b}{d}}}\right )}{4 b}\right )}{8 b}+\frac {d \left (d x +c \right )^{\frac {5}{2}} \cos \left (\frac {4 b \left (d x +c \right )}{d}+\frac {4 a d -4 c b}{d}\right )}{32 b}-\frac {5 d \left (\frac {d \left (d x +c \right )^{\frac {3}{2}} \sin \left (\frac {4 b \left (d x +c \right )}{d}+\frac {4 a d -4 c b}{d}\right )}{8 b}-\frac {3 d \left (-\frac {d \sqrt {d x +c}\, \cos \left (\frac {4 b \left (d x +c \right )}{d}+\frac {4 a d -4 c b}{d}\right )}{8 b}+\frac {d \sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (\frac {4 a d -4 c b}{d}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {2}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )-\sin \left (\frac {4 a d -4 c b}{d}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {2}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{32 b \sqrt {\frac {b}{d}}}\right )}{8 b}\right )}{32 b}}{d}\) | \(470\) |
[In]
[Out]
none
Time = 0.30 (sec) , antiderivative size = 406, normalized size of antiderivative = 1.00 \[ \int (c+d x)^{5/2} \cos (a+b x) \sin ^3(a+b x) \, dx=\frac {15 \, \sqrt {2} \pi d^{3} \sqrt {\frac {b}{\pi d}} \cos \left (-\frac {4 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {C}\left (2 \, \sqrt {2} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) - 15 \, \sqrt {2} \pi d^{3} \sqrt {\frac {b}{\pi d}} \operatorname {S}\left (2 \, \sqrt {2} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) \sin \left (-\frac {4 \, {\left (b c - a d\right )}}{d}\right ) - 480 \, \pi d^{3} \sqrt {\frac {b}{\pi d}} \cos \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {C}\left (2 \, \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) + 480 \, \pi d^{3} \sqrt {\frac {b}{\pi d}} \operatorname {S}\left (2 \, \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) \sin \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) + 4 \, {\left (320 \, b^{3} d^{2} x^{2} + 640 \, b^{3} c d x + 320 \, b^{3} c^{2} + 8 \, {\left (64 \, b^{3} d^{2} x^{2} + 128 \, b^{3} c d x + 64 \, b^{3} c^{2} - 15 \, b d^{2}\right )} \cos \left (b x + a\right )^{4} - 255 \, b d^{2} - 8 \, {\left (128 \, b^{3} d^{2} x^{2} + 256 \, b^{3} c d x + 128 \, b^{3} c^{2} - 75 \, b d^{2}\right )} \cos \left (b x + a\right )^{2} - 160 \, {\left (2 \, {\left (b^{2} d^{2} x + b^{2} c d\right )} \cos \left (b x + a\right )^{3} - 5 \, {\left (b^{2} d^{2} x + b^{2} c d\right )} \cos \left (b x + a\right )\right )} \sin \left (b x + a\right )\right )} \sqrt {d x + c}}{8192 \, b^{4}} \]
[In]
[Out]
Timed out. \[ \int (c+d x)^{5/2} \cos (a+b x) \sin ^3(a+b x) \, dx=\text {Timed out} \]
[In]
[Out]
Result contains complex when optimal does not.
Time = 0.40 (sec) , antiderivative size = 551, normalized size of antiderivative = 1.35 \[ \int (c+d x)^{5/2} \cos (a+b x) \sin ^3(a+b x) \, dx=-\frac {{\left (640 \, {\left (d x + c\right )}^{\frac {3}{2}} b^{3} \sin \left (\frac {4 \, {\left ({\left (d x + c\right )} b - b c + a d\right )}}{d}\right ) - 5120 \, {\left (d x + c\right )}^{\frac {3}{2}} b^{3} \sin \left (\frac {2 \, {\left ({\left (d x + c\right )} b - b c + a d\right )}}{d}\right ) - 16 \, {\left (\frac {64 \, {\left (d x + c\right )}^{\frac {5}{2}} b^{4}}{d} - 15 \, \sqrt {d x + c} b^{2} d\right )} \cos \left (\frac {4 \, {\left ({\left (d x + c\right )} b - b c + a d\right )}}{d}\right ) + 256 \, {\left (\frac {16 \, {\left (d x + c\right )}^{\frac {5}{2}} b^{4}}{d} - 15 \, \sqrt {d x + c} b^{2} d\right )} \cos \left (\frac {2 \, {\left ({\left (d x + c\right )} b - b c + a d\right )}}{d}\right ) - 240 \, {\left (\left (i - 1\right ) \cdot 4^{\frac {1}{4}} \sqrt {2} \sqrt {\pi } b d^{2} \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \cos \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) + \left (i + 1\right ) \cdot 4^{\frac {1}{4}} \sqrt {2} \sqrt {\pi } b d^{2} \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \sin \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right )\right )} \operatorname {erf}\left (\sqrt {d x + c} \sqrt {\frac {2 i \, b}{d}}\right ) - 15 \, {\left (-\left (i - 1\right ) \, \sqrt {2} \sqrt {\pi } b d^{2} \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \cos \left (-\frac {4 \, {\left (b c - a d\right )}}{d}\right ) - \left (i + 1\right ) \, \sqrt {2} \sqrt {\pi } b d^{2} \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \sin \left (-\frac {4 \, {\left (b c - a d\right )}}{d}\right )\right )} \operatorname {erf}\left (2 \, \sqrt {d x + c} \sqrt {\frac {i \, b}{d}}\right ) - 15 \, {\left (\left (i + 1\right ) \, \sqrt {2} \sqrt {\pi } b d^{2} \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \cos \left (-\frac {4 \, {\left (b c - a d\right )}}{d}\right ) + \left (i - 1\right ) \, \sqrt {2} \sqrt {\pi } b d^{2} \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \sin \left (-\frac {4 \, {\left (b c - a d\right )}}{d}\right )\right )} \operatorname {erf}\left (2 \, \sqrt {d x + c} \sqrt {-\frac {i \, b}{d}}\right ) - 240 \, {\left (-\left (i + 1\right ) \cdot 4^{\frac {1}{4}} \sqrt {2} \sqrt {\pi } b d^{2} \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \cos \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) - \left (i - 1\right ) \cdot 4^{\frac {1}{4}} \sqrt {2} \sqrt {\pi } b d^{2} \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \sin \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right )\right )} \operatorname {erf}\left (\sqrt {d x + c} \sqrt {-\frac {2 i \, b}{d}}\right )\right )} d}{32768 \, b^{5}} \]
[In]
[Out]
Result contains complex when optimal does not.
Time = 1.14 (sec) , antiderivative size = 2435, normalized size of antiderivative = 5.98 \[ \int (c+d x)^{5/2} \cos (a+b x) \sin ^3(a+b x) \, dx=\text {Too large to display} \]
[In]
[Out]
Timed out. \[ \int (c+d x)^{5/2} \cos (a+b x) \sin ^3(a+b x) \, dx=\int \cos \left (a+b\,x\right )\,{\sin \left (a+b\,x\right )}^3\,{\left (c+d\,x\right )}^{5/2} \,d x \]
[In]
[Out]